USER GUIDE TO THE UNC PROCESS AND THREE 

UTILITY PROGRAMS FOR COMPUTATION OF 

NONLINEAR CONFIDENCE AND PREDICTION 

INTERVALS USING MODFLOW-2000 

Techniques and Methods Report 6-A10

U.S. Department of the Interior 

U.S. Geological Survey 
U.S. Department of the Interior 

Gale A. Norton, Secretary 

U.S. Geological Survey 

Charles G. Groat, Director 

U.S. Geological Survey, Reston, Virginia 2004 

For more information about the USGS and its products: 

Telephone: 1-888-ASK-USGS 

World Wide Web: http://www.usgs.gov/ 

Although this report is in the public domain, permission must be secured from the individual 

copyright owners to reproduce any copyrighted material contained within 

this report. 
PREFACE 

This report introduces and documents the Uncertainty (UNC) Process, which is a 

new Process in MODFLOW-2000 that can be used to compute regression-based 

confidence or prediction intervals for parameters of the Parameter-Estimation Process, 

and for most types of predictions that can be computed by a MODFLOW-2000 model 

calibrated by the Parameter-Estimation Process. The report also documents three 

programs, RESAN2-2k, BEALE2-2k, and CORFAC-2k that are valuable for the 

evaluation of results from the Parameter-Estimation Process and for the preparation of 

input values for the UNC Process. The performance of the programs has been tested in a 

variety of applications; however, there may be errors that were not detected in the test 

simulations. Users are requested to notify the U.S. Geological Survey of any errors found 

in this user guide by using the address on the back of the report title page. Updates might 

occasionally be made both to the user guide and to the programs. Users can check for 

updates on the Internet at URL http://water.usgs.gov/software/ground_water.html/.

iii

Contents 

Abstract............................................................................................................................. 1 

Chapter 1. Introduction...................................................................................................... 2 

Purpose and Scope................................................................................................... 4 

Acknowledgments ..................................................................................................... 5 

Chapter 2. Theory ............................................................................................................. 6

The UNC Process ..................................................................................................... 8 

Confidence Intervals.......................................................................................... 8 

Weighted Residuals ........................................................................................ 11 

Prediction Intervals.......................................................................................... 11 

The RESAN2-2k Program....................................................................................... 12 

A Test for Model Intrinsic Nonlinearity............................................................. 13 

A Test for Model and System Types of Intrinsic Nonlinearity.......................... 15 

A Test for Normality of Weighted Residuals.................................................... 15 

The BEALE2-2k Program........................................................................................ 18 

A Measure of Average Total Model Nonlinearity............................................. 18 

A Measure of Average Model Intrinsic Nonlinearity......................................... 19 

A Measure of Model Combined Intrinsic Nonlinearity for Confidence 

Intervals ............................................................................................... 20 

A Measure of Model Combined Intrinsic Nonlinearity for Prediction Intervals. 22 

The CORFAC-2k Program ...................................................................................... 24 

Calculation of the Correction Factors When Ω Is Known ................................ 24 

Approximate Calculation of the Correction Factors When Ω Is Unknown....... 26 

Approximate Calculation of the Correction Factors When Ω Is Partly Known. 28 

Chapter 3. User Guide to The UNC Process .................................................................. 30 

When to Use the UNC Process............................................................................... 30 

Changes to MODFLOW-2000 to Include the UNC Process.................................... 31 

Activation of the UNC Process................................................................................ 32 

Types of Parameter-Dependent Functions For Which Intervals Can Be Computed 

By the UNC Process.................................................................................... 32 

Determination of the Critical Value.......................................................................... 33 

Setting the Starting Values for the Parameters....................................................... 34 

Setting the Scaling Vector....................................................................................... 34 

Searching for the Global Extreme Value................................................................. 34 

Addressing Convergence Problems........................................................................ 35 

Checking for Intrinsic Nonlinearity Using Weighted Residuals Produced By the UNC 

Process........................................................................................................ 36 

Hydraulic Heads and Head-Dependent Flows ........................................................ 36 

Input Instructions..................................................................................................... 38 

 Instructions for the UNC Input File .............................................................. 38 

 Example UNC Input File ............................................................................. 39 

Explanation of Variables in UNC Input File ............................................................. 39 

v
Chapter 4. User Guide to RESAN2-2k............................................................................ 43 

Testing Weighted Residuals for Indications of Non-Normality ................................ 43 

Testing for Intrinsic Nonlinearity.............................................................................. 44 

Input Instructions..................................................................................................... 45 

Output ..................................................................................................................... 45 

Content of the _rs Input File Needed By RESAN2-2k............................................. 45 

Content of the _ws Input File Needed By RESAN2-2k ........................................... 47 

Chapter 5. User Guide to BEALE2-2k............................................................................. 48 

Input Instructions..................................................................................................... 49 

Output from BEALE2-2k.......................................................................................... 49 

Content of the _b1 Input File Needed By BEALE2-2k............................................. 50 

Content of the _b2 Input File Needed By BEALE2-2k............................................. 52 

Content of the _b3 Input File Needed By BEALE2-2k............................................. 52 

Content of the _b4 Input File Needed By BEALE2-2k............................................. 53 

Chapter 6. User Guide to CORFAC-2k........................................................................... 55 

Input Instructions..................................................................................................... 55 

Output from CORFAC-2k ........................................................................................ 56 

Content of the _cf Input File Needed By CORFAC-2k ............................................ 57 

References...................................................................................................................... 60 

Appendix A. Solutions for Extreme Values .................................................................... 63 

Confidence Intervals................................................................................................ 63 

Prediction Intervals.................................................................................................. 64 

Appendix B. Computation of Q and R for Testing for Equality of Constrained and 

Unconstrained Residuals .................................................................................... 65 

Appendix C. Example Simulations.................................................................................. 67 

Parameter Estimation.............................................................................................. 68 

Computation of Correction Factors Using CORFAC-2k .......................................... 69 

Computation of Nonlinearity Measures Using BEALE2-2k ..................................... 70 

Residuals Analysis And Detection of Nonlinearity Using RESAN2-2k .................... 71 

Calculation of Confidence Intervals Using MODFLOW-2000 With the UNC 

Process ...................................................................................................... 71 

Parameter Estimation – Input Files ......................................................................... 77 

Parameter Estimation – GLOBAL Output File......................................................... 93 

Modified Input Files For Generation of CORFAC-2k Input File............................. 110 

Modified _cf Input File to CORFAC-2k.................................................................. 112 

Output File from CORFAC-2k ............................................................................... 117 

Input Files Used to Generate BEALE2-2k Input Files ........................................... 122 

BEALE2-2k Input Files .......................................................................................... 123 

BEALE2-2k Output File ......................................................................................... 129 

RESAN2-2k Input Files ......................................................................................... 144 

RESAN2-2k Output Files....................................................................................... 150 

MODFLOW-2000 Input Files For Confidence Interval Calculation........................ 156 

GLOBAL Output File For Confidence Interval Calculation .................................... 158 

Summary And Weighted Residuals Output Files .................................................. 175 

vi
Figures 

Figure 1. Ordered weighted residuals plotted with ± 2vd confidence band 

(approximate 95 percent confidence intervals)................................................ 44 

Figure C1. Model domain, boundary conditions, zonation, locations, and values of head 

observations, and locations of head predictions.............................................. 67 

Figure C2. Ordered weighted residuals (dots) plotted with approximate 95 percent 

confidence band (bars).................................................................................... 72 

Figure C3. Weighted residuals at confidence limits in relation to weighted residuals from 

unconstrained regression (parameter estimation)........................................... 75 

Tables 

Table C1. Correction factors, CC, and correction factor bounds, CCB, computed by 

CORFAC-2k for six head predictions (G1 to G6) and two parameters (HK_1 

and HK_2) ....................................................................................................... 69 

Table C2. Combined intrinsic nonlinearity measures computed by BEALE2-2k............ 70 

vii
USER GUIDE TO THE UNC PROCESS AND 

THREE UTILITY PROGRAMS FOR COMPUTATION 

OF NONLINEAR CONFIDENCE AND PREDICTION 

INTERVALS USING MODFLOW-2000 

By Steen Christensen1 and Richard L. Cooley2

Abstract 

This report introduces and documents the Uncertainty (UNC) Process, a new Process in 

MODFLOW-2000 that calculates uncertainty measures for model parameters and for predictions 

produced by the model. Uncertainty measures can be computed by various methods, but when 

regression is applied to calibrate a model (for example when using the Parameter-Estimation 

Process of MODFLOW-2000) it is advantageous to also use regression-based methods to 

quantify uncertainty. For this reason the UNC Process computes (1) confidence intervals for 

parameters of the Parameter-Estimation Process and (2) confidence and prediction intervals for 

most types of functions that can be computed by a MODFLOW-2000 model calibrated by the 

Parameter-Estimation Process. The types of functions for which the Process works include 

hydraulic heads, hydraulic head differences, head-dependent flows computed by the headdependent flow packages for drains (DRN6), rivers (RIV6), general-head boundaries (GHB6), 

streams (STR6), drain-return cells (DRT1), and constant-head boundaries (CHD), and for 

differences between flows computed by any of the mentioned flow packages. The UNC Process 

does not allow computation of intervals for the difference between flows computed by two 

different flow packages. 

The report also documents three programs, RESAN2-2k, BEALE2-2k, and CORFAC-2k, 

which are valuable for the evaluation of results from the Parameter-Estimation Process and for 

the preparation of input values for the UNC Process. RESAN2-2k and BEALE2-2k are 

significant updates of the residual analysis and modified Beale’s measure programs first 

published by Cooley and Naff (1990) and later modified for use with MODFLOWP (Hill, 1994) 

and MODFLOW-2000 (Hill and others, 2000). CORFAC-2k is a new program that computes 

correction factors to be used by UNC. 

 1 Department of Earth Sciences, University of Aarhus, Aarhus, Denmark. 

 2 U.S. Geological Survey, Lakewood, CO.

1
Chapter 1. INTRODUCTION 

Chapter 1. INTRODUCTION 

Ground-water models almost never perfectly represent real systems or fit the data used 

for their calibration. This results in uncertainties in both estimates of parameters and predictions 

of model-related variables. The uncertainty of estimated parameters and predicted variables can 

be quantified by using either confidence or prediction intervals. However, computation of such 

intervals is not straightforward because model-computed hydraulic heads and quantities such as 

flows that are functions of hydraulic heads generally are nonlinear functions of the model 

parameters. 

Vecchia and Cooley (1987) and Clark (1987) independently derived similar 

methodologies that can be used for the computation of confidence and prediction intervals for 

both the estimated parameters and the output from a nonlinear regression model with normally 

distributed observation errors. Vecchia and Cooley (1987) showed that the same computer 

program can be used both to estimate the parameters by nonlinear regression and to compute 

confidence limits from which the desired confidence intervals are obtained. Computation of each 

confidence limit is a separate optimization problem in which the limit is obtained as an extreme 

value on the edge of the parameter likelihood region. Beven and Binley (1992) and Brooks and 

others (1994) presented methods that are similar in intent to the method of Vecchia and Cooley 

(1987). For brevity we will term confidence intervals computed using the nonlinear regression 

model as nonlinear confidence intervals. 

The nonlinear regression model can be linearized using a first-order Taylor series 

expansion. Confidence intervals computed using the linearized model are the standard ones 

found in most regression texts (for example, Seber and Wild, 1989, chapter 5) and are termed 

here linear confidence intervals. Hill (1994) presented two computer codes, BCINT and YCINT, 

that can be used to compute linear confidence intervals for parameters and linear confidence and 

prediction intervals for hydraulic heads and flows from MODFLOWP. Hill and others (2000) 

developed YCINT-2000 as a modification of YCINT to be used with MODFLOW-2000 to 

compute confidence and prediction intervals for hydraulic heads and flows; linear confidence 

intervals for the parameters are computed by MODFLOW-2000 at the end of the ParameterEstimation Process. 

Synthetic case studies (Cooley and Vecchia, 1987; Vecchia and Cooley, 1987; Hill, 1989; 

and Cooley, 1993a, b) show the following. Corresponding linear and nonlinear confidence 

intervals often are offset, and the nonlinear intervals generally are larger (Hill, 1989; Cooley, 

1993b); the variability in sizes of nonlinear confidence intervals generally is larger than the 

variability in sizes of corresponding linear confidence intervals (Hill, 1989; Cooley, 1993b); the 

differences between the sizes of corresponding nonlinear and linear confidence intervals increase 

as the sizes of the intervals increase (Cooley, 1993a); and use of prior information can have a 

significant effect on the sizes of confidence intervals (Cooley, 1993b). Field case studies 

(Christensen and Cooley, 1996; Christensen and others 1998; and Christensen and Cooley, 

2
Chapter 1. INTRODUCTION 

1999a) support the conclusions from the synthetic studies. However, the studies of Christensen 

and Cooley (1996; 1999a) show that linear confidence intervals sometimes are larger than the 

corresponding nonlinear intervals. 

Donaldson and Schnabel (1987) and Cooley (1997) present synthetic case studies that 

investigate the coverage probability of nonlinear confidence intervals as well as of linear 

confidence intervals. In the studied cases the coverage probability of nonlinear confidence 

intervals is correct, whereas the coverage probability of corresponding linear confidence 

intervals often is incorrect. 

Christensen and Cooley (1999b) tested the accuracy of 95 percent individual prediction 

intervals for hydraulic heads, streamflow gains, and effective transmissivities computed by 

ground-water models of two Danish aquifers. An individual prediction interval is for a random 

variable , where is a model-related variable (for example, hydraulic head, 

flux, or a model parameter), and is a normally distributed error that is a composite of smallscale model error and other errors including measurement errors. Small-scale model errors result 

from smaller-scale variability in geology and hydrology than those represented in the model. 

Such errors often are significant in ground-water modeling because the model commonly is 

constructed using model parameters that represent only important, or large-scale, geologic and 

hydrologic features. Christensen and Cooley (1999b) tested the accuracy of the prediction 

intervals by a cross-validation method and by comparing new field measurements, which were 

not used to develop or calibrate the models, to corresponding prediction intervals. They 

concluded that for the ground-water models of the two real aquifers the linear as well as the 

nonlinear individual prediction intervals appear to be accurate. However, their results indicate 

that for the predicted values of streamflow gains some of the nonlinear prediction intervals may 

be conservative without using a correction factor. In a somewhat similar study Hamilton and 

Wiens (1987) found that for a confidence interval the correction factor can be either larger or 

smaller than unity. 

( ) p p gY β += ε g( ) β

p ε

Inspired by the experiences from the previously mentioned studies, Cooley (2004) 

developed and tested a regression theory for modeling ground-water flow in heterogeneous 

systems. The theory addresses the small-scale model error that usually is present in ground-water 

models of heterogeneous systems where the model only uses average system characteristics 

having the same form as the drift. Important aspects of the theory are summarized and further 

tested by Christensen and Cooley (2003) and Cooley and Christensen (written communication, 

2003). Cooley (2004) showed, among other things, that small-scale model error can result in 

significant correlations among the true errors of the regression model; that the model error often 

contributes bias in estimated model parameters, whereas predictions of hydraulic head often are 

unbiased; that the statistic used to compute a confidence or prediction interval often should be 

multiplied by a correction factor that depends on covariances and interrelations among all of the 

types of system characteristics that give rise to model parameters. The synthetic case studies by 

Cooley (2004), Christensen and Cooley (2003), and Cooley and Christensen (written 

3
Chapter 1. INTRODUCTION 

communication, 2003) show that correction factors for confidence intervals can be very large, 

whereas the correction factors for prediction intervals appear to be relatively close to one. 

The results of Cooley (1997), Christensen and Cooley (2003), Cooley and Christensen 

(written communication, 2003), and Cooley (2004) strongly indicate that linear confidence 

intervals for estimated parameters and predictions of ground-water flow models may be 

inaccurate in many cases, whereas corresponding nonlinear confidence intervals seem to be 

correct or nearly correct in at least the synthetic cases without small-scale model error. Nonlinear 

prediction intervals seem to be correct for the field cases. This motivated development of the 

Uncertainty (UNC) Process (or simply UNC for simplicity) that can be incorporated into 

MODFLOW-2000 (Harbaugh and others, 2000) and can be used to compute nonlinear 

confidence and prediction intervals for any model parameter and for most types of data that can 

be used as regression data by MODFLOW-2000 (hydraulic head, differences between hydraulic 

heads, head dependent flow, and differences between head-dependent flows). UNC implements 

the computation method of Vecchia and Cooley (1987) into MODFLOW-2000. 

Use of UNC is intended to be straightforward and flexible. This is accomplished by 

minimizing the amount of extra input needed for the computation of intervals compared to the 

input needed by MODFLOW-2000 for parameter estimation. Options have been added to 

facilitate the search for a global extreme value within the likelihood region (corresponding to the 

upper or lower limit of a confidence or prediction interval), and so convergence problems 

resulting from parameter correlations can sometimes be avoided. Any number and any type of 

confidence or prediction intervals can be computed in the same run. The output from UNC is 

similar to the output from the Parameter-Estimation Process of MODFLOW-2000. However, the 

results from the computation of interval limits also can be summarized in a separate easy-tooverview output file. 

The procedures used by the UNC Process to compute nonlinear confidence and 

prediction intervals assume the model errors to be random and normally distributed. Whether or 

not this is the case is indicated by residuals analysis. The work of Cooley (2004) and Cooley and 

Christensen (written communication, 2003) indicate that some deviation from these ideals can be 

tolerated. The graphical procedures by Cooley and Naff (1990, p. 167-171) together with the 

results produced by the RESAN2-2k program are valuable for the analysis. Intrinsic nonlinearity 

also is assumed to be small for the UNC Process. Whether or not this may be the case can be 

investigated by using the RESAN2-2k and BEALE2-2k programs and by analyzing weighted 

residuals computed by UNC. 

Purpose and Scope 

This report introduces and documents the UNC Process, which is a new Process in 

MODFLOW-2000 that calculates uncertainty measures for model parameters and for predictions 

produced by the model. Uncertainty measures can be computed by various methods, but when 

regression is applied to calibrate a model (for example, when using the Parameter-Estimation 

4
Chapter 1. INTRODUCTION 

Process of MODFLOW-2000), it is advantageous to also use regression-based methods to 

quantify uncertainty. For this reason the UNC Process computes (1) confidence intervals for 

parameters of the Parameter-Estimation Process and (2) confidence and prediction intervals for 

most types of functions that can be computed by a MODFLOW-2000 model calibrated by the 

Parameter-Estimation Process. The report also documents two programs, RESAN2-2k and 

BEALE2-2k, that are valuable for the evaluation of results from the Parameter-Estimation 

Process and a third program, CORFAC-2k, that computes correction factors, including 

approximations that require minimal geostatistical information, for UNC. The theory behind 

UNC and the RESAN2-2k, BEALE2-2k, and CORFAC-2k programs is summarized in chapter 2 

and appendixes A and B. (For a thorough description the reader is referred to Cooley, 2004.) 

Guidance in application of, and input instructions for, the programs are given in chapters 3 to 6. 

Appendix C includes input and output files for an example problem. Source files for 

MODFLOW-2000 (including the programs documented in this report) are available at the 

Internet address listed in the preface of the report. 

Users of this report need to be familiar with the Ground-Water Flow Process as well as 

the Observation, Sensitivity, and Parameter-Estimation Processes of MODFLOW-2000. The 

Ground-Water Flow Process is described by Harbaugh and others (2000), and the three other 

Processes are described by Hill and others (2000). The user also should be familiar with basic 

statistics and the application of nonlinear regression. The book by Cooley and Naff (1990) 

discusses these subjects with the focus on application in ground-water modeling. 

Acknowledgments 

The Danish Natural Science Research Council provided support for Steen Christensen to 

travel to the United States in 1996 and 2001 to work with Richard Cooley to develop the 

methods presented in this report. Support for Richard Cooley after June, 2001, was provided by a 

post retirement contract with the U.S. Geological Survey. Subroutines RANUM and RNDV in 

RESAN2-2k were modified from programs given by Press and others (1986). 

5
Chapter 2. THEORY 

Chapter 2. THEORY 

The modeling concepts and terminology described in the this section have been more 

thoroughly described by Cooley (2004) and Christensen and Cooley (2003). 

A ground-water flow system can be characterised by hydrogeologic variables (termed 

system characteristics) such as hydraulic head, hydraulic conductivity, recharge, discharge, 

hydraulic heads and fluxes along internal and external boundaries, and point sources and sinks. 

In a model these system characteristics can be conceptualized as being discretely variable (in 

space and/or time) because discrete variation can be at as small a scale as desired (depending 

only on model discretization), allowing it to be virtually the same as continuous variation. All of 

these system characteristics can be assembled into an m-vector β. Because β includes all scales 

of variation necessary to construct an accurate model, any model function of β is almost free of 

model error, assuming that the model accurately represents the physical processes. 

Seen from a practical modeling perspective, vector β represents small-scale variability 

that cannot be explicitly represented in a model and larger scale variability (the drift) that can. In 

addition, vector β is unknown. Because β has much too large a dimension to be estimated from a 

small number of uncertain observations, the vector of model parameters to be estimated is 

reduced from one of large dimension, β, to one of much smaller dimension, . The p-vector 

represents the spatial and temporal average of β and has the same form as the drift 

θ*

θ* θ ; it does 

not represent the small-scale variation. A model function f(β), which contains large- and smallscale variation from β, is represented in terms of as θ* ( ) * f γθ , where γ is an interpolation 

or averaging matrix. 

× pm

In ground-water modeling we are interested in estimating rather than θ* θ because 

characterizes the actual realization β. An estimate of , can be found by minimizing 

θ*

θ* ,ˆ

θ

θ ( )( ) fY ωγθ − fY γθ)()()( ′ S −= , (1) 

where Y is an n-vector of observations (m>>n>p), f (γθ) is the corresponding n-vector of values 

simulated using general vector γθ instead of β, ω is an × nn weight matrix, and the prime 

indicates transpose. 

Predictions can be made with the ground-water model. In practice, variables of interest 

g(β) are predicted by ). ˆ g( θγ

Different types of observations, Y, can be used in the Parameter Estimation Process. 

Cooley (2004) does not distinguish among the several possible types of observations such as 

hydraulic heads, measured streamflow gains and losses, measured spring and well discharges, 

and measured direct (often called “prior”) information on model parameters. It is convenient to 

divide the types of information into two groups: (1) measured direct observations of parameters 

or linear combinations of them and (2) observations corresponding to model functions of the 

parameters such as hydraulic heads and fluxes. The various matrices and vectors pertaining to 

6
Chapter 2. THEORY 

observations have a subscript d for the partition of direct information and a subscript m for the 

partition for model-function information. Corresponding to this notation, the number of direct 

observations is designated nd and the number of model-function observations is designated nm, for 

a total of nd+nm=n observations. The matrices and vectors are used in the partitioned form in an 

equation only when the form of the result for each partition is different. The difference is caused 

by the forms of the observation second-moment matrix, , and 

the weight matrix, , used as an approximation for in the weighted regression analysis. As 

in standard regression analysis, is a generally unknown scalar multiplier that is estimated by 

the regression. We assume that the partition of the observation second-moment matrix for the 

model functions depends on an unknown model error and often may not be known, but that the 

partition for the direct information would be well estimated using the same techniques used to 

acquire the direct information. We also assume that the two partitions are not correlated. (This is 

a standard assumption dating back to Theil (1963). Should it prove to be inaccurate in any 

application, the correction factors computed by CORFAC-2k will be somewhat in error. If 

n

−=σ − ))())((( ′ ε * *

2 Ω E fY γθ fY γθ

ω −1 Ω

2 σε

d

<<nm, the errors should be small.) Thus, the forms for Ω and ω are 

 (2) ⎥

⎤ ⎢⎡ = 0

Ω ,

where 0 is a matrix of zeros, and 

 (3) 

The partitioning also can be used for the n×n matrix R defined as 

 (4) 

d ⎦⎣

m

0 Ω

Ω

⎢

⎣ −

d

m

0 Ω

ω

⎣

⎡ ′′ −

ddm

mdm

RR

RR

ωDffDDf fD ω 2/1 1 2/1

. 1 ⎥

⎦

⎤⎡ = 0

ω

( ) ⎢ ⎥ R = ω =

⎦

⎤

in which Df is the n×p sensitivity matrix of derivatives of f with respect to θ , 

 

 (5) ⎥

⎦

⎡ ∂ ⎤

Df = ⎢

⎣∂ j

i

θ

f

theoretically evaluated at the drift set of parameters ( ) = E θθ * , and 

7
Chapter 2. THEORY 

( )

( )

( )

( ) ⎪

⎪

⎭

⎪

⎪

⎬

⎫

= ′′

= ′′

= ′′

= ′′

−

−

−

−

2/1 1 2/1

2/1 1 2/1

2/1 1 2/1

2/1 1 2/1

ddd dd

dddm mm

md mm dd

mmm mm

R ω ωDffDDf fD ω

R ω ωDffDDf fD ω

R ω ωDffDDf fD ω

R ω ωDffDDf fD ω

, (6) 

where fm and fd are model-function and direct-information partitions of f, and D(...) is the 

derivative operator applied as in equation 5 to yield the nm×p and the nd×p matrices Dfm and Dfd. 

Similar partitioning is used for the n-vector Q, defined as 

 

 (7) ( ) ( ) ⎥

⎤ ⎢

′′ ⎡ = ω D Q ω Df ( ) , 2/1 1

2/1 1

2/1 1

⎣ ⎦ ⎥ =

⎦

⎤ ⎢

⎣

⎡

′′ ′′ = −

−

−

d

m

dd

m m

g

g

g Q

Q

ω ωDffDDf D

ω DffDDf ωDffD D

where [ gg θ∂∂= i D ′ ]

)

 is the column p-vector of sensitivities of the prediction g γθ).(

The UNC Process described in ‘The UNC Process’ section can be used to compute 

confidence or prediction intervals for parameters (θ ) of the Parameter-Estimation Process, and 

for most types of predictions ( or g(β)+ε where ε is a random observation error) that can 

be computed by a MODFLOW-2000 model calibrated by the Parameter-Estimation Process. The 

programs RESAN2-2k, BEALE2-2k, and CORFAC-2k that also are described in this report are 

valuable in the evaluation of results from the Parameter-Estimation Process and in the 

preparation of input values for UNC.

ˆ

)( * g γθ

The UNC Process 

The purpose of UNC is to compute confidence and prediction intervals for variables of 

the form ( . Both the model functions * g γθ ( ) * f γθ and predictions ( ) * g γθ can be nonlinear, and 

the intervals can be individual, Scheffé type, or, with correction factors not given in this report, 

other simultaneous types (Hill, 1994, p. 26-33). Calculation procedures are the same for all types 

of confidence intervals and for all types of prediction intervals; only the critical values described 

after equation 8 are different. Extensive information on the theory, calculation procedures, and 

use of confidence and prediction intervals can be found in Graybill (1976, chapters 6-10), Seber 

and Wild (1989, chapter 5), Hill (1994, p. 26-37), Cooley and Vecchia (1987), Vecchia and 

Cooley (1987), Cooley (1997), Christensen and Cooley (1999a, b), Cooley and Naff (1990, 

chapter 5), and Cooley (2004). Cooley (2004) summarizes much of the information found in the 

other references. 

Confidence Intervals 

A confidence interval for is defined by two confidence limits, the maximum and 

minimum values of over a likelihood region. The definition of the region depends on the 

type of confidence interval being computed. For example, for a Scheffé interval the likelihood 

( * g γθ )

g( ) γθ

8
Chapter 2. THEORY 

region is the standard confidence region given in terms of (θ)ˆ S , and for an individual confidence 

interval the likelihood region is of the same form but has a smaller diameter (Cooley, 2004, 

p. 53). If we assume that the maximum and minimum values of g(γθ) occur on the boundary of 

the likelihood region as they do for linear problems, then these extreme values can be found 

using the method of Lagrange multipliers (Vecchia and Cooley, 1987). Cooley (1999, p. 118) 

states that this assumption is almost non-restrictive; the assumption has never been violated in 

any of the calculations made by the authors of this report. Therefore, confidence and prediction 

intervals computed by the UNC Process are based on the method of Lagrange multipliers. 

A percent confidence interval is computed by finding extreme values (the 

confidence limits) of the Lagrange function 

( ) 1 α ×− 100

 ( ) () γθθ ( ( ) (θθ )) ˆ

2λ

1 , λ 2 gL α +−+= SSd , (8) 

where λ-1 is the Lagrange multiplier and is the critical value that depends on the type of 

confidence interval being computed. For an individual confidence interval 

2 dα

( ) ( pnStcd ) = c θ − ˆ 2

α 2/

2

α and for a Scheffé interval ( ) ( ) ( pnSpnpFcd ) α = r α , − θ

ˆ − 2 . Here cc

 and 

cr are correction factors defined in ‘The RESAN2-2k Program’ section, is the α 2/ t ( ) 1− α ×1002

percentile of the cumulative t distribution, ( − pnpF ) α , is the upper α point of the F distribution 

with p and n − p degrees of freedom, p is the number of model parameters, and n is the total 

number of observations used to estimate . Theoretically, the equations to compute extreme 

values are obtained simply by taking the derivatives of 

θ

ˆ

L(θ,λ) with respect to θ and λ-1, then 

setting the results to zero. However, this yields a set of nonlinear equations to be solved 

whenever or (or both) are nonlinear. Another method was developed by Vecchia 

and Cooley (1987) and involves linearizing the nonlinear functions 

g( ) γθ f ( ) γθ

g(γθ) and (or) first, 

then taking the derivatives of and setting the results to zero. The resulting linear system 

of equations then can be written to yield an iterative solution to the nonlinear problem. The 

iterative solution is derived in appendix A; the solution at iteration r+1 is given in terms of 

quantities computed at iteration r as 

f ( ) γθ

L(θ,λ)

θr+1

 

( ) ( ) ( ) () ( ) ( )

21

2 2/12/1

1

ˆ

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

− ′ −++− ±=λ α

+

rr

r r r r

r

SSd

QQ

θθ fY ωγθ R ω fY γθ (9) 

and 

λ ( ) ( ) ( ( )), 1 1

r 1 rr 1 rr gr rr r r θθ += ′ωDffD ′ + ′ωDffDD fD ′ω − fY γθ − −+ + (10) 

where subscript r indicates evaluation using parameter set and is the initial, user supplied, 

set. Use of the positive sign in equation 9 yields the maximum and use of the negative sign yields 

θr θ0

9
Chapter 2. THEORY 

the minimum. At convergence of the process θθθ

~ rr +1 ≈≈ , the set of parameters defining either 

limit ( ) θγ

~ g . 

To control round-off errors resulting from large variations in magnitude of the elements 

of and to put equation 10 into a form for the Marquardt-like solution scheme 

developed by Vecchia and Cooley (1987), Df

ωDffD rr′

r

, Dgr, θr , and θr+1 need to be scaled. Let Ar be a 

diagonal matrix composed of the square roots of the inverses of the diagonal elements of 

. Then let ωDffD rr′

= ADfS rrr , (11) 

rrr = DAZ g′ , (12) 

and 

( ) r Ad + −= θθ rrr

−

+ 1

1

1 . (13) 

With the above scaled matrix and vectors, equation 10 transforms to 

( ) ( ) ( ( )) r rrrr r d = ′ωSS + ωSSZ S′′ ω − fY γθ − −

+ +

1 1

1 1 λrr r . (14) 

Note that Qr and Rr in equation 9 are invariant under the scaling used for equations 11 and 12, so 

can be written in terms of Sr and Zr. 

Vecchia and Cooley (1987) found that a Marquardt-like modification (Seber and Wild, 

1989, p. 624) of equation 14 often was effective for solving ill-conditioned problems. We have 

used numerous tests to confirm this finding. The modification is simply to add μr+1I to rωSS r ′ in 

equation 14 to obtain 

( ) ( ) ( ( )) rr r rrrr r r d = r

′ωSS + r + ′ωSSZI + SI ′ω − fY γθ −

+

−

++ +

1

1

1 λ 11 μ 1 μ , (15) 

where μr+1 is a parameter obtained using the algorithm given by Cooley and Naff (1990, p. 71-

72). 

Another modification involves damping of the parameter displacement vector dr+1. This 

often is necessary to achieve convergence because dr+1 is frequently too large (Cooley and 

Vecchia, 1987). Thus, parameter set is computed using a damped form of equation 13: θr+1

θrrrrr θ = ρ dA +++ 111 + . (16) 

Damping parameter ρr+1 is computed using the algorithm given by Cooley (1993c, p. 2-3) except 

that tr+1 used by that algorithm is defined in the present report as the value of sj

r

j θδ +1 for which 

sj

r

r jj t max θδ 1

1

+

+ = , where is an element of , is an element of , and is a +1 δr

j rr +1 dA sj θ θs θs

10
Chapter 2. THEORY 

user-supplied scaling vector. The scaling vector adjusts for differing sizes of elements of , so 

should reflect the user’s best knowledge of , except that no element of can be zero. 

θ

θ* θs

We consider the iteration scheme to have converged if either r+1 t or 

( ) () ( ) () ( ) ( ) r+1 − rr r −+ r−1 SSSSSS θθθθθθ r−1 is small enough. The convergence criteria for 

r + 1 t and ( ) () ( ) ( ) ( ) ( ) r+1 − rr r −+ r−1 SSSSSS θθθθθθ r−1 are user supplied. Another test that 

overrides the above two also is used, if specified by the user. Sometimes can stabilize 

even though one or more elements of have not converged. To address this case the user can 

specify that the process has converged whenever 

( r+1 g γθ )

θ

( ( ) ( ))( ) () ( ) r r r gggg r 2 +1 −× +1 + γθγθγθγθ is 

smaller than another user-supplied criterion. This criterion is not used when 

( ) ( ) () .10 12

1

−

r+ gg γθγθ r <+

Weighted Residuals 

Cooley (2004, p. 56-57) showed that weighted residuals from a regression constrained so 

that ( ) ( * ) ~ = gg γθθγ (where θ

~ is the constrained regression estimate) can be defined as 

( ) QQQQI ω ( fY ( ) θγ ) 21 ~ − ′′ − and are equal to weighted residuals ω ( fY ( θγ )) ˆ 21 − from the 

unconstrained regression if model intrinsic nonlinearity and model combined intrinsic 

nonlinearity as defined by Cooley (2004, p. 35-36) both are small. Parameter set θ

~ can be the 

set computed for either confidence limit because minimization of S(θ), subject to the constraint 

() ( ), is the same as finding an extreme value of * = gg γθγθ g(γθ), subject to the constraint 

( ) ( ) 2 ˆ

θθ =− dSS α , if assumed for the constrained regression is the confidence limit for 

 computed using . Thus, the constrained weighted residuals for both confidence limits 

for are computed and printed if specified by the user. Vector Q is computed using the 

initial set of parameters because if θ

( * g γθ )

)

)

( * g γθ 2

α d

( * g γθ

θ0

~ were used, the term in ( QQQQI )ω ( ) fY ( θγ ) 21 ~ − ′′ −

expressing model combined intrinsic nonlinearity, that is( QQQQR )ω ( ) fY ( ) θγ 21 ~ − ′′ − , always 

would be zero, as shown in appendix B. 

Prediction Intervals

As for a confidence interval, a prediction interval for a predicted observation of g(ß), Yp, 

is defined by two limits, the maximum and minimum values of (γθ) + vg over a likelihood 

region, where v is the predicted error (Cooley, 2004, p. 59-60). The form of the likelihood region 

is developed in Cooley (2004, p. 61) and the prediction limits are assumed to lie on the edge of 

the likelihood region so that the method of Lagrange multipliers can again be used to find the 

extreme values. To obtain the Lagrange function we assume that the weight matrix incorporating 

the prediction, of Cooley (2004, p. 63), is block diagonal of the form ωa

 (17) ⎥

⎦

⎤ ⎢

⎣

⎡ = p

a 0 ω

ω 0

ω

11
Chapter 2. THEORY 

as described by Cooley (2004, p. 63). In equation 17, ωp is the weight for the prediction that is 

analogous to diagonal elements of ω. The form given by equation 17 not only simplifies 

calculations, but also does not require the user to estimate second moments between observations 

and the prediction, which often would be unknown. However, if such second moments actually 

exist so that the second-moment matrix that includes the prediction is of the form (Cooley, 2004, 

equation 5-87) 

⎥

⎦

⎤ ⎢

⎣

⎡

′ ω = −1 ˆ p

a C b

Ω C

Ω , (18) 

then a correction factor cp

, described in ‘The RESAN2-2k Program’ section, must be used. In 

equation 18, ( ) ( ) ( ( )) 2 −= * * σ− ε E fYC γθ p gY γθ , ( ( ))

2 2

*

1 ωˆ ε

−

p p −= gYE γθ σ , and 

trb ( ) n

2121 = Ωωω , where tr( ) K signifies matrix trace. 

On the basis of the assumed structure of , Cooley (2004, p. 65) stated the Lagrange 

function as 

ωa

 ( ) () θ γθ ( ( ) θ (θ)) ˆ

2

1 ,, 2 2

p +ω−− SvSdvgvL λ

++=λ α , (19) 

where critical value is equal to 2

α d ( ) ( ) ( pnSpntc ) p α − θ

ˆ − 2

2 for an individual prediction interval. 

As explained by Cooley (2004, p. 65), sometimes the second moment of , 

, can be estimated more easily than for use in equation 19. In this case, the 

term can replace the general weight ω

( ) * p − gY γθ

( ( ) 2

* p − gYE γθ ) )

ωp ˆ

( ( ) 2 2

* p sgYE − − γθ p in equation 19. 

Iterative solution of the extreme value problem based on equation 19 is derived in 

appendix A using the same method as used for equation 8. The solution is given as 

( ) ( ) ( ) () ( ) ( )

21

1

2 2/12/1

1

ˆ

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′ ω+

− ′ −++− ±=λ −

α

+

prr

r r r r

r

SSd

QQ

θθ fY ωγθ R ω fY γθ , (20) 

, (21) 1

1

1 +

−

r+ λω= rp v

and equation 10. All of the developments following equation 10 for a confidence interval are 

used for a prediction interval. 

The RESAN2-2k Program 

The purpose of analyzing residuals is to test whether or not the assumptions made for 

nonlinear regression and uncertainty analysis seem to be violated (Cooley and Naff, 1990, p. 

167). RESAN2-2k focuses on detection of model and system types of intrinsic nonlinearity as 

defined by Cooley (2004, p. 35-36), as well as the traditional graphical examination of residuals 

12
Chapter 2. THEORY 

plots for lack of model fit and testing of residuals for indications of non-normality. More on the 

purposes, theory, and methods of residuals analysis is given in Draper and Smith (1998, chapters 

2, 7, and 8), Cooley and Naff (1990, chapter 5), Hill (1992, p. 56-66, 88-89), Hill (1994), Cooley 

(2004), and references cited in these works. 

A Test for Model Intrinsic Nonlinearity

Model intrinsic nonlinearity causes confidence regions, confidence intervals, and 

prediction intervals to be too small unless they are corrected with a correction factor. Cooley 

(2004, section 5) showed that (1) the size of a confidence region is proportional to 

( pnpFc , where c r α , − )

)) r

 is the correction factor; (2) the size of a Scheffé interval is proportional 

to ( ( 21 , pnpFcr α − ; (3) the size of an individual confidence interval is proportional to 

( ) ( ) 21 2

2 pntcc α − , where cc

 is the correction factor; and (4) the size of an individual prediction 

interval is proportional to ( ) ( ) 21 2

2 pntcp α − , where cp is the correction factor. The correction 

factors are defined by 

( )

( ) ( ) pn

p

c

w I

wr Ir

r −σγ+σγ+σ

σγ+σγ+σ =

ε β ε

ε β ε

2 2 4

2 2 4

ˆˆ , (22) 

( ) ( pn ) c

w I

wc Ic

c −σγ+σγ+σ

σγ+σγ+σ =

ε β ε

ε β ε

2 2 4

2 2 4

ˆˆ , (23) 

and, for the form of prediction interval computed by UNC, 

( ) ( ) pn

c

w I

wa Ia

p −σγ+σγ+σ

σγ+σγ+σ =

ε β ε

ε β ε

2 2 4

2 2 4

ˆˆ . (24) 

Variables contained in equations 22-24 are defined as follows: is a scalar multiplier for the 

variance of the vector of system characteristics, , defined by 

2 σβ

β ( ) 2 Var β V σ= ββ ; is a scalar 

multiplier for the variance of the observation-error vector, , of order n, defined by 

; and are component 

correction factors defined by Cooley (2004, section 5). (The subscripts r and c were added in the 

present report.) 

2 σε

ε

( ) 2 V σ= εε Var ε ˆ , ˆ , , , , , , 42 42 42 2

Iw εβ wr Ir εβ wc Ic εβ waσγσγσγσγσγσγσγ β

4 and Iaσγ ε

The factors examined by RESAN2-2k, , measure the importance of model 

intrinsic nonlinearity. Cooley (2004, p. 50) showed that . Thus, only needs 

to be computed, and this factor is expressed by Cooley (2004, equation 6-3) as 

4 4 ˆ and I ε Irσγσγ ε

4 4 ˆ Ir ε Iσγ−=σγ ε4 ˆ Iσγ ε

13
Chapter 2. THEORY 

( () () ( ) ( )) ( ) ( ) ( )

( ) () () () () ( )

( ) () () () () ( ) ( ) ( ) ( )

( ) () () () ()

( ) ( ) ( ) ( ) ( ) ( )

( ) () () () () ( )

( ) () () () () ( ) ( ) ( ) ( ), ˆˆ ˆ ˆ

ˆˆ

ˆˆ 2

ˆˆ ˆ ˆ

ˆˆ

ˆˆ ˆ σγ 2

2121

*0* 0

21 21

*0* 0

*

21

*

21

21

*0* 0

2121

*0* 0

21 21

*0* 0

*

21 21

*0* 0

4

ε

f γθ f γθ f θγ f θγ fY ωωθγ fYR θγ

f γθ f γθ f θγ f ωθγ RI ω

RI ω fY γθ R ω fY γθ

f γθ f γθ f θγ f ωθγ

f γθ f γθ f θγ f θγ fY ωωθγ fYR θγ

f γθ f γθ f θγ f ωθγ RI ω

f γθ f γθ f θγ f ωθγ RI ω fY γθ

− ′ −++−−

− ′ + +−−

− −− −

′ = +−−

− ′ −++−−

− ′ + +−−

− − ′ = +−−

E

E

E

E

E

E

m m m m

m m m m mmmm

mmmmm dddmd

m m m m m

I

⋅

⋅

⋅

 (25) 

where ( is the linear-model approximation of *0 f γθ ) ( )* f γθ defined in Cooley (2004, equation 

5-104 and text preceding equation 6-2), f ( θγ )ˆ 0 is the linear-model approximation of f ( θγ )ˆ

defined in Cooley (2004, equation 5-104 and text preceding equation 6-2), and I is the identity 

matrix. 

The last term of equation 25 is the expected value of the product of row vector 

 and its transpose. If Df in R is evaluated using , then is 

always 0 (Cooley, 2004, p. 39). Also, if model intrinsic nonlinearity is absent, then R is constant 

(that is, the same for any set ) (Cooley, 2004, p. 39). Thus, a test for the importance of intrinsic 

nonlinearity in the denominators of the correction factors is to compare the estimate of the last 

term of equation 25, , where R is not computed using θ , with 

another appropriate term in equations 22-24. This term is obtained from equation 5-16 of Cooley 

(2004, p. 48), which shows that model intrinsic nonlinearity is negligible in the denominator of 

the correction factors if 

fY ωθγ R

2/1 ))ˆ − (( ′ θ

ˆ )) ˆ (( 2/1 ω − fYR θγ

θ

)) ˆ )) (( ˆ (( 2/12/1 − fY ′ ωωθγ − fYR θγ ˆ

( ) ( ) 4 2 2 2 ˆ ˆ ε ε β σ−=σγ+σ−<<σγ ε pn apnb I w , where a and b are factors that 

equal 1 if , the inverse of the correct weight matrix for the Gauss-Markov method of 

weighted regression to find θ (Cooley, 2004, section 4). Specifically, 

= Ωω−1 b

ˆ

= (R( ) ( ) ωΩω ) = ( (R ( mm btrpbbtra ) (ωΩω mm ) )+ ( ) Rd ) pbtrb 21 21 21 21 // / / (26) 

and 

trb ( ) (trn ( ) ) nn = = + dmmm

2121 2121 Ωωω ωΩω . (27) 

Because is estimated by s 2 bσε

2

 defined by (Cooley, 2004, p. 50) as 

( )

apn

S

s − = θ

ˆ 2 , (28) 

( ) 2 σ− ε apnb is estimated by (θ)ˆ S , where 

14
Chapter 2. THEORY 

( ) ( )) ˆ ()ˆ ()ˆ (θ fY ωθγ − fY θγ ′ S −= . (29) 

Thus, the test is that )) should be much smaller than ˆ )) (( ˆ (( 2/12/1 − fY ′ ωωθγ − fYR θγ (θ)ˆ S if R

is computed using some reasonable set of parameters not equal to θ . A set of direct 

observations or initial estimates for should work. If desired, vector 

θ ˆ

θ* ( )) ˆ ( 21 ω − fYR θγ can be 

examined to determine where in observation space the model intrinsic nonlinearity is large. 

Because , the test also can be used to assess the importance of model 

intrinsic nonlinearity in the numerator of c

4 4 ˆ Ir ε Iσγ−=σγ ε

r

, except that the test criterion should be changed. 

From Cooley (2004, p. 48) model intrinsic nonlinearity is negligible in the numerator of cr when 

4 2 2 2

ε ε β σ=σγ+σ<<σγ ε p bap Ir w . The term is estimated by 2 bapσε = (θ) ( − apnapSaps ) ˆ 2 , so the 

test is that )) should be much smaller than ˆ )) (( ˆ (( 2/12/1 − fY ′ ωωθγ − fYR θγ (θ) ( ) − apnapS ˆ . 

When a is unknown, the more conservative criterion (θ) ( − pnpS ) ˆ should be used. 

A Test for Model and System Types of Intrinsic Nonlinearity 

Cooley (2004, p. 38-39) showed that both the mean weighted residual 

n (where i i/)) ˆ Σ (( 2/1 ω − fY θγ 21 21 ωi i of row is ω ) and the slope of the plot of weighted residuals 

)) ˆ (( 21 ω − fY θγ i in relation to weighted model functions ω f ( θγ )ˆ 21

i should be small if the model 

 is adequate and both model and system types of intrinsic nonlinearity are small. Formal t

tests could be used to determine the significance of these measures, but graphical examination as 

performed by Cooley (2004, section 7) on test problems and comparisons with synthetic residual 

sets as described next should be adequate to detect significant model and system types of 

intrinsic nonlinearity as well as significant model inadequacy as described in the references cited 

at the beginning of this section. 

)ˆ f ( θγ

A Test for Normality of the Weighted Residuals 

Examination of the probability plot of weighted residuals ω ( fY ( θγ )) ˆ 21

i − will sometimes 

reveal obvious departures from normality. However, as discussed by Cooley and Naff (1990, p. 

168-170), departures from a standard normal distribution are expected for weighted residuals 

because they always are correlated and heteroscedastic. Under ideal conditions of no model or 

system intrinsic nonlinearity and zero-mean normal distributions of , Cooley (2004, p. 

38) showed that the weighted residuals have the normal distribution 

 and εβ

( ( )) ( ( )( ) ( ) ( ) ) 21 21 21 2 − θγ

ˆ N ,~ − RI0 ωΩω RI bbb σ− ε ω fY . (30) 

If Gauss-Markov estimation is used so that = Ωω , then equation 30 becomes −1 b

( ( )) ( ( ) ) 21 2 − θγ

ˆ N ,~ RI0 bσ− ε ω fY , (31) 

15
Chapter 2. THEORY 

which is of the form of the distribution of standard linear-regression weighted residuals. 

The weighted residual distribution can be graphically compared with synthetic weighted 

residual distributions generated according to equation 30 (or 31) as described by Cooley and 

Naff (1990, p. 168-171) to determine whether or not the weighted residuals appear to have the 

specified normal distribution. However, a more precise test also can be used. To make the test, 

first synthetic weighted residuals are generated from equation 30 or 31 (with the estimate 

) as follows. 2 2 replacing ε s bσ

 1. Standard normal deviates are generated from ( ) 2,~ I0u sN . Vector is initialized as 

. 

uT

T = uu

2. If specified in the input, u is modified to be the synthetic, weighted, true error vector 

= , Luu where L is the lower triangular Cholesky factor of T ( ) ( ) 21 21 ωΩω bb . (See 

Kitanidis (1997, p. 237-238).) 

 3. Vector is modified according to equation 30 (or 31) to give , the set of 

synthetic, weighted residuals for realization k. 

uT ( ) T

k −= uRId

 4. Steps 1-3 are repeated for a large number of realizations, M. 

Note that skipping step 2 yields synthetic weighted residuals distributed according to equation 31 

(with the estimate ). This step would be skipped if Gauss-Markov estimation 

were used or if 

2 2 replacing ε s bσ

Ω were unknown, in which case would have to be estimated by 2

ε bσ

(θ) ( − pnS ) ˆ or a in the estimate would have to be approximated using equation 76 in ‘The 

CORFAC-2k Program’ section of this report. 

2

s

Next, the following quantities are computed: 

= ∑ , k

k dd M (32)

the mean n-vector over all M realizations, 

, (33) ⎥

⎦

⎤ ⎢

⎣

⎡ = ∑

k

k

i

k s i dd

the vector of sum of squared values of over all realizations, and k

i d

v = [ ( ) − iii (MddMs −1) ], (34) 

the sample standard deviation n-vector. 

A normal probability plot of vectors ± 2vd gives a band within which the weighted 

residuals usually might be expected to lie (Cooley, 2004, section 7). Even though ii ± 2vd

defines an approximate 95 percent individual confidence interval, interpretation of the plot is 

16
Chapter 2. THEORY 

limited because each interval is an individual one so that all intervals in the band cannot be 

interpreted simultaneously. A better test for normality is obtained as follows as a generalization 

of a test given by Shapiro and Francia (1972) and used by Hill (1992, p. 63). First, the grand 

mean is computed: 

= ∑

i

i ndd , (35) 

Next the square of the correlation between synthetic weighted residuals for each realization k and 

the means over all realizations is obtained: 

 

( )

( ) .

2 2

2

∑ ∑ ⎟ − ⎠

⎞ ⎜

⎝

⎛ − ∑

⎟

⎠

⎞ ⎜

⎝

⎛

∑ ⎟ − ⎠

⎞ ⎜

⎝

⎛ − ∑

=

i i

i

j

k

j

k

i

i

i

j

k

j

k

i

k

ddndd

ddndd

c (36) 

Then, all values of c

k

 are ordered from smallest to largest. 

By definition, the probability that c

k

 has some specified value cs or a smaller value is 

∑ ( ≤= )

k

k

s MccIP , (37) 

where ( ) k

s

k

s

k

s ≤ ccI indicator theis = => when1 and when0 function ≤ ccIccI . Therefore, a 

95 percentile is given by ( ) int and 950int where, (L) 1 1 M.kcc k

s = = indicates truncation to the 

nearest integer. Similar 90 and 99 percentiles also can be computed. These percentiles should be 

compared to cd, the square of the correlation between weighted residuals ( ( )) ˆ ˆ 21 e ω −= fY θγ ii and 

the means i d , which is 

( )

( ) . 

ˆˆ

ˆˆ

2

2

2

∑ ∑ ∑

∑ ∑

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −

=

i i

i

j

i j

i

i

j

i j

d

ddnee

ddnee

c (38) 

The comparison will tell if the weighted residuals are significantly different from the specified 

normal distribution at some predetermined level of significance equal to 100 minus the selected 

percentile. Also, by letting , the probability P of c ds = cc d

 or a smaller value can be obtained, 

which is a more direct test. 

17
Chapter 2. THEORY 

The BEALE2-2k Program

The purpose of BEALE2-2k is to compute measures of total model nonlinearity, model 

intrinsic nonlinearity, and model combined intrinsic nonlinearity in the vicinity of θ . These 

measures relate to component correction factors that compose 

correction factors c

ˆ

44 4 4 ˆ and , , , IrI εε Ic ε Iaσγσγσγσγ ε

r

, cc, and cp. 

A Measure of Average Total Model Nonlinearity 

Total model nonlinearity is the sum of model intrinsic nonlinearity and parameter effects 

nonlinearity, the latter of which is nonlinearity that can be eliminated by some generally 

unknown transformation of parameters (θφ ) (Draper and Smith, 1998, p. 528-529). In effect, 

intrinsic nonlinearity is the smallest possible total model nonlinearity for any transformation of 

parameters. The measure of total nonlinearity was developed to determine when linear theory 

can be used to compute confidence regions (Beale, 1960; Guttman and Meeter, 1965; Cooley and 

Naff, 1990, p. 187-188). 

Cooley and Naff (1990, p. 187-188) defined a measure of average total nonlinearity as a 

combination of measures developed by Beale (1960) and Linssen (1975). For reasons discussed 

by Cooley (2004, p. 85) the measure used in this report was altered slightly from the one defined 

by Cooley and Naff (1990). It is defined as the average weighted sum of squared discrepancies 

between nonlinear and linear model values on the edge of the linear probability region having 

diameter and centered on the drift set of parameters 2 σε apb θ . Therefore, the measure of 

average total model nonlinearity is calculated using estimates as (Cooley, 2004, p. 87) 

( () ()) () () ( ) ( )

( () ()) () () ( ) ( ),2 1

2 1 ˆ

2

1 2 0 0

2

1 2 0 0

∑ − ′ = −

∑ − ′ = −

=

=

p

l

lm lmmlm lm

p

l

l l l l

p

aps

p

aps

N

f γθ f ωγθ f γθ f γθ

f γθ f ωγθ f γθ f γθ

 (39) 

where () () L L indicate linear model values, and the sets 0 0m and ff l p,2,,2,1 , θl = L are 

calculated as given by Cooley and Naff (1990, p. 189): 

θθ λ+= ( ωDfD ) lf ′′ −1 ˆˆˆ l , (40) 

in which 

( )

21

1

2

ˆˆ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′′ ±=λ −

ωDfDl lf

aps (41) 

18
Chapter 2. THEORY 

and l is the row p-vector 

l = [ LL 0 1 0 0 ], (42) 

with the 1 being in column l. Sensitivity matrix . Note that the diameter of 

the linear confidence region can be written in terms of c

fD θ

ˆ at evaluated is ˆ

2

aps r as evaluated in CORFAC-2k 

(which assumes no model intrinsic nonlinearity) as 

( ) ( ) ( ) ( )

pn

S

pc

pn

S

p

apn

apn

apn

apS aps r − = −−− = − = θ θ θ ˆ ˆ ˆ 2 . (43) 

If Ω is unknown so that a cannot be calculated using equation 26, then a may be estimated using 

equation 76 derived in ‘The CORFAC-2k Program’ section. 

By analogy with the standard criteria for ranking nonlinearity first given by Beale (1960, 

p. 60) and later augmented by Cooley and Naff (1990, p. 189), we consider the model to be 

highly nonlinear if so that the numerator of is greater than the diameter of the 

confidence region; nonlinear if ; moderately nonlinear if ; and 

essentially linear if . Linear theory for computing confidence regions seems to produce 

good approximate results when (Cooley and Naff, 1990, p. 189). 

1 Nˆ > Nˆ 2

aps

09.0 ˆ 1 N >≥ 01.0 ˆ 09.0 N >≥

01.0 Nˆ ≤

09.0 Nˆ ≤

A Measure of Average Model Intrinsic Nonlinearity

Average model intrinsic nonlinearity is assessed by the measure modified by 

Cooley (2004, p. 85) from forms given earlier by Beale (1960) and Linssen (1975). As for the 

measure of total nonlinearity, this measure is defined in terms of the linear probability region 

centered on 

min Nˆ

θ and having diameter . It is calculated using the same estimates and linear 

confidence region as used for . Therefore, the model intrinsic nonlinearity measure is 

calculated as (Cooley, 2004, p. 85) 

2 σε apb

Nˆ

( ) () () ( () () ) ( )

(( ) () () ( ) () ()

() ) ,)2( ˆ ˆ

ˆ ˆ 1

2 ˆ ˆ 1 ˆ

2

1 2 0 0

2

1

min 2 0 0

p

aps

p

aps

N

lddld

p

i

lm lm lmmlm lm lm

p

l

l l l l l l

fD ωψ fD ψ

f γθ f γθ fD ωψ f γθ f γθ fD ψ

f γθ f γθ fD ωψ f γθ f γθ fD ψ

′ +

∑ −− ′ = −−

∑ −− ′ = −−

=

=

 (44) 

where is computed using equations 40-42, and θl

 (45) ( ) ( ) () ()

( ) ( ) () () . ˆˆˆ

ˆˆˆ

0

1

0

1

lmmm lm

l l l

ωDfD fDf ω f γθ f γθ

ψ ωDfD fDf ω f γθ f γθ

= ′′ −

= ′′ −

−

−

19
Chapter 2. THEORY 

The measure is similar to the second expected value in given by equation 

25. Hence, the measure provides another test of the importance of model intrinsic nonlinearity in 

c

min

2

Naps ˆ 4 ˆ Iσγ ε

r

, cc, and cp in addition to the test using the term given after 

equation 29. By analogy with the former test, should be much smaller than 

)) ˆ )) (( ˆ (( 2/12/1 − fY ′ ωωθγ − fYR θγ

min

2

Naps ˆ (θ)ˆ S when 

applied as a test of the size of and should be much smaller than when applied as a 

test of the size of . Therefore, the test criterion for is that should be much 

smaller than (n−ap)/(ap), and the test criterion for is that should be much smaller 

than 1. The ranking used to classify nonlinearity for also can be used for because 

is just the smallest possible value for . 

4 ˆ Iσγ ε

2

aps

4

Irσγ ε

4 ˆ Iσγ ε min Nˆ

4

Irσγ ε min Nˆ

Nˆ min Nˆ min Nˆ

Nˆ

A Measure of Model Combined Intrinsic Nonlinearity for Confidence Intervals 

Model combined intrinsic nonlinearity affects cc because is a component of this 

correction factor. Cooley (2004, equation 6-20) expressed the component correction factor 

as 

4

Icσγ ε

4

Icσγ ε

( ) () () ( ) ( )

( ) () () () () ( ) ( )

( ) () () () ()

( ) () () () ()

( ) () () () () ( ) () () () ()

( ) ( ) ( ) ( ) ˆ σγ , ~ ~

~~ ~~

~~

~~

~~

~~ σγ 2

4

ε

21 21

*0* 0 *0* 0

21

*0* 0

21

*0* 0

*

21

*0* 0

21

*0* 0

4

ε

I

Ic

E

gggg

gggg

E

gggg

E

−⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

′ − ′ −+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+−− ′ −+−−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

′ +− ⎟

⎟

⎠

⎞

′

′ +−−−

⎜

⎝

⎛ ′ + +−−

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

′ +− ⎟

⎟

⎠

⎞

′

′ +−−−

⎜

⎝

⎛ ′ = +−−

ω fY θγ QQ

QQ fY ωθγ R

θγθγγθγθ QQ

Q ω f γθ f γθ f θγ f θγ

QQ

QQ RI

QQ

Q θγθγγθγθ

f γθ f γθ f θγ f ωθγ

ω fY γθ QQ

QQ RI

QQ

Q θγθγγθγθ

f γθ f γθ f θγ f ωθγ

⋅

 (46) 

where ( ) ,

~ or with, g = * θθθγθ is a model prediction for which a confidence interval is to be 

computed, g0 ( ) γθ is the linearized approximation of it analogous to f0 γθ)( , and θ

~ is the 

weighted regression estimate of that is constrained so that θ* ( ) ( θγγθ ) ~ * = gg . 

Cooley (2004, p. 86-87) developed a measure of model combined intrinsic nonlinearity, 

, similar to the second expected value in equation 46. As for , the measure is defined 

in terms of a likelihood region centered on 

min Mˆ min Nˆ

θ , but this time having diameter , where 2 bσξ ε

=ξ Q′( ) ( ) ωΩω bb ′ = (QQQQ ′ ( mm b) (ωΩω mm b) + ddm b) ′′ QQQQQ 21 21 21 21 . (47) 

20
Chapter 2. THEORY 

Thus, it is calculated using estimates as 

( ) () () ( () () )

(( ) () () ( ) () ()

() ) ,2 ˆ ˆ

ˆ ˆ

ξ

1

2 ˆ ˆ

ξ

1 ˆ

2

1 2 0 0

2

1

min 2 0 0

lddld

l

lm lm lmmlm lm lm

l

l l l l l l

s

s

M

fD ωψ fD ψ

f γθ f γθ fD ωψ f γθ f γθ fD ψ

f γθ f γθ fD ωψ f γθ f γθ fD ψ

′ +

∑ −− ′ = −−

∑ −− ′ = −−

=

=

 (48) 

where is given by Cooley (2004, p. 87) as θl

( ) g

s

l ˆ ˆˆ ˆˆ ˆ 1

21 2

′′ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

ξ ±= −

ωDfD Df QQθθ , (49) 

in which the carets over f, g, and Q indicate evaluation using θθ , = ˆ

( ) ( () () ll l gg l

g γθγθ QQ

ωDfD Df ψψ 0

1

0

ˆˆ

ˆ ˆˆ − ′

′′ +=

−

) (50) 

and 

( ) ( ) ( ) () ()

( ) ( ) ( ) () () . ˆˆˆ ˆˆ

ˆ ˆ ˆˆ

ˆˆˆ ˆˆ

ˆ ˆ ˆˆ

0

2/121 1

1

0

2121 1

1

0

lmmmm lm

m

l l l

g

g

ωDfD fDf ωω f γθ f γθ

QQ

ωDfD QDf

ωDfD fDf ωω f γθ f γθ

QQ

ωDfD QDf ψ

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− ′′ ′

′′′ −=

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ′′ ′

′′′ −=

−

−

−

−

 (51) 

Note that the diameter of the linear likelihood region can be written in terms of c

2 ξs c

 as 

evaluated in CORFAC-2k (which assumes no model combined intrinsic nonlinearity) as 

( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) . ˆ ˆˆ ξ

21 21 21 21

2

pn

S

c

pn

S

apn

bbpn

apn

bb S

s c − = ′ − −

− ′ = ′ −

′ = θθ

QQ

θ Q ωΩω Q

QQ

Q ωΩω Q (52) 

If Ω is unknown so that cc

 cannot be exactly computed, then the approximate bound for cc given 

by equation 83 derived in the sub-section titled ‘The CORFAC-2k Program’ can be used. In all 

cases, Q is replaced with Q to make the calculations. ˆ

Note that as defined by equation 46 contains . A measure similar to the sum 

 uses two components in addition to . These are (Cooley, 2004, p. 86) 

4

Icσγ ε

4 ˆ Iσγ ε

( ) 4 ˆIIc σγ+γ ε min Mˆ

21
Chapter 2. THEORY 

∑( ) () () ( ) =

− ′ ξ =

2

1

2

2 0 ˆˆ 2 1 ˆ

l

U l l gg

s

B γθγθ QQ (53) 

and 

( ) () () ( () () )

(( ) () () ( ) () ()

() ) .2 ˆ ˆ

ˆ ˆ

ξ

1

2 ˆ ˆ

ξ

1 ˆ

0 0

2

1

0

0

0

2 0

2

1

0

0

0

2 0

lddld

l

lm lm lmmlm lm lm

l

L l l l l l l

s

s

B

fD ωψ fD ψ

f γθ f γθ fD ωψ f γθ f γθ fD ψ

f γθ f γθ fD ωψ f γθ f γθ fD ψ

′ +

∑ −− ′ = −−

∑ −− ′ = −−

=

=

 (54) 

The measure of model combined intrinsic nonlinearity related to the sum is given by the 

maximum in absolute value of and (Cooley, 2004, p. 87). BM U

ˆ 2 ˆ min + BM L ˆ 2 ˆ min −

From Cooley (2004, equation 5-49), model combined intrinsic nonlinearity is negligible 

in the correction factor cc

 if ( ) ( ) 4 2 2 21 21 2 2 / ε ε β ε σξ= ε =σγ+σ<<σγ ′ bbb σ ′ b Ic wc Q ωΩω QQQ . Thus, 

( ) ( ) 4 2 ˆ ε σξσγ+γ ε b IIc should be much greater than ( ) ( ) ( ) ξ+ξ=σξσ+σξ ε ε ε bapbb ap

2 2 2 , which has 

a value greater than 1, so that, as a conservative test criterion, and BM U

ˆ 2 ˆ min + BM L ˆ 2 ˆ min −

should both be much less than 1. It is likely that the same ranking used to classify nonlinearity 

for also can be used to classify model intrinsic nonlinearity for its measure, although this 

ranking might be conservative in the present instance. 

Nˆ

Cooley (2004, p. 66) showed that when model intrinsic nonlinearity and model combined 

intrinsic nonlinearity both are small, standard linear confidence intervals are accurate 

approximations of nonlinear intervals. Thus, because measures the magnitude of these two 

sources of nonlinearity, can be used to indicate when standard linear confidence intervals 

can be used. On the basis of preliminary results given by Cooley (2004, section 7), linear 

confidence intervals should be good approximations when . 

min Mˆ

min Mˆ

01.0 ˆMmin ≤

A Measure of Model Combined Intrinsic Nonlinearity for Prediction Intervals 

The development and measure of model combined intrinsic nonlinearity for prediction 

intervals is analogous to the development and measure for confidence intervals. The measure for 

prediction intervals was not explicitly derived by Cooley (2004) but follows from the equations 

derived for confidence intervals with some added terms obtained from equations 6-50 - 6-55 of 

Cooley (2004, p. 84-85). For the prediction intervals calculated by UNC, the diameter of the 

likelihood region analogous to , where 2 2 is ss a ξξ a ξ is defined using equation 7-12 from Cooley 

(2004, p. 113) as 

22
Chapter 2. THEORY 

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) . ω

2 ωω ωˆ

ω

2 ωω ωˆ ξ

1

21 21 21 21 21 1

1

21 21 21 21 21 1

−

− −

−

− −

′ +

′ + ′ − ′ + =

′ +

′ − ′ + =

p

mm mm ddm pp mm pm

p

pp p

a

b b b b b

bb bb

QQ

Q ωΩω QQQ C ω Q

QQ

Q ωΩω Q C ω Q

 (55) 

We assume in this report that errors ( )* p − gY γθ in predictions are not correlated with errors 

( in direct information so that * − fY dd γθ ) d = 0C . The three measures analogous to equations 

48, 53, and 54 are 

(( ) () () ( () () )

)

(( ) () () ( ) () ()

( ) ψωψ ) ,2 ˆ ˆ

ˆ ˆ

ξ

1

ψωψ 2

ˆ ˆ

ξ

1 ˆ

2

1 2 0 0

2

1

min 2 0 0

plpplalddald

l

lm lm lmmalm lm alm

a

plppl

l

l l al l l al

a

a

s

s

M

+ ′ +

∑ −− ′ = −−

+

∑ −− ′ = −−

=

=

fD ωψ fD ψ

f γθ f γθ fD ωψ f γθ f γθ fD ψ

f γθ f γθ fD ωψ f γθ f γθ fD ψ

 (56) 

∑( ) () () ( ( )) =

− − ′ ω+

ξ =

2

1

2 1

2 0 ˆˆ 2 1 ˆ

l

l l p

a

a

U gg

s

B γθγθ QQ , (57) 

and 

(( ) () () ( () () ) )

(( ) () () ( ) () ()

( ) ψωψ ) ,2 ˆ ˆ

ˆ ˆ

ξ

1

ψωψ 2 ˆ ˆ

ξ

1 ˆ

0 000

2

1

0

0

0

2 0

2

1

000

0

0

2 0

plpplalddald

l

lm lm lmmalm lm alm

a

l

l l al l l al plppl

a

a

L

s

s

B

+ ′ +

∑ −− ′ = −−

∑ +−− ′ = −−

=

=

fD ωψ fD ψ

f γθ f γθ fD ωψ f γθ f γθ fD ψ

f γθ f γθ fD ωψ f γθ f γθ fD ψ

 (58) 

where is obtained from equation H-12 in Cooley (2004, p. 205) as θl

( ) g

s

p

a

l ˆ ˆˆ ˆˆ ˆ 1

21

1

2

′′ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′ ω+

ξ ±= −

− ωDfD Df QQ

θθ , (59) 

and remaining variables are obtained straightforwardly from equations 6-51 - 6-55 in Cooley 

(2004, p. 84-85) as 

23
Chapter 2. THEORY 

( ) ( () () l l

p

alal gg g γθγθ QQ

ωDfD Df ψψ 1 0

1

0

ˆˆ

ˆ ˆˆ − ′ ω+

′′ += −

−

), (60) 

( ) ( ) ( ) () ()

( ) ( ) ( ) () () , ˆˆˆ ˆˆ ω

ˆ ˆ ˆˆ

ˆˆˆ ˆˆ ω

ˆ ˆ ˆˆ

0

2/121 1

1

1

0

2121 1

1

1

0

lmlmmmm

p

m

l l

p

al

g

g

ωDfD fDf ωω f γθ f γθ QQ

ωDfD QDf

ωDfD fDf ωω f γθ f γθ QQ

ωDfD QDf ψ

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ′′ ′ +

′′′ −=

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ′′ ′ +

′′′ −=

−

−

−

−

−

−

 (61) 

( () () l l

p

p

plpl gg γθγθ

QQ 0 1

1

0

ˆˆ ω

ω

ψψ − ′ + −= −

−

), (62) 

( ) () ()

( ) () () . ˆ ˆˆ ω

ω

ˆ ˆˆ ω

ω

ψ

0

2/1

1

1

0

21

1

1

0

lmmm lm

p

p

l l

p

p

pl

Q ω f γθ f γθ

QQ

Q ω f γθ f γθ

QQ

′ − ′ + =

′ − ′ + =

−

−

−

−

 (63) 

The criteria for assessing the importance of model combined intrinsic nonlinearity in cp

are completely analogous to the criteria pertaining to cc; equations 5-93 and 7-12 of Cooley 

(2004) replace equation 5-49 of Cooley (2004) in the development. Thus, and a

U

a BM ˆ 2 ˆ min + a

L

a ˆ 2BM ˆ min − should be much less than 1, and the ranking used to classify nonlinearity for 

can again be used to classify model combined nonlinearity for its measures. 

Nˆ

The measure can be used in the same way as was used for confidence 

intervals to indicate when standard linear prediction intervals should be accurate approximations 

of nonlinear ones. Thus, linear prediction intervals should be good approximations of nonlinear 

ones when . 

a M min ˆ min Mˆ

ˆ min ≤ 01.0 a M

The CORFAC-2k Program

The purpose of CORFAC-2k is to compute correction factors cr, cc, and cp, together with 

components of them, assuming that model intrinsic nonlinearity and model combined intrinsic 

nonlinearity are negligible. The component correction factors , , , and 

are equal to zero when weight matrix is equal to the inverse of the matrix 

2 ˆ wσγ β2

wrσγ β 2

wcσγ β

2

waσγ β

ω Ω . When Ω is 

unknown, approximations and bounds are developed for the correction factors. 

Calculation of the Correction Factors When Ω Is Known

Component correction factor is defined using equations 5-10 and F-133 of Cooley 

(2004) as 

2 ˆ wσγ β

24
Chapter 2. THEORY 

(( ) ( ) ) 2 2 ˆ w β 11 pbanb σ−−−=σγ ε , (64) 

and is defined using equations 5-12 and F-135 of Cooley (2004) as 2

wrσγ β

( ) 2 2

wr β 1 pba σ−=σγ ε , (65) 

where a and b are given by equations 26 and 27 and (Cooley, 2004, p. 52). Note that if 

, and if . Correction factor c

a ≥1

 then ,1 ˆ σγ 0 2 b ≥ w β ≤ 2

β

2 then ,1 ˆ β σγσγ b = w −= wr r

 is obtained by 

combining equations 64 and 65 according to equation 22, assuming and are both 

zero, to get 

4 ˆ Iσγ ε4

Irσγ ε

( )

apn

apn cr −

− = . (66) 

Note from equations 26 and 27 that if ω is proportional to, but not equal to, , then −1 Ω a = 1, but 

 and component correction factors from equations 64 and 65 are both nonzero. However, b

cancels when obtaining equation 66 so that only has to be proportional to for c

b ≠1

ω −1 Ω r

 to be 

equal to 1. Also note that at least when model intrinsic nonlinearity is negligible, cr≥1, so that 

uncorrected confidence regions and uncorrected Scheffé intervals would be too small unless 

. To compute c −1 ∝ Ωω r, a and thus Ω have to be known. 

Component correction factor is defined using equation F-146 of Cooley (2004, p. 

187) and equation 47 as 

2

wcσγ β

. (67) 2 2 )1( wc β b σ−ξ=σγ ε

Variable given by equation 52 is not necessarily greater than 1, so is not necessarily 

greater than zero. Correction factor c

ξ 2

wcσγ β

c

 is obtained by combining equations 64 and 67 according to 

equation 23, assuming and are both zero, to get 4 ˆ Iσγ ε

4

Icσγ ε

( )

apn

pn cc −

− ξ = , (68) 

where again ω only has to be proportional to Ω for cc to equal 1. Matrix Ω has to be known in 

order to compute cc. 

Finally, component correction factor is defined using equation 5-93 of Cooley 

(2004, p. 63) and equation 55 as 

2

waσγ β

( ) 2 2

wa β b a 1 σ−ξ=σγ ε . (69) 

25
Chapter 2. THEORY 

As for equation 67, is not necessarily greater than 1, so is not necessarily greater than 

zero. Correction factor c

a ξ 2

waσγ β

p is obtained by combining equations 64 and 69 according to equation 

24, assuming and are both zero, to get 4 ˆ Iσγ ε

4

Iaσγ ε

( )

apn

pn c a

p −

− ξ = , (70) 

which behaves like cc does. However, cp can differ from 1 even when is equal to because 

of the term involving C in equation 55. 

ω −1 Ω

Approximate Calculation of the Correction Factors When Ω Is Unknown

When m a a Ω ξ and , , unknown, is ξ must be approximated. To obtain the approximations, 

we set equal to the diagonal weight matrix composed of the inverses of the diagonals of 

, we let , and we set 

ωm ωmˆ

Ωm

1 ˆ − == Ωωω ddd ωp equal to ωp ˆ . (Usually, will only be an 

approximation of the inverse of the diagonals of . This is discussed briefly later in this 

section.) Then we make use of approximations from equations 5-20, 5-56, and 5-117 of Cooley 

(2004). To approximate a we use equation 5-20 in equation 5-10 of Cooley (2004), noting that 

b=1 and so that equation 5-20 need only be applied to the model function partition. 

Therefore, 

ωmˆ

Ωm

−1 = Ωω dd

tr(( )( ) ( ) ) trcc ( ) ( ) apntcstc ˆ

mm 1 mm RI1IRI

ˆ

dd 1 dmm −=++−=−++−− , (71) 

where c is an effective correlation, is an 1m ×nn mm matrix of 1’s, partitions of Rˆ are partitions 

of R computed using ωˆ , and the traces are evaluated as 

( ) ( ) ∑( ) ∑ , (72) = =

−=−=−= m nm

i

m ii

n

i

m mmm ii trt rnr

1 1

1 ˆ ˆ ˆ IRI

( ) ( ) ∑∑( ) ∑∑ , (73) = = = =

−=−δ=−= m m nm m

i

n

j

m ij

n

i

n

j

m mmm ijij trs rnr

1 1 1 1

ˆ ˆ ˆ 1RI

( ) ∑( ) ∑ . (74) += +=

−=−=−=

n

ni

d ii

n

ni

d dd ii

m m

trt rnr

1 1

1 ˆ ˆ RI

ˆ

In equations 72-74, is an element of ij rˆ Rˆ and ij δ is the Kronecker delta ( ji ij =δ when1 = and 

ji ). ij when0 ≠=δ

Equation 71 can be solved for c by making use of the fact that pntt + dm −= . The result 

is 

26
Chapter 2. THEORY 

( )

mm st

pa c −

− = 1 , (75) 

which can serve as a definition of c. Note that if . The condition has always 

been found to hold. If c is presumed known, then equation 75 can be solved for a to get 

cst ≥> 0 , mm mm > st

( ) st mm +−= 1

p

c

a , (76) 

which has a lower bound of a=1 when c=0 and an upper bound of ( ) pst 1+ − mm when c=1. By 

assuming a reasonable upper limit for c, ce

, an approximate upper limit for a can be computed 

using equation 76. 

The variable ξ can be bounded using equation 5-56 from Cooley (2004, p. 56). However, 

a more refined approximation can be obtained as follows. From equation 5-56 from Cooley 

(2004) and the fact that the approximation need not be applied for the direct information, 

(Q ωΩω ) ( ) QQQQQQQQQ ˆˆˆˆˆˆˆˆˆ ˆˆ ˆ 2121 =ξ ′ + ′′ +≤ ′′ ddmmmmm V ddmx , (77) 

where 

 (78) 

( )

2

max ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∑

si

mi s

Vmx Q

and the notation indicates the maximum of either the squared sum of positive values of the Qmi’s 

or the squared sum of negative values of the Qmi’s (Cooley, 2004, p. 55-56). Also, if is 

diagonal so that , then so that . Therefore, a 

value of always exists so that 

Ωm

mmm m ωΩω = I 2/12/1 ˆˆ Q ωΩω QQQ mmmmmmm

ˆˆˆ ˆˆ ˆ 2/12/1 ′ = ′ =ξ 1

ce ≤ 1

(( ) QQ ˆˆ ) QQQQ ˆˆˆˆ 1−≤ξ ′ ++ ′′ Vcc ddmxemme . (79) 

It is almost certain that equation 79 with ce

<1 would produce a closer bound than equation 77. 

Finally, is bounded using equation 5-117 of Cooley (2004, p. 68) and a development 

completely analogous to the one used to obtain equation 79 to obtain 

a ξ

(( )( ) ) ( ) 1 1 ˆ ˆˆˆˆ ˆ ˆˆ 1 − −

a −≤ξ ′ Vc mxaepmme ++ω+ dd ′′ ω+ p c QQ QQQQ , (80) 

where Vmxa is evaluated using equation 78 as 

27
Chapter 2. THEORY 

( )

( )

⎪

⎭

⎪

⎬

⎫

+= +

=

∑

∑

− − 2ωˆ ωˆ negative of suma as for values

positive of suma as for values

12121

si

mxa mx mxp p mi

si

mxa mx mi

VV V Q

VV Q

. (81) 

Correction factors cr, cc, and cp are approximated by letting c = ce and using equations 71, 

76, 79, and 80 to get 

( )

( ) dmeme

mme

r tsctc

pstc

p

pn c

++−

− − + ≈ 1 , (82) 

( )

( ) dmeme

ddmxemme

c tsctc

pn c Vc

c

++−

− ′ ++ ′

′

− ≈ 1

ˆˆ ˆˆ 1

ˆˆ

QQ QQ

QQ

, (83) 

and 

( )( )

( ) dmeme

mxaepmme dd

p

p tsctc

pn c Vc

c

++−

− ′ +++ ′

′ +

− ≈

−

− 1

ω ˆˆ ˆ ˆˆ 1

ωˆ ˆˆ

1

1

QQ QQ

QQ

. (84) 

Note that if ce=0 is correct, then cr=1, cc=1, and cp=1, which also are correct. If ce=1, then cr, cc, 

and cp, attain their maximum values, which could be very large because the denominators could 

be very small. For the correction factors examined thus far in tests by the authors of this report, 

ce=0.8 has yielded bounds for all values of cc and cp. This value of ce also has yielded bounds for 

most values of cr and has yielded good approximations of cr for the remainder. 

For field problems will only be an approximation of the matrix of inverses of the 

diagonal elements of . The results of example 2 of Cooley (2004, section 7) suggest that a 

reasonable estimate of yields very little error in the correction factors. Further work by the 

senior author suggests that use of the approximation for obtained for the Tude aa case 

studied by Christensen and Cooley (1999b) also yields very little error. It appears that careful 

hydrogeologic work can yield a satisfactory estimate for . 

ωmˆ

Ωm

ωmˆ

ωmˆ

ωmˆ

Approximate Calculation of the Correction Factors When Ω Is Partly Known

In order to make the approximations in equations 71, 76, 79, and 80, we assumed the 

matrix to be similar to a correlation matrix so that all elements have magnitudes ≤1 

(Cooley, 2004, p. 48-49). This implies that is diagonal with elements equal to the inverses of 

the diagonal elements of . Matrix has the form 

2/12/1 ˆˆ ωΩω mmm

ωmˆ

Ωm Ωm ε +VV mm , where is the diagonal block 

in for model functions and , which is the 

model-error second-moment matrix for model functions divided by (Cooley, 2004, p. 18-19). 

Observations such as streamflow gains and losses may have measurement errors that have 

Vεm

Vε

2

* * /))()())(()(( σε = fV β − f γθ f β − f γθ ′ m E m m m m

2 σε

28
Chapter 2. THEORY 

known correlation so that is not diagonal. In this case, assuming to be diagonal would 

result in a loss of information. If is known, then to preserve this information should be 

written in the form 

Vεm ωmˆ

Vεm ωmˆ

, (85) 1 ˆ )(

− ωm ε += DV mm

which gives the form , where D 2/12/1 ˆˆ ωΩω mmm

2/1

ε ε

2/1

ε ))(()( − −

m += mm ++ DVVVDVC mmmm m is 

the diagonal matrix of diagonal elements of Vm. Unless Vεm is diagonal, Cm does not have a 

correlation-like form. However, as Vεm approaches 0, Cm approaches the matrix , 

which is similar to a correlation matrix; as V

− − 2/12/1 DVD mmm

m approaches0, Cm approaches I, the identity matrix; 

and if Vεm is diagonal, Cm has the standard form. Thus, Cm has the form assumed for the 

approximations in the limits. We assume that elements of Cm have magnitudes ≤1 (at least in 

final effect) for all matrices , so that equations 82, 83, and 84 can be used to compute 

correction factors when is defined by equation 85. (This assumption held in tests made 

during development of the computer code for this report.) Note, however, that as in the standard 

case where is diagonal, D

Vεm

ωmˆ

ωmˆ m is approximate because Vm is unknown. 

29
Chapter 3. USER GUIDE TO THE UNC PROCESS 

Chapter 3. USER GUIDE TO THE UNC PROCESS 

When to Use the UNC Process 

If one is confident that a linear confidence or prediction interval will be an accurate 

approximation for the actual interval, then a parameter confidence interval can be computed by 

the Parameter-Estimation Process of MODFLOW-2000, and a confidence or prediction interval 

for dependent variables like hydraulic heads or head-dependent flows can be computed by 

YCINT-2000. Otherwise the UNC Process should be used with MODFLOW-2000 to compute 

nonlinear approximations for the confidence or prediction intervals for both parameters and 

dependent variables. 

Significant bias in the model can cause a non-random distribution of residuals. To check 

residuals for randomness, the graphical procedures described by Cooley and Naff (1990, p. 167-

171) can be used. If these analyses show that the residuals do not appear to be random, the model 

is likely to be biased. In this case the model should be modified in order to better describe the 

properties and processes of the actual physical system. 

To check for a normal distribution of residuals, standard statistical procedures such as the 

normal probability plot can be used. In this context it is important to note that residuals usually 

are correlated and have unequal variances because of the regression method (Cooley and Naff, 

1990, p. 167-168). Therefore, if residuals analysis indicates that the residuals are correlated and 

(or) have unequal variances (for example, if the normal probability plot for the weighted 

residuals does not exhibit a linear trend), a control group should be developed for comparison 

with the weighted residuals. For this purpose Cooley and Naff (1990, p. 168-170) described a 

graphical procedure, but it is better to use the measure of weighted residuals correlation and the 

percentiles computed by the RESAN2-2k program described in this report. 

In ground-water modeling whether or not a linear confidence or prediction interval is a 

good approximation of the actual interval depends on the degree of total nonlinearity of the 

regression model (also called the total model nonlinearity). If the total model nonlinearity is 

significant, then linearized methods (of, for example, YCINT-2000) should not be used to 

compute confidence or prediction intervals, and MODFLOW-2000 with the UNC Process should 

be used to compute the more accurate nonlinear intervals. However, Cooley (2004, p. 66) has 

shown that, if model intrinsic nonlinearity and model combined intrinsic nonlinearity both are 

small, then linear intervals should be good approximations. (See remark near the end of this subsection on the need to use correction factors for calculation of linear intervals.) 

Beale’s (1960, p. 54-55) empirical measure of nonlinearity as corrected by Linssen 

(1975) quantifies the total model nonlinearity. Cooley and Naff (1990) published a code that can 

be used to compute the modified (corrected) Beale’s measure, and Hill (1994) and Hill and 

others (2000) modified the code of Cooley and Naff (1990) to work with MODFLOWP and 

30
Chapter 3. USER GUIDE TO THE UNC PROCESS 

MODFLOW-2000 (termed BEALE-2000), respectively. The present report describes a 

significantly expanded version called BEALE2-2k that also works with MODFLOW-2000. 

BEALE2-2k not only computes measures of total model nonlinearity, but also measures of 

intrinsic and combined intrinsic nonlinearity that are important for the computation of the critical 

value that is described by equation 86 given in this section. A comparison of the modified 

Beale’s measure of total model nonlinearity with the ranges summarized in chapter 2 will 

indicate whether or not the regression model is approximately linear. If the model appears to be 

approximately linear, then linear intervals might not differ much from intervals computed by 

UNC. However, note from Hill (1994, p. 47) that, if the quantities for which confidence or 

prediction intervals are to be calculated are very different from the observations used in the 

regression, then the modified Beale’s measure may be an inadequate measure of model linearity. 

Therefore, if, for example, hydraulic head observations were the only type of data used for 

nonlinear regression, then linear intervals for model parameters or head dependent flows may not 

be good approximations of the actual intervals. A better indication than Beale’s measure of total 

model nonlinearity is whether or not the intrinsic nonlinearity and combined intrinsic 

nonlinearity both are small. 

Model intrinsic nonlinearity and model combined intrinsic nonlinearity should be 

quantified using both RESAN2-2k and BEALE2-2k. If the tests indicate that both types of 

intrinsic nonlinearity are small, then linear confidence or prediction intervals can be used instead 

of nonlinear intervals calculated by UNC. However, it must be remembered that correction 

factors defined by equations 22-24 also must be used to correct the likelihood region from which 

linear confidence or prediction intervals are calculated. (For individual confidence intervals, 

 should be used instead of ; for Scheffé type confidence intervals, 

should be used instead of ; for individual prediction intervals, should be used 

instead of .) 

2

α 2/ tcc

2

α 2/ t ( ) pnpFcr α , −

α ( , − pnpF ) 2

α 2/ tcp

2

α 2/ t

Changes to MODFLOW-2000 to Include the UNC Process 

The UNC Process consists of a number of subroutines (called modules in MODFLOW) 

necessary to compute confidence and prediction intervals by the Vecchia and Cooley (1987) 

method. UNC subroutines that are called from the MAIN program of MODFLOW-2000 have 

names that begin with UNC1NLI1. Subroutines called from within the UNC1NLI1 subroutines 

have names beginning with SUNC1NLI1. 

The UNC Process has been incorporated into MODFLOW-2000 by making a few 

changes to the MAIN program: “UNC” was added to element 50 of the CUNIT array; calls to ten 

UNC1NLI1 subroutines were added; and calls to OBS1BAS6SS and OBS1BAS6OT are 

cancelled when the UNC process is active. Minor modifications were made in the following 

seven MODFLOW-2000 subroutines: OBS1BAS6HAL, OBS1BAS6FAL, OBS1DRN6AL, 

OBS1DRT1AL, OBS1GHB6AL, OBS1RIV6AL, and OBS1STR6AL. For each of these 

subroutines two calling arguments were added that have to do with the reading of dependent 

31
Chapter 3. USER GUIDE TO THE UNC PROCESS 

variables for which confidence or prediction intervals are to be computed. This is explained in 

the ‘Hydraulic Heads and Head-Dependent Flows’ section. 

Activation of the UNC Process 

To activate the UNC Process of MODFLOW-2000 the name file must include a short 

input file of type UNC that contains information necessary for the uncertainty calculations. The 

input instructions are given in the ‘Input Instructions’ section. When the UNC Process is 

activated, the Observations (OBS), Sensitivity (SEN), and Parameter-Estimation (PES) Processes 

also must be activated. 

The UNC Process is used for three different actions that should be carried out in the 

following sequence: 

 1. Generation of the input file for the CORFAC-2k program. 

 2. Generation of the input files for the BEALE2-2k program. 

 3. Computation of the confidence or prediction intervals. 

BEALE2-2k (and RESAN2-2k) should be used to study the significance of total nonlinearity, 

model intrinsic nonlinearity, and model combined intrinsic nonlinearity. The correction factors 

computed by CORFAC-2k are based on the assumption that both types of intrinsic nonlinearity 

are small. If testing using BEALE2-2k and RESAN2-2k indicates that this is not the case, then 

the correction factors computed by CORFAC-2k can be inaccurate. The correction factors are 

used in action 3. The desired action of UNC is set by the input variable IACT described in the 

‘Explanation of Variables in UNC Input File’ section. 

Types of Parameter-Dependent Functions For Which Intervals Can Be 

Computed By the UNC Process 

MODFLOW-2000 with the UNC Process can be used to compute confidence or 

prediction intervals for parameters of the Parameter-Estimation Process, and for most types of 

predictions that can be computed by a MODFLOW-2000 model calibrated by the ParameterEstimation Process. The types of predictions for which the package works include hydraulic 

heads, hydraulic head differences, head-dependent flows computed by various head-dependent 

flow packages, and differences between flows computed by a flow package. The computation of 

intervals for the difference between flows computed by two different flow packages is not 

allowed. The UNC Process is prepared for the head-dependent flow packages for drains (DRN6), 

rivers (RIV6), general-head boundaries (GHB6), streams (STR6), drain-return cells (DRT1), and 

constant-head boundaries (CHD). 

If MODFLOW-2000 with the UNC Process is used to compute confidence or prediction 

intervals for predictions, and if the calibration simulation period and/or stresses are different 

from the prediction simulation period and/or stresses, then MODFLOW-2000 must be set up to 

32
Chapter 3. USER GUIDE TO THE UNC PROCESS 

run the calibration simulation and the prediction simulation in sequence. (This is for example 

seen from equation 8 where g(γθ) is the prediction modeled by the prediction simulation and 

S(θ), which is defined in equation 1, quantifies the fit of the model to the observation data from 

the calibration period.) 

Determination of the Critical Value 

As mentioned in chapter 2, a confidence or prediction interval for a parameter-dependent 

function is defined by two limits, the maximum and minimum values of the function over a 

likelihood region. Definition of the likelihood region depends on the type of interval being 

computed. UNC needs an input value that defines the likelihood region for each interval to be 

computed. This input value is called the critical value and is computed as 

() () ( ) ( ) , Σ

1 Σ ˆ ˆ ˆ ˆ ,, α

,, α

2

1 α

2

1 α ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− += − − − +=+=

pn

C

SC

pn

S SdSD pn

pn θ θ θ θ (86) 

where (θ)ˆ S is the weighted sum of squared residuals computed by MODFLOW-2000 using the 

weighted least-squares estimated parameter values, ; n is the number of observations used to 

estimate θ ; p is the number of estimated parameters (the number of elements in θ ); C is a 

correction factor explained below; and 

θ

ˆ

ˆ ˆ

( pnt ) Σ pn αα − 2

2/,, is for individual confidence and 

prediction intervals and is ( − pnpF for Scheffé-type confidence intervals. α , )

As described in chapter 2, a correction factor is necessary to correct the likelihood region 

for intrinsic nonlinearity and for the difference between the inverse of the observation covariance 

matrix, , and the weight matrix, , used in the regression to compute the parameter 

estimates, . The CORFAC-2k program can be used to compute this correction factor (labeled C

in equation 86) assuming that model intrinsic nonlinearity and model combined intrinsic 

nonlinearity are negligible. For an individual confidence interval the correction factor C is C=c

−1 Ω ω

θ

ˆ

c

; 

for a Scheffé-type confidence interval, C=cr

; and for an individual prediction interval, C=cp. The 

correction factors are defined by equations 22-24, calculated exactly by equations 66, 68, and 70, 

and approximated by equations 82-84. 

Three steps are used to determine whether or not the assumption of negligible model and 

model combined types of intrinsic nonlinearity is plausible for a specific case. The first step is to 

use BEALE2-2k to compute measures of intrinsic nonlinearity and combined intrinsic 

nonlinearity. Use of these measures is explained in the user guide to BEALE2-2k. The second 

step is to use RESAN2-2k to compute intrinsic nonlinearity test statistics. Use of these statistics 

is explained in the user guide to RESAN2-2k. The first two steps should be carried out before 

UNC is used to compute the confidence or prediction interval. If these results indicate that 

intrinsic nonlinearity and combined intrinsic nonlinearity are insignificant, then the third step is 

to use MODFLOW-2000 with UNC to compute the confidence or prediction interval with UNC 

33
Chapter 3. USER GUIDE TO THE UNC PROCESS 

set to print the weighted residuals from the constrained regression. (How to set UNC to do this is 

explained in the ‘Checking for Intrinsic Nonlinearity Using Weighted Residuals Produced By the 

UNC Process’ section.) After the computation, the weighted residuals from the constrained 

regression should be compared with the weighted residuals from the Parameter Estimation 

Process of MODFLOW-2000. The two sets of weighted residuals should be nearly the same if 

the intrinsic nonlinearity and combined intrinsic nonlinearity both are negligible. The model 

intrinsic nonlinearity and model combined intrinsic nonlinearity have been found to be negligible 

for all the ground-water models the authors have analyzed. 

Setting the Starting Values for the Parameters 

Starting values for the parameters used in the computation of a limit of a confidence or 

prediction interval can be specified in two ways. If NIB>0 in the UNC input file, then the 

starting values also are read as the B2 array from this file. (See the ‘Input Instructions’ section.) 

Otherwise the starting parameter values equal the values given as B in item 3 of the SEN file. 

Normally it is advantageous to use the parameter values estimated by nonlinear 

regression as starting values for the UNC computations. However, as described in the ‘Searching 

for the Global Extreme Value’ section, other starting values sometimes may be preferred in a 

search for a global extreme value. 

Setting the Scaling Vector 

As discussed in chapter 2, a user-supplied scaling vector is used in the computation of the 

damping parameter that is used to damp the parameter displacements computed by the Vecchia 

and Cooley (1987) procedure. If IBSC≠0 in the UNC input file, the scaling vector is read as the 

BS array from the UNC input file, otherwise the scaling vector is set equal to the starting 

parameter values given as B in item 3 of the SEN file. Also, if LN>0 for a parameter (in item 3 

of the SEN file), then the scaling value is log-transformed. If the scaling value for a parameter 

equals zero, then UNC automatically sets the scaling value equal to 1. 

Searching for the Global Extreme Value 

Vecchia and Cooley’s (1987) implementation of the likelihood method is based on the 

assumption that the set of parameters producing a confidence limit for a parameter-dependent 

function is located on the edge of the likelihood region. The validity of this assumption was 

investigated in the field case study of Christensen and Cooley (1996; 1999a) by using the 

following two-stage approach. First, each confidence limit was computed by using nine different 

sets of starting parameter values. The starting values were obtained by adding or subtracting as 

much as two linearized standard deviations to or from the values estimated by nonlinear 

regression. In the cases tested by Christensen and Cooley (1996; 1999a) the variation of starting 

values did not produce significant differences in the computed confidence limits for either the 

34
Chapter 3. USER GUIDE TO THE UNC PROCESS 

parameters or the simulated hydraulic heads. Next, the extreme values corresponding to the 

confidence limits were searched for inside the likelihood region by reducing the critical value, 

 Reduced critical values were obtained from equation 86 by varying between 0 and 

the theoretical value corresponding to the edge of the likelihood region. In all cases tested by 

Christensen and Cooley (1996; 1999a) the parameter sets giving the confidence limits were 

found on the edge of the region. 

. 2 D1 α− CΣ pn ,, α

The UNC Process includes facilities making it easy to search for global extreme values 

by the described approach. Various starting parameter values can be specified and used to 

compute interval limits in a single model run. This is done by setting the input value NIB equal 

to the number of different parameter starting values and by specifying the various starting values 

in the B2 input array (see ‘Input Instructions’). Similarly, interval limits for a parameterdependent function corresponding to nc

 various critical values, , can be computed in the 

same model run by doing the following: (1) Repeat the necessary information about the 

parameter-dependent function n

2 D1 α−

c

 times. For the hydraulic head at a specific point, for example, 

the MODFLOW-2000 Hydraulic-Head Observations input must be modified so that in item 1 

NH is increased by nc and NHI is set equal to nc, and so that the list of Hydraulic-Head 

Observations in item 3 is appended (at the end) by nc identical records of information for the 

hydraulic head at the point. (2) Set the value of IDSQ in the UNC input file equal to 1. (3) Set 

the nc elements of the DSQ array in the UNC input file equal to the various values of . 2 D1 α−

Addressing Convergence Problems 

The studies of Vecchia and Cooley (1987), Christensen and others (1998) and 

Christensen and Cooley (1996; 1999a) show that convergence problems sometimes occur when 

computing confidence intervals, even if convergence problems were not observed during the 

nonlinear regression estimation process. The convergence problems were caused either by 

parameter correlations or by lack of sensitivity to one or more parameters. Adjusting one 

parameter manually and computing the others by the regression algorithm often could rectify the 

problem. This procedure can be invoked automatically by UNC by allowing incremental 

adjustment of one parameter in an outer iteration loop, while the other parameters are computed 

in an inner iteration loop by the Vecchia and Cooley (1987) algorithm. A guide to how this can 

be done is given in the following paragraph. 

Suppose that the ground-water flow model has NPE estimated parameters, and suppose 

that a confidence limit cannot be computed within a reasonable number of iterations. If ITPR≠1, 

then set ITPR to 1 in the UNC input file, and repeat the computation to print the iteration log in 

the output file. Check the iteration log to see if the convergence problem seems to be caused by a 

specific parameter or a specific pair of parameters. If this is the case, modify the SEN input file 

(Hill and others, 2000, p. 72-75) so the problem parameter (or one of the pair) is specified in 

item 3 as the last parameter with ISENS>0. In the UNC input file specify IMAN, BDEL and 

TOLDSQ according to the instructions given in ‘Explanation of Variables in UNC Input File.’ If 

35
Chapter 3. USER GUIDE TO THE UNC PROCESS 

IMAN is not equal to 0, then the NPE’th parameter (with ISENS>0) is adjusted incrementally 

during an outer iteration loop, while the other NPE-1 parameters are computed in an inner 

iteration loop using the Vecchia and Cooley (1987) algorithm. A confidence or prediction 

interval cannot be computed in this case for the NPE’th parameter. BDEL is the initial increment 

by which the NPE’th parameter is changed. The algorithm reduces BDEL when approaching the 

solution. TOLDSQ is a percentage of deviation from the critical value that is accepted for 

sufficient convergence of the incremental computation procedure. 

When IMAN≠0 and incremental adjustment of one parameter is used, then weighted 

residuals at the confidence or prediction limit should not be computed, printed, and used to check 

for model intrinsic nonlinearity and model combined intrinsic nonlinearity. (The reason is that 

the sensitivity of the NPE’th parameter is not computed when IMAN≠0, and the weighted 

residuals can therefore not be computed correctly.) To avoid this UNC automatically sets 

IWRP=0 when IMAN≠0. 

Checking for Intrinsic Nonlinearity Using Weighted Residuals Produced By 

the UNC Process 

As described in chapter 2, a comparison of weighted residuals computed by UNC when 

computing a confidence or prediction limit with the weighted residuals from the unconstrained 

regression made by MODFLOW-2000 during the Parameter-Estimation Process will indicate 

whether or not model intrinsic nonlinearity and model combined intrinsic nonlinearity both are 

small. If the two sets of residuals are nearly equal, it indicates that the two types of intrinsic 

nonlinearity both are small. If the test on the contrary indicates that model intrinsic nonlinearity 

and model combined intrinsic nonlinearity may not be small, then the correction factor for the 

confidence or prediction interval computed by the CORFAC-2k may be inaccurate, which also 

would make the corresponding computed interval limits inaccurate. 

The weighted residuals from the constrained regression are printed to a file when 

IWRP>0 is specified in the UNC input file described in the ‘Input Instructions’ section. The 

weighted residuals are printed to this file for all computed interval limits. 

Hydraulic Heads and Head-Dependent Flows 

An interval for a hydraulic head computed by the Vecchia and Cooley (1987) method 

depends on the sensitivity of the hydraulic head to each of the model parameters. An easy way to 

have MODFLOW-2000 compute these sensitivities is to append information about the hydraulic 

heads to the list of hydraulic head observations in item 3 of the input to the Head-Observation 

Package (Hill and others, 2000, p. 38), and to increase NH in item 1 by one. If the OBS and SEN 

processes are active, the sensitivities are calculated for all hydraulic heads specified in the HeadObservation Package input file. To indicate that the last hydraulic head was not used to estimate 

the model parameters, but is a head for which an interval is to be computed, the input value of a 

new fourth variable, NHI, of integer type is appended to item 1 of the Head-Observation Package 

36
Chapter 3. USER GUIDE TO THE UNC PROCESS 

input file. The input value of NHI in this case must be equal to 1. For the head information 

appended to item 3, STATISTIC and STAT-FLAG define the weight ωp used to compute a 

hydraulic head prediction interval (see equations 17-21), but they are not used for computation of 

a confidence interval; HOBS is not used for either type of interval. 

If intervals are to be computed for more than one hydraulic head, the corresponding 

information must be appended to item 3 (and, optionally, items 4 through 6), and NH and NHI 

(and, optionally, MOBS and MAXM) must be changed correspondingly. If intervals are to be 

computed for differences between hydraulic heads, then the information for the second head of a 

pair must follow immediately after the information for the first head of the pair. That is, the NHNHI+1’th and NH-NHI+2’th heads of the input file constitute the first pair, the NH-NHI+3’th 

and NH-NHI+4’th heads of the input file constitute the second pair, and so on. For the prediction 

interval of a head difference the weight, ωp

, is computed by UNC as ωp=(ωp1

-1+ωp2-1)-1, where ωp1

and ωp2 are the weights defined by STATISTIC and STAT-FLAG in item 3 for each of the heads 

in the difference, respectively. 

If intervals are to be computed for flows (or differences between flows) of one of the 

head-dependent flow packages, information about these flows is appended similarly to the 

corresponding flow-observation input file. For example, for an interval for a General-Head 

Boundary (GHB) flow, the General-Head Boundary Flow Observation input file is modified by 

increasing NQTGB in item 1 by one, and either a new cell group is created by increasing NQGB 

in item 1 by one and appending new items 3, 4, and 5, or the last group is enlarged by increasing 

NQOBGB of the last group’s item 3 by one and appending a record of information to the 

corresponding item 4. If IOWTQGB in item 2 is greater than 0, then the variance-covariance 

matrix (WTQ) specified in item 7 must be expanded by a line and a column. Because only the 

diagonal value is used to compute the weight, ωp, for the added flow (and only for a prediction 

interval), all off-diagonal values can be set equal to 0. Finally, to indicate that the last input value 

of flow is not used in the regression but is used for the computation of a confidence or prediction 

interval, an input value of a new fourth variable, NGHBI, of integer type must be appended to 

item 1 of the General-Head Boundary Flow Observation input file. The input value of NGHBI 

for only one GHB flow interval must be equal to 1. If more than one GHB flow interval is to be 

computed, or if intervals are to be computed for differences between GHB flows, then 

information must be appended to and changed in the General-Head Boundary Flow Observation 

input file in a manner similar to that explained for the hydraulic head intervals. 

For intervals for head-dependent flows computed by other packages, flow information is 

appended, values of variables are changed, and input of a fourth variable to item 1 (indicating the 

number of appended flows that are to be used for computation of confidence or prediction 

intervals) are appended similarly for the flow observation input file of each of the flow packages. 

The total number of head-dependent flows used to define variables for which a confidence or 

prediction interval is computed equals NQI = NGHBI + NDRNI + NRIVI + NSTRI + NCHDI + 

NDRTI, where NGHBI, NDRNI, NRIVI, NSTRI, NCHDI, and NDRTI are the numbers of 

37
Chapter 3. USER GUIDE TO THE UNC PROCESS 

appended flows defined in the GHB, DRN, RIV, STR, CHD, and DRT Flow Observation input 

files, respectively. 

Input Instructions 

Computation of an upper or lower confidence or prediction limit is similar to the iterative 

solution of the nonlinear regression problem solved by the Parameter Estimation (PES) Process 

of MODFLOW-2000. The solution algorithm for interval limits therefore requires input values 

for variables also used by the solution algorithm of the PES Process. This includes MAX-ITER, 

MAX-CHANGE, IOSTAR, CSA and information about prior information (Hill and others, 2000, 

p. 78-83). UNC uses the values for these variables specified in the PES file. Other values 

important to the solution algorithm of UNC are read from the UNC input file described in the 

following section. 

Instructions for the UNC Input File 

Input in addition to that just explained is read from a file that is specified with file type 

“UNC” in the MODFLOW-2000 name file. Meanings of the following variables are given in the 

‘Explanation of Variables in UNC Input File’ section. 

0. [#Text] 

Item 0 is optional and can include as many lines as desired. Each line needs to begin with 

the “#” character in the first column. 

1. IACT NPI IDIF ISGN ITYP NIB IOIN IDSQ ITPR IMAN IWRP IBS (free 

format)

2. TOLP TOLS TOLY (free format) 

If IMAN is not equal to 0, then read item 3. 

3. BDEL TOLDSQ (free format)

4. DSQ(1), ... , DSQ(ndsq) (list directed input)

If IDSQ is equal to 0 then ndsq = 1. Otherwise ndsq = NHI + NQI + NPI when intervals 

are for heads and flows, or ndsq = NHI/2 + NQI/2 + NPI when intervals are for head and 

flow differences. (NHI and NQI are defined in the ‘Hydraulic Heads and HeadDependent Flows’ section.) 

Read NPI repetitions of item 5. 

5. PARNAMVAR (list directed input)

Read NIB repetitions of item 6. 

6. B2(1), B2(2), B2(3), ... , B2(NPLIST) (list directed input) 

NPLIST is the number of named parameters listed in the Sensitivity Process input file 

(Hill and others, 2000, p. 72). 

38
Chapter 3. USER GUIDE TO THE UNC PROCESS 

If IBS>0 read item 7. 

7. BS(1), BS(2), BS(3), ... , BS(NPLIST) (list directed input) 

If IACT=1 read items 8 and 9. 

8. CFR (free format)

9. CFI(1), ... , CFI(ncfi) (list directed input) 

Variable ncfi = NHI + NQI + NPI when intervals are for heads and flows, or ncfi = 

NHI/2 + NQI/2 + NPI when intervals are for head and flow differences. 

Example UNC Input File 

# Example UNC file 

# 

 0 4 0 0 1 2 61 0 1 1 0 1 ITEM 1: IACT NPI IDIF ISGN ITYP NIB IOIN 

IDSQ ITPR IMAN IWRP IBS 

0.01 0.001 0.0 ITEM 2: TOLP TOLS TOLY 

 0.1 0.05 ITEM 3: BDEL TOLDSQ 

 42.4383 ITEM 4: DSQ 

 RCH_ZONE_1 0.0 ITEMS 5: PARNAM VAR 

 RIVERS 0.0 

 SS_1 0.0 

 VERT_K_CB 0.0 

-1. 34. 50. 0.001 0.001 0.4E-3 0.2E-6 0.6E-4 0.5E-4 ITEM 6: B2

-.9 24. 40. 0.002 0.001 0.6E-3 0.3E-6 0.5E-4 0.6E-4 ITEM 6: B2

-1. 34. 50. 0.001 0.001 0.4E-3 0.2E-6 0.6E-4 0.5E-4 ITEM 7: BS 

In this example RCH_ZONE_1, RIVERS, SS_1, and VERT_K_CB are parameter names that 

need to be defined in Ground-Water Flow Process input files and need to be listed with ISENS>0 

in the Sensitivity Process input file (Hill and others, 2000, p. 72-75). The program matches 

parameter names among the various input files in a case-insensitive manner. Because IMAN is 1, 

the last parameter listed with ISENS>0 in the Sensitivity Process input file (not shown here) is 

incrementally adjusted during the calculation of confidence intervals for RCH_ZONE_1, 

RIVERS, SS_1, and VERT_K_CB. The incrementally adjusted parameter must therefore not be 

identical to RCH_ZONE_1, RIVERS, SS_1, and VERT_K_CB as indicated above. (In the 

example it is thus assumed that more parameters than RCH_ZONE_1, RIVERS, SS_1, and 

VERT_K_CB have been estimated.) 

Explanation of Variables in UNC Input File 

Text-----is a character string (maximum of 79 characters) that starts in column 2. Any characters 

can be included in the text. The “#” character needs to be in column 1. Text is printed 

39
Chapter 3. USER GUIDE TO THE UNC PROCESS 

when the file is read and provides an opportunity for the user to include information 

about the model both in the input file and in the associated output file. 

IACT---is a flag controlling the action of UNC. For IACT=0 confidence or prediction intervals 

are computed. For IACT=1 the _b1 and _b3 files for the BEALE2-2k program are 

generated. For IACT=2 the _b2 and _b4 files for the BEALE2-2k program are generated. 

For IACT=3 the input file for the CORFAC-2k program is generated. 

NPI-----is the number of intervals computed for estimated parameters (NPI≤NPE, where NPE is 

the number of parameters listed with ISENS>0 in the Sensitivity Process input file). 

IDIF----is a flag that controls whether the computed intervals pertaining to heads and flows are 

for differences of heads and differences of flows (IDIF=1), or for heads and flows 

(IDIF≠1). 

ISGN---is a flag that controls whether only lower limits (ISGN<0), or only upper limits 

(ISGN>0), or both lower and upper limits (ISGN=0) are computed for the intervals. 

ITYP---is a flag that controls whether confidence intervals (ITYP≠2) or prediction intervals 

(ITYP=2) are computed. 

NIB-----is the number of sets of parameter values (B2) to be read and used as starting parameter 

values for the computation of each limit of all confidence or prediction intervals. 

IOIN----is the unit number of an output file defined in the MODFLOW-2000 name file to be 

used for summary output of computed confidence or prediction limits. No other output 

should be directed to this file. If IOIN<2, no output summary is written; output is written 

only in the GLOBAL and/or LIST output files of MODFLOW-2000 (Harbaugh and 

others, 2000, p. 6-7). 

IDSQ---is a flag that controls whether the same critical value is to be used for the computation of 

all confidence or prediction intervals (IDSQ=0), or if individual critical values are to be 

read and used for calculation of each confidence or prediction interval (IDSQ≠0). 

ITPR----is a flag that controls the printing of information from iterations to the screen and to the 

LIST output file. If ITPR = 1 data from each iteration is printed; otherwise it is not. 

IMAN---is a flag that controls when the intervals are computed by an approximate procedure 

where one parameter is computed by incremental adjustment in an outer iteration loop, 

while the other parameters are computed in an inner iteration loop by the Vecchia and 

Cooley (1987) algorithm. The incrementally adjusted parameter is the last parameter 

listed with ISENS>1 in the Sensitivity-Process input file (Hill and others, 2000, p. 72-75). 

The approximate procedure is used when IMAN≠0. 

IWRP---is a flag that controls the printing of weighted residuals. If IWRP>0, they are printed in 

an output file defined in the MODFLOW-2000 name file with the unit number being 

equal to IWRP. If IMAN≠0, IWRP is automatically set equal to 0, and the weighted 

40
Chapter 3. USER GUIDE TO THE UNC PROCESS 

residuals will not be printed. When IWRP>0 then a scratch file with the unit number 

equal to IWRP+1 is generated automatically by the program. Other files used by the 

program must not use this unit number.

IBS------is a flag that controls setting the scaling values used in each iteration to compute the 

damping parameter used in equation 16. If IBSC≠0, the scaling vector, BS, is read as item 

6 of the UNC input file, otherwise BS is set equal to the starting parameter values given 

as B in item 3 of the Sensitivity Process input file (Hill and others, 2000, p. 72-75). 

TOLP----is the convergence criterion for the fractional parameter change. Convergence is 

achieved when max TOLP 1

1r <δ= ++ j

r

j j

t BS , where is the computed change for 

parameter j and BS

+1 δr

j

j

 is the scaling value for the parameter. 

TOLS----is the convergence criterion for the change of the objective function over two iterations. 

Convergence is achieved when ( ) ( ) ( ) ( ) ( ) ( ) r+1 − rr r −+ r−1 SSSSSS θθθθθθ r−1 < TOLS. 

A value of one tenth of TOLP has often been found to work well. 

TOLY---is the convergence criterion for the change of the value of the computed confidence or 

prediction limit. Convergence is achieved when 

( ) ( ) () ( ( ) ( )) r r r gggg r 2 +1 −× +1 + γθγθγθγθ < TOLY. This criterion is not applied to check 

for convergence of incremental parameter adjustment when IMAN≠0. 

BDEL---is the initial increment by which the NPE’th parameter is changed when IMAN≠0. The 

algorithm reduces BDEL when approaching the solution. 

TOLDSQ—is a percentage of deviation from the critical value that is accepted for sufficient 

convergence of the incremental computation procedure. 

DSQ-----is the critical value (equal to in equation 86) that determines the size of the 

likelihood region on the edge of which it is assumed that parameter sets producing 

maximum and minimum limits of a confidence or prediction interval are found. 

2 D1−α

PARNAM—is a set of parameter names. The names also must be specified in the Sensitivity 

Process input file and have values of ISENS>0. 

VAR-----is a value that defines the weight, , which is used to compute the 

prediction interval for the parameter (see equations 80-84). VAR is not used for the 

computation of a confidence interval, but a value must still be specified (for example 0). 

If the parameter was log-transformed during the Parameter-Estimation Process, VAR 

must be specified for the log

-1 ˆ p =ω VAR

10-transformed parameter value. 

B2-------is a set of parameter values used as starting values for the computation of confidence 

and prediction intervals. The specified values must not be log-transformed because this is 

done internally when necessary. If more sets (as specified by NIB) are read, then each 

interval is computed NIB times, each time using a different set of B2 as starting 

41
Chapter 3. USER GUIDE TO THE UNC PROCESS 

parameter values. If NIB=0, then B2 is automatically set equal to values given as B in 

item 3 of the Sensitivity Process input file (Hill and others, 2000, p. 72-75). 

BS-------is the scaling vector used in the computation of the damping parameter used in equation 

16. The scaling vector adjusts for differing sizes of elements of the parameter vector 

(defined as item 3 of the Sensitivity Process input file; Hill and others, 2000, p. 72-75) 

and should reflect the user’s best knowledge of the parameter values. For a logtransformed parameter (with LN>0 in item 3 of the Sensitivity Process input file), the 

scaling value is log-transformed internally by UNC. If the (possibly log-transformed) 

scaling value equals zero, then UNC automatically sets the scaling value equal to 1. 

42

)

CFR-----is the correction factor for the confidence region used to generate the _b1 and _b2 input 

files for the BEALE2-2k program when IACT equals 1. These input files are used by 

BEALE2-2k to compute measures of total model nonlinearity and model intrinsic 

nonlinearity. If CFR = α ( , − pnpF , then the generated _b1 and _b2 files will be identical 

to the corresponding files generated by MODFLOW-2000 for BEALE-2000. 

CFI------are correction factors for confidence intervals used to generate the _b3 and _b4 input 

files for the BEALE2-2k program when IACT equals 1. These input files are used by 

BEALE2-2k to compute model combined intrinsic nonlinearity measures.
Chapter 4. USER GUIDE TO RESAN2-2k 

Chapter 4. USER GUIDE TO RESAN2-2k 

RESAN2-2k performs two functions. The first function is to test residuals for indications 

of non-normality. This purpose is shared by earlier codes like RESAN (Cooley and Naff, 1990) 

and RESAN-2000 (Hill and others, 2000), but the analysis is somewhat different and more 

precise when using RESAN2-2k. The second function is to compute measures that can be used to 

detect model and system types of intrinsic nonlinearity. 

Testing Weighted Residuals for Indications of Non-Normality 

Like earlier versions of the residuals analysis program, RESAN2-2k generates sets of 

synthetic weighted residuals having the distribution given by equation 30 or 31. A normal 

probability plot of the actual weighted residuals distribution together with the synthetic sets will 

give a preliminary idea as to whether or not the weighted residuals appear to have the specified 

normal distribution, as described by Cooley and Naff (1990, p. 168-171). If a large number of 

synthetic sets of weighted residuals are generated, then the mean values, d (equation 32), and 

standard deviation, v (equation 34), over all realizations can be computed for the synthetic 

weighted residuals. The necessary number to obtain stable results is usually from several 

hundred to several thousand. A normal probability plot of ± 2vd gives a band within which the 

weighted residuals might usually be expected to lie. Figure 1 shows an example of a normal 

probability plot with weighted residuals plotted together with the corresponding band given by 

± 2vd computed from 1,000 realizations. The results used to produce this kind of plot are all 

written by RESAN2-2k to a file with extension _md. 

RESAN2-2k computes a measure of the correlation, cd (equation 38), between weighted 

residuals and the means of the synthetic residuals, and it computes the probability P that cd or a 

smaller value can be obtained. The larger the probability P, the larger is the probability that the 

weighted residuals have the theoretical distribution given by equation 30 or 31. RESAN2-2k also 

computes and prints 99, 95, and 90 percent confidence limits for the correlation. These measures 

are written by RESAN2-2k to the file with extension #nr, where they are listed the following 

way: 

CORRELATION (CED) --------------- = 0.98984 

PROBABILITY OF CORRELATION (PROB) = 0.71300 

99% CONFIDENCE LIMIT (CL99) ----- = 0.99624 

95% CONFIDENCE LIMIT (CL95) ----- = 0.99423 

90% CONFIDENCE LIMIT (CL90) ----- = 0.99304 

 

CED corresponds to cd, PROB corresponds to P, CL99 corresponds to the 99 percent confidence 

limit, CL95 corresponds to the 95 percent confidence limit, and CL90 corresponds to the 90 

percent confidence limit. 

43
Chapter 4. USER GUIDE TO RESAN2-2k 

-4 -2 0 2 4

WEIGHTED RESIDUAL

0.01

0.02

0.05

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.95

0.98

0.99

PROB

ABILITY

Weighted residual

Figure 1. Ordered weighted residuals plotted with ± 2vd confidence band (approximate 

95 percent confidence intervals). 

Testing for Intrinsic Nonlinearity 

As described in chapter 2, RESAN2-2k computes three measures that can be used to 

detect model intrinsic nonlinearity: the mean weighted residual, the slope of weighted residuals 

in relation to weighted model functions, and )) ˆ (())ˆ (( 2121 − fY ′ ωωθγ − fYR θγ . If the first two 

measures are small (close to zero), and if the third measure is much smaller than (θ)ˆ S (the 

weighted sum of squared residuals computed using the optimum parameter set, ), then the 

model intrinsic nonlinearity probably is small. In the RESAN2-2k output file with extension #nr, 

the measures are listed as 

θ

ˆ

MEAN WEIGHTED RESIDUAL (EM) --------- = 0.99903E-01 

SLOPE OF WEIGHTED RESIDUAL PLOT (SLP) = 0.56149E-05 

INTRINSIC NONLINEARITY MEASURE (QINT) = 0.56149E-05 (SHOULD BE<<36.506) 

EM and SLP are both close to 0, and QINT is much smaller than 36.506, which is the 

weighted sum of squared residuals at the optimum in this example. The three measures thus all 

indicate that model intrinsic nonlinearity is small. 

44
Chapter 4. USER GUIDE TO RESAN2-2k 

Input Instructions 

RESAN2-2k needs two input files that can be generated by MODFLOW-2000, the _rs 

file and the _ws file (Hill and others, 2000, tables 5 and 8). The first file is the input file also 

used by the RESAN-2000 program of Hill and others (2000). The content of the second file is 

weighted simulated functions and weighted residuals. To generate the files MODFLOW-2000 

must be run in the Sensitivty Analysis or Parameter-Estimation mode, and OUTNAM in the 

Observation Process input file must be specified as a string other than “NONE.” It is important 

that the two files be generated in two different steps using MODFLOW-2000: 

 1. The _rs file must be generated using a parameter set other than the optimal, θ . ˆ

2. The _ws file must be generated using the optimal parameter set, θ . ˆ

Before running RESAN2-2k the following changes must be made to the _rs file. First, 

NSETS in item 1 defines the number of sets of synthetic weighted residuals generated to test for 

normality of the weighted residuals. NSETS must be set to a value of several hundred or a few 

thousand. Second, values of additional input variables used by RESAN2-2k (but not by RESAN2000) can be added to item 1 as described in the ‘Content of the _rs Input File Needed by 

RESAN2-2k’ section if special features of the program are wanted. Third, depending on the 

value of two of the variables added to item 1, it may be necessary to append two new arrays, ET 

and V, to the file. The content of the input files is described in the ‘Content of the _rs Input File 

Needed by RESAN2-2k’ and the ‘Content of the _ws Input File Needed by RESAN2-2k’ 

sections. 

Output 

The RESAN2-2k program produces two files with extensions #nr and _md. The #nr file 

contains input values and summarizes the results from testing for nonlinearity and from testing 

weighted residuals for indications of non-normality. The summary of results is described in the 

‘Testing Weighted Residuals for Indications of Non-Normality’ and the ‘Testing for Intrinsic 

Nonlinearity’ sections. The _md file contains results that can be used to produce a probability 

plot of ordered weighted residuals together with the ordered means of simulated weighted 

residuals and a confidence band (for example, fig. 1). The content of the columns in the file is: 

(1) ordered weighted residuals; (2) ordered means of simulated weighted residuals; (3) standard 

deviations of simulated weighted residuals; (4) two times the standard deviations of simulated 

weighted residuals; (5) frequency/probability; probability plot position; (6) observation name; 

and (7) plot symbol identifier. 

Content of the _rs Input File Needed by RESAN2-2k 

RESAN2-2k reads the following from the _rs file. Explanations of the variables follow 

this list. 

45
Chapter 4. USER GUIDE TO RESAN2-2k 

1. NPAR NOBS NH NQT MPR IPR NSETS NRAN VAR STDV IPCF IPRN IERT ICOV 

(format: 7I5, I10, F25.0, F10.0, 4I5)

2. PARNAM(NPAR) (format: 6(A10,1X)) 

3. COV(NPAR,NPAR)(format: 16F25.0)

4. WT(NH) (format: 16F15.0)

5. WQ(NQT,NQT) (format: 16F15.0)

6. X(NPAR,NOBS) (format: 16F15.0) 

7. PRM(NP,I), WP(I), I=1,MPR (format: 8F15.0) 

8. NIPR(IPR) (format: 16I5) 

9. WTPS(IPR,IPR) (format: 8F15.0) 

If IERT is greater than zero, read item 10. 

10. ET(NOBS) (free format) 

If ICOV is greater than zero, read item 11. 

11. V(NOBS,NOBS) (free format) 

The variables have the following explanations. 

NPAR---is the number of estimated parameters. 

NOBS---is the number of model function observations used for estimation. NOBS must thus be 

equal to NH+NQT. 

NH-------is the number of head observations. 

NQT-----is the number of observations other than heads. 

MPR-----is the number of prior information equations

IPR------is the number of prior information observations with a full weight matrix. 

NSETS—is the number of sets of synthetic weighted residuals generated to test for normality of 

the weighted residuals. NSETS must be increased from 4 (the default value set by 

MODFLOW-2000) to a value of several hundred or a few thousand. 

NRAN---is the seed for the random number generator. 

VAR-----is the calculated error variance, (θ) ( −= pnSs ) ˆ ˆ

2 . 

STDV---is the theoretical error variance, , used to generate synthetic weighted residuals 

distributed as given by equation 30. If a value of 0 is specified, RESAN2-2k 

automatically sets it equal to , the calculated error variance from the least squares 

estimation, before the synthetic weighted residuals are generated. 

2 σε b

2

sˆ

IPCF-----is a flag. If IPCF is greater than zero, the program prints the I–R matrix. 

IPRN----is a printing flag. Weighted simulated residuals are printed when IPRN is greater than 

zero. 

IERT----is a flag. If IERT is greater than zero, the program reads the ET array described below. 

46
Chapter 4. USER GUIDE TO RESAN2-2k 

ICOV----is a flag. If ICOV is greater than zero, the program reads the V array described below. 

PARNAM—is the parameter name list. 

COV-----is the parameter covariance matrix. 

WT------are the square roots of weights for heads. 

WQ------is the square root of the full weight matrix for observations other than heads. 

X---------is the sensitivity matrix for all parameters and observations. (For log-transformed 

parameters the sensitivities are with respect to natural log-transformed parameters, not 

with respect to log10-transformed parameters.) 

PRM----are coefficients for the prior information equations. 

WP------are weights for prior information equations. 

NIPR----is the number list for parameters with prior information observations that have a full 

weight matrix. 

WTPS---is the square root of the full weight matrix for correlated prior information. (For logtransformed parameters the weights are for natural log-transformed parameters, not for 

log10-transformed parameters.) 

ET-------are the expected values of the true errors for the model function observations, 

, which are used to generate the synthetic, weighted, true error vectors 

used to test weighted residuals for normality. If IERT=0, then ET is not read but is 

assumed to be all zeros. 

( ( ) * E − fY mm γθ )

V------ is the matrix ( ) () ( ) ( ) 2

* * σε

′ Ωm E −= fY mm γθ − fY mm γθ for model function observations 

that is used to generate the synthetic, weighted, true error vectors from equation 30. If 

ICOV=0, then V is not read and equation 31 is used instead of equation 30. 

Content of the _ws Input File Needed By RESAN2-2k

 RESAN2-2k reads the following items from the _ws file NTOT times, where NTOT= 

NOBS+MPR+IPR. That is, the items are read once for each observation that was used to 

estimate the model parameters. 

1. F E ISYM DID (format: 2(G15.7,1X),I5,2X,A) 

The variables have the following explanations. 

F---------is the weighted simulated equivalent to the observation. 

E---------is the corresponding weighted residual. 

ISYM---is a plot symbol identifier (an integer value). 

DID-----is the observation name.

47
Chapter 5. USER GUIDE TO BEALE2-2k 

Chapter 5. USER GUIDE TO BEALE2-2k 

BEALE2-2k computes measures of total model nonlinearity, model intrinsic nonlinearity, 

and model combined intrinsic nonlinearity. The measures can be used as a guide for whether 

confidence or prediction intervals can be computed using linearized approximations (for 

example, using YCINT-2000 of Hill and others, 2000), or if nonlinear intervals should be 

computed using MODFLOW-2000 with UNC. The measures also indicate whether correction 

factors computed by CORFAC-2k may be inaccurate because of model intrinsic nonlinearity or 

model combined intrinsic nonlinearity. BEALE2-2k extends the capabilities of BEALE (Cooley 

and Naff, 1990) and BEALE-2000 (Hill and others, 2000) to include the measures of model 

intrinsic nonlinearity and model combined intrinsic nonlinearity in addition to the measure of 

total nonlinearity, which was all that was computed in BEALE and BEALE-2000. BEALE2-2k 

also redefines the measure of total model nonlineariy slightly to make it more directly useful for 

evaluating the accuracy of a linearized confidence region (Cooley, 2004, p. 85-87). 

Total model nonlinearity is quantified by calculating (equation 39), which is called 

BNT in the BEALE2-2k output file. The model is considered to be highly nonlinear if , 

nonlinear if , moderately nonlinear if , and essentially linear if 

. Linear theory for computing confidence regions seems to produce good approximate 

results when (Cooley and Naff, 1990, p. 189). 

Nˆ

1 Nˆ >

09.0 ˆ 1 N >≥ 01.0 ˆ 09.0 N >≥

01.0 Nˆ ≤

09.0 Nˆ ≤

Model intrinsic nonlinearity is quantified by calculating (equation 44), which is 

called BNI in the BEALE2-2k output file. The same criteria as used to rank values of are 

used for . In particular, if , then the model intrinsic nonlinearity is so small that 

correction factors for confidence regions, confidence intervals, and prediction intervals are 

probably not significantly dependent on such nonlinearity. 

min Nˆ

Nˆ

min Nˆ 09.0 ˆNmin ≤

Model combined intrinsic nonlinearity is computed either for confidence intervals or for 

prediction intervals. For a confidence interval this is quantified by (equation 48), 

(equation 53), and (equation 54). In the output file these variables are called BMI, BMF0, 

and BMG0, respectively. If the larger of and 

min Mˆ BU

ˆ

BL

ˆ

BM U

ˆ 2 ˆ min + BM L ˆ 2 ˆ min − (that is, BMI+2BMF0 

and the absolute value of BMI-2BMG0) is less than 0.09, then model combined intrinsic 

nonlinearity probably does not contribute significantly to the correction factor for the confidence 

interval although, as indicated in chapter 2, this criterion is apt to be conservative. The larger of 

 and BM U

ˆ 2 ˆ min + BM L ˆ 2 ˆ min − is called BMIMAX in the output file. A standard linear confidence 

interval should be a good approximation if 01.0 (Cooley, 2004, section 7). ˆM min ≤

For a prediction interval BMI, BMF0, and BMG0 in the output file correspond to 

(equation 56), (equation 57), and (equation 58), respectively. If the larger of 

and 

a Mmin ˆ

a BU

ˆ a BL

ˆ a

U

a BM ˆ 2 ˆ min + a

L

a ˆ 2BM ˆ min − (that is, BMI+2BMF0 and the absolute value of BMI-2BMG0) is less than 

0.09, then model combined intrinsic nonlinearity probably does not contribute significantly to the 

correction factor for the prediction interval although again this criterion is apt to be conservative. 

48
Chapter 5. USER GUIDE TO BEALE2-2k 

A standard linear prediction interval should be a good approximation if (Cooley, 

2004, section 7). 

ˆ min ≤ 01.0 a M

Input Instructions 

The nonlinearity measures are calculated by BEALE2-2k using four input files that can 

be generated by MODFLOW-2000 using UNC. The generated files have the extensions _b1, 

_b2, _b3, and _b4, and they are generated in two steps like the _b1 and _b2 files needed by 

BEALE-2000 (Hill and others, 2000, p. 92). Actually, the _b1 and _b2 files needed by BEALE2-

2k are almost identical to the _b1 and _b2 files needed by BEALE-2000 (Hill and others, 2000, 

tables 12 and 13), and they can be read and used by BEALE-2000. To generate the four files, 

MODFLOW-2000 must be run in the Uncertainty Process mode. OUTNAM in the Observation 

Process input file must be specified as a string other than “NONE.” 

First generate the _b1 and _b3 files as follows: 

1. In the Uncertainty Process input file, set IACT=1 in item 1 and specify CFR and CFI in 

items 8 and 9, respectively. 

 2. Substitute the final calibrated parameter values from the _b file into the Sensitivity 

Process input file. 

 3. Execute MODFLOW-2000. This run will generate the _b1 and _b3 files. 

Next, generate the _b2 and _b4 files and execute BEALE2-2k as follows: 

 4. In the Uncertainty Process input file, set IACT=2. 

 5. Execute MODFLOW-2000. This run will generate the _b2 and _b4 files. 

 6. Finally, execute BEALE2-2k, which directs output to a file with extension #be. It will 

overwrite a possible output file already produced by BEALE-2000. 

The content of the four input files is described in the four sections following the next section, 

‘Output from BEALE2-2k.’ 

Output from BEALE2-2k 

The BEALE2-2k program produces an output file with the extension #be. This file will 

overwrite a possible output file with the same name already produced by BEALE-2000 (Hill and 

others, 2000). The #be file contains input values and computed measures of total model 

nonlinearity, model intrinsic nonlinearity, and model combined intrinsic nonlinearity. The 

measures of total model nonlinearity and model intrinsic nonlinearity are listed in the file as 

TOTAL NONLINEARITY (BNT).......... = 0.70303E-01 

INTRINSIC NONLINEARITY (BNI)...... = 0.26567E-04 

49
Chapter 5. USER GUIDE TO BEALE2-2k 

The measures of model combined intrinsic nonlinearity for confidence (or prediction) 

intervals are listed (here for 7 intervals) as 

COMBINED INTRINSIC NONLINEARITY 

 INTERVAL NO. BMI INTERVAL NO. BMI 

 1 G1 0.91136E-01 5 G5 0.92165E-01 

 2 G2 0.13976E-04 6 G6 0.89142E-01 

 3 G3 0.80532E-01 7 HK_1 0.37137E-01 

 4 G4 0.92049E-01 8 HK_2 0.22188E-01 

 COMBINED INTRINSIC NONLINEARITY AS IF F WERE LINEAR 

 INTERVAL NO. BMF0 INTERVAL NO. BMF0 

 1 G1 0.12232E-01 5 G5 0.12491E-01 

 2 G2 0.37404E-02 6 G6 0.11832E-01 

 3 G3 0.99148E-02 7 HK_1 0.58584E-12 

 4 G4 0.12468E-01 8 HK_2 0.59462E-10 

 COMBINED INTRINSIC NONLINEARITY AS IF G WERE LINEAR 

 INTERVAL NO. BMG0 INTERVAL NO. BMG0 

 1 G1 0.36826E-01 5 G5 0.37051E-01 

 2 G2 0.40559E-02 6 G6 0.36261E-01 

 3 G3 0.34126E-01 7 HK_1 0.37137E-01 

 4 G4 0.37003E-01 8 HK_2 0.22186E-01 

 COMBINED INTRINSIC NONLINEARITY - MAX. SUM 

 INTERVAL NO. BMIMAX INTERVAL NO. BMIMAX 

 1 G1 0.11560 5 G5 0.11715 

 2 G2 0.80979E-02 6 G6 0.11280 

 3 G3 0.10036 7 HK_1 0.37137E-01 

 4 G4 0.11699 8 HK_2 0.22188E-01 

The various measures and their uses are explained in the beginning of this chapter. 

Content of the _b1 Input File Needed By BEALE2-2k 

 The _b1 file contains the following items. 

1. NPE NOBS NQ MPR IPR VAR CR (format: 5I10,2F14.0) 

2. PARNAM(NPE) (format: 6(A10,1X)) 

3. BOPT(NPE) (format: 6F13.0) 

50
Chapter 5. USER GUIDE TO BEALE2-2k 

4. OBSNAM(NOBS) (format: 6(A12,1X))

5. H(NOBS) (format: 6F13.0)

6. HOBS(NOBS) (format: 6F13.0)

7. WH(NH) (NH=NOBS-NQ) (format: 8F15.0)

8. WQ(NQ,NQ) (format: 8F15.0)

9. X(NPE,NOBS) (format: 6F13.0)

10. PRM(NPE+1,J), WP(J), J=1,MPR (format: 8F15.0)

11. NIPR(IPR) (format: 8I10)

12. BPRI(IPR) (format: 6F13.0)

13. WTP(IPR,IPR) (format: 6F13.0)

14. LN(NPE) (format: 8I10)

15. A line with the text: THE PARAMETER SETS FOLLOW 

Read 2×NPE repetitions of item 16.

16. B(NPE) (format: 8F13.0)

The explanation of variables follows: 

NPE-----is the number of estimated parameters. 

NOBS---is the number of observations used for estimation. 

NQ-------is the number observations other than heads. 

MPR-----is the number of prior information equations. 

IPR-------is the number of prior information observations with a full weight matrix. 

VAR------is the calculated error variance, (θ) ( −= pnSs ) ˆ ˆ

2 . 

CR--------is the correction factor for the confidence region (computed using CORFAC-2k). 

PARNAM—is the parameter name list. 

BOPT----are the optimized parameter values. 

OBSNAM—are the observation names. 

H---------are the simulated equivalents of the observations using the optimized parameter values. 

HOBS---are the observed values of hydraulic heads and head dependent flows (not used by 

BEALE2-2k). 

WH------are the weights for hydraulic heads. 

WQ------is the weight matrix for the observations other than heads. 

X---------is the sensitivity matrix for all parameters and observations. (For log-transformed 

parameters the sensitivities are with respect to natural log-transformed parameters, not 

with respect to log10-transformed parameters.) 

51
Chapter 5. USER GUIDE TO BEALE2-2k 

PRM----are coefficients for the prior information equations. 

WP------are weights for prior information equations. 

NIPR----is the number list for parameters with prior information observations that have a full 

weight matrix. 

BPRI----are the prior information observations (not used by BEALE2-2k). 

WTP-----is the full weight matrix for correlated prior information observations. (For logtransformed parameters the weights are for natural log-transformed parameters, not for 

log10-transformed parameters.) 

LN-------flag indicating whether each parameter is log-transformed. The parameter is logtransformed if LN≠0. 

B---------is a set of parameter values used to calculate measures of total model nonlinearity and 

model intrinsic nonlinearity. 

Content of the _b2 Input File Needed By BEALE2-2k 

The following two items are repeated 2 × NPE times. 

1. B(NPE) (format: 8F13.0) 

2. F(NOBS) (format: 6F13.0) 

These variables are: 

B---------are parameter values for one of the 2 × NPE sets of parameter values used to calculate 

measures of total model nonlinearity and model intrinsic nonlinearity. 

F---------are simulated equivalents of the observations, calculated using the preceding set of 

parameter values. 

Content of the _b3 Input File Needed By BEALE2-2k 

The _b3 file contains the following items. 

1. NOINT NHI NQI NPI ITYP IDIF (format: 6I10) 

2. GNAM(NOINT) (format: 6(A12,1X))

3. GOPT(NOINT) (format: 6F13.0)

4. WG(NOINT) (format: 8F15.0)

5. Z(NPE,NOINT) (format: 6F13.0)

6. CC(NOINT) (format: 6(1X,F13.0))

52
Chapter 5. USER GUIDE TO BEALE2-2k 

The variables are defined as: 

NOINT—is the number of intervals for which model combined intrinsic nonlinearity measure is 

calculated. 

NHI------is the number of intervals for hydraulic heads. 

NQI------is the number of intervals for head dependent flows. 

NPI------is the number of intervals for parameters. 

ITYP----is a flag for the type of interval: ITYP=1 is for confidence intervals; ITYP=2 is for 

prediction intervals. 

IDIF-----is a flag. When IDIF=1 it indicates in the output that intervals pertaining to heads and 

head dependent flows are for differences of heads or flows. When IDIF=0 it indicates that 

the intervals pertain to heads and head dependent flows. 

GNAM—is the name of each prediction for which model combined intrinsic nonlinearity 

measure is calculated. 

GOPT---is the predicted value using the optimized parameter values. 

WG------are weights for the predictions corresponding to ωp . Values must always be specified 

for WG whenever NOINT is greater than 0, but WG is only used in calculations 

pertaining to prediction intervals. For prediction intervals for log-transformed parameters 

WG must be specified for the natural log-transforms of the parameters, not for the log10-

transforms. 

Z---------is the sensitivity matrix for all parameters and predictions. (For log-transformed 

parameters the sensitivities are with respect to natural log-transformed parameters, not 

with respect to log10-transformed parameters.) 

CC-------are correction factors for each of the confidence or prediction intervals. 

Content of the _b4 Input File Needed By BEALE2-2k 

 The following two items are repeated 2 × NOINT times. 

1. B(NPE) (format: 8F13.0) 

2. F(NOBS) (format: 6F13.0) 

3. G (free format) 

The variables are defined as: 

B---------are parameter values for one of the 2 × NOINT sets of parameter values used to 

calculate measures of model combined intrinsic nonlinearity for confidence or prediction 

intervals. 

53
Chapter 5. USER GUIDE TO BEALE2-2k 

F---------are simulated equivalents of the observations, calculated using the preceding set of 

parameter values. 

G---------is simulated equivalent of the prediction, calculated using the preceding set of 

parameter values. 

The first two repetitions of items 1 and 3 are used by BEALE2-2k to compute a measure of 

model combined intrinsic nonlinearity for the first confidence interval. For these repetitions G is 

the simulated value of the first prediction. The third and fourth repetitions of items 1 and 3 are 

used to compute the same kind of measure for the second confidence interval, and G is the 

simulated value of the second prediction. This logic is continued for the remaining repetitions.

54
Chapter 6. USER GUIDE TO CORFAC-2k 

Chapter 6. USER GUIDE TO CORFAC-2k 

The purpose of CORFAC-2k is to compute correction factors for confidence regions, 

confidence intervals, and prediction intervals. In the development of the procedures implemented 

in CORFAC-2k, Cooley (2004, section 5) assumed that the model intrinsic nonlinearity and 

model combined intrinsic nonlinearity are negligible. As described in previous chapters, the 

validity of this assumption can be investigated using BEALE2-2k and RESAN2-2k, and by 

analyzing weighted residuals computed by the UNC Process. 

The correction factors are to be used to calculate the critical value ( , equation 86) 

needed by UNC for the calculation of confidence and prediction intervals, and by BEALE2-2k 

for computation of nonlinearity measures. CORFAC-2k is used first to compute correction 

factors, and then BEALE2-2k, RESAN2-2k, and weighted residuals computed by UNC are used 

to analyze for model intrinsic nonlinearity and model combined intrinsic nonlinearity. If this 

indicates that model intrinsic nonlinearity and/or model combined intrinsic nonlinearity are not 

small then the correction factors computed by CORFAC-2k, and thus the confidence or 

prediction intervals computed by UNC, may be inaccurate. 

2 D1−α

When the true error covariance structure, Ω , is known, cr is computed from equation 66, 

cc is computed from equation 68, and cp is computed from equation 70. In the output file these 

correction factors are called CR, CC, and CP, respectively. 

When Ω is unknown, the approximate bound for cr is computed from equation 82, the 

approximate bound for cc is computed from equation 83, and the approximate bound for cp is 

computed from equation 84. In the output file these approximations are called CRB, CCB, and 

CPB, respectively. CRB, CCB and CPB also are written in the output file when Ω is known. 

Input Instructions 

The correction factors are calculated by CORFAC-2k using an input file with the 

extension _cf that can be generated by MODFLOW-2000 using the UNC Process. To generate 

the file, MODFLOW-2000 must be run in the Uncertainty-Process mode (with IACT=3), and 

OUTNAM in the Observation Process input file must be specified as a string other than 

“NONE.” Before running MODFLOW-2000, the final calibrated parameter values must be 

substituted from the _b file into the Sensitivity Process input file. The content of the input file is 

described in detail in the ‘Content of the _cf Input File Needed By CORFAC-2k’ section. 

If the effective correlation ce

 and/or is known, it is necessary to manually make the 

following changes to the _cf file produced by MODFLOW-2000 before running CORFAC-2k: 

Ω

1. Change the value for the effective correlation, EC, from -1 (which is the value always 

written by MODFLOW-2000) to a value in the range from 0.0 to 1.0. If the value for EC 

is specified to be less than 0.0 or larger than 1.0, then CORFAC-2k automatically 

changes the value to 0.8. 

55
Chapter 6. USER GUIDE TO CORFAC-2k 

 2. Set ITRN to a value larger than 0 if is known, and append V, VP and CP to the file. 

VP and CP are only needed if correction factors are to be computed for prediction 

intervals. 

Ω

Output From CORFAC-2k 

The CORFAC-2k program produces an output file with the extension #cf. When is 

known, the output for a confidence region has the form 

Ω

CORRECTION FACTOR FOR N-P (A) ---- = 6.0528 

SCALING FACTOR (B) --------------- = 1.0000 

EFFECTIVE CORRELATION (C) -------- = 0.77891 

(N-P)/(N-A*P) VALUE (FAC) -------- = 4.1308 

CORRECTION FACTOR (CR) ----------- = 25.003 

 

APPROXIMATE N-A*P VALUE (DFB) ---- = 4.4313 

APPROX. (N-P)/(N-A*P) VALUE (FACB) = 4.5134 

APPROXIMATE BOUND FOR CR (CRB) --- = 27.936 

 

where the number of observations, n, used for parameter estimation is called N; the number of 

estimated parameters, p, is called P; a computed from equation 26 is called A; b computed from 

equation 27 is called B; c computed from equation 75 is called C; cr computed from equation 66 

is called CR; n − ap computed from equation 71 using the input value ce (EC in input file) for c is 

called DFB; and cb

 approximated from equation 82 is called CRB. If is unknown, then only 

DFB, FACB, and CRB are printed. 

Ω

 For each confidence interval the output is 

CORRECTION FACTOR XI FOR C. I. ------ = 2.9713 

CORRECTION FACTOR (CC) -------------- = 12.274 

 

ESTIMATED VARIANCE OF PREDICTION (VE) = 0.39250 

BOUND FOR XI USING POS. VALUES (XIPB) = 4.8727 

BOUND FOR XI USING NEG. VALUES (XINB) = 0.38389E-01 

APPROXIMATE BOUND FOR CC (CCB) ------ = 18.497 

 

where ξ computed from equation 47 is called XI; cc

 computed from equation 68 is called CC; VE 

is computed as = ′QQ (θ) ( − pnS ) VE ˆ ; XIPB is the squared sum of positive values of the Qmi’s 

used in equation 78; XINB is the squared sum of negative values of the Qmi’s used in equation 

78; and cc

 approximated from equation 83 is called CCB. If is unknown, then only VE, XIPB, 

XINB, and CCB are printed. 

Ω

56
Chapter 6. USER GUIDE TO CORFAC-2k 

 For a prediction interval the output is 

 

CORRECTION FACTOR XI FOR P. I. ------ = 0.26079 

CORRECTION FACTOR (CP) -------------- = 1.0773 

 

ESTIMATED VARIANCE OF PREDICTION (VE) = 0.15901 

BOUND FOR XI USING POS. VALUES (XIPB) = 0.77966 

BOUND FOR XI USING NEG. VALUES (XINB) = 0.93736 

APPROXIMATE BOUND FOR CP (CPB) ------ = 4.2872 

 

where ξa computed from equation 55 is called XI, cp computed from equation 70 is called CP; 

VE is again computed as = ′QQ (θ) ( − pnS ) VE ˆ ; XIPB is the squared sum of positive values of 

the Qmi’s used in equation 81; XINB is the squared sum of negative values of the Qmi’s used in 

equation 81; and cp approximated from equation 84 is called CPB. If is unknown, only VE, 

XIPB, XINB, and CPB are printed. 

Ω

Content of the _cf Input File Needed By CORFAC-2k 

The _cf file contains the following items. 

1. NPE NOBS NQ MPR IPR VAR EC ITRN (format: 5I10,2F14.0,I10) 

2. PARNAM(NPE) (format: 6(A10,1X)) 

3. OBSNAM(NOBS) (format: 6(A12,1X))

4. WH(NH) (NH=NOBS-NQ) (format: 8F15.0)

5. WQ(NQ,NQ) (format: 8F15.0)

6. X(NPE,NOBS) (format: 6F13.0)

7. PRM(NPE,J), WP(J), J=1,MPR (format: 8F15.0)

8. NIPR(IPR) (format: 8I10)

9. WTP(IPR,IPR) (format: 6F13.0)

10. NOINT NHI NQI NPI ITYP IDIF (format: 6I10)

11. GNAM(NOINT) (format: 6(A12,1X))

12. WG(NOINT) (format: 8F15.0)

13. Z(NPE,NOINT) (format: 6F13.0)

If ITRN is greater than zero, read the following item. 

14. V(NOBS,NOBS) (free format)

If ITRN is greater than zero and ITYP is greater then 1, read the following item NOINT

times.

15. VP CP(NOBS) (free format)

The variables are defined as: 

. 

57
Chapter 6. USER GUIDE TO CORFAC-2k 

NPE-----is the number of estimated parameters. 

NOBS---is the number of observations used for estimation. 

NQ-------is the number observations other than hydraulic heads. 

MPR-----is the number of prior information equations. 

IPR-------is the number of prior information observations with a full weight matrix. 

VAR------is the calculated error variance, . 2

sˆ

EC--------is the effective correlation, ce. 

ITRN----is a flag that when specified to be greater than zero directs CORFAC-2k to read and use 

V, VE, and CP (defined below). 

PARNAM—are the parameter names. 

OBSNAM—are the observation names. 

WH-----are the square root of the weights for hydraulic heads. 

WQ-----is the square root of the full weight matrix for the observations other than heads. 

X--------is the sensitivity matrix for all parameters and observations. (For log-transformed 

parameters the sensitivities are with respect to natural log-transformed parameters, not 

with respect to log10-transformed parameters.) 

PRM----are coefficients for the prior information equations. 

WP------are the square root of weights for prior information equations. 

NIPR----is the number list for parameters with prior information observations that have a full 

weight matrix. 

WTP----is the square root of the full weight matrix for correlated prior information. (For logtransformed parameters the weights are for natural log-transformed parameters, not for 

log10-transformed parameters.) 

NOINT—is the number of intervals for which a correction factor is calculated. 

NHI-----is the number of intervals for hydraulic heads. 

NQI-----is the number of intervals for head-dependent flows. 

NPI-----is the number of intervals for parameters. 

ITYP---is a flag for the type of interval: ITYP=1 is for confidence intervals; ITYP=2 is for 

prediction intervals. 

IDIF-----is a flag. When IDIF=1 it indicates in the output that intervals pertaining to heads and 

head-dependent flows are for differences of heads or flows. When IDIF=0 it indicates 

that the intervals pertain to heads and head-dependent flows. 

GNAM—is the name of each prediction for which an interval correction factor is calculated. 

58
Chapter 6. USER GUIDE TO CORFAC-2k 

WG------are the square roots of the weights for the predictions corresponding to . Values must 

always be specified for WG whenever NOINT is greater than 0, but WG is only used in 

calculations pertaining to prediction intervals. For prediction intervals for logtransformed parameters WG must be specified for the natural log-transforms of the 

parameters, not for the log

ωp

10-transforms. 

Z---------is the sensitivity matrix for all parameters and predictions. (For log-transformed 

parameters the sensitivities are with respect to natural log-transformed parameters, not 

with respect to log10-transformed parameters.) 

V---------is equal to ( ) () ( ) ( ) 2

* * σε

′ Ωm E −= fY mm γθ − fY mm γθ . 

VP-------is equal to ( ) ( ) 2 2

*

1 ˆ ε

−

p p −=ω gYE γθ σ . 

CP-------is equal to ( ) () ( ) ( ) 2

* * σε

′ E −= fYC mm γθ p − gY γθ . 

59
REFERENCES 

REFERENCES

Beale, E.M.L., 1960, Confidence regions in non-linear estimation: Journal of the Royal 

Statistical Society, series B, v. 22, no. 1, p. 41-76. 

Beven, K., and Binley, A., 1992, The future of distributed models: Model calibration and 

uncertainty prediction: Hydrological Processes, v. 6, p. 279-298. 

Brooks, R. J., Lerner, D. N., and Tobias, A. M., 1994, Determining the range of predictions of a 

groundwater model which arises from alternative calibrations: Water Resources 

Research, v. 30, no. 11, p. 2993-3000. 

Christensen, S., and Cooley, R.L., 1996, Simultaneous confidence intervals for a steady-state 

leaky aquifer groundwater flow model: In Calibration and Reliability in Groundwater 

Modelling, ed. by K. Kovar and P. van der Heide, IAHS pub. no. 237, p. 561-569. 

Christensen, S., and Cooley, R.L., 1999a, Evaluation of confidence intervals for a steady-state 

leaky aquifer model: Advances in Water Resources, v. 22, no. 8, p. 807-817. 

Christensen, S., and Cooley, R.L., 1999b, Evaluation of prediction intervals for expressing 

uncertainties in groundwater flow model predictions: Water Resources Research, v. 35, 

no. 9, p. 2627-2639. 

Christensen, S., and Cooley, R.L., 2003, Experiences gained in testing a theory for modelling 

groundwater flow in heterogeneous media: In Calibration and Reliability in Groundwater 

Modelling: A Few Steps Closer to Reality, edited by K. Kovar and Z. Hrkal, IAHS pub. 

no. 277, p. 22-27. 

Christensen, S., Rasmussen, K.R., and Møller, K., 1998, Prediction of regional ground-water 

flow to streams: Ground Water, v. 36, no. 2, p. 351-360. 

Clarke, G.P.Y., 1987, Approximate confidence limits for a parameter function in nonlinear 

regression: Journal of the American Statistical Association, v. 82, no. 397, p. 221-230. 

Cooley, R.L., 1993a, Exact Scheffé-type confidence interval for output from groundwater flow 

models, 1, Use of hydrogeologic information: Water Resources Research, v. 29, no. 1, p. 

17-33. 

Cooley, R.L., 1993b, Exact Scheffé-type confidence intervals for output from groundwater flow 

models, 2, Combined use of hydrogeologic information and calibration data: Water 

Resources Research, v. 29, no. 1, p. 35-50. 

Cooley, R.L., 1993c, Regression modeling of ground-water flow. Supplement 1—modifications 

to the computer code for nonlinear regression solution of steady-state ground-water flow 

problems: U.S. Geological Survey Techniques of Water Resources Investigations, book 

3, chap. B4, supp. 1, 8 p. 

Cooley, R.L., 1997, Confidence intervals for ground-water models using linearizaton, likelihood, 

and bootstrap methods: Ground Water, v. 35, no. 5, p. 869-880. 

Cooley, R.L., 1999, Practical Scheffé-type credibility intervals for variables of a groundwater 

model: Water Resources Research, v. 35, no. 1, p. 113-126. 

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Cooley, R.L., 2004, A theory for modeling ground-water flow in heterogeneous media: U.S. 

Geological Survey Professional Paper 1679, 220 p. 

Cooley, R.L., and Hill, M.C., 1992, A comparison of three Newton-like nonlinear least-squares 

methods for estimating parameters of ground-water flow models, in Russell, T.F., Ewing, 

R.E., Brebbia, C.A., Gray, W.G., and Pinder, G.F.eds., Computational Methods in Water 

Resources IX, v. 1, , Numerical methods in water resources − Proceedings of the 

International Conference on Computational Methods in Water Resources, 9th, Denver, 

Colorado, 1992: Oxford, U. K., Computational Mechanics Publications, p. 379-386. 

Cooley, R.L., and Naff, R.L., 1990, Regression modeling of ground-water flow: U.S. Geological 

Survey Techniques of Water-Resources Investigations, book 3, chap. B4, 232 p. 

Cooley, R.L., and Vecchia, A.V., 1987, Calculation of nonlinear confidence and prediction 

intervals for ground-water flow models: Water Resources Bulletin, v. 23, no. 4, p. 581-

599. 

Draper, N.R., and Smith, Harry, 1998, Applied regression analysis (3rd ed.): New York, John 

Wiley, 706 p. 

Donaldson, J.R., and Schnabel, R.B., 1987, Computational experience with confidence regions 

and confidence intervals for nonlinear least squares: Technometrics, v. 29, no, 1, p. 67-

82. 

Graybill, F. A., 1976, Theory and application of the linear model: Pacific Grove, Calif., 

Wadsworth and Brooks/Cole, 704 p. 

Guttman, Irwin, and Meeter, D. A., 1965, On Beale’s measures of non-linearity: Technometrics, 

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Hamilton, D., and Wiens, D., 1987, Correction factors for F ratios in nonlinear regression: 

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Harbaugh, A.W., Banta, E.R, Hill, M.C., and McDonald, M.G., 2000, MODFLOW-2000, the 

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Hill, M.C., 1992, A computer program (MODFLOWP) for estimating parameters of a transient, 

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62
APPENDIX A. SOLUTIONS FOR EXTREME VALUES 

APPENDIX A. SOLUTIONS FOR EXTREME VALUES 

Confidence Intervals 

In this section we derive equations for extreme values of g(γθ) for a linearized form of 

 given by equation 8. These equations become iteration equations to solve the nonlinear 

problem. 

L(θ,λ)

First, we linearize g(γθ) and f (γθ) using a truncated Taylor series around parameter set 

 obtained at the rth iteration: θr

( ) ( ) ( ) r r r ≈ γθγθ + Dggg − θθ , (A-1) 

( ) ( ) ( ) r Dfr θθ r f γθ ≈ f γθ + − , (A-2) 

where subscript r indicates evaluation using . Second, we take derivatives of written 

using equations A-1 and A-2 and set the results to zero to obtain 

θr L(θ,λ)

( ) ( ( )) rr r r r ′ DffD − θθω = λ g′ + fDD ′ω − fY γθ , (A-3) 

( ) (θθ )ˆ 2

α −= SSd . (A-4) 

Third, we write S( ) θ using equation A-2, then substitute − θθ r from equation A-3 into the 

result to get 

() ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ( ))

 (A-5) ( ) ( ) ( ) ( ) ( ( ) ( ))

Fourth, we put equation A-5 into equation A-4 and solve for λ to get 

 (A-6) 

To obtain the solution for the (r+1)th iteration, we write equations A-6 and A-3 in the forms 

() ( ) ( ) ( ) () ( ) ( )

( ) λ ( ) ( ) ( ) ( ) .

λ

λ λ

2 λ

2

2 2121

2 1 1

1

1

r rr rr r rr

r rrr r rr rrr r rr

rr rr rr rr rr

r rr rr rr r rr

r rr rr rrr r

S

S g g

g g

S g

SS

θ fYQQ ωγθ R ω fY γθ

θ ωDffDD fYD γθ ωDf ωDffD fD ω fY γθ

fDD ω fY γθ ωDffD fDD ω fY γθ

θ fDD ω fY γθ ωDffD fD ω fY γθ

θθθθ fD ω fY θθγθ DffD θθω

− ′ += ′ − −

′ ′ − ′ += ′ ′ −−

′ ′ + ′ − ′ + ′ + ′ −

′ ′ − ′ −= ′ + ′ −

′ − ′ ′ −+− ′ −−=

− −

−

−

( ) ( ) ( ) ( ) ( ) ( ) λ . − ±=

Sd ˆ 21

2 2/12/1

α

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

− ′ −++

rr

r r r r S

QQ

θθ fY ωγθ R ω fY γθ

63
APPENDIX A. SOLUTIONS FOR EXTREME VALUES 

 (A-7) ( ) ( ) ( ) ( ) ( ) ( )

21

2/12/1

1

ˆ

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

− ′ α −++ +

rr

r r r r

r

S

QQ

θθ fY ωγθ R ω fY γθ 2 − ±=λ Sd

and 

( ) ( ) ( ( )) rrrrr r rrr r θθ λ+= ωDffD g′′ + ωDffDD fD ′′ ω − fY γθ − −

+ +

1 1

1 1 . (A-8) 

Prediction Intervals 

Extreme values of are derived using the same general method as used for 

. First, we use equations A-1 and A-2 in equation 19 and take derivatives with respect to 

. Then we set the results to zero to obtain 

( ) γθ + vg

g( ) γθ

θ v and , , λ

( ) ( ( )) rr r rr r ′ DffD − θθω = λ g′ + fDD ′ω − fY γθ , (A-9) 

ωpv = λ , (A-10) 

( ) ( ) 2 2 ˆ vSSdα θθ ω+−= p . (A-11) 

Second, we use equations A-5 and A-10 in equation A-11 to get 

 

( ) ( ) ( ) () ( ( ) ( ) r SSd prr r r r θθ QQ fY ωγθ R ω − fY γθ ) ′ λ+−= ′ −−ω+ −

α

2 2 1 2/12/1 ˆ ,

from which 

 ( ) ( ) ( ) (A-12) 

(A-13) 

( ) ( ) ( ) λ . − ±=

Sd

Iteration equations are obtained directly from equations A-12, A-10, and A-9 and have the form 

ω

ˆ 21

1

2 2/12/1

α

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′ +

− ′ −++

−

prr

r r r r S

QQ

θθ fY ωγθ R ω fY γθ

( ) ( ) ( ) ( ) ( ) ( ) λ , − ±=

Sd

ω

ˆ 21

1

2 2/12/1

α

1 ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′ +

− ′ −++ + −

prr

r r r r

r

S

QQ

θθ fY ωγθ R ω fY γθ

, (A-14) 1

1

1 +

−

r+ λω= rp v

and equation A-8. 

64
APPENDIX B. COMPUTATION OF Q AND R 

APPENDIX B. COMPUTATION OF Q AND R FOR TESTING 

FOR EQUALITY OF CONSTRAINED AND UNCONSTRAINED 

RESIDUALS 

To show that Q and R must be computed using θθ

~ ≠ for the test of equality of 

constrained and unconstrained weighted residuals, we need to use results from the constrained 

regression. The constrained regression involves finding the minimum value of 

subject to the constraint This is accomplished using the Lagrange function given 

as equation E-7 in Cooley (2004, p. 145): 

( ) ( ), ~ , SS θθ

() ( ). * = gg γθγθ

( ) λ = (θθ ) + λ( ( ) − ggSL (γθγθ )) * , 2 . (B-1) 

The method used is the same one used for equation 8. First, we substitute equations A-1 and A-2, 

appendix A, into equation B-1, take the derivatives of L(θ,λ) with respect to and λ, and set 

the results to zero to yield, for iteration r+1 

θ

( ) ( ) ( ( )) rrrrr r rrr r θθ λ=− ωDffD g′′ + ωDffDD fD ′′ ω − fY γθ − −

+ +

1 1

1 1 (B-2) 

and 

( ) ( ) ( ) r rrr * = γθγθ + Dggg +1 − θθ . (B-3) 

Second, we put equation B-2 into equation B-3 and solve for λ to obtain for iteration r+1 

(B-4) ( ) ( ) ( ) () ( ) λ . r g D ′ =

1

*

1

rr

r rr r r

r

gg

QQ

γθγθ ωDffD fD ω fY γθ

′

−− ′ − −

+

At convergence so that equation B-4 can be substituted into equation B-2 

to give 

θθθ

~ +1 rr ≈≈

( ) () ( ) ( ) ( ) ( ) ( ) ~ ~ ~ ~ ~ ~

 (B-5) ( ) ( ) ( ) ( )

,

~ ~ ~~

~~ ~~ ~~

~~

~~ ~~~ ~

1

1

1 1

1

0

fD ω fY θγ QQ

ωDfDDD f ωDfD If

QQ

ωDfD ωDfDDDf fDf ω fY θγ ωDfD fDf ω fY θγ

=

′ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

′′ = ′ −

′

′′′′ − ′′ −−

−

−

− −

−

gg

gg

65
APPENDIX B. COMPUTATION OF Q AND R 

where the tildes over variables indicate evaluation using θ

~ . Cooley (2004, equation E-14) 

showed that equation B-5 can be written in the form 

 (B-6) ( ) ( ) ( ) . ~ ~~

~ ~ ~~~ 21 ~ 21 1

ω fY θγ 0

QQ

QQ ωDfD fDf ω R =− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

′ ′′ − −

Premultiplication of equation B-6 by ω fD21 ~ and use of the definition and properties of R

~ lead 

to the result needed to complete the proof: 

 (B-7) ( )

To derive the equations needed to show that weighted residuals from the constrained 

regression, defined as ( ) QQQQI ω ( fY ( θγ )) 21 ~ − ′′ − , equal standard weighted residuals 

ω ( fY ( θγ )) ˆ 21 − when model intrinsic and model combined intrinsic types of nonlinearity are 

negligible, we substitute equation 6-17 of Cooley (2004) into the constrained weighted residuals 

to get 

 (B-8) 

If model intrinsic nonlinearity is negligible, then from Cooley (2004, equations 6-2 and I-1 - I-5) 

the first term on the right side of equation B-8 is equal to the weighted residual vector and the 

second term is negligible. If model combined intrinsic nonlinearity is negligible, then from 

Cooley (2004, equation I-21) the last term is negligible. However, from equation B-7, if R

~ and 

Q were used in place of R and Q, then this term always would be zero. ~

( ) . ~ ~~

~ ~ ~ 1 ω fY θγ 0

QQ

Q =− ⎟

⎟

⎠

⎞

⎜

⎝ ′

′ Q 2 ⎜R

⎛ −

( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ). ~ ~

~

21

*

21

*

21

21

ω fY θγ QQ

QQ RI ω fY γθ RI ω f γθ f θγ R

ω fY θγ ⎝ Q Q ⎠

QQ I

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

′ −= −+− −+−

− ⎟

⎟

⎞

⎜

⎜

⎛

′

′ −

66
APPENDIX C. EXAMPLE SIMULATIONS 

APPENDIX C. EXAMPLE SIMULATIONS 

The test case presented in this section provides example input and output files for an 

entire modeling process including parameter estimation using MODFLOW-2000 in the 

Parameter-Estimation Process (PES) mode, calculation of correction factors using CORFAC-2k, 

calculation of nonlinearity measures using BEALE2-2k, calculation of quantities needed for 

residuals and nonlinearity analyses using RESAN2-2k, and calculation of confidence intervals 

and residuals analysis using MODFLOW-2000 in Uncertainty Process (UNC) mode. 

The test example is taken from the synthetic case study of Christensen and Cooley 

(2003). The dimensions of the two-dimensional flow domain (fig. C1) are 18 by 8, divided into 

90 by 40 uniform structural elements. The transmissivity is constant within each structural 

element. There is a pumping well in the center of the domain where ground water is withdrawn 

at a rate of 1. Boundary conditions include no flow across the top and bottom boundaries, a 

constant head of zero along the right boundary, and a constant flux across the left boundary 

(simulated as recharge with rate equal to 3.1076 over the left-most column of cells). The vector β

of unknown system characteristics consists of spatially varying log-transmissivity, which is 

0 18

0

8

F1: 14.88

F2: 14.78

F3: 14.86

F4: 13.87

F5: 13.01

F6: 12.97

F7: 5.61

F8: 5.63

F9: 5.94

F10: 8.83

F11: 11.08

F12: 11.35

F13: 3.70

F14: 3.75

F15: 3.52

F16: 3.35

F17: 3.42

F18: 3.71

F19: 2.37

F20: 1.91

F21: 1.76

F22: 1.96

F23: 1.82

F24: 2.07

F25: 1.54

F26: 1.66

F27: 1.78

F28: 1.54

F29: 1.70

F30: 1.55

F31: 0.94

F32: 1.41

F33: 1.22

F34: 1.47

F35: 1.08

F36: 0.88

G1 G2 G3 G4 G5

G6

Head observation Predicted head Pumping well

Constant flux

Constant head

HK_1

HK_2

Figure C1. Model domain, boundary conditions, zonation, locations, and values of head 

observations, and locations of head predictions. 

67
APPENDIX C. EXAMPLE SIMULATIONS 

known to be normally distributed with a constant mean of zero and with exponential covariance 

with a correlation scale of 1.0 and a variance of 4.0. 

The observations used for parameter estimation consist of 36 observations of hydraulic 

head observed at almost uniformly distributed grid points (fig. C1). The observation errors are 

known to be normally distributed with a variance of 0.01. (In the study of Christensen and 

Cooley (2003) the variance was 1.0.) The true error second-moment matrix, , was computed 

by the same Monte-Carlo method as used by Christensen and Cooley (2003). Weights for the 

observations are obtained as the inverses of the diagonal elements of 

Ω

2 σε

Ω . 

The numerical model used for parameter estimation and computation of confidence 

intervals has 91 by 41 elements. The width of the left-most and right-most columns is 0.1, while 

the remaining column widths are all 0.2. The width of the upper-most and lower-most rows is 

0.1, while all other row-widths are 0.2. Based on the available 36 head observations, we divided 

the transmissivity field of the model into two transmissivity zones (fig. C1), one of presumably 

higher transmissivity (low hydraulic head gradient), and one of presumably lower transmissivity 

(high hydraulic head gradient). The model parameters are the log-transmissivities of each of the 

two zones and are named HK_1 and HK_2, respectively. 

Parameter Estimation 

The input files for, and the GLOBAL output file from, the parameter estimation are listed 

in the ‘Parameter Estimation – Input Files’ and the ‘Parameter Estimation – Global Output File’ 

sections. The diagonal elements of the true error second-moment matrix, , have values 

between 2.0 and 7.3. Because equals 0.01, the values of the diagonal elements of 

Ω

2 σε

2 σε Ω vary 

between 200 and 730. These values are input to MODFLOW-2000 as STATISTIC in item 6 of 

the head observation input where STAT-FLAG is set equal to 0 (Hill and others, 2000). Thus the 

weights for the head observations used in the regression are calculated as the inverses of the 

diagonal elements of Ω . 

OUTNAM in the Observation Process input file is specified to be “example.” This 

defines the names of a number of MODFLOW-2000 output files as explained by Hill and others 

(2000, p. 27) and in this report. 

The initial values of two zonal transmissivity parameters are both set to 1.0, which equals 

the known expected value of the transmissivity. The estimated parameters HK_1 and HK_2 are 

both log-transformed zonal transmissivity. HK_1 and HK_2 are estimated by MODFLOW-2000 

to be 1.46 and 9.94×10-2, respectively. The high estimate of HK_1 and the low estimate of HK_2 

conform with the expectations from the head observations. 

The value of the weighted least-squares objective function at the minimum is 

( ) .0140390 ˆ

θ = .S This value is used later to calculate the critical values needed for the 

calculation of confidence intervals. 

68
APPENDIX C. EXAMPLE SIMULATIONS 

Computation of Correction Factors Using CORFAC-2k 

The first step to compute correction factors is to make MODFLOW-2000 create the input 

file example._cf for CORFAC-2k. This is done by the following. 

 1. The Uncertainty (UNC) process is activated by adding specification of the UNC input file 

to the Name File (example.nam). 

 2. The UNC file (unc.dat) where IACT is set equal to 3 is created. Confidence intervals are 

specified to be calculated for the two parameters HK_1 and HK_2. 

 3. The final calibrated parameter values from the example._b file are substituted into the 

SEN file. 

 4. Information about the six hydraulic head predictions is added to the HOB file (hob.dat). 

 5. MODFLOW-2000 is run. 

The MODFLOW-2000 input files are listed in the ‘Modified Input Files for Generation of 

CORFAC-2k Input File’ section. 

The created file, example._cf, is in this case modified by setting ITRN in item 1 equal to 

1 and by setting V in item 14 equal to Ω . This modified file is listed in the ‘Modified _cf Input 

File to CORFAC-2k’ section. Running CORFAC-2k produces the output file example.#cf, which 

is listed in the ‘Output File from CORFAC-2k’ section. 

In the output file, example.#cf, the computed correction factor for the confidence region 

is listed as CR=33.243, whereas the corresponding approximate bound is computed to be 

CRB=46.859. The approximate bound was calculated using the default effective correlation of 

0.8 and not the actual effective correlation C=0.7149 computed by CORFAC-2k. If Ω had been 

unknown, CRB would have to be used instead of CR to compute Scheffé type confidence 

intervals using MODFLOW-2000 with the UNC Process. Using CRB would produce wider 

Scheffé intervals than using CR. 

In example.#cf the correction factors CC computed for the individual confidence 

intervals for the six head predictions and the two parameters, and the corresponding approximate 

bounds CCB, have the values given in table C1. Using the CCB would in all these cases produce 

wider individual confidence intervals than using the CC values. 

Table C1. Correction factors, CC, and correction factor bounds, CCB, computed by 

CORFAC-2k for six head predictions (G1 to G6) and two parameters (HK_1 and HK_2). 

G1 G2 G3 G4 G5 G6 HK_1 HK_2 

CC 45.638 15.989 46.966 45.492 45.477 45.875 45.473 10.668 

CCB 71.613 25.130 73.142 71.470 71.456 71.842 71.452 37.083 

69
APPENDIX C. EXAMPLE SIMULATIONS 

Computation of Nonlinearity Measures Using BEALE2-2k 

BEALE2-2k needs four input files, example._b1, example._b2, example._b3, and 

example._b4, that are generated by MODFLOW-2000 using UNC. The _b1 and _b3 files are 

generated as follows: 

 1. In the UNC file (unc.dat) IACT in item 1 is set equal to 1 and CFR and CFI are specified 

in items 8 and 9, respectively. The values of CFR and CFI equal the CR and CC values 

found in the example.#cf output file from CORFAC-2k. 

 2. The final calibrated parameter values from the example._b file are substituted into the 

SEN input file (sen.dat). 

 3. MODFLOW-2000 is run. 

After the example._b1 and example._b3 files are generated, the example._b2 and 

example._b4 files are generated by changing the value of IACT to 2 in the UNC file (unc.dat), 

and then re-running MODFLOW-2000. Finally, BEALE2-2k is executed to produce the output 

file example.#be. The unc.dat, example._b1, example._b2, example._b3, example._b4, and 

example.#be files are listed in the ‘Input Files Used to Generate BEALE2-2k Input Files’, the 

‘BEALE2-2k Input Files’, and the ‘BEALE2-2k Output File’ sections. 

In the example.#be output file the computed measure of total nonlinearity, BNT, is 

approximately 0.064, and the measure of intrinsic nonlinearity, BNI, is approximately 0.45×10-3. 

This indicates that the regression model is moderately nonlinear, and the intrinsic nonlinearity is 

insignificant. The combined intrinsic nonlinearity measures for confidence intervals are listed in 

table C2. These values indicate that the combined intrinsic nonlinearity is insignificant for the 

G2 head prediction confidence interval, but moderate for all the other intervals. Because the 

correction factors computed by CORFAC-2k for the latter seven intervals are only slightly larger 

than 0.09 (see chapter 2) and are all much less than one, correction factors CC should not be 

effected much by the omitted intrinsic nonlinearity components, and factors CCB should be 

conservative. 

Table C2. Combined intrinsic nonlinearity measures computed by BEALE2-2k. 

G1 G2 G3 G4 G5 G6 HK_1 HK_2 

BMI 0.91136E-01 0.13976E-04 0.80532E-01 0.92049E-01 0.92165E-01 0.89142E-01 0.37137E-01 0.22188E-01 

BMF0 0.12232E-01 0.37404E-02 0.99148E-02 0.12468E-01 0.12491E-01 0.11832E-01 0.58584E-12 0.59462E-10 

BMG0 0.36826E-01 0.40559E-02 0.34126E-01 0.37003E-01 0.37051E-01 0.36261E-01 0.37137E-01 0.22186E-01 

BMIMAX 0.11560 0.80979E-02 0.10036 0.11699 0.11715 0.11280 0.37137E-01 0.22188E-01 

70
APPENDIX C. EXAMPLE SIMULATIONS 

Residuals Analysis And Detection of Nonlinearity Using RESAN2-2k 

RESAN2-2k needs two input files that for the example are named example._ws and 

example._rs. The example._ws file was generated during parameter estimation and contains 

weighted simulated equivalents and residuals (Hill and others, 2000, table 5). The example._rs 

file is generated in a separate step where the B values for HK_1 and HK_2 are both set to 1.0 in 

item 3 of the SEN file (sen.dat), and where MAX-ITER is set to 0 in item 1 of the PES file 

(pes.dat). This means that the example._rs file is generated using values of 1.0 for both 

parameters but parameter estimation is not carried out (Hill and others, 2000, p. 72, 75, 78, and 

79). 

Before running RESAN2-2k, the example._rs file is modified by increasing NSETS in 

item 1 from 4 to 1,000, setting STDV in items 1 to 0.01, setting IERT in item 1 to 1, setting 

ICOV in item 1 to 1, specifying ET in item 10 to equal the expected values of the true errors, and 

specifying V in item 11 to equal Ω m. RESAN2-2k is then run to generate the example.#nr and 

example._md output files. The input files for, and output files from, RESAN2-2k are listed in the 

‘RESAN2-2k Input Files’ and ‘RESAN2-2k Output Files’ sections. 

The example.#nr output file contains results of the analysis of residusls. The mean 

weighted residual, EM, is approximately –0.0005, and the slope of the weighted residual plot, 

SLP, is approximately 0.0036. Because both values are close to zero, they both indicate that 

model and system types of intrinsic nonlinearity are small. The intrinsic nonlinearity measure, 

QINT, is approximately 0.0002, which is much smaller than ( ) 0140390 ˆ

θ = .S . This also indicates 

small model intrinsic nonlinearity. The correlation between weighted residuals and the means of 

synthetic residuals, CED, is approximately 0.996, and that the probability, PROB, of this or a 

smaller correlation is 0.988. Figure C2 shows a probability plot of ordered weighted residuals 

with an approximate 95 percent confidence band. This plot was generated from the results in the 

example._md file. The weighted residuals fall inside the confidence band. This, together with the 

high probability of CED, indicates normality of the weighted residuals. 

Calculation of Confidence Intervals Using MODFLOW-2000 With the UNC 

Process 

Finally, individual 95 percent confidence intervals are calculated for the six predicted 

heads, G1 to G6, and for the two model parameters, HK_1 and HK_2. This is accomplished with 

the following steps: 

 1. The UNC Process is activated by specifying the UNC file (unc.dat) in the Name File 

(example.nam). Two DATAGLO data files are specified to be opened in the Name file, 

one called confint.dat with unit number equal to 61 and the other called weight-res.dat 

with unit number equal to 62. The summary output of computed confidence limits are to 

be printed in the former data file, whereas weighted residuals from the constrained 

regression are to be printed in the latter. 

71
APPENDIX C. EXAMPLE SIMULATIONS 

-0.2 -0.1 0 0.1 0.2

ORDERED WEIGHTED RESIDUALS

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

PROBABILITY

Figure C2. Ordered weighted residuals (dots) plotted with approximate 95 percent 

confidence band (bars). 

 2. IACT is set equal to 0, NPI is set equal to 2, IOIN is set equal to 61, IDSQ is set equal to 

1, and IWRP is set equal to 62 in item 1 of the UNC file. 

 3. A critical value is computed from equation 86 for each individual 95 percent confidence 

interval using the correction factors, CC, in table C1, t 5.2 ( ) = ,032.234 and 

( ) 0140390 ˆ

θ = .S . The elements of DSQ in item 4 of the UNC file are set equal to the 

critical values. 

72
APPENDIX C. EXAMPLE SIMULATIONS 

 4. The two parameters for which confidence intervals are to be computed are specified in 

item 5 of the UNC file as HK_1 and HK_2, respectively. 

 5. The initial parameter values in the SEN file (sen.dat) are set equal to the final calibrated 

values. 

 6. MODFLOW-2000 is run. 

Besides producing the GLOBAL and LIST output files with details concerning the computation 

of confidence intervals, MODFLOW-2000 in this run also produces the two files mentioned 

above named confint.dat and weight-res.dat. The input and output files are all listed in the 

‘MODFLOW-2000 Input Files For Confidence Interval Calculation’, the ’GLOBAL Output File 

For Confidence Interval Calculation’, and the ’Summary and Weighted Residuals Output Files’ 

sections. 

The output file named confint.dat contains the following: 

C.I. 1: FL= 4.1361 ; SSE=0.91894E-01; ITER=4; PAR: 1.007 0.1538 

C.I. 2: FL= 13.439 ; SSE=0.41351E-01; ITER=3; PAR: 1.519 0.7981E-01 

C.I. 3: FL= 6.4088 ; SSE=0.94127E-01; ITER=4; PAR: 1.003 0.1496 

C.I. 4: FL= 3.1090 ; SSE=0.91639E-01; ITER=4; PAR: 1.008 0.1541 

C.I. 5: FL= 1.8458 ; SSE=0.91618E-01; ITER=4; PAR: 1.007 0.1542 

C.I. 6: FL= 4.6167 ; SSE=0.92338E-01; ITER=4; PAR: 1.006 0.1531 

C.I. 7: FL=0.42100 ; SSE=0.91608E-01; ITER=4; PAR: 2.636 0.6966E-01 

C.I. 8: FL=-0.87902 ; SSE=0.32298E-01; ITER=4; PAR: 1.264 0.1321 

C.I. -1: FL= 1.5895 ; SSE=0.91861E-01; ITER=4; PAR: 2.641 0.6964E-01 

C.I. -2: FL= 10.581 ; SSE=0.41295E-01; ITER=4; PAR: 1.436 0.1282 

C.I. -3: FL= 2.5599 ; SSE=0.94083E-01; ITER=4; PAR: 2.671 0.6945E-01 

C.I. -4: FL= 1.1892 ; SSE=0.91594E-01; ITER=4; PAR: 2.636 0.6967E-01 

C.I. -5: FL= 0.70517 ; SSE=0.91630E-01; ITER=4; PAR: 2.636 0.6967E-01 

C.I. -6: FL= 1.7864 ; SSE=0.92292E-01; ITER=4; PAR: 2.646 0.6961E-01 

C.I. -7: FL= 0.32539E-02; SSE=0.91624E-01; ITER=4; PAR: 1.008 0.1542 

C.I. -8: FL= -1.1052 ; SSE=0.32343E-01; ITER=3; PAR: 1.750 0.7849E-01 

Each line summarizes the results for one computed confidence limit: C.I. 1 is for the upper limit 

of the first confidence interval, C.I. 2 is for the upper limit of the second interval, C.I. –1 is for 

the lower limit of the first interval, and so on. In this case the first six intervals are for head 

predictions (G1 to G6), whereas the last two intervals are for the parameters (HK_1 and HK_2). 

The column of numbers preceded by “FL=” is the confidence limit value. The column of 

numbers preceded by “SSE=” is the value of the objective function at the parameter values 

corresponding to the confidence limit. The SSE value should be close to the critical value, DSQ, 

specified for the confidence interval in the UNC file. This is fulfilled for all confidence limits 

computed in this example. The column of numbers above preceded by “ITER=” is the number of 

73
APPENDIX C. EXAMPLE SIMULATIONS 

iterations necessary to compute the confidence limit. Finally, the two columns of numbers 

preceded by “PAR:” are the parameter values corresponding to the computed confidence limit. 

The output file named weight-res.dat contains weighted residuals for all computed 

confidence limits. These weighted residuals are defined as ( QQQQI )ω ( ) fY ( ) θγ 21 ~ − ′′ − in 

chapter 2. In figure C3 the weighted residuals at the confidence limits are plotted against the 

weighted residuals from unconstrained regression (parameter estimation), ω ( fY ( θγ )) ˆ 21 − . In all 

cases the points fall close to the identity line. This is especially true for the head prediction G2, 

whereas there is a little more scatter for the other head predictions and for the two parameters. 

This may indicate small combined intrinsic nonlinearity for the confidence interval of G2, and 

larger combined intrinsic nonlinearity for the other seven confidence intervals. This conforms 

with the indications of the combined intrinsic nonlinearity measures computed by BEALE2-2k 

listed in table C2. 

74
APPENDIX C. EXAMPLE SIMULATIONS 

 

-0.04 -0.02 0 0.02 0.04

WEIGHTED RESIDUALS AT CONFIDENCE LIMITS

-0.04

-0.02

0

0.02

0.04

WEIGHTED RESIDUALS FROM UNCONSTRAINED REGRESSION

Residual plot for G1

-0.04 -0.02 0 0.02 0.04

WEIGHTED RESIDUALS AT CONFIDENCE LIMITS

-0.04

-0.02

0

0.02

0.04

WEIGHTED RESIDUALS FROM UNCONSTRAINED REGRESSION

Residual plot for G2

-0.04 -0.02 0 0.02 0.04

WEIGHTED RESIDUALS AT CONFIDENCE LIMITS

-0.04

-0.02

0

0.02

0.04

WEIGHTED RESIDUALS FROM UNCONSTRAINED REGRESSION

Residual plot for G3

-0.04 -0.02 0 0.02 0.04

WEIGHTED RESIDUALS AT CONFIDENCE LIMITS

-0.04

-0.02

0

0.02

0.04

WEIGHTED RESIDUALS FROM UNCONSTRAINED REGRESSION

Residual plot for G4

 

Figure C3. Weighted residuals at confidence limits in relation to weighted residuals 

from unconstrained regression (parameter estimation). The line is the one-to-one 

correlation line. 

75
APPENDIX C. EXAMPLE SIMULATIONS 

-0.04 -0.02 0 0.02 0.04

WEIGHTED RESIDUALS AT CONFIDENCE LIMITS

-0.04

-0.02

0

0.02

0.04

WEIGHTED RESIDUALS FROM UNCONSTRAINED REGRESSION

Residual plot for G5

-0.04 -0.02 0 0.02 0.04

WEIGHTED RESIDUALS AT CONFIDENCE LIMITS

-0.04

-0.02

0

0.02

0.04

WEIGHTED RESIDUALS FROM UNCONSTRAINED REGRESSION

Residual plot for G6

-0.04 -0.02 0 0.02 0.04

WEIGHTED RESIDUALS AT CONFIDENCE LIMITS

-0.04

-0.02

0

0.02

0.04

WEIGHTED RESIDUALS FROM UNCONSTRAINED REGRESSION

Residual plot for HK_1

-0.04 -0.02 0 0.02 0.04

WEIGHTED RESIDUALS AT CONFIDENCE LIMITS

-0.04

-0.02

0

0.02

0.04

WEIGHTED RESIDUALS FROM UNCONSTRAINED REGRESSION

Residual plot for HK_2

 

Figure C3 (continued). Weighted residuals at confidence limits in relation to weighted 

residuals from unconstrained regression (parameter estimation). The line is the one-toone correlation line. 

76
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

Parameter Estimation – Input Files 

Name file (example.nam): 

GLOBAL 4 mf2k.global_listing 

LIST 3 output.dat 

DIS 95 discret.dat 

BAS6 1 bas6.dat 

ZONE 93 zone.dat 

MULT 94 multiple.dat 

LPF 31 lpf6.dat 

RCH 11 rech6.dat 

WEL 12 wel6.dat 

PCG 23 pcg2.dat 

OC 22 oc.dat 

DATA 10 inithead.dat 

OBS 40 obs.dat 

HOB 41 hob.dat 

PES 47 pes.dat 

SEN 46 sen.dat 

DIS file (discret.dat): 

# Discretization file for MODFLOW-2000, UNC1NLI1 Report Example 

 1 41 91 1 1 0 

 0 

 95 1(16G5.0) -1 3. DELR(NCOL) 

 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 

 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 

 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 

 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 

 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 

 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 

 95 1(16G5.0) -1 4. DELC(NROW) 

 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 

 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 

 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 

 0 1(20G14.0) -1 5. TOP OF SYSTEM 

 0 0(20G14.0) -1 5. BOTTOM OF OF LAYER 1 

 1 1 1 SS

BAS6 file (bas6.dat): 

# Basic package file for MODFLOW-2000, UNC1NLI1 Report Example 

FREE 

 1 1(40I2) -1 2. IBOUND Array for Layer 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

77
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

78
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

79
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

 1 1 1 1 1 1 1 1 1 1-1 

 -999.99 

 10 1(10F8.3) -1 4. Starting Heads in Layer 1

ZONE file (zone.dat): 

# Zone file for MODFLOW-2000, UNC1NLI1 Report Example 

 2 

ZHK1 

 93 1(91I1) -1 HK zone array for layer 1 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111111111111111222222222111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

80
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

1111111111111222222222222111111111111111111111111111111111111111111111111111111111111111111 

ZVK1 

 0 999(20I4) -1 VK zone array for layer 1

MULT file (multiple.dat): 

# Multiplier file for MODFLOW-2000, UNC1NLI1 Report Example 

 2 

HK1 

 0 1(20G14.0) -1 4. HK Mult. array 

VK1 

 0 1(20G14.0) -1 4. VK Mult. array

LPF file (lpf6.dat): 

# Layer-Property Flow file for MODFLOW-2000, UNC1NLI1 Report Example 

 0 999. 3 

 0 

 0 

 1 

 0 

 0 

HK_1 HK 1 1 

 1 HK1 ZHK1 1 

HK_2 HK 1 1 

 1 HK1 ZHK1 2 

VK_0 VK 1 1 

 1 VK1 ZVK1 999 

 31 1(20G14.0) -1 10. HK of layer 1 

 31 1(20G14.0) -1 11. HANI of layer 1 

 31 1(20G14.0) -1 12. VK of layer 1

RCH file (rech6.dat): 

# RECH file for MODFLOW-2000, UNC1NLI1 Report Example 

1 0 NRCHOP IRCHCB 

1 INRECH 

 11 6.2152(91G2.0) -1 10. RCH of layer 1 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

81
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

82
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0. 

1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0

.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.

WEL file (wel6.dat): 

# WELL file for MODFLOW-2000, UNC1NLI1 Report Example 

# 

PARAMETER 1 1 

 1 00 

WEL_0_1 Q 1 1 

 1 21 46 -1 

 0 1 

WEL_0_1

PCG file (pcg2.dat): 

 50 30 1 

 .001 .001 1 1 1 0 1

OC file (oc.dat): 

# Output control file for MODFLOW-2000, UNC1NLI1 Report Example 

 0 0 0 0 

 1 0 0 0 PERIOD: 1 TIME STEP: 1 

 0 0 0 0 LAYER: 1 

DATA file with initial heads (inithead.dat): 

 14.549 14.484 14.397 14.309 14.219 14.128 14.036 13.942 13.848 13.753 

 13.657 13.561 13.465 13.369 13.274 13.180 13.088 12.996 12.907 12.821 

 12.737 12.656 12.579 12.505 12.435 11.913 10.940 9.971 9.005 8.042 

 7.082 6.125 5.170 4.216 3.707 3.642 3.577 3.512 3.447 3.381 

 3.316 3.250 3.183 3.117 3.051 2.984 2.918 2.851 2.785 2.718 

 2.652 2.585 2.519 2.453 2.386 2.320 2.254 2.188 2.122 2.055 

83

 1.989 1.923 1.857 1.790 1.724 1.658 1.591 1.525 1.458 1.391 
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 1.325 1.258 1.191 1.124 1.057 0.990 0.923 0.856 0.789 0.722 

 0.655 0.588 0.521 0.454 0.387 0.319 0.252 0.185 0.118 0.050 

 0.000 

 14.549 14.485 14.398 14.309 14.220 14.128 14.036 13.942 13.848 13.753 

 13.657 13.561 13.465 13.369 13.274 13.180 13.087 12.996 12.906 12.820 

 12.736 12.655 12.577 12.503 12.433 11.912 10.938 9.969 9.002 8.040 

 7.081 6.124 5.170 4.217 3.707 3.643 3.578 3.513 3.447 3.382 

 3.316 3.250 3.184 3.117 3.051 2.984 2.918 2.851 2.785 2.718 

 2.652 2.585 2.519 2.453 2.386 2.320 2.254 2.188 2.122 2.055 

 1.989 1.923 1.857 1.790 1.724 1.658 1.591 1.525 1.458 1.391 

 1.325 1.258 1.191 1.124 1.057 0.990 0.923 0.856 0.789 0.722 

 0.655 0.588 0.521 0.454 0.387 0.319 0.252 0.185 0.118 0.050 

 0.000 

 14.551 14.487 14.400 14.311 14.222 14.130 14.038 13.944 13.849 13.754 

 13.658 13.561 13.465 13.368 13.273 13.178 13.084 12.992 12.903 12.816 

 12.731 12.650 12.572 12.498 12.427 11.906 10.932 9.963 8.997 8.035 

 7.077 6.122 5.169 4.216 3.708 3.643 3.578 3.513 3.448 3.382 

 3.316 3.250 3.184 3.117 3.051 2.984 2.918 2.851 2.785 2.718 

 2.652 2.585 2.519 2.453 2.386 2.320 2.254 2.188 2.122 2.055 

 1.989 1.923 1.857 1.790 1.724 1.658 1.591 1.525 1.458 1.391 

 1.325 1.258 1.191 1.124 1.057 0.990 0.923 0.856 0.789 0.722 

 0.655 0.588 0.521 0.454 0.387 0.319 0.252 0.185 0.118 0.050 

 0.000 

 14.555 14.491 14.404 14.315 14.225 14.133 14.040 13.946 13.851 13.755 

 13.658 13.561 13.464 13.367 13.270 13.174 13.080 12.987 12.897 12.808 

 12.723 12.641 12.563 12.488 12.418 11.896 10.922 9.953 8.988 8.027 

 7.070 6.117 5.166 4.215 3.708 3.644 3.579 3.514 3.448 3.383 

 3.317 3.251 3.184 3.118 3.051 2.984 2.918 2.851 2.784 2.718 

 2.651 2.585 2.519 2.453 2.386 2.320 2.254 2.188 2.122 2.055 

 1.989 1.923 1.857 1.790 1.724 1.658 1.591 1.525 1.458 1.391 

 1.325 1.258 1.191 1.124 1.057 0.991 0.924 0.857 0.789 0.722 

 0.655 0.588 0.521 0.454 0.387 0.320 0.252 0.185 0.118 0.050 

 0.000 

 14.560 14.496 14.409 14.320 14.230 14.138 14.044 13.950 13.854 13.757 

 13.660 13.562 13.463 13.365 13.267 13.170 13.074 12.980 12.888 12.798 

 12.712 12.629 12.550 12.475 12.404 11.881 10.908 9.939 8.975 8.016 

 7.061 6.110 5.161 4.214 3.709 3.644 3.580 3.515 3.449 3.384 

 3.318 3.251 3.185 3.118 3.051 2.985 2.918 2.851 2.784 2.718 

 2.651 2.585 2.519 2.452 2.386 2.320 2.254 2.188 2.122 2.055 

 1.989 1.923 1.857 1.791 1.724 1.658 1.591 1.525 1.458 1.392 

 1.325 1.258 1.191 1.125 1.058 0.991 0.924 0.857 0.790 0.723 

 0.655 0.588 0.521 0.454 0.387 0.319 0.252 0.185 0.118 0.050 

 0.000 

 14.567 14.503 14.416 14.326 14.236 14.143 14.050 13.954 13.858 13.760 

 13.661 13.562 13.462 13.362 13.263 13.164 13.066 12.970 12.877 12.786 

 12.698 12.613 12.533 12.457 12.385 11.862 10.888 9.920 8.957 8.000 

 7.048 6.100 5.156 4.213 3.710 3.645 3.581 3.516 3.451 3.385 

 3.319 3.252 3.185 3.119 3.052 2.985 2.918 2.851 2.784 2.717 

 2.651 2.585 2.518 2.452 2.386 2.320 2.254 2.188 2.122 2.056 

 1.989 1.923 1.857 1.791 1.724 1.658 1.592 1.525 1.458 1.392 

 1.325 1.258 1.192 1.125 1.058 0.991 0.924 0.857 0.790 0.723 

 0.655 0.588 0.521 0.454 0.387 0.319 0.252 0.185 0.118 0.050 

 0.000 

 14.576 14.511 14.424 14.335 14.244 14.151 14.056 13.960 13.863 13.764 

 13.664 13.563 13.461 13.360 13.258 13.157 13.057 12.959 12.863 12.769 

84
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 12.679 12.593 12.511 12.434 12.362 11.839 10.863 9.895 8.934 7.979 

 7.031 6.087 5.148 4.211 3.711 3.647 3.583 3.518 3.452 3.386 

 3.320 3.253 3.186 3.119 3.052 2.985 2.917 2.850 2.784 2.717 

 2.650 2.584 2.518 2.452 2.386 2.320 2.254 2.188 2.122 2.056 

 1.990 1.923 1.857 1.791 1.725 1.658 1.592 1.525 1.459 1.392 

 1.325 1.259 1.192 1.125 1.058 0.991 0.924 0.857 0.790 0.723 

 0.656 0.588 0.521 0.454 0.387 0.319 0.252 0.185 0.118 0.050 

 0.000 

 14.586 14.522 14.434 14.345 14.253 14.160 14.065 13.968 13.869 13.769 

 13.667 13.564 13.461 13.357 13.253 13.149 13.046 12.945 12.846 12.750 

 12.657 12.569 12.485 12.407 12.334 11.809 10.833 9.864 8.904 7.952 

 7.008 6.070 5.138 4.208 3.712 3.649 3.585 3.520 3.454 3.388 

 3.322 3.254 3.187 3.120 3.052 2.985 2.917 2.850 2.783 2.716 

 2.650 2.584 2.517 2.451 2.385 2.320 2.254 2.188 2.122 2.056 

 1.990 1.924 1.858 1.791 1.725 1.659 1.592 1.526 1.459 1.392 

 1.326 1.259 1.192 1.125 1.058 0.991 0.924 0.857 0.790 0.723 

 0.656 0.589 0.521 0.454 0.387 0.320 0.252 0.185 0.118 0.050 

 0.000 

 14.598 14.534 14.446 14.356 14.264 14.171 14.075 13.977 13.877 13.775 

 13.672 13.567 13.461 13.354 13.247 13.140 13.034 12.929 12.827 12.727 

 12.631 12.540 12.454 12.373 12.300 11.773 10.794 9.825 8.865 7.917 

 6.978 6.048 5.125 4.205 3.715 3.652 3.588 3.523 3.457 3.391 

 3.324 3.256 3.188 3.120 3.052 2.984 2.917 2.849 2.782 2.716 

 2.649 2.583 2.517 2.451 2.385 2.319 2.254 2.188 2.122 2.056 

 1.990 1.924 1.858 1.792 1.725 1.659 1.593 1.526 1.459 1.393 

 1.326 1.259 1.192 1.125 1.059 0.992 0.924 0.857 0.790 0.723 

 0.656 0.589 0.521 0.454 0.387 0.320 0.252 0.185 0.118 0.050 

 0.000 

 14.613 14.548 14.460 14.370 14.278 14.183 14.087 13.988 13.886 13.783 

 13.677 13.570 13.461 13.351 13.241 13.130 13.020 12.911 12.804 12.700 

 12.601 12.505 12.416 12.334 12.258 11.729 10.745 9.774 8.817 7.872 

 6.941 6.021 5.109 4.201 3.717 3.655 3.591 3.527 3.461 3.394 

 3.326 3.258 3.189 3.121 3.052 2.984 2.916 2.848 2.781 2.714 

 2.648 2.582 2.516 2.450 2.385 2.319 2.254 2.188 2.122 2.056 

 1.990 1.924 1.858 1.792 1.726 1.660 1.593 1.527 1.460 1.393 

 1.326 1.260 1.193 1.126 1.059 0.992 0.925 0.858 0.790 0.723 

 0.656 0.589 0.522 0.454 0.387 0.320 0.252 0.185 0.118 0.050 

 0.000 

 14.629 14.565 14.477 14.386 14.293 14.198 14.101 14.001 13.898 13.793 

 13.685 13.575 13.463 13.350 13.235 13.120 13.005 12.891 12.779 12.670 

 12.565 12.465 12.372 12.286 12.209 11.674 10.684 9.710 8.753 7.815 

 6.893 5.985 5.088 4.196 3.721 3.660 3.596 3.532 3.465 3.398 

 3.329 3.260 3.191 3.121 3.052 2.983 2.914 2.847 2.780 2.713 

 2.647 2.581 2.515 2.450 2.384 2.319 2.254 2.188 2.122 2.057 

 1.991 1.925 1.859 1.793 1.726 1.660 1.594 1.527 1.460 1.394 

 1.327 1.260 1.193 1.126 1.059 0.992 0.925 0.858 0.791 0.724 

 0.656 0.589 0.522 0.454 0.387 0.320 0.253 0.185 0.118 0.050 

 0.000 

 14.648 14.584 14.495 14.405 14.312 14.216 14.118 14.016 13.912 13.805 

 13.695 13.582 13.467 13.349 13.229 13.109 12.988 12.868 12.750 12.635 

 12.524 12.419 12.321 12.231 12.151 11.608 10.608 9.628 8.672 7.740 

 6.831 5.939 5.061 4.191 3.726 3.666 3.603 3.538 3.471 3.403 

 3.333 3.263 3.192 3.121 3.051 2.981 2.913 2.845 2.778 2.711 

 2.645 2.580 2.514 2.449 2.384 2.319 2.254 2.188 2.123 2.057 

 1.991 1.926 1.860 1.793 1.727 1.661 1.594 1.528 1.461 1.394 

85
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 1.328 1.261 1.194 1.127 1.060 0.993 0.925 0.858 0.791 0.724 

 0.657 0.589 0.522 0.455 0.387 0.320 0.253 0.185 0.118 0.051 

 0.000 

 14.669 14.605 14.516 14.425 14.332 14.236 14.137 14.035 13.930 13.821 

 13.708 13.592 13.473 13.350 13.225 13.098 12.971 12.844 12.719 12.596 

 12.478 12.365 12.260 12.165 12.082 11.528 10.510 9.521 8.565 7.643 

 6.750 5.880 5.027 4.183 3.733 3.674 3.611 3.546 3.478 3.409 

 3.338 3.266 3.194 3.121 3.050 2.979 2.910 2.842 2.775 2.709 

 2.643 2.578 2.513 2.449 2.384 2.319 2.254 2.189 2.123 2.058 

 1.992 1.926 1.860 1.794 1.728 1.662 1.595 1.528 1.462 1.395 

 1.328 1.261 1.194 1.127 1.060 0.993 0.926 0.859 0.791 0.724 

 0.657 0.589 0.522 0.455 0.387 0.320 0.253 0.185 0.118 0.051 

 0.000 

 14.693 14.628 14.540 14.449 14.355 14.259 14.160 14.057 13.951 13.840 

 13.725 13.605 13.481 13.353 13.222 13.089 12.954 12.819 12.685 12.553 

 12.426 12.304 12.191 12.088 11.999 11.428 10.385 9.382 8.426 7.516 

 6.645 5.805 4.985 4.175 3.743 3.685 3.622 3.557 3.488 3.416 

 3.343 3.269 3.195 3.121 3.047 2.976 2.906 2.838 2.771 2.706 

 2.641 2.576 2.512 2.448 2.384 2.319 2.254 2.189 2.124 2.059 

 1.993 1.927 1.861 1.795 1.729 1.662 1.596 1.529 1.462 1.396 

 1.329 1.262 1.195 1.128 1.061 0.993 0.926 0.859 0.792 0.724 

 0.657 0.590 0.522 0.455 0.388 0.320 0.253 0.185 0.118 0.051 

 0.000 

 14.719 14.654 14.566 14.475 14.382 14.285 14.186 14.083 13.975 13.863 

 13.746 13.623 13.494 13.360 13.222 13.080 12.936 12.792 12.649 12.506 

 12.368 12.234 12.110 11.997 11.901 11.302 10.219 9.196 8.241 7.350 

 6.510 5.709 4.931 4.165 3.756 3.699 3.637 3.570 3.499 3.426 

 3.350 3.273 3.196 3.119 3.043 2.970 2.900 2.832 2.766 2.702 

 2.638 2.575 2.511 2.447 2.384 2.319 2.255 2.190 2.125 2.060 

 1.994 1.928 1.862 1.796 1.730 1.663 1.597 1.530 1.463 1.396 

 1.330 1.263 1.196 1.128 1.061 0.994 0.927 0.859 0.792 0.725 

 0.657 0.590 0.523 0.455 0.388 0.320 0.253 0.186 0.118 0.051 

 0.000 

 14.747 14.682 14.594 14.504 14.410 14.315 14.215 14.113 14.005 13.892 

 13.773 13.647 13.513 13.371 13.224 13.073 12.920 12.765 12.610 12.456 

 12.304 12.157 12.017 11.889 11.781 11.137 9.992 8.942 7.994 7.132 

 6.337 5.588 4.866 4.156 3.775 3.719 3.656 3.587 3.514 3.437 

 3.358 3.277 3.196 3.116 3.037 2.962 2.891 2.825 2.760 2.697 

 2.635 2.573 2.510 2.447 2.384 2.320 2.256 2.191 2.126 2.061 

 1.995 1.929 1.863 1.797 1.731 1.664 1.598 1.531 1.464 1.397 

 1.330 1.263 1.196 1.129 1.062 0.995 0.927 0.860 0.793 0.725 

 0.658 0.590 0.523 0.455 0.388 0.321 0.253 0.186 0.118 0.051 

 0.000 

 14.777 14.713 14.625 14.535 14.442 14.347 14.249 14.147 14.040 13.927 

 13.807 13.678 13.538 13.388 13.231 13.069 12.905 12.739 12.572 12.404 

 12.236 12.071 11.911 11.763 11.636 10.911 9.667 8.588 7.660 6.849 

 6.119 5.441 4.790 4.148 3.801 3.746 3.681 3.609 3.532 3.451 

 3.368 3.282 3.196 3.110 3.027 2.950 2.879 2.815 2.753 2.692 

 2.632 2.571 2.510 2.447 2.384 2.321 2.257 2.192 2.128 2.062 

 1.997 1.931 1.865 1.799 1.732 1.666 1.599 1.532 1.465 1.398 

 1.331 1.264 1.197 1.130 1.062 0.995 0.928 0.860 0.793 0.726 

 0.658 0.591 0.523 0.456 0.388 0.321 0.253 0.186 0.118 0.051 

 0.000 

 14.809 14.745 14.658 14.568 14.477 14.383 14.286 14.186 14.081 13.970 

 13.851 13.720 13.574 13.412 13.241 13.067 12.891 12.713 12.534 12.351 

86
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 12.167 11.980 11.794 11.615 11.456 10.571 9.178 8.083 7.210 6.483 

 5.848 5.265 4.706 4.146 3.839 3.782 3.713 3.636 3.554 3.468 

 3.379 3.288 3.195 3.102 3.012 2.930 2.862 2.801 2.744 2.687 

 2.629 2.570 2.510 2.448 2.386 2.323 2.259 2.194 2.129 2.064 

 1.999 1.933 1.867 1.800 1.734 1.667 1.600 1.533 1.466 1.399 

 1.332 1.265 1.198 1.130 1.063 0.996 0.928 0.861 0.794 0.726 

 0.659 0.591 0.523 0.456 0.388 0.321 0.253 0.186 0.118 0.051 

 0.000 

 14.844 14.780 14.693 14.604 14.514 14.422 14.327 14.229 14.128 14.021 

 13.905 13.777 13.626 13.444 13.256 13.068 12.879 12.690 12.498 12.301 

 12.098 11.888 11.670 11.446 11.231 9.968 8.391 7.356 6.612 6.026 

 5.524 5.066 4.622 4.162 3.893 3.831 3.754 3.669 3.579 3.487 

 3.392 3.295 3.194 3.091 2.989 2.896 2.836 2.785 2.735 2.683 

 2.628 2.570 2.511 2.450 2.388 2.325 2.261 2.197 2.132 2.066 

 2.001 1.935 1.868 1.802 1.735 1.669 1.602 1.535 1.468 1.401 

 1.333 1.266 1.199 1.131 1.064 0.996 0.929 0.862 0.794 0.726 

 0.659 0.591 0.524 0.456 0.389 0.321 0.254 0.186 0.118 0.051 

 0.000 

 14.879 14.815 14.729 14.641 14.553 14.463 14.371 14.277 14.180 14.080 

 13.974 13.856 13.711 13.481 13.270 13.068 12.869 12.669 12.466 12.257 

 12.038 11.805 11.551 11.267 10.951 8.573 7.064 6.335 5.859 5.484 

 5.157 4.854 4.553 4.223 3.977 3.895 3.802 3.705 3.608 3.509 

 3.408 3.305 3.196 3.080 2.955 2.831 2.800 2.769 2.729 2.681 

 2.629 2.572 2.513 2.453 2.391 2.328 2.264 2.199 2.134 2.069 

 2.003 1.937 1.870 1.804 1.737 1.670 1.603 1.536 1.469 1.402 

 1.334 1.267 1.200 1.132 1.065 0.997 0.930 0.862 0.795 0.727 

 0.659 0.592 0.524 0.457 0.389 0.321 0.254 0.186 0.118 0.051 

 0.000 

 14.916 14.852 14.766 14.680 14.593 14.505 14.417 14.327 14.237 14.145 

 14.053 13.961 13.880 13.631 13.317 13.048 12.791 12.533 12.268 11.986 

 11.675 11.313 10.847 10.128 8.629 6.255 5.939 5.656 5.402 5.169 

 4.950 4.739 4.532 4.325 4.110 3.970 3.852 3.743 3.638 3.533 

 3.428 3.319 3.205 3.079 2.921 2.671 2.763 2.762 2.730 2.685 

 2.633 2.577 2.518 2.457 2.395 2.331 2.267 2.203 2.137 2.072 

 2.005 1.939 1.873 1.806 1.739 1.672 1.605 1.538 1.470 1.403 

 1.336 1.268 1.201 1.133 1.065 0.998 0.930 0.863 0.795 0.727 

 0.660 0.592 0.524 0.457 0.389 0.321 0.254 0.186 0.118 0.051 

 0.000 

 14.953 14.889 14.805 14.719 14.634 14.549 14.464 14.379 14.294 14.211 

 14.131 14.058 13.999 13.758 13.362 12.998 12.644 12.286 11.912 11.506 

 11.046 10.494 9.782 8.774 7.225 5.976 5.763 5.541 5.324 5.115 

 4.915 4.720 4.530 4.345 4.169 4.022 3.895 3.778 3.666 3.558 

 3.450 3.340 3.227 3.108 2.979 2.852 2.819 2.787 2.745 2.696 

 2.642 2.584 2.524 2.463 2.400 2.336 2.271 2.206 2.141 2.075 

 2.008 1.942 1.875 1.808 1.741 1.674 1.607 1.539 1.472 1.404 

 1.337 1.269 1.202 1.134 1.066 0.999 0.931 0.863 0.796 0.728 

 0.660 0.593 0.525 0.457 0.389 0.322 0.254 0.186 0.119 0.051 

 0.000 

 14.990 14.927 14.843 14.759 14.675 14.592 14.510 14.429 14.350 14.274 

 14.203 14.140 14.089 13.832 13.376 12.936 12.501 12.056 11.589 11.080 

 10.508 9.836 9.012 7.962 6.613 5.751 5.596 5.420 5.237 5.054 

 4.873 4.696 4.524 4.356 4.199 4.056 3.927 3.807 3.692 3.582 

 3.473 3.365 3.255 3.145 3.037 2.939 2.874 2.820 2.767 2.712 

 2.654 2.594 2.532 2.470 2.406 2.341 2.276 2.211 2.145 2.078 

 2.012 1.945 1.878 1.810 1.743 1.676 1.608 1.541 1.473 1.406 

87
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 1.338 1.270 1.203 1.135 1.067 0.999 0.932 0.864 0.796 0.728 

 0.661 0.593 0.525 0.457 0.390 0.322 0.254 0.186 0.119 0.051 

 0.000 

 15.028 14.964 14.881 14.798 14.716 14.635 14.556 14.478 14.404 14.333 

 14.267 14.209 14.161 13.882 13.372 12.870 12.367 11.849 11.305 10.719 

 10.069 9.330 8.467 7.450 6.265 5.572 5.449 5.306 5.150 4.990 

 4.829 4.669 4.511 4.358 4.213 4.076 3.949 3.829 3.715 3.604 

 3.496 3.390 3.285 3.181 3.083 2.993 2.919 2.853 2.791 2.730 

 2.668 2.605 2.542 2.478 2.413 2.347 2.281 2.215 2.149 2.082 

 2.015 1.948 1.880 1.813 1.745 1.678 1.610 1.543 1.475 1.407 

 1.339 1.272 1.204 1.136 1.068 1.000 0.932 0.865 0.797 0.729 

 0.661 0.593 0.526 0.458 0.390 0.322 0.254 0.186 0.119 0.051 

 0.000 

 15.064 15.001 14.918 14.836 14.755 14.676 14.599 14.525 14.453 14.386 

 14.323 14.268 14.220 13.918 13.361 12.806 12.245 11.669 11.065 10.421 

 9.721 8.945 8.078 7.105 6.033 5.427 5.324 5.203 5.069 4.928 

 4.783 4.638 4.494 4.353 4.217 4.088 3.964 3.847 3.734 3.624 

 3.518 3.414 3.312 3.213 3.119 3.032 2.954 2.882 2.814 2.748 

 2.683 2.618 2.552 2.486 2.420 2.354 2.287 2.220 2.153 2.086 

 2.018 1.951 1.883 1.816 1.748 1.680 1.612 1.544 1.476 1.408 

 1.341 1.273 1.205 1.137 1.069 1.001 0.933 0.865 0.797 0.730 

 0.662 0.594 0.526 0.458 0.390 0.322 0.254 0.187 0.119 0.051 

 0.000 

 15.099 15.036 14.954 14.872 14.793 14.716 14.640 14.568 14.499 14.433 

 14.373 14.318 14.270 13.948 13.348 12.748 12.140 11.516 10.866 10.180 

 9.447 8.654 7.792 6.858 5.862 5.309 5.218 5.112 4.994 4.868 

 4.738 4.606 4.474 4.343 4.216 4.093 3.974 3.859 3.749 3.641 

 3.537 3.435 3.336 3.240 3.149 3.063 2.983 2.907 2.836 2.766 

 2.698 2.630 2.563 2.495 2.428 2.360 2.293 2.225 2.158 2.090 

 2.022 1.954 1.886 1.818 1.750 1.682 1.614 1.546 1.478 1.410 

 1.342 1.274 1.206 1.138 1.070 1.002 0.934 0.866 0.798 0.730 

 0.662 0.594 0.526 0.458 0.390 0.323 0.255 0.187 0.119 0.051 

 0.000 

 15.133 15.070 14.988 14.907 14.829 14.753 14.679 14.608 14.540 14.476 

 14.417 14.362 14.313 13.972 13.336 12.697 12.051 11.389 10.704 9.988 

 9.233 8.431 7.578 6.675 5.729 5.210 5.128 5.033 4.927 4.813 

 4.695 4.574 4.452 4.330 4.210 4.093 3.979 3.868 3.760 3.655 

 3.553 3.454 3.357 3.263 3.174 3.088 3.007 2.929 2.855 2.783 

 2.712 2.642 2.573 2.504 2.436 2.367 2.299 2.231 2.162 2.094 

 2.026 1.957 1.889 1.821 1.753 1.684 1.616 1.548 1.480 1.411 

 1.343 1.275 1.207 1.139 1.071 1.003 0.935 0.867 0.799 0.731 

 0.663 0.595 0.527 0.459 0.391 0.323 0.255 0.187 0.119 0.051 

 0.000 

 15.165 15.102 15.020 14.940 14.863 14.787 14.715 14.645 14.578 14.515 

 14.456 14.401 14.351 13.993 13.325 12.655 11.976 11.284 10.572 9.834 

 9.065 8.259 7.415 6.535 5.624 5.127 5.051 4.964 4.867 4.764 

 4.655 4.543 4.429 4.316 4.202 4.091 3.981 3.874 3.769 3.667 

 3.567 3.469 3.375 3.283 3.194 3.109 3.027 2.948 2.872 2.798 

 2.725 2.654 2.583 2.513 2.443 2.374 2.305 2.236 2.167 2.098 

 2.029 1.961 1.892 1.823 1.755 1.686 1.618 1.550 1.481 1.413 

 1.344 1.276 1.208 1.140 1.072 1.003 0.935 0.867 0.799 0.731 

 0.663 0.595 0.527 0.459 0.391 0.323 0.255 0.187 0.119 0.051 

 0.000 

 15.195 15.132 15.051 14.971 14.894 14.819 14.747 14.678 14.612 14.549 

 14.490 14.435 14.384 14.012 13.317 12.620 11.916 11.200 10.467 9.712 

88
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 8.933 8.125 7.288 6.425 5.539 5.058 4.986 4.904 4.815 4.719 

 4.618 4.514 4.407 4.300 4.193 4.086 3.981 3.877 3.775 3.675 

 3.578 3.482 3.389 3.299 3.211 3.126 3.044 2.964 2.887 2.811 

 2.737 2.664 2.593 2.521 2.451 2.381 2.311 2.241 2.171 2.102 

 2.033 1.964 1.895 1.826 1.757 1.688 1.620 1.551 1.483 1.414 

 1.346 1.277 1.209 1.141 1.072 1.004 0.936 0.868 0.800 0.732 

 0.663 0.595 0.527 0.459 0.391 0.323 0.255 0.187 0.119 0.051 

 0.000 

 15.223 15.160 15.079 15.000 14.923 14.849 14.777 14.708 14.643 14.580 

 14.521 14.465 14.413 14.029 13.312 12.593 11.867 11.132 10.382 9.615 

 8.828 8.020 7.189 6.337 5.469 4.999 4.930 4.853 4.769 4.679 

 4.585 4.487 4.386 4.284 4.182 4.080 3.979 3.878 3.780 3.682 

 3.587 3.493 3.401 3.312 3.225 3.140 3.058 2.978 2.900 2.823 

 2.748 2.674 2.601 2.529 2.458 2.387 2.316 2.246 2.176 2.106 

 2.036 1.967 1.898 1.829 1.759 1.691 1.622 1.553 1.484 1.416 

 1.347 1.278 1.210 1.142 1.073 1.005 0.937 0.868 0.800 0.732 

 0.664 0.596 0.528 0.459 0.391 0.323 0.255 0.187 0.119 0.051 

 0.000 

 15.249 15.186 15.105 15.026 14.949 14.876 14.804 14.736 14.670 14.608 

 14.548 14.492 14.439 14.045 13.309 12.572 11.829 11.078 10.315 9.539 

 8.746 7.936 7.110 6.267 5.411 4.949 4.882 4.809 4.729 4.644 

 4.554 4.462 4.366 4.269 4.172 4.073 3.976 3.878 3.782 3.687 

 3.594 3.502 3.411 3.323 3.237 3.152 3.070 2.990 2.911 2.834 

 2.758 2.683 2.609 2.537 2.464 2.393 2.321 2.251 2.180 2.110 

 2.040 1.970 1.900 1.831 1.762 1.692 1.623 1.554 1.486 1.417 

 1.348 1.280 1.211 1.142 1.074 1.006 0.937 0.869 0.801 0.732 

 0.664 0.596 0.528 0.460 0.392 0.323 0.255 0.187 0.119 0.051 

 0.000 

 15.272 15.210 15.128 15.050 14.973 14.900 14.829 14.760 14.695 14.632 

 14.572 14.516 14.462 14.060 13.308 12.556 11.799 11.035 10.262 9.478 

 8.681 7.871 7.047 6.210 5.363 4.906 4.842 4.771 4.694 4.613 

 4.528 4.439 4.348 4.255 4.161 4.067 3.972 3.877 3.784 3.691 

 3.599 3.509 3.420 3.332 3.247 3.163 3.080 3.000 2.921 2.843 

 2.766 2.691 2.617 2.543 2.470 2.398 2.326 2.255 2.184 2.113 

 2.043 1.973 1.903 1.833 1.764 1.694 1.625 1.556 1.487 1.418 

 1.349 1.281 1.212 1.143 1.075 1.006 0.938 0.869 0.801 0.733 

 0.665 0.596 0.528 0.460 0.392 0.324 0.255 0.187 0.119 0.051 

 0.000 

 15.293 15.231 15.150 15.071 14.995 14.921 14.850 14.782 14.716 14.653 

 14.593 14.536 14.482 14.073 13.309 12.544 11.775 11.002 10.221 9.430 

 8.630 7.818 6.996 6.164 5.323 4.870 4.807 4.738 4.665 4.586 

 4.504 4.419 4.332 4.242 4.151 4.060 3.968 3.876 3.784 3.693 

 3.603 3.514 3.426 3.340 3.255 3.171 3.089 3.008 2.929 2.851 

 2.774 2.698 2.623 2.549 2.476 2.403 2.331 2.259 2.188 2.117 

 2.046 1.975 1.905 1.835 1.766 1.696 1.627 1.557 1.488 1.419 

 1.350 1.281 1.213 1.144 1.075 1.007 0.938 0.870 0.802 0.733 

 0.665 0.597 0.528 0.460 0.392 0.324 0.256 0.187 0.119 0.051 

 0.000 

 15.312 15.249 15.168 15.090 15.014 14.940 14.869 14.801 14.735 14.672 

 14.612 14.554 14.499 14.085 13.310 12.535 11.757 10.976 10.188 9.393 

 8.589 7.777 6.956 6.127 5.290 4.840 4.778 4.711 4.639 4.563 

 4.484 4.402 4.317 4.230 4.142 4.053 3.964 3.874 3.784 3.695 

 3.606 3.518 3.431 3.346 3.261 3.178 3.096 3.015 2.936 2.858 

 2.780 2.704 2.629 2.554 2.480 2.407 2.335 2.262 2.191 2.119 

 2.048 1.978 1.907 1.837 1.767 1.698 1.628 1.559 1.489 1.420 

89
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 1.351 1.282 1.213 1.145 1.076 1.007 0.939 0.870 0.802 0.734 

 0.665 0.597 0.529 0.460 0.392 0.324 0.256 0.188 0.119 0.051 

 0.000 

 15.328 15.265 15.184 15.106 15.030 14.956 14.885 14.817 14.751 14.688 

 14.627 14.569 14.514 14.095 13.312 12.529 11.744 10.955 10.162 9.363 

 8.557 7.744 6.924 6.097 5.263 4.815 4.754 4.688 4.618 4.544 

 4.467 4.387 4.305 4.220 4.134 4.047 3.960 3.872 3.784 3.696 

 3.608 3.522 3.436 3.351 3.267 3.184 3.102 3.021 2.942 2.863 

 2.786 2.709 2.634 2.559 2.484 2.411 2.338 2.266 2.194 2.122 

 2.051 1.980 1.909 1.839 1.769 1.699 1.629 1.560 1.490 1.421 

 1.352 1.283 1.214 1.145 1.077 1.008 0.939 0.871 0.802 0.734 

 0.666 0.597 0.529 0.461 0.392 0.324 0.256 0.188 0.119 0.051 

 0.000 

 15.342 15.279 15.198 15.120 15.044 14.970 14.899 14.831 14.765 14.701 

 14.641 14.582 14.526 14.104 13.314 12.525 11.734 10.940 10.143 9.341 

 8.533 7.719 6.899 6.073 5.242 4.795 4.734 4.669 4.600 4.528 

 4.453 4.374 4.294 4.211 4.127 4.042 3.956 3.870 3.783 3.696 

 3.610 3.524 3.439 3.354 3.271 3.188 3.107 3.026 2.946 2.868 

 2.790 2.713 2.638 2.562 2.488 2.414 2.341 2.268 2.196 2.124 

 2.053 1.982 1.911 1.840 1.770 1.700 1.630 1.561 1.491 1.422 

 1.353 1.284 1.215 1.146 1.077 1.008 0.940 0.871 0.803 0.734 

 0.666 0.597 0.529 0.461 0.392 0.324 0.256 0.188 0.119 0.051 

 0.000 

 15.353 15.290 15.209 15.131 15.055 14.981 14.910 14.842 14.776 14.712 

 14.651 14.593 14.536 14.111 13.317 12.522 11.727 10.929 10.128 9.323 

 8.514 7.699 6.879 6.054 5.224 4.779 4.718 4.654 4.586 4.515 

 4.441 4.364 4.285 4.204 4.122 4.038 3.953 3.868 3.782 3.696 

 3.611 3.526 3.441 3.357 3.274 3.192 3.110 3.030 2.950 2.872 

 2.794 2.717 2.641 2.565 2.491 2.417 2.343 2.270 2.198 2.126 

 2.054 1.983 1.912 1.842 1.771 1.701 1.631 1.561 1.492 1.423 

 1.353 1.284 1.215 1.146 1.077 1.009 0.940 0.871 0.803 0.734 

 0.666 0.598 0.529 0.461 0.393 0.324 0.256 0.188 0.119 0.051 

 0.000 

 15.362 15.299 15.218 15.139 15.063 14.990 14.919 14.850 14.784 14.721 

 14.660 14.601 14.544 14.117 13.319 12.520 11.722 10.921 10.118 9.311 

 8.500 7.685 6.864 6.040 5.211 4.767 4.706 4.643 4.576 4.505 

 4.432 4.356 4.278 4.199 4.117 4.034 3.950 3.866 3.781 3.696 

 3.612 3.527 3.443 3.359 3.277 3.194 3.113 3.033 2.953 2.874 

 2.797 2.720 2.643 2.568 2.493 2.419 2.345 2.272 2.200 2.127 

 2.056 1.984 1.913 1.843 1.772 1.702 1.632 1.562 1.493 1.423 

 1.354 1.285 1.216 1.147 1.078 1.009 0.940 0.872 0.803 0.735 

 0.666 0.598 0.529 0.461 0.393 0.324 0.256 0.188 0.119 0.051 

 0.000 

 15.368 15.305 15.224 15.146 15.070 14.996 14.925 14.857 14.790 14.727 

 14.665 14.607 14.550 14.121 13.320 12.520 11.719 10.916 10.111 9.302 

 8.490 7.674 6.854 6.030 5.202 4.758 4.698 4.635 4.568 4.498 

 4.426 4.351 4.274 4.195 4.114 4.032 3.949 3.865 3.781 3.696 

 3.612 3.528 3.444 3.361 3.278 3.196 3.115 3.035 2.955 2.876 

 2.799 2.721 2.645 2.570 2.495 2.420 2.347 2.273 2.201 2.128 

 2.057 1.985 1.914 1.843 1.773 1.702 1.632 1.563 1.493 1.423 

 1.354 1.285 1.216 1.147 1.078 1.009 0.941 0.872 0.803 0.735 

 0.666 0.598 0.529 0.461 0.393 0.324 0.256 0.188 0.120 0.051 

 0.000 

 15.372 15.309 15.228 15.149 15.073 15.000 14.929 14.860 14.794 14.730 

 14.669 14.610 14.553 14.124 13.322 12.519 11.717 10.913 10.107 9.297 

90
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

 8.484 7.668 6.848 6.024 5.196 4.753 4.693 4.630 4.563 4.494 

 4.422 4.347 4.271 4.192 4.112 4.030 3.948 3.864 3.780 3.696 

 3.612 3.528 3.445 3.362 3.279 3.197 3.116 3.036 2.956 2.878 

 2.800 2.723 2.646 2.571 2.496 2.421 2.347 2.274 2.201 2.129 

 2.057 1.986 1.915 1.844 1.773 1.703 1.633 1.563 1.493 1.424 

 1.354 1.285 1.216 1.147 1.078 1.009 0.941 0.872 0.803 0.735 

 0.666 0.598 0.530 0.461 0.393 0.324 0.256 0.188 0.120 0.051 

 0.000 

 15.373 15.310 15.229 15.150 15.074 15.001 14.930 14.861 14.795 14.731 

 14.670 14.611 14.554 14.125 13.322 12.519 11.716 10.911 10.105 9.295 

 8.482 7.666 6.846 6.022 5.195 4.751 4.692 4.629 4.562 4.493 

 4.421 4.347 4.270 4.191 4.111 4.030 3.947 3.864 3.780 3.696 

 3.612 3.529 3.445 3.362 3.279 3.198 3.116 3.036 2.957 2.878 

 2.800 2.723 2.646 2.571 2.496 2.421 2.348 2.274 2.201 2.129 

 2.057 1.986 1.915 1.844 1.773 1.703 1.633 1.563 1.493 1.424 

 1.354 1.285 1.216 1.147 1.078 1.009 0.941 0.872 0.803 0.735 

 0.666 0.598 0.530 0.461 0.393 0.324 0.256 0.188 0.120 0.051 

 0.000

OBS file (obs.dat): 

# OBS file for MODFLOW-2000, UNC1NLI1 Report Example 

Example 1

HOB file (hob.dat): 

# HOBS file for MODFLOW-2000, UNC1NLI1 Report Example 

 36 0 0 

 1 1 

F1 1 37 13 1 0 0 0 14.88 7.2928 0 1 1 

F2 1 30 13 1 0 0 0 14.78 6.9068 0 1 1 

F3 1 25 13 1 0 0 0 14.86 6.6802 0 1 1 

F4 1 20 13 1 0 0 0 13.87 6.9935 0 1 1 

F5 1 14 13 1 0 0 0 13.01 7.2859 0 1 1 

F6 1 8 13 1 0 0 0 12.97 7.4680 0 1 1 

F7 1 37 26 1 0 0 0 5.61 6.4327 0 1 1 

F8 1 30 26 1 0 0 0 5.63 6.6651 0 1 1 

F9 1 25 26 1 0 0 0 5.94 6.8781 0 1 1 

F10 1 20 26 1 0 0 0 8.83 7.2128 0 1 1 

F11 1 14 26 1 0 0 0 11.08 7.1766 0 1 1 

F12 1 8 26 1 0 0 0 11.35 7.3150 0 1 1 

F13 1 37 39 1 0 0 0 3.7 6.4067 0 1 1 

F14 1 30 39 1 0 0 0 3.75 6.4828 0 1 1 

F15 1 25 39 1 0 0 0 3.52 6.1912 0 1 1 

F16 1 20 39 1 0 0 0 3.35 6.0780 0 1 1 

F17 1 14 39 1 0 0 0 3.42 6.3866 0 1 1 

F18 1 8 39 1 0 0 0 3.71 6.2947 0 1 1 

F19 1 37 52 1 0 0 0 2.37 5.4804 0 1 1 

F20 1 30 52 1 0 0 0 1.91 5.3105 0 1 1 

F21 1 25 52 1 0 0 0 1.76 5.2649 0 1 1 

F22 1 20 52 1 0 0 0 1.96 5.3133 0 1 1 

F23 1 14 52 1 0 0 0 1.82 5.4639 0 1 1 

F24 1 8 52 1 0 0 0 2.06 5.4263 0 1 1 

91
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Input Files 

F25 1 37 65 1 0 0 0 1.54 4.2383 0 1 1 

F26 1 30 65 1 0 0 0 1.66 4.2794 0 1 1 

F27 1 25 65 1 0 0 0 1.78 4.2922 0 1 1 

F28 1 20 65 1 0 0 0 1.54 4.2612 0 1 1 

F29 1 14 65 1 0 0 0 1.7 4.3214 0 1 1 

F30 1 8 65 1 0 0 0 1.55 4.4493 0 1 1 

F31 1 37 78 1 0 0 0 .94 1.5312 0 1 1 

F32 1 30 78 1 0 0 0 1.41 1.9728 0 1 1 

F33 1 25 78 1 0 0 0 1.22 1.9722 0 1 1 

F34 1 20 78 1 0 0 0 1.47 2.0206 0 1 1 

F35 1 14 78 1 0 0 0 1.08 2.1253 0 1 1 

F36 1 8 78 1 0 0 0 .88 2.0226 0 1 1

PES file (pes.dat): 

# PES file for MODFLOW-2000, UNC1NLI1 Report Example 

# 

 50 2 .01 0 Item 1 

 0 0 0 0 0 0 .001 1.5 0 Item 2 

 1 0 0 Item 3 

 .08 0 1 Item 4 

 0 0 0 Item 5

SEN file (sen.dat): 

# SEN file for MODFLOW-2000, UNC1NLI1 Report Example 

# 

 2 0 0 2 Item 1 

 1 0 0 0 Item 2 

HK_1 1 1 1.00000 .01 100 1E-10 Item 3 

HK_2 1 1 1.00000 .01 100 1E-10 Item 3 

92
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

Parameter Estimation – GLOBAL Output File 

GLOBAL Output File (mf2k.global_listing): 

 U.S. GEOLOGICAL SURVEY MODULAR FINITE-DIFFERENCE GROUND-WATER FLOW MODEL 

 VERSION 1.6SC1  ̈25/02/2003 

 This model run produced both GLOBAL and LIST files. This is the GLOBAL file. 

 GLOBAL LISTING FILE: mf2k.global_listing 

 UNIT 4 

 OPENING output.dat 

 FILE TYPE:LIST UNIT 3 STATUS:REPLACE 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING discret.dat 

 FILE TYPE:DIS UNIT 95 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING bas6.dat 

 FILE TYPE:BAS6 UNIT 1 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING zone.dat 

 FILE TYPE:ZONE UNIT 93 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING multiple.dat 

 FILE TYPE:MULT UNIT 94 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING lpf6.dat 

 FILE TYPE:LPF UNIT 31 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING rech6.dat 

 FILE TYPE:RCH UNIT 11 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wel6.dat 

 FILE TYPE:WEL UNIT 12 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING pcg2.dat 

 FILE TYPE:PCG UNIT 23 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING oc.dat 

 FILE TYPE:OC UNIT 22 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

93
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 OPENING inithead.dat 

 FILE TYPE:DATA UNIT 10 STATUS:UNKNOWN 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING obs.dat 

 FILE TYPE:OBS UNIT 40 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING hob.dat 

 FILE TYPE:HOB UNIT 41 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING pes.dat 

 FILE TYPE:PES UNIT 47 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING sen.dat 

 FILE TYPE:SEN UNIT 46 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

#UNC 60 unc.dat 

#DATA 61 ci-res.dat 

 THE FREE FORMAT OPTION HAS BEEN SELECTED 

 DISCRETIZATION INPUT DATA READ FROM UNIT 95 

 # Discretization file for MODFLOW-2000, UNC1NLI1 Report Example 

 1 LAYERS 41 ROWS 91 COLUMNS 

 1 STRESS PERIOD(S) IN SIMULATION 

 MODEL TIME UNIT IS SECONDS 

 MODEL LENGTH UNIT IS UNDEFINED 

 THE GROUND-WATER TRANSPORT PROCESS IS INACTIVE 

 THE OBSERVATION PROCESS IS ACTIVE 

 THE SENSITIVITY PROCESS IS ACTIVE 

 THE PARAMETER-ESTIMATION PROCESS IS ACTIVE 

 MODE: PARAMETER ESTIMATION 

 ZONE OPTION, INPUT READ FROM UNIT 93 

 # Zone file for MODFLOW-2000, UNC1NLI1 Report Example 

 2 ZONE ARRAYS 

 MULTIPLIER OPTION, INPUT READ FROM UNIT 94 

 # Multiplier file for MODFLOW-2000, UNC1NLI1 Report Example 

 2 MULTIPLIER ARRAYS 

 Confining bed flag for each layer: 

 0 

 41173 ELEMENTS OF GX ARRAY USED OUT OF 41173 

 3731 ELEMENTS OF GZ ARRAY USED OUT OF 3731 

 11193 ELEMENTS OF IG ARRAY USED OUT OF 11193 

 DELR 

94
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 READING ON UNIT 95 WITH FORMAT: (16G5.0) 

 DELC 

 READING ON UNIT 95 WITH FORMAT: (16G5.0) 

 TOP ELEVATION OF LAYER 1 = 1.00000 

 MODEL LAYER BOTTOM EL. = 0.00000 FOR LAYER 1 

 STRESS PERIOD LENGTH TIME STEPS MULTIPLIER FOR DELT SS FLAG 

 ---------------------------------------------------------------------------- 

 1 1.000000 1 1.000 SS 

 STEADY-STATE SIMULATION 

 MULT. ARRAY: HK1 = 1.00000 

 MULT. ARRAY: VK1 = 1.00000 

 ZONE ARRAY: ZHK1 

 READING ON UNIT 93 WITH FORMAT: (91I1) 

 ZONE ARRAY: ZVK1 = 999 

 LPF1 -- LAYER PROPERTY FLOW PACKAGE, VERSION 1, 1/11/2000 

 INPUT READ FROM UNIT 31 

 # Layer-Property Flow file for MODFLOW-2000, UNC1NLI1 Report Example 

 HEAD AT CELLS THAT CONVERT TO DRY= 999.00 

 3 Named Parameters 

 LAYER FLAGS: 

 LAYER LAYTYP LAYAVG CHANI LAYVKA LAYWET 

 --------------------------------------------------------------------------- 

 1 0 0 1.000E+00 0 0 

 INTERPRETATION OF LAYER FLAGS: 

 INTERBLOCK HORIZONTAL DATA IN 

 LAYER TYPE TRANSMISSIVITY ANISOTROPY ARRAY VKA WETTABILITY 

 LAYER (LAYTYP) (LAYAVG) (CHANI) (LAYVKA) (LAYWET) 

 --------------------------------------------------------------------------- 

 1 CONFINED HARMONIC 1.000E+00 VERTICAL K NON-WETTABLE 

 7462 ELEMENTS IN X ARRAY ARE USED BY LPF 

 6 ELEMENTS IN IX ARRAY ARE USED BY LPF 

 PCG2 -- CONJUGATE GRADIENT SOLUTION PACKAGE, VERSION 2.4, 12/29/98 

 MAXIMUM OF 50 CALLS OF SOLUTION ROUTINE 

95
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 MAXIMUM OF 30 INTERNAL ITERATIONS PER CALL TO SOLUTION ROUTINE 

 MATRIX PRECONDITIONING TYPE : 1 

 14193 ELEMENTS IN X ARRAY ARE USED BY PCG 

 10500 ELEMENTS IN IX ARRAY ARE USED BY PCG 

 7462 ELEMENTS IN Z ARRAY ARE USED BY PCG 

 SEN1BAS6 -- SENSITIVITY PROCESS, VERSION 1.0, 10/15/98 

 INPUT READ FROM UNIT 46 

 # SEN file for MODFLOW-2000, UNC1NLI1 Report Example 

 # 

 NUMBER OF PARAMETER VALUES TO BE READ FROM SEN FILE: 2 

 ISENALL............................................: 0 

 SENSITIVITIES WILL BE STORED IN MEMORY 

 FOR UP TO 2 PARAMETERS 

 3745 ELEMENTS IN X ARRAY ARE USED FOR SENSITIVITIES 

 3731 ELEMENTS IN Z ARRAY ARE USED FOR SENSITIVITIES 

 4 ELEMENTS IN IX ARRAY ARE USED FOR SENSITIVITIES 

 PES1BAS6 -- PARAMETER-ESTIMATION PROCESS, VERSION 1.0, 07/22/99 

 INPUT READ FROM UNIT 47 

 # PES file for MODFLOW-2000, UNC1NLI1 Report Example 

 # 

 MAXIMUM NUMBER OF PARAMETER-ESTIMATION ITERATIONS (MAX-ITER) = 50 

 MAXIMUM PARAMETER CORRECTION (MAX-CHANGE) ------------------- = 2.0000 

 CLOSURE CRITERION (TOL) ------------------------------------- = 0.10000E-01 

 SUM OF SQUARES CLOSURE CRITERION (SOSC) --------------------- = 0.0000 

 FLAG TO GENERATE INPUT NEEDED BY BEALE-2000 (IBEFLG) -------- = 0 

 FLAG TO GENERATE INPUT NEEDED BY YCINT-2000 (IYCFLG) -------- = 0 

 OMIT PRINTING TO SCREEN (IF = 1) (IOSTAR) ------------------- = 0 

 ADJUST GAUSS-NEWTON MATRIX WITH NEWTON UPDATES (IF = 1)(NOPT) = 0 

 NUMBER OF FLETCHER-REEVES ITERATIONS (NFIT) ----------------- = 0 

 CRITERION FOR ADDING MATRIX R (SOSR) ------------------------ = 0.0000 

 VALUE USED TO INCREMENT MARQUARDT PARAMETER (RMAR) ---------- = 0.10000E-02 

 MARQUARDT PARAMETER MULTIPLIER (RMARM) ---------------------- = 1.5000 

 APPLY MAX-CHANGE IN REGRESSION SPACE (IF = 1) (IAP) --------- = 0 

 FORMAT CODE FOR COVARIANCE AND CORRELATION MATRICES (IPRCOV) = 1 

 PRINT PARAMETER-ESTIMATION STATISTICS 

 EACH ITERATION (IF > 0) (IPRINT) ----------------------- = 0 

 PRINT EIGENVALUES AND EIGENVECTORS OF 

 COVARIANCE MATRIX (IF > 0) (LPRINT) -------------------- = 0 

 SEARCH DIRECTION ADJUSTMENT PARAMETER (CSA) ----------------- = 0.80000E-01 

 MODIFY CONVERGENCE CRITERIA (IF > 0) (FCONV) ---------------- = 0.0000 

 CALCULATE SENSITIVITIES USING FINAL 

 PARAMETER ESTIMATES (IF > 0) (LASTX) -------------------- = 1 

 NUMBER OF USUALLY POS. PARAMETERS THAT MAY BE NEGATIVE (NPNG) = 0 

 NUMBER OF PARAMETERS WITH CORRELATED PRIOR INFORMATION (IPR) = 0 

 NUMBER OF PRIOR-INFORMATION EQUATIONS (MPR) ----------------- = 0 

96
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 479 ELEMENTS IN X ARRAY ARE USED FOR PARAMETER ESTIMATION 

 81 ELEMENTS IN Z ARRAY ARE USED FOR PARAMETER ESTIMATION 

 52 ELEMENTS IN IX ARRAY ARE USED FOR PARAMETER ESTIMATION 

 OBS1BAS6 -- OBSERVATION PROCESS, VERSION 1.0, 4/27/99 

 INPUT READ FROM UNIT 40 

 # OBS file for MODFLOW-2000, UNC1NLI1 Report Example 

 OBSERVATION GRAPH-DATA OUTPUT FILES 

 WILL BE PRINTED AND NAMED USING THE BASE: Example 

 DIMENSIONLESS SCALED OBSERVATION SENSITIVITIES WILL BE PRINTED 

 HEAD OBSERVATIONS -- INPUT READ FROM UNIT 41 

 # HOBS file for MODFLOW-2000, UNC1NLI1 Report Example 

 NUMBER OF HEADS....................................: 36 

 NUMBER OF MULTILAYER HEADS.......................: 0 

 MAXIMUM NUMBER OF LAYERS FOR MULTILAYER HEADS....: 0 

 718 ELEMENTS IN X ARRAY ARE USED FOR OBSERVATIONS 

 2 ELEMENTS IN Z ARRAY ARE USED FOR OBSERVATIONS 

 363 ELEMENTS IN IX ARRAY ARE USED FOR OBSERVATIONS 

 COMMON ERROR VARIANCE FOR ALL OBSERVATIONS SET TO: 1.000 

 26597 ELEMENTS OF X ARRAY USED OUT OF 26597 

 11276 ELEMENTS OF Z ARRAY USED OUT OF 11276 

 10925 ELEMENTS OF IX ARRAY USED OUT OF 10925 

 7462 ELEMENTS OF XHS ARRAY USED OUT OF 7462 

 INFORMATION ON PARAMETERS LISTED IN SEN FILE 

 LOWER UPPER ALTERNATE 

 VALUE IN SEN REASONABLE REASONABLE SCALING 

 NAME ISENS LN INPUT FILE LIMIT LIMIT FACTOR 

 ---------- ----- -- ------------ ------------ ------------ ------------ 

 HK_1 1 1 1.0000 0.10000E-01 100.00 0.10000E-09 

 HK_2 1 1 1.0000 0.10000E-01 100.00 0.10000E-09 

 ----------------------------------------------------------------------------- 

 FOR THE PARAMETERS LISTED IN THE TABLE ABOVE, PARAMETER VALUES IN INDIVIDUAL 

 PACKAGE INPUT FILES ARE REPLACED BY THE VALUES FROM THE SEN INPUT FILE. THE 

 ALTERNATE SCALING FACTOR IS USED TO SCALE SENSITIVITIES IF IT IS LARGER THAN 

 THE PARAMETER VALUE IN ABSOLUTE VALUE AND THE PARAMETER IS NOT LOG-TRANSFORMED. 

 ISENS IS GREATER THAN ZERO FOR 2 PARAMETERS 

 HEAD OBSERVATION VARIANCES ARE MULTIPLIED BY: 1.000 

 OBSERVED HEAD DATA -- TIME OFFSETS ARE MULTIPLIED BY: 1.0000 

 REFER. 

 OBSERVATION STRESS TIME STATISTIC PLOT 

 OBS# NAME PERIOD OFFSET OBSERVATION STATISTIC TYPE SYM. 

 1 F1 1 0.000 14.88 729.3 VARIANCE 1 

 2 F2 1 0.000 14.78 690.7 VARIANCE 1 

 3 F3 1 0.000 14.86 668.0 VARIANCE 1 

 4 F4 1 0.000 13.87 699.3 VARIANCE 1 

97
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 5 F5 1 0.000 13.01 728.6 VARIANCE 1 

 6 F6 1 0.000 12.97 746.8 VARIANCE 1 

 7 F7 1 0.000 5.610 643.3 VARIANCE 1 

 8 F8 1 0.000 5.630 666.5 VARIANCE 1 

 9 F9 1 0.000 5.940 687.8 VARIANCE 1 

 10 F10 1 0.000 8.830 721.3 VARIANCE 1 

 11 F11 1 0.000 11.08 717.7 VARIANCE 1 

 12 F12 1 0.000 11.35 731.5 VARIANCE 1 

 13 F13 1 0.000 3.700 640.7 VARIANCE 1 

 14 F14 1 0.000 3.750 648.3 VARIANCE 1 

 15 F15 1 0.000 3.520 619.1 VARIANCE 1 

 16 F16 1 0.000 3.350 607.8 VARIANCE 1 

 17 F17 1 0.000 3.420 638.7 VARIANCE 1 

 18 F18 1 0.000 3.710 629.5 VARIANCE 1 

 19 F19 1 0.000 2.370 548.0 VARIANCE 1 

 20 F20 1 0.000 1.910 531.0 VARIANCE 1 

 21 F21 1 0.000 1.760 526.5 VARIANCE 1 

 22 F22 1 0.000 1.960 531.3 VARIANCE 1 

 23 F23 1 0.000 1.820 546.4 VARIANCE 1 

 24 F24 1 0.000 2.060 542.6 VARIANCE 1 

 25 F25 1 0.000 1.540 423.8 VARIANCE 1 

 26 F26 1 0.000 1.660 427.9 VARIANCE 1 

 27 F27 1 0.000 1.780 429.2 VARIANCE 1 

 28 F28 1 0.000 1.540 426.1 VARIANCE 1 

 29 F29 1 0.000 1.700 432.1 VARIANCE 1 

 30 F30 1 0.000 1.550 444.9 VARIANCE 1 

 31 F31 1 0.000 0.9400 153.1 VARIANCE 1 

 32 F32 1 0.000 1.410 197.3 VARIANCE 1 

 33 F33 1 0.000 1.220 197.2 VARIANCE 1 

 34 F34 1 0.000 1.470 202.1 VARIANCE 1 

 35 F35 1 0.000 1.080 212.5 VARIANCE 1 

 36 F36 1 0.000 0.8800 202.3 VARIANCE 1 

 HEAD CHANGE 

 REFERENCE 

 OBSERVATION ROW COL OBSERVATION 

 OBS# NAME LAY ROW COL OFFSET OFFSET (IF > 0) 

 1 F1 1 37 13 0.000 0.000 0 

 2 F2 1 30 13 0.000 0.000 0 

 3 F3 1 25 13 0.000 0.000 0 

 4 F4 1 20 13 0.000 0.000 0 

 5 F5 1 14 13 0.000 0.000 0 

 6 F6 1 8 13 0.000 0.000 0 

 7 F7 1 37 26 0.000 0.000 0 

 8 F8 1 30 26 0.000 0.000 0 

 9 F9 1 25 26 0.000 0.000 0 

 10 F10 1 20 26 0.000 0.000 0 

 11 F11 1 14 26 0.000 0.000 0 

 12 F12 1 8 26 0.000 0.000 0 

 13 F13 1 37 39 0.000 0.000 0 

 14 F14 1 30 39 0.000 0.000 0 

 15 F15 1 25 39 0.000 0.000 0 

 16 F16 1 20 39 0.000 0.000 0 

 17 F17 1 14 39 0.000 0.000 0 

 18 F18 1 8 39 0.000 0.000 0 

98
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 19 F19 1 37 52 0.000 0.000 0 

 20 F20 1 30 52 0.000 0.000 0 

 21 F21 1 25 52 0.000 0.000 0 

 22 F22 1 20 52 0.000 0.000 0 

 23 F23 1 14 52 0.000 0.000 0 

 24 F24 1 8 52 0.000 0.000 0 

 25 F25 1 37 65 0.000 0.000 0 

 26 F26 1 30 65 0.000 0.000 0 

 27 F27 1 25 65 0.000 0.000 0 

 28 F28 1 20 65 0.000 0.000 0 

 29 F29 1 14 65 0.000 0.000 0 

 30 F30 1 8 65 0.000 0.000 0 

 31 F31 1 37 78 0.000 0.000 0 

 32 F32 1 30 78 0.000 0.000 0 

 33 F33 1 25 78 0.000 0.000 0 

 34 F34 1 20 78 0.000 0.000 0 

 35 F35 1 14 78 0.000 0.000 0 

 36 F36 1 8 78 0.000 0.000 0 

 SOLUTION BY THE CONJUGATE-GRADIENT METHOD 

 ------------------------------------------- 

 MAXIMUM NUMBER OF CALLS TO PCG ROUTINE = 50 

 MAXIMUM ITERATIONS PER CALL TO PCG = 30 

 MATRIX PRECONDITIONING TYPE = 1 

 RELAXATION FACTOR (ONLY USED WITH PRECOND. TYPE 1) = 0.10000E+01 

 PARAMETER OF POLYNOMIAL PRECOND. = 2 (2) OR IS CALCULATED : 1 

 HEAD CHANGE CRITERION FOR CLOSURE = 0.10000E-02 

 RESIDUAL CHANGE CRITERION FOR CLOSURE = 0.10000E-02 

 PCG HEAD AND RESIDUAL CHANGE PRINTOUT INTERVAL = 1 

 PRINTING FROM SOLVER IS LIMITED(1) OR SUPPRESSED (>1) = 0 

 DAMPING PARAMETER = 0.10000E+01 

 CONVERGENCE CRITERIA FOR SENSITIVITIES 

 PARAMETER HCLOSE RCLOSE 

 ---------- ------------ ------------ 

 HK_1 0.10000E-04 0.10000E-04 

 HK_2 0.10000E-04 0.10000E-04 

 -------------------------------------- 

 WETTING CAPABILITY IS NOT ACTIVE IN ANY LAYER 

 PARAMETERS DEFINED IN THE LPF PACKAGE 

 PARAMETER NAME:HK_1 TYPE:HK CLUSTERS: 1 

 Parameter value from package file is: 1.0000 

 LAYER: 1 MULTIPLIER ARRAY: HK1 ZONE ARRAY: ZHK1 

 ZONE VALUES: 1 

 PARAMETER NAME:HK_2 TYPE:HK CLUSTERS: 1 

 Parameter value from package file is: 1.0000 

 LAYER: 1 MULTIPLIER ARRAY: HK1 ZONE ARRAY: ZHK1 

 ZONE VALUES: 2 

99
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 PARAMETER NAME:VK_0 TYPE:VK CLUSTERS: 1 

 Parameter value from package file is: 1.0000 

 LAYER: 1 MULTIPLIER ARRAY: VK1 ZONE ARRAY: ZVK1 

 ZONE VALUES: 999 

 HYD. COND. ALONG ROWS FOR LAYER 1 WILL BE DEFINED BY PARAMETERS 

 (PRINT FLAG= 31) 

 VERTICAL HYD. COND. FOR LAYER 1 WILL BE DEFINED BY PARAMETERS 

 (PRINT FLAG= 31) 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

 WELL NO. LAYER ROW COL STRESS FACTOR 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 

OBSERVATION SENSITIVITY TABLE(S) FOR PARAMETER-ESTIMATION ITERATION 1 

 DIMENSIONLESS SCALED SENSITIVITIES (SCALED BY B*(WT**.5)) 

 PARAMETER: HK_1 HK_2 

 OBS # OBSERVATION 

 1 F1 -0.261 -0.556E-01 

 2 F2 -0.269 -0.564E-01 

 3 F3 -0.275 -0.557E-01 

 4 F4 -0.275 -0.482E-01 

 5 F5 -0.274 -0.425E-01 

 6 F6 -0.272 -0.408E-01 

 7 F7 -0.270 -0.346E-02 

 8 F8 -0.263 -0.518E-02 

 9 F9 -0.255 -0.899E-02 

 10 F10 -0.236 -0.218E-01 

 11 F11 -0.226 -0.324E-01 

 12 F12 -0.221 -0.350E-01 

 13 F13 -0.209 -0.244E-02 

 14 F14 -0.206 -0.228E-02 

 15 F15 -0.210 -0.158E-02 

 16 F16 -0.213 0.431E-03 

 17 F17 -0.212 0.221E-02 

 18 F18 -0.215 0.245E-02 

 19 F19 -0.165 -0.106E-02 

 20 F20 -0.166 -0.771E-03 

 21 F21 -0.165 -0.381E-03 

100
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 22 F22 -0.164 0.100E-03 

 23 F23 -0.164 0.625E-03 

 24 F24 -0.168 0.968E-03 

 25 F25 -0.124 -0.433E-03 

 26 F26 -0.123 -0.296E-03 

 27 F27 -0.123 -0.141E-03 

 28 F28 -0.124 0.364E-04 

 29 F29 -0.123 0.238E-03 

 30 F30 -0.122 0.379E-03 

 31 F31 -0.102 -0.229E-03 

 32 F32 -0.900E-01 -0.138E-03 

 33 F33 -0.900E-01 -0.656E-04 

 34 F34 -0.890E-01 0.165E-04 

 35 F35 -0.869E-01 0.107E-03 

 36 F36 -0.892E-01 0.178E-03 

 COMPOSITE SCALED SENSITIVITIES ((SUM OF THE SQUARED VALUES)/ND)**.5 

 0.196 0.224E-01 

 PARAMETER COMPOSITE SCALED SENSITIVITY 

 ---------- ---------------------------- 

 HK_1 1.95517E-01 

 HK_2 2.24057E-02 

 STARTING VALUES OF REGRESSION PARAMETERS : 

 HK_1 HK_2 

 1.000 1.000 

 SUMS OF SQUARED, WEIGHTED RESIDUALS: 

 ALL DEPENDENT VARIABLES: 0.40901 

 DEP. VARIABLES PLUS PARAMETERS: 0.40901 

 ----------------------------------------------------------------------- 

 PARAMETER VALUES AND STATISTICS FOR ALL PARAMETER-ESTIMATION ITERATIONS 

 ----------------------------------------------------------------------- 

 MODIFIED GAUSS-NEWTON CONVERGES IF THE ABSOLUTE VALUE OF THE MAXIMUM 

 FRACTIONAL PARAMETER CHANGE (MAX CALC. CHANGE) IS LESS THAN TOL OR IF THE 

 SUM OF SQUARED, WEIGHTED RESIDUALS CHANGES LESS THAN SOSC OVER TWO 

 PARAMETER-ESTIMATION ITERATIONS. 

 MODIFIED GAUSS-NEWTON PROCEDURE FOR PARAMETER-ESTIMATION ITERATION NO. = 1 

 VALUES FROM SOLVING THE NORMAL EQUATION : 

 MARQUARDT PARAMETER ------------------- = 0.0000 

 MAX. FRAC. PAR. CHANGE (TOL= 0.100E-01) = -.99783 

 OCCURRED FOR PARAMETER "HK_2 " TYPE N 

 CALCULATION OF DAMPING PARAMETER 

 MAX-CHANGE SPECIFIED: 2.0 USED: 2.0 

 OSCILL. CONTROL FACTOR (1, NO EFFECT)-- = 1.0000 

 DAMPING PARAMETER (RANGE 0 TO 1) ------ = 1.0000 

 CONTROLLED BY PARAMETER "HK_2 " TYPE N 

101
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 UPDATED ESTIMATES OF REGRESSION PARAMETERS : 

 HK_1 HK_2 

 1.404 2.1694E-03 

 SUMS OF SQUARED, WEIGHTED RESIDUALS: 

 ALL DEPENDENT VARIABLES: 1022.7 

 DEP. VARIABLES PLUS PARAMETERS: 1022.7 

 MODIFIED GAUSS-NEWTON PROCEDURE FOR PARAMETER-ESTIMATION ITERATION NO. = 2 

 VALUES FROM SOLVING THE NORMAL EQUATION : 

 MARQUARDT PARAMETER ------------------- = 0.0000 

 MAX. FRAC. PAR. CHANGE (TOL= 0.100E-01) = 1.6663 

 OCCURRED FOR PARAMETER "HK_2 " TYPE P 

 CALCULATION OF DAMPING PARAMETER 

 MAX-CHANGE SPECIFIED: 2.0 USED: 2.0 

 OSCILL. CONTROL FACTOR (1, NO EFFECT)-- = 0.98277 

 DAMPING PARAMETER (RANGE 0 TO 1) ------ = 0.98277 

 CONTROLLED BY PARAMETER "HK_2 " TYPE P 

 UPDATED ESTIMATES OF REGRESSION PARAMETERS : 

 HK_1 HK_2 

 1.476 5.6874E-03 

 SUMS OF SQUARED, WEIGHTED RESIDUALS: 

 ALL DEPENDENT VARIABLES: 139.07 

 DEP. VARIABLES PLUS PARAMETERS: 139.07 

 MODIFIED GAUSS-NEWTON PROCEDURE FOR PARAMETER-ESTIMATION ITERATION NO. = 3 

 VALUES FROM SOLVING THE NORMAL EQUATION : 

 MARQUARDT PARAMETER ------------------- = 0.0000 

 MAX. FRAC. PAR. CHANGE (TOL= 0.100E-01) = 1.5827 

 OCCURRED FOR PARAMETER "HK_2 " TYPE P 

 CALCULATION OF DAMPING PARAMETER 

 MAX-CHANGE SPECIFIED: 2.0 USED: 2.0 

 OSCILL. CONTROL FACTOR (1, NO EFFECT)-- = 1.0000 

 DAMPING PARAMETER (RANGE 0 TO 1) ------ = 1.0000 

 CONTROLLED BY PARAMETER "HK_2 " TYPE P 

 UPDATED ESTIMATES OF REGRESSION PARAMETERS : 

 HK_1 HK_2 

 1.477 1.4689E-02 

 SUMS OF SQUARED, WEIGHTED RESIDUALS: 

 ALL DEPENDENT VARIABLES: 17.395 

102
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 DEP. VARIABLES PLUS PARAMETERS: 17.395 

 MODIFIED GAUSS-NEWTON PROCEDURE FOR PARAMETER-ESTIMATION ITERATION NO. = 4 

 VALUES FROM SOLVING THE NORMAL EQUATION : 

 MARQUARDT PARAMETER ------------------- = 0.0000 

 MAX. FRAC. PAR. CHANGE (TOL= 0.100E-01) = 1.3735 

 OCCURRED FOR PARAMETER "HK_2 " TYPE P 

 CALCULATION OF DAMPING PARAMETER 

 MAX-CHANGE SPECIFIED: 2.0 USED: 2.0 

 OSCILL. CONTROL FACTOR (1, NO EFFECT)-- = 1.0000 

 DAMPING PARAMETER (RANGE 0 TO 1) ------ = 1.0000 

 CONTROLLED BY PARAMETER "HK_2 " TYPE P 

 UPDATED ESTIMATES OF REGRESSION PARAMETERS : 

 HK_1 HK_2 

 1.472 3.4864E-02 

 SUMS OF SQUARED, WEIGHTED RESIDUALS: 

 ALL DEPENDENT VARIABLES: 1.8804 

 DEP. VARIABLES PLUS PARAMETERS: 1.8804 

 MODIFIED GAUSS-NEWTON PROCEDURE FOR PARAMETER-ESTIMATION ITERATION NO. = 5 

 VALUES FROM SOLVING THE NORMAL EQUATION : 

 MARQUARDT PARAMETER ------------------- = 0.0000 

 MAX. FRAC. PAR. CHANGE (TOL= 0.100E-01) = 0.94318 

 OCCURRED FOR PARAMETER "HK_2 " TYPE P 

 CALCULATION OF DAMPING PARAMETER 

 MAX-CHANGE SPECIFIED: 2.0 USED: 2.0 

 OSCILL. CONTROL FACTOR (1, NO EFFECT)-- = 1.0000 

 DAMPING PARAMETER (RANGE 0 TO 1) ------ = 1.0000 

 CONTROLLED BY PARAMETER "HK_2 " TYPE P 

 UPDATED ESTIMATES OF REGRESSION PARAMETERS : 

 HK_1 HK_2 

 1.465 6.7748E-02 

 SUMS OF SQUARED, WEIGHTED RESIDUALS: 

 ALL DEPENDENT VARIABLES: 0.14038 

 DEP. VARIABLES PLUS PARAMETERS: 0.14038 

 MODIFIED GAUSS-NEWTON PROCEDURE FOR PARAMETER-ESTIMATION ITERATION NO. = 6 

 VALUES FROM SOLVING THE NORMAL EQUATION : 

 MARQUARDT PARAMETER ------------------- = 0.0000 

 MAX. FRAC. PAR. CHANGE (TOL= 0.100E-01) = 0.38419 

 OCCURRED FOR PARAMETER "HK_2 " TYPE P 

103
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 CALCULATION OF DAMPING PARAMETER 

 MAX-CHANGE SPECIFIED: 2.0 USED: 2.0 

 OSCILL. CONTROL FACTOR (1, NO EFFECT)-- = 1.0000 

 DAMPING PARAMETER (RANGE 0 TO 1) ------ = 1.0000 

 CONTROLLED BY PARAMETER "HK_2 " TYPE P 

 UPDATED ESTIMATES OF REGRESSION PARAMETERS : 

 HK_1 HK_2 

 1.462 9.3776E-02 

 SUMS OF SQUARED, WEIGHTED RESIDUALS: 

 ALL DEPENDENT VARIABLES: 0.16299E-01 

 DEP. VARIABLES PLUS PARAMETERS: 0.16299E-01 

 MODIFIED GAUSS-NEWTON PROCEDURE FOR PARAMETER-ESTIMATION ITERATION NO. = 7 

 VALUES FROM SOLVING THE NORMAL EQUATION : 

 MARQUARDT PARAMETER ------------------- = 0.0000 

 MAX. FRAC. PAR. CHANGE (TOL= 0.100E-01) = 0.59154E-01 

 OCCURRED FOR PARAMETER "HK_2 " TYPE P 

 CALCULATION OF DAMPING PARAMETER 

 MAX-CHANGE SPECIFIED: 2.0 USED: 2.0 

 OSCILL. CONTROL FACTOR (1, NO EFFECT)-- = 1.0000 

 DAMPING PARAMETER (RANGE 0 TO 1) ------ = 1.0000 

 CONTROLLED BY PARAMETER "HK_2 " TYPE P 

 UPDATED ESTIMATES OF REGRESSION PARAMETERS : 

 HK_1 HK_2 

 1.463 9.9323E-02 

 SUMS OF SQUARED, WEIGHTED RESIDUALS: 

 ALL DEPENDENT VARIABLES: 0.14039E-01 

 DEP. VARIABLES PLUS PARAMETERS: 0.14039E-01 

 MODIFIED GAUSS-NEWTON PROCEDURE FOR PARAMETER-ESTIMATION ITERATION NO. = 8 

 VALUES FROM SOLVING THE NORMAL EQUATION : 

 MARQUARDT PARAMETER ------------------- = 0.0000 

 MAX. FRAC. PAR. CHANGE (TOL= 0.100E-01) = 0.11102E-02 

 OCCURRED FOR PARAMETER "HK_2 " TYPE P 

 CALCULATION OF DAMPING PARAMETER 

 MAX-CHANGE SPECIFIED: 2.0 USED: 2.0 

 OSCILL. CONTROL FACTOR (1, NO EFFECT)-- = 1.0000 

 DAMPING PARAMETER (RANGE 0 TO 1) ------ = 1.0000 

 CONTROLLED BY PARAMETER "HK_2 " TYPE P 

 UPDATED ESTIMATES OF REGRESSION PARAMETERS : 

 HK_1 HK_2 

104
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 1.464 9.9433E-02 

 *** PARAMETER ESTIMATION CONVERGED BY SATISFYING THE TOL CRITERION *** 

 

OBSERVATION SENSITIVITY TABLE(S) FOR PARAMETER-ESTIMATION ITERATION 8 

 DIMENSIONLESS SCALED SENSITIVITIES (SCALED BY B*(WT**.5)) 

 PARAMETER: HK_1 HK_2 

 OBS # OBSERVATION 

 1 F1 -0.258 -0.281 

 2 F2 -0.262 -0.287 

 3 F3 -0.262 -0.289 

 4 F4 -0.243 -0.276 

 5 F5 -0.232 -0.268 

 6 F6 -0.229 -0.264 

 7 F7 -0.189 -0.950E-04 

 8 F8 -0.190 -0.366E-02 

 9 F9 -0.197 -0.101E-01 

 10 F10 -0.185 -0.135 

 11 F11 -0.181 -0.246 

 12 F12 -0.186 -0.251 

 13 F13 -0.148 -0.182E-02 

 14 F14 -0.146 -0.228E-02 

 15 F15 -0.148 -0.225E-02 

 16 F16 -0.146 -0.706E-03 

 17 F17 -0.140 0.169E-02 

 18 F18 -0.141 0.299E-02 

 19 F19 -0.115 -0.974E-03 

 20 F20 -0.115 -0.800E-03 

 21 F21 -0.114 -0.496E-03 

 22 F22 -0.112 -0.399E-04 

 23 F23 -0.111 0.567E-03 

 24 F24 -0.112 0.105E-02 

 25 F25 -0.858E-01 -0.425E-03 

 26 F26 -0.849E-01 -0.305E-03 

 27 F27 -0.843E-01 -0.160E-03 

 28 F28 -0.843E-01 0.170E-04 

 29 F29 -0.835E-01 0.233E-03 

 30 F30 -0.823E-01 0.396E-03 

 31 F31 -0.703E-01 -0.230E-03 

 32 F32 -0.618E-01 -0.141E-03 

 33 F33 -0.617E-01 -0.700E-04 

 34 F34 -0.608E-01 0.131E-04 

 35 F35 -0.591E-01 0.107E-03 

 36 F36 -0.605E-01 0.184E-03 

 COMPOSITE SCALED SENSITIVITIES ((SUM OF THE SQUARED VALUES)/ND)**.5 

 0.154 0.130 

 PARAMETER COMPOSITE SCALED SENSITIVITY 

 ---------- ---------------------------- 

 HK_1 1.53635E-01 

105
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 HK_2 1.29572E-01 

 FINAL PARAMETER VALUES AND STATISTICS: 

 PARAMETER NAME(S) AND VALUE(S): 

 HK_1 HK_2 

 1.464 9.9433E-02 

 SUMS OF SQUARED WEIGHTED RESIDUALS: 

 OBSERVATIONS PRIOR INFO. TOTAL 

 0.140E-01 0.00 0.140E-01 

 ----------------------------------------------------------------------- 

 SELECTED STATISTICS FROM MODIFIED GAUSS-NEWTON ITERATIONS 

 MAX. PARAMETER CALC. CHANGE MAX. CHANGE DAMPING 

 ITER. PARNAM MAX. CHANGE ALLOWED PARAMETER 

 ----- ---------- ------------- ------------- ------------ 

 1 HK_2 -0.997831 2.00000 1.0000 

 2 HK_2 1.66634 2.00000 0.98277 

 3 HK_2 1.58271 2.00000 1.0000 

 4 HK_2 1.37349 2.00000 1.0000 

 5 HK_2 0.943184 2.00000 1.0000 

 6 HK_2 0.384188 2.00000 1.0000 

 7 HK_2 0.591537E-01 2.00000 1.0000 

 8 HK_2 0.111021E-02 2.00000 1.0000 

 SUMS OF SQUARED WEIGHTED RESIDUALS FOR EACH ITERATION 

 SUMS OF SQUARED WEIGHTED RESIDUALS 

 ITER. OBSERVATIONS PRIOR INFO. TOTAL 

 1 0.40901 0.0000 0.40901 

 2 1022.7 0.0000 1022.7 

 3 139.07 0.0000 139.07 

 4 17.395 0.0000 17.395 

 5 1.8804 0.0000 1.8804 

 6 0.14038 0.0000 0.14038 

 7 0.16299E-01 0.0000 0.16299E-01 

 8 0.14039E-01 0.0000 0.14039E-01 

 FINAL 0.14039E-01 0.0000 0.14039E-01 

 *** PARAMETER ESTIMATION CONVERGED BY SATISFYING THE TOL CRITERION *** 

 ----------------------------------------------------------------------- 

 VARIANCE-COVARIANCE MATRIX FOR THE PARAMETERS 

 --------------------------------------------- 

 HK_1 HK_2 

 ................................ 

 HK_1 1.083E-03 -9.540E-04 

 HK_2 -9.540E-04 1.523E-03 

106
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 _________________ 

 PARAMETER SUMMARY 

 _________________ 

 ________________________________________________________________________ 

 PARAMETER VALUES IN "REGRESSION" SPACE --- LOG TRANSFORMED AS APPLICABLE 

 ________________________________________________________________________ 

 PARAMETER: HK_1 HK_2 

 * = LOG TRNS: * * 

 UPPER 95% C.I. 1.95E-01 -9.68E-01 

 FINAL VALUES 1.65E-01 -1.00E+00 

 LOWER 95% C.I. 1.36E-01 -1.04E+00 

 STD. DEV. 1.43E-02 1.69E-02 

 COEF. OF VAR. (STD. DEV. / FINAL VALUE); "--" IF FINAL VALUE = 0.0 

 8.64E-02 1.69E-02 

 ------------------------------------------------------------------------ 

 ------------------------------------------------------------------------ 

 ________________________________________________________________________ 

 PHYSICAL PARAMETER VALUES --- EXP10 OF LOG TRANSFORMED PARAMETERS 

 ________________________________________________________________________ 

 PARAMETER: HK_1 HK_2 

 * = LOG TRNS: * * 

 UPPER 95% C.I. 1.57E+00 1.08E-01 

 FINAL VALUES 1.46E+00 9.94E-02 

 LOWER 95% C.I. 1.37E+00 9.18E-02 

 REASONABLE 

 UPPER LIMIT 1.00E+02 1.00E+02 

 REASONABLE 

 LOWER LIMIT 1.00E-02 1.00E-02 

 ESTIMATE ABOVE (1) 

 BELOW(-1)LIMITS 0 0 

 ENTIRE CONF. INT. 

 ABOVE(1)BELOW(-1) 0 0 

 ------------------------------------- 

 CORRELATION MATRIX FOR THE PARAMETERS 

 ------------------------------------- 

107
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 HK_1 HK_2 

 ................................ 

 HK_1 1.00 -0.743 

 HK_2 -0.743 1.00 

 THE CORRELATION OF THE FOLLOWING PARAMETER PAIRS >= .95 

 PARAMETER PARAMETER CORRELATION 

 THE CORRELATION OF THE FOLLOWING PARAMETER PAIRS IS BETWEEN .90 AND .95 

 PARAMETER PARAMETER CORRELATION 

 THE CORRELATION OF THE FOLLOWING PARAMETER PAIRS IS BETWEEN .85 AND .90 

 PARAMETER PARAMETER CORRELATION 

 LEAST-SQUARES OBJ FUNC (DEP.VAR. ONLY)- = 0.14039E-01 

 LEAST-SQUARES OBJ FUNC (W/PARAMETERS)-- = 0.14039E-01 

 CALCULATED ERROR VARIANCE-------------- = 0.41292E-03 

 STANDARD ERROR OF THE REGRESSION------- = 0.20320E-01 

 CORRELATION COEFFICIENT---------------- = 0.99322 

 W/PARAMETERS---------------------- = 0.99322 

 ITERATIONS----------------------------- = 8 

 MAX LIKE OBJ FUNC = 289.19 

 AIC STATISTIC---- = 293.19 

 BIC STATISTIC---- = 296.36 

 SMALLEST AND LARGEST WEIGHTED RESIDUALS 

 SMALLEST WEIGHTED RESIDUALS | LARGEST WEIGHTED RESIDUALS 

 WEIGHTED PERCENT OF | WEIGHTED PERCENT OF 

 NAME RESIDUAL OBJ FUNC | NAME RESIDUAL OBJ FUNC 

 F21 -0.376E-01 10.05 | F34 0.427E-01 12.96 

 F20 -0.334E-01 7.93 | F32 0.384E-01 10.52 

 F23 -0.325E-01 7.54 | F7 0.325E-01 7.51 

 F22 -0.267E-01 5.09 | F33 0.251E-01 4.50 

 F24 -0.226E-01 3.65 | F3 0.242E-01 4.19 

 STATISTICS FOR ALL RESIDUALS : 

 AVERAGE WEIGHTED RESIDUAL :-0.500E-03 

 # RESIDUALS >= 0. : 16 

 # RESIDUALS < 0. : 20 

 NUMBER OF RUNS : 9 IN 36 OBSERVATIONS 

 INTERPRETING THE CALCULATED RUNS STATISTIC VALUE OF -3.18 

 NOTE: THE FOLLOWING APPLIES ONLY IF 

 # RESIDUALS >= 0 . IS GREATER THAN 10 AND 

 # RESIDUALS < 0. IS GREATER THAN 10 

 THE NEGATIVE VALUE MAY INDICATE TOO FEW RUNS: 

 IF THE VALUE IS LESS THAN -1.28, THERE IS LESS THAN A 10 PERCENT 

 CHANCE THE VALUES ARE RANDOM, 

 IF THE VALUE IS LESS THAN -1.645, THERE IS LESS THAN A 5 PERCENT 

 CHANCE THE VALUES ARE RANDOM, 

 IF THE VALUE IS LESS THAN -1.96, THERE IS LESS THAN A 2.5 PERCENT 

 CHANCE THE VALUES ARE RANDOM. 

108
APPENDIX C. EXAMPLE SIMULATIONS – Parameter Estimation – Global Output File 

 CORRELATION BETWEEN ORDERED WEIGHTED RESIDUALS AND NORMAL ORDER STATISTICS 

 FOR OBSERVATIONS = 0.988 

 -------------------------------------------------------------------------- 

 COMMENTS ON THE INTERPRETATION OF THE CORRELATION BETWEEN 

 WEIGHTED RESIDUALS AND NORMAL ORDER STATISTICS: 

 The critical value for correlation at the 5% significance level is 0.944 

 IF the reported CORRELATION is GREATER than the 5% critical value, ACCEPT 

 the hypothesis that the weighted residuals are INDEPENDENT AND NORMALLY 

 DISTRIBUTED at the 5% significance level. The probability that this 

 conclusion is wrong is less than 5%. 

 IF the reported correlation IS LESS THAN the 5% critical value REJECT the, 

 hypothesis that the weighted residuals are INDEPENDENT AND NORMALLY 

 DISTRIBUTED at the 5% significance level. 

 The analysis can also be done using the 10% significance level. 

 The associated critical value is 0.953 

 -------------------------------------------------------------------------- 

 *** PARAMETER ESTIMATION CONVERGED BY SATISFYING THE TOL CRITERION ***

109
APPENDIX C. EXAMPLE SIMULATIONS – Modified Input Files For Generation of CORFAC-2k Input File 

Modified Input Files For Generation of CORFAC-2k Input File 

Name file (example.nam): 

GLOBAL 4 mf2k.global_listing 

LIST 3 output.dat 

DIS 95 discret.dat 

BAS6 1 bas6.dat 

ZONE 93 zone.dat 

MULT 94 multiple.dat 

LPF 31 lpf6.dat 

RCH 11 rech6.dat 

WEL 12 wel6.dat 

PCG 23 pcg2.dat 

OC 22 oc.dat 

DATA 10 inithead.dat 

OBS 40 obs.dat 

HOB 41 hob.dat 

PES 47 pes.dat 

SEN 46 sen.dat 

UNC 60 unc.dat

UNC file (example.unc): 

 3 2 0 0 1 0 61 0 1 0 4 0 

.01 .1 .1 

 0.0 

HK_1 .0 

HK_2 .0

SEN file (sen.dat): 

# SEN file for MODFLOW-2000, UNC1NLI1 Report Example 

# 

 2 0 0 2 Item 1 

 1 0 0 0 Item 2 

HK_1 1 1 1.46378 .01 100 1E-10 Item 3 

HK_2 1 1 .0994330 .01 100 1E-10 Item 3

HOB file (hob.dat): 

# HOBS file for MODFLOW-2000, UNC1NLI1 Report Example 

 42 0 0 6 

 1 1 

F1 1 37 13 1 0 0 0 14.88 729.28 0 1 

F2 1 30 13 1 0 0 0 14.78 690.68 0 1 

F3 1 25 13 1 0 0 0 14.86 668.02 0 1 

F4 1 20 13 1 0 0 0 13.87 699.35 0 1 

F5 1 14 13 1 0 0 0 13.01 728.59 0 1 

110
APPENDIX C. EXAMPLE SIMULATIONS – Modified Input Files For Generation of CORFAC-2k Input File 

F6 1 8 13 1 0 0 0 12.97 746.80 0 1 

F7 1 37 26 1 0 0 0 5.61 643.27 0 1 

F8 1 30 26 1 0 0 0 5.63 666.51 0 1 

F9 1 25 26 1 0 0 0 5.94 687.81 0 1 

F10 1 20 26 1 0 0 0 8.83 721.28 0 1 

F11 1 14 26 1 0 0 0 11.08 717.66 0 1 

F12 1 8 26 1 0 0 0 11.35 731.50 0 1 

F13 1 37 39 1 0 0 0 3.7 640.67 0 1 

F14 1 30 39 1 0 0 0 3.75 648.28 0 1 

F15 1 25 39 1 0 0 0 3.52 619.12 0 1 

F16 1 20 39 1 0 0 0 3.35 607.80 0 1 

F17 1 14 39 1 0 0 0 3.42 638.66 0 1 

F18 1 8 39 1 0 0 0 3.71 629.47 0 1 

F19 1 37 52 1 0 0 0 2.37 548.04 0 1 

F20 1 30 52 1 0 0 0 1.91 531.05 0 1 

F21 1 25 52 1 0 0 0 1.76 526.49 0 1 

F22 1 20 52 1 0 0 0 1.96 531.33 0 1 

F23 1 14 52 1 0 0 0 1.82 546.39 0 1 

F24 1 8 52 1 0 0 0 2.06 542.63 0 1 

F25 1 37 65 1 0 0 0 1.54 423.83 0 1 

F26 1 30 65 1 0 0 0 1.66 427.94 0 1 

F27 1 25 65 1 0 0 0 1.78 429.22 0 1 

F28 1 20 65 1 0 0 0 1.54 426.12 0 1 

F29 1 14 65 1 0 0 0 1.7 432.14 0 1 

F30 1 8 65 1 0 0 0 1.55 444.93 0 1 

F31 1 37 78 1 0 0 0 .94 153.12 0 1 

F32 1 30 78 1 0 0 0 1.41 197.28 0 1 

F33 1 25 78 1 0 0 0 1.22 197.22 0 1 

F34 1 20 78 1 0 0 0 1.47 202.06 0 1 

F35 1 14 78 1 0 0 0 1.08 212.53 0 1 

F36 1 8 78 1 0 0 0 .88 202.26 0 1 

G1 1 22 46 1 0 0 0 0 1037.4 0 1 

G2 1 21 20 1 0 0 0 0 723.55 0 1 

G3 1 21 33 1 0 0 0 0 620.74 0 1 

G4 1 21 59 1 0 0 0 0 488.07 0 1 

G5 1 21 72 1 0 0 0 0 329.80 0 1 

G6 1 40 46 1 0 0 0 0 597.63 0 1

111
APPENDIX C. EXAMPLE SIMULATIONS – Modified _cf Input File to CORFAC-2k 

Modified _cf Input File to CORFAC-2k 

Modified _cf Input File to CORFAC-2k (example._cf): 

 2 36 0 0 0 0.412962E-03 -1.00000 

1 

HK_1 HK_2 

F1 F2 F3 F4 F5 F6 

F7 F8 F9 F10 F11 F12 

F13 F14 F15 F16 F17 F18 

F19 F20 F21 F22 F23 F24 

F25 F26 F27 F28 F29 F30 

F31 F32 F33 F34 F35 F36 

 0.37029926E-01 0.38050603E-01 0.38690582E-01 0.37814010E-01 0.37047457E-01 

0.36592986E-01 0.39427873E-01 0.38734384E-01 

 0.38129907E-01 0.37234716E-01 0.37328508E-01 0.36973692E-01 0.39507795E-01 

0.39275225E-01 0.40189497E-01 0.40562026E-01 

 0.39569918E-01 0.39857723E-01 0.42716324E-01 0.43394260E-01 0.43581776E-01 

0.43382823E-01 0.42780772E-01 0.42928737E-01 

 0.48574030E-01 0.48340213E-01 0.48268080E-01 0.48443336E-01 0.48104730E-01 

0.47408275E-01 0.80813520E-01 0.71196474E-01 

 0.71207300E-01 0.70349306E-01 0.68594590E-01 0.70314519E-01 

 -6.95528 -7.59430 

 -6.88616 -7.54031 

 -6.76804 -7.46494 

 -6.43567 -7.28740 

 -6.26939 -7.22403 

 -6.24739 -7.22516 

 -4.78414 -0.240649E-02 

 -4.91206 -0.944998E-01 

 -5.17005 -0.265524 

 -4.96155 -3.61991 

 -4.84161 -6.59591 

 -5.02165 -6.79714 

 -3.74236 -0.460639E-01 

 -3.72761 -0.580034E-01 

 -3.68348 -0.559502E-01 

 -3.59604 -0.174140E-01 

 -3.53574 0.426478E-01 

 -3.53499 0.750806E-01 

 -2.69891 -0.227933E-01 

 -2.66038 -0.184378E-01 

 -2.61047 -0.113816E-01 

 -2.57549 -0.919775E-03 

 -2.59367 0.132558E-01 

 -2.61204 0.244984E-01 

 -1.76582 -0.875923E-02 

 -1.75638 -0.630512E-02 

 -1.74753 -0.331463E-02 

 -1.74041 0.349868E-03 

 -1.73644 0.484277E-02 

 -1.73592 0.835407E-02 

-0.870245 -0.284664E-02 

112
APPENDIX C. EXAMPLE SIMULATIONS – Modified _cf Input File to CORFAC-2k 

-0.868096 -0.198254E-02 

-0.865989 -0.983839E-03 

-0.863890 0.186636E-03 

-0.862079 0.156564E-02 

-0.861022 0.261434E-02 

 8 6 0 2 1 0 

G1 G2 G3 G4 G5 G6 

 HK_1 HK_2 

 0.31047517E-01 0.37176263E-01 0.40137023E-01 0.45264628E-01 0.55064879E-01 

0.40905699E-01 0.0000000 0.0000000 

 -2.84418 -0.123538E-01 

 -5.29811 -6.69752 

 -4.37059 -0.168617 

 -2.13973 -0.106125E-02 

 -1.27031 -0.137881E-03 

 -3.16943 -0.333658E-01 

 1.00000 0.00000 

 0.00000 1.00000 

729.28 667.57 615.68 586.33 557.59 534.25 584.85 598.77 

591.88 569.46 549.34 541.11 477.64 485.80 477.81 474.03 

490.18 486.31 326.73 325.31 323.54 332.79 344.61 339.91 

208.67 221.75 226.29 225.34 240.21 245.61 68.870 101.52 

103.49 110.91 118.31 98.075 

667.57 690.68 654.63 620.72 586.71 561.51 589.59 601.63 

595.99 577.41 559.76 549.42 478.84 483.95 474.92 467.99 

483.29 478.43 321.55 318.49 315.94 322.14 333.98 331.21 

214.14 224.01 227.44 222.64 233.67 236.83 72.997 106.27 

108.19 114.37 119.57 96.373 

615.68 654.63 668.02 651.43 618.29 587.79 577.61 590.09 

588.78 576.27 558.71 553.40 467.96 471.16 460.22 452.06 

465.32 460.42 308.79 303.50 300.50 304.75 314.61 312.42 

202.47 210.94 215.14 210.02 220.51 221.90 72.489 104.27 

105.26 110.19 114.24 92.439 

586.33 620.72 651.43 699.35 686.13 649.67 562.03 574.93 

574.79 566.42 550.59 553.37 452.96 452.53 440.17 430.63 

441.05 439.34 293.55 290.00 287.19 289.52 297.87 293.66 

193.48 201.20 204.83 198.65 206.20 206.93 69.506 99.734 

99.477 103.72 107.11 87.286 

557.59 586.71 618.29 686.13 728.59 707.08 543.25 559.94 

561.06 555.06 538.05 538.75 443.24 442.05 428.49 417.27 

424.38 421.98 288.46 285.79 281.38 283.14 288.82 282.64 

192.57 200.60 204.67 197.13 203.52 203.09 69.115 99.001 

98.459 104.13 107.79 87.132 

534.25 561.51 587.79 649.67 707.08 746.80 525.38 543.55 

547.06 545.23 529.14 527.88 437.16 436.68 423.31 411.63 

418.82 415.31 289.42 286.38 280.51 281.62 286.86 279.57 

191.49 200.05 202.37 193.86 198.61 197.71 65.800 95.801 

95.834 100.16 103.22 82.084 

584.85 589.59 577.61 562.03 543.25 525.38 643.27 632.46 

614.20 600.57 577.59 562.10 507.35 507.86 491.01 477.72 

483.99 476.30 331.83 326.31 325.01 329.89 338.91 334.53 

216.78 225.84 232.23 227.82 238.01 242.82 71.272 107.29 

108.79 114.71 123.51 104.54 

598.77 601.63 590.09 574.93 559.94 543.55 632.46 666.51 

113

510.39 501.63 346.45 338.31 333.05 337.30 349.23 345.68 

657.18 644.11 618.41 599.20 524.83 525.59 509.64 499.58 
APPENDIX C. EXAMPLE SIMULATIONS – Modified _cf Input File to CORFAC-2k 

221.67 230.89 237.21 232.60 242.72 247.01 73.206 109.94 

110.31 115.26 124.50 105.37 

591.88 595.99 588.78 574.79 561.06 547.06 614.20 657.18 

687.81 686.21 659.29 637.88 537.68 541.81 526.18 517.05 

531.07 522.43 362.45 354.14 347.07 349.82 361.59 358.58 

232.67 240.53 246.07 242.48 250.68 254.24 78.717 114.03 

114.30 119.04 127.26 106.76 

569.46 577.41 576.27 566.42 555.06 545.23 600.57 644.11 

686.21 721.28 701.42 676.61 554.47 562.74 551.08 543.76 

558.91 552.38 379.87 372.12 363.43 363.69 375.96 372.75 

242.19 249.08 254.99 254.22 261.06 263.36 84.298 120.19 

120.19 124.31 131.48 110.02 

549.34 559.76 558.71 550.59 538.05 529.14 577.59 618.41 

659.29 701.42 717.66 702.17 553.51 566.83 558.53 551.51 

566.99 560.49 382.08 375.64 366.97 367.21 378.36 374.44 

245.30 252.08 257.29 255.75 261.15 262.41 88.611 121.30 

120.89 124.28 131.52 112.32 

541.11 549.42 553.40 553.37 538.75 527.88 562.10 599.20 

637.88 676.61 702.17 731.50 549.20 564.90 560.42 557.96 

574.98 566.97 382.48 378.58 371.67 371.59 381.56 374.77 

248.91 256.43 261.51 258.12 261.74 261.68 91.934 123.40 

123.28 127.11 134.16 113.32 

477.64 478.84 467.96 452.96 443.24 437.16 507.35 524.83 

537.68 554.47 553.51 549.20 640.67 626.22 591.51 563.61 

555.08 535.82 453.45 447.14 440.73 438.31 441.10 423.05 

295.30 305.66 312.49 311.97 316.94 320.88 105.64 144.94 

148.93 153.49 165.66 142.97 

485.80 483.95 471.16 452.53 442.05 436.68 507.86 525.59 

541.81 562.74 566.83 564.90 626.22 648.28 618.24 588.02 

580.79 558.88 460.96 454.52 448.96 446.67 449.88 432.98 

303.50 313.53 320.28 319.66 325.02 328.34 111.31 149.19 

151.82 155.30 165.42 142.83 

477.81 474.92 460.22 440.17 428.49 423.31 491.01 509.64 

526.18 551.08 558.53 560.42 591.51 618.24 619.12 597.36 

588.69 566.62 452.11 449.96 444.33 443.98 453.68 439.92 

300.25 308.91 315.03 313.82 319.76 324.34 108.06 145.56 

148.64 152.55 161.93 139.60 

474.03 467.99 452.06 430.63 417.27 411.63 477.72 499.58 

517.05 543.76 551.51 557.96 563.61 588.02 597.36 607.80 

599.81 577.83 445.39 443.84 438.99 440.63 456.72 446.56 

296.70 305.68 312.57 310.46 316.16 321.64 106.61 144.37 

146.88 150.81 159.16 137.11 

490.18 483.29 465.32 441.05 424.38 418.82 483.99 510.39 

531.07 558.91 566.99 574.98 555.08 580.79 588.69 599.81 

638.66 617.67 438.33 438.82 437.99 440.62 458.55 449.80 

292.93 303.57 309.94 308.99 313.48 315.66 105.79 143.49 

145.72 148.31 155.68 131.83 

486.31 478.43 460.42 439.34 421.98 415.31 476.30 501.63 

522.43 552.38 560.49 566.97 535.82 558.88 566.62 577.83 

617.67 629.47 428.62 428.57 430.59 437.01 456.18 447.76 

288.83 299.85 307.41 307.76 311.41 313.08 105.58 143.63 

146.56 150.52 156.68 132.63 

326.73 321.55 308.79 293.55 288.46 289.42 331.83 346.45 

362.45 379.87 382.08 382.48 453.45 460.96 452.11 445.39 

114

379.15 390.62 392.18 383.88 378.94 379.17 137.98 179.29 

438.33 428.62 548.04 522.35 500.04 484.78 471.74 454.53 
APPENDIX C. EXAMPLE SIMULATIONS – Modified _cf Input File to CORFAC-2k 

184.21 190.21 201.84 174.92 

325.31 318.49 303.50 290.00 285.79 286.38 326.31 338.31 

354.14 372.12 375.64 378.58 447.14 454.52 449.96 443.84 

438.82 428.57 522.35 531.05 515.27 499.46 486.02 465.12 

377.99 389.36 391.95 384.08 380.66 379.82 139.14 177.36 

182.01 187.59 198.06 171.04 

323.54 315.94 300.50 287.19 281.38 280.51 325.01 333.05 

347.07 363.43 366.97 371.67 440.73 448.96 444.33 438.99 

437.99 430.59 500.04 515.27 526.49 517.52 502.76 477.30 

381.07 395.46 399.39 394.44 392.14 390.23 145.44 183.03 

188.11 193.42 204.07 177.28 

332.79 322.14 304.75 289.52 283.14 281.62 329.89 337.30 

349.82 363.69 367.21 371.59 438.31 446.67 443.98 440.63 

440.62 437.01 484.78 499.46 517.52 531.33 522.51 497.03 

371.13 389.59 396.26 394.86 398.34 402.69 143.86 182.63 

188.43 193.53 205.17 179.10 

344.61 333.98 314.61 297.87 288.82 286.86 338.91 349.23 

361.59 375.96 378.36 381.56 441.10 449.88 453.68 456.72 

458.55 456.18 471.74 486.02 502.76 522.51 546.39 526.83 

360.58 381.27 390.93 393.82 401.50 409.36 141.58 181.57 

187.82 192.83 203.97 177.63 

339.91 331.21 312.42 293.66 282.64 279.57 334.53 345.68 

358.58 372.75 374.44 374.77 423.05 432.98 439.92 446.56 

449.80 447.76 454.53 465.12 477.30 497.03 526.83 542.63 

346.27 367.35 378.05 383.54 394.49 404.22 134.42 174.09 

180.86 186.84 198.00 171.15 

208.67 214.14 202.47 193.48 192.57 191.49 216.78 221.67 

232.67 242.19 245.30 248.91 295.30 303.50 300.25 296.70 

292.93 288.83 379.15 377.99 381.07 371.13 360.58 346.27 

423.83 409.60 393.71 366.08 338.05 324.23 170.89 210.96 

207.57 202.51 198.52 162.88 

221.75 224.01 210.94 201.20 200.60 200.05 225.84 230.89 

240.53 249.08 252.08 256.43 305.66 313.53 308.91 305.68 

303.57 299.85 390.62 389.36 395.46 389.59 381.27 367.35 

409.60 427.94 417.14 392.68 365.92 350.74 173.99 216.39 

216.19 214.18 214.04 179.48 

226.29 227.44 215.14 204.83 204.67 202.37 232.23 237.21 

246.07 254.99 257.29 261.51 312.49 320.28 315.03 312.57 

309.94 307.41 392.18 391.95 399.39 396.26 390.93 378.05 

393.71 417.14 429.22 411.90 388.36 372.38 172.20 216.59 

217.49 217.96 221.48 188.33 

225.34 222.64 210.02 198.65 197.13 193.86 227.82 232.60 

242.48 254.22 255.75 258.12 311.97 319.66 313.82 310.46 

308.99 307.76 383.88 384.08 394.44 394.86 393.82 383.54 

366.08 392.68 411.90 426.12 411.10 395.28 165.65 211.41 

215.82 218.80 226.66 197.82 

240.21 233.67 220.51 206.20 203.52 198.61 238.01 242.72 

250.68 261.06 261.15 261.74 316.94 325.02 319.76 316.16 

313.48 311.41 378.94 380.66 392.14 398.34 401.50 394.49 

338.05 365.92 388.36 411.10 432.14 425.39 160.29 207.36 

214.39 219.04 231.88 204.96 

245.61 236.83 221.90 206.93 203.09 197.71 242.82 247.01 

254.24 263.36 262.41 261.68 320.88 328.34 324.34 321.64 

315.66 313.08 379.17 379.82 390.23 402.69 409.36 404.22 

115

210.36 217.26 232.57 206.41 

324.23 350.74 372.38 395.28 425.39 444.93 153.77 201.63 
APPENDIX C. EXAMPLE SIMULATIONS – Modified _cf Input File to CORFAC-2k 

68.870 72.997 72.489 69.506 69.115 65.800 71.272 73.206 

78.717 84.298 88.611 91.934 105.64 111.31 108.06 106.61 

105.79 105.58 137.98 139.14 145.44 143.86 141.58 134.42 

170.89 173.99 172.20 165.65 160.29 153.77 153.12 155.86 

140.83 128.13 111.47 93.066 

101.52 106.27 104.27 99.734 99.001 95.801 107.29 109.94 

114.03 120.19 121.30 123.40 144.94 149.19 145.56 144.37 

143.49 143.63 179.29 177.36 183.03 182.63 181.57 174.09 

210.96 216.39 216.59 211.41 207.36 201.63 155.86 197.28 

185.59 173.70 156.95 128.51 

103.49 108.19 105.26 99.477 98.459 95.834 108.79 110.31 

114.30 120.19 120.89 123.28 148.93 151.82 148.64 146.88 

145.72 146.56 184.21 182.01 188.11 188.43 187.82 180.86 

207.57 216.19 217.49 215.82 214.39 210.36 140.83 185.59 

197.22 190.73 175.32 147.00 

110.91 114.37 110.19 103.72 104.13 100.16 114.71 115.26 

119.04 124.31 124.28 127.11 153.49 155.30 152.55 150.81 

148.31 150.52 190.21 187.59 193.42 193.53 192.83 186.84 

202.51 214.18 217.96 218.80 219.04 217.26 128.13 173.70 

190.73 202.06 192.15 162.93 

118.31 119.57 114.24 107.11 107.79 103.22 123.51 124.50 

127.26 131.48 131.52 134.16 165.66 165.42 161.93 159.16 

155.68 156.68 201.84 198.06 204.07 205.17 203.97 198.00 

198.52 214.04 221.48 226.66 231.88 232.57 111.47 156.95 

175.32 192.15 212.53 188.12 

98.075 96.373 92.439 87.286 87.132 82.084 104.54 105.37 

106.76 110.02 112.32 113.32 142.97 142.83 139.60 137.11 

131.83 132.63 174.92 171.04 177.28 179.10 177.63 171.15 

162.88 179.48 188.33 197.82 204.96 206.41 93.066 128.51 

147.00 162.93 188.12 202.26 

116
APPENDIX C. EXAMPLE SIMULATIONS – Output File from CORFAC-2k 

Output File from CORFAC-2k 

Output File from CORFAC-2k (example.#cf): 

 CORFAC 

 MODFLOW-2000 POST-PROCESSING PROGRAM TO CALCULATE 

 CORRECTION FACTORS FOR CONFIDENCE REGION AND CONFIDENCE AND PREDICTION 

INTERVALS 

 Version 1.0.0 31/10/2003 

 NUMBER OF ESTIMATED PARAMETERS...................: 2 

 NUMBER OF HEAD OBSERVATIONS......................: 36 

 NUMBER OF ALL OTHER OBSERVATIONS.................: 0 

 TOTAL NUMBER OF OBSERVATIONS.....................: 36 

 NUMBER OF PRIOR-INFORMATION EQUATIONS............: 0 

 NUMBER OF PRIOR INFORMATION W/ FULL WEIGHT MATRIX: 0 

 CALCULATED ERROR VARIANCE........................: 0.41296E-03 

 EFFECTIVE CORRELATION (EC).......................: 0.80000 (WAS RESET 

BECAUSE IN INPUT FILE EC<0.0) 

 TRANSFORM WITH OBS. COV. MATRIX? (YES,NO)=(1,0)..: 1 

 SQUARE-ROOT OF WEIGHTS FOR SAMPLE INFORMATION 

 NO. OBSERVATION W NO. OBSERVATION W 

 HEADS: 

 1 F1 0.37030E-01 19 F19 0.42716E-01 

 2 F2 0.38051E-01 20 F20 0.43394E-01 

 3 F3 0.38691E-01 21 F21 0.43582E-01 

 4 F4 0.37814E-01 22 F22 0.43383E-01 

 5 F5 0.37047E-01 23 F23 0.42781E-01 

 6 F6 0.36593E-01 24 F24 0.42929E-01 

 7 F7 0.39428E-01 25 F25 0.48574E-01 

 8 F8 0.38734E-01 26 F26 0.48340E-01 

 9 F9 0.38130E-01 27 F27 0.48268E-01 

 10 F10 0.37235E-01 28 F28 0.48443E-01 

 11 F11 0.37329E-01 29 F29 0.48105E-01 

 12 F12 0.36974E-01 30 F30 0.47408E-01 

 13 F13 0.39508E-01 31 F31 0.80814E-01 

 14 F14 0.39275E-01 32 F32 0.71196E-01 

 15 F15 0.40189E-01 33 F33 0.71207E-01 

 16 F16 0.40562E-01 34 F34 0.70349E-01 

 17 F17 0.39570E-01 35 F35 0.68595E-01 

 18 F18 0.39858E-01 36 F36 0.70315E-01 

 SENSITIVITIES FOR OPTIMUM PARAMETERS 

 PARAMETER F1 F2 F3 F4 F5 

 ---------- ------------ ------------ ------------ ------------ ------------ 

117

 HK_1 -6.9553 -6.8862 -6.7680 -6.4357 -6.2694 
APPENDIX C. EXAMPLE SIMULATIONS – Output File from CORFAC-2k 

 HK_2 -7.5943 -7.5403 -7.4649 -7.2874 -7.2240 

 PARAMETER F6 F7 F8 F9 F10 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -6.2474 -4.7841 -4.9121 -5.1701 -4.9616 

 HK_2 -7.2252 -0.24065E-02 -0.94500E-01 -0.26552 -3.6199 

 PARAMETER F11 F12 F13 F14 F15 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -4.8416 -5.0216 -3.7424 -3.7276 -3.6835 

 HK_2 -6.5959 -6.7971 -0.46064E-01 -0.58003E-01 -0.55950E-01 

 PARAMETER F16 F17 F18 F19 F20 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -3.5960 -3.5357 -3.5350 -2.6989 -2.6604 

 HK_2 -0.17414E-01 0.42648E-01 0.75081E-01 -0.22793E-01 -0.18438E-01 

 PARAMETER F21 F22 F23 F24 F25 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -2.6105 -2.5755 -2.5937 -2.6120 -1.7658 

 HK_2 -0.11382E-01 -0.91977E-03 0.13256E-01 0.24498E-01 -0.87592E-02 

 PARAMETER F26 F27 F28 F29 F30 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -1.7564 -1.7475 -1.7404 -1.7364 -1.7359 

 HK_2 -0.63051E-02 -0.33146E-02 0.34987E-03 0.48428E-02 0.83541E-02 

 PARAMETER F31 F32 F33 F34 F35 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -0.87024 -0.86810 -0.86599 -0.86389 -0.86208 

 HK_2 -0.28466E-02 -0.19825E-02 -0.98384E-03 0.18664E-03 0.15656E-02 

 PARAMETER F36 

 ---------- ------------ 

 HK_1 -0.86102 

 HK_2 0.26143E-02 

 SQUARE-ROOT OF WEIGHTS FOR INTERVALS 

 NO. INTERVAL W NO. INTERVAL W 

 1 G1 0.0000 5 G5 0.0000 

 2 G2 0.0000 6 G6 0.0000 

 3 G3 0.0000 7 HK_1 0.0000 

 4 G4 0.0000 8 HK_2 0.0000 

 SENSITIVITIES FOR OPTIMUM PARAMETERS 

 PARAMETER G1 G2 G3 G4 G5 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -2.8442 -5.2981 -4.3706 -2.1397 -1.2703 

 HK_2 -0.12354E-01 -6.6975 -0.16862 -0.10612E-02 -0.13788E-03 

 PARAMETER G6 HK_1 HK_2 

 ---------- ------------ ------------ ------------ 

 HK_1 -3.1694 1.0000 0.0000 

 HK_2 -0.33366E-01 0.0000 1.0000 

118
APPENDIX C. EXAMPLE SIMULATIONS – Output File from CORFAC-2k 

 ****** CORRECTION FACTOR COMPUTED FOR CONFIDENCE REGION 

 CORRECTION FACTOR FOR N-P (A) ---- = 11.910 

 SCALING FACTOR (B) --------------- = 1.0000 

 EFFECTIVE CORRELATION (C) -------- = 0.71490 

 (N-P)/(N-A*P) VALUE (FAC) -------- = 2.7913 

 CORRECTION FACTOR (CR) ----------- = 33.243 

 APPROXIMATE N-A*P VALUE (DFB) ---- = 9.5836 

 APPROX. (N-P)/(N-A*P) VALUE (FACB) = 3.5477 

 APPROXIMATE BOUND FOR CR (CRB) --- = 46.859 

 ****** CORRECTION FACTORS COMPUTED FOR CONFIDENCE INTERVALS 

 TOTAL NUMBER OF INTERVALS (NOINT)......................: 8 

 NUMBER OF INTERVALS FOR HEAD (NHI).....................: 6 

 NUMBER OF INTERVALS FOR HEAD DEPENDENT FLOWS (NQI).....: 0 

 NUMBER OF INTERVALS FOR PARAMETERS (NPI)...............: 2 

 INTERVAL TYPE (1=CONFIDENCE,2=PREDICTION)..............: 1 

 HEAD AND FLOW INTERVALS ON DIFFERENCES (0,1)=(NO,YES)..: 0 

 SQUARE-ROOT OF WEIGHTS FOR INTERVALS 

 NO. INTERVAL W NO. INTERVAL W 

 1 G1 0.0000 5 G5 0.0000 

 2 G2 0.0000 6 G6 0.0000 

 3 G3 0.0000 7 HK_1 0.0000 

 4 G4 0.0000 8 HK_2 0.0000 

 SENSITIVITIES FOR OPTIMUM PARAMETERS 

 PARAMETER G1 G2 G3 G4 G5 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -2.8442 -5.2981 -4.3706 -2.1397 -1.2703 

 HK_2 -0.12354E-01 -6.6975 -0.16862 -0.10612E-02 -0.13788E-03 

 PARAMETER G6 HK_1 HK_2 

 ---------- ------------ ------------ ------------ 

 HK_1 -3.1694 1.0000 0.0000 

 HK_2 -0.33366E-01 0.0000 1.0000 

** CONFIDENCE INTERVAL NO. 1, FOR G1 

 CORRECTION FACTOR XI FOR C. I. ------ = 16.350 

 CORRECTION FACTOR (CC) -------------- = 45.638 

 ESTIMATED VARIANCE OF PREDICTION (VE) = 0.86984E-02 

119

 BOUND FOR XI USING NEG. VALUES (XINB) = 0.15332E-01 

 BOUND FOR XI USING POS. VALUES (XIPB) = 24.982 
APPENDIX C. EXAMPLE SIMULATIONS – Output File from CORFAC-2k 

 APPROXIMATE BOUND FOR CC (CCB) ------ = 71.613 

** CONFIDENCE INTERVAL NO. 2, FOR G2 

 CORRECTION FACTOR XI FOR C. I. ------ = 5.7280 

 CORRECTION FACTOR (CC) -------------- = 15.989 

 ESTIMATED VARIANCE OF PREDICTION (VE) = 0.31039E-01 

 BOUND FOR XI USING POS. VALUES (XIPB) = 8.6042 

 BOUND FOR XI USING NEG. VALUES (XINB) = 0.27022 

 APPROXIMATE BOUND FOR CC (CCB) ------ = 25.130 

** CONFIDENCE INTERVAL NO. 3, FOR G3 

 CORRECTION FACTOR XI FOR C. I. ------ = 16.826 

 CORRECTION FACTOR (CC) -------------- = 46.966 

 ESTIMATED VARIANCE OF PREDICTION (VE) = 0.19335E-01 

 BOUND FOR XI USING POS. VALUES (XIPB) = 25.521 

 BOUND FOR XI USING NEG. VALUES (XINB) = 0.91738E-02 

 APPROXIMATE BOUND FOR CC (CCB) ------ = 73.142 

** CONFIDENCE INTERVAL NO. 4, FOR G4 

 CORRECTION FACTOR XI FOR C. I. ------ = 16.298 

 CORRECTION FACTOR (CC) -------------- = 45.492 

 ESTIMATED VARIANCE OF PREDICTION (VE) = 0.49567E-02 

 BOUND FOR XI USING POS. VALUES (XIPB) = 24.932 

 BOUND FOR XI USING NEG. VALUES (XINB) = 0.16355E-01 

 APPROXIMATE BOUND FOR CC (CCB) ------ = 71.470 

** CONFIDENCE INTERVAL NO. 5, FOR G5 

 CORRECTION FACTOR XI FOR C. I. ------ = 16.292 

 CORRECTION FACTOR (CC) -------------- = 45.477 

 ESTIMATED VARIANCE OF PREDICTION (VE) = 0.17482E-02 

 BOUND FOR XI USING POS. VALUES (XIPB) = 24.927 

 BOUND FOR XI USING NEG. VALUES (XINB) = 0.16459E-01 

 APPROXIMATE BOUND FOR CC (CCB) ------ = 71.456 

** CONFIDENCE INTERVAL NO. 6, FOR G6 

 CORRECTION FACTOR XI FOR C. I. ------ = 16.435 

 CORRECTION FACTOR (CC) -------------- = 45.875 

 ESTIMATED VARIANCE OF PREDICTION (VE) = 0.10685E-01 

 BOUND FOR XI USING POS. VALUES (XIPB) = 25.063 

120

 APPROXIMATE BOUND FOR CC (CCB) ------ = 71.842 

 BOUND FOR XI USING NEG. VALUES (XINB) = 0.13742E-01 
APPENDIX C. EXAMPLE SIMULATIONS – Output File from CORFAC-2k 

** CONFIDENCE INTERVAL NO. 7, FOR HK_1 

 CORRECTION FACTOR XI FOR C. I. ------ = 16.291 

 CORRECTION FACTOR (CC) -------------- = 45.473 

 ESTIMATED VARIANCE OF PREDICTION (VE) = 0.10836E-02 

 BOUND FOR XI USING POS. VALUES (XIPB) = 0.16488E-01 

 BOUND FOR XI USING NEG. VALUES (XINB) = 24.925 

 APPROXIMATE BOUND FOR CC (CCB) ------ = 71.452 

** CONFIDENCE INTERVAL NO. 8, FOR HK_2 

 CORRECTION FACTOR XI FOR C. I. ------ = 3.8218 

 CORRECTION FACTOR (CC) -------------- = 10.668 

 ESTIMATED VARIANCE OF PREDICTION (VE) = 0.15234E-02 

 BOUND FOR XI USING POS. VALUES (XIPB) = 12.816 

 BOUND FOR XI USING NEG. VALUES (XINB) = 3.8470 

 APPROXIMATE BOUND FOR CC (CCB) ------ = 37.083 

121
APPENDIX C. EXAMPLE SIMULATIONS – Input Files Used to Generate BEALE2-2k Input Files 

Input Files Used to Generate BEALE2-2k Input Files 

Name file (example.nam): 

GLOBAL 4 mf2k.global_listing 

LIST 3 output.dat 

DIS 95 discret.dat 

BAS6 1 bas6.dat 

ZONE 93 zone.dat 

MULT 94 multiple.dat 

LPF 31 lpf6.dat 

RCH 11 rech6.dat 

WEL 12 wel6.dat 

PCG 23 pcg2.dat 

OC 22 oc.dat 

DATA 10 inithead.dat 

OBS 40 obs.dat 

HOB 41 hob.dat 

PES 47 pes.dat 

SEN 46 sen.dat 

UNC 60 unc.dat

UNC file (unc.dat): 

 1 2 0 00 1 0 61 0 1 0 4 0 

.01 .1 .1 

 0.0 

HK_1 .0 

HK_2 .0 

33.243 

45.638 15.989 46.966 45.492 45.477 45.875 45.473 10.668

SEN file (sen.dat): 

# SEN file for MODFLOW-2000, UNC1NLI1 Report Example 

# 

 2 0 0 2 Item 1 

 1 0 0 0 Item 2 

HK_1 1 1 1.46378 .01 100 1E-10 Item 3 

HK_2 1 1 .0994330 .01 100 1E-10 Item 3

122
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Input Files 

BEALE2-2k Input Files 

_b1 file (example._b1): 

 2 36 0 0 0 0.412962E-03 33.2430 

HK_1 HK_2 

 1.46378 0.994330E-01 

F1 F2 F3 F4 F5 F6 

F7 F8 F9 F10 F11 F12 

F13 F14 F15 F16 F17 F18 

F19 F20 F21 F22 F23 F24 

F25 F26 F27 F28 F29 F30 

F31 F32 F33 F34 F35 F36 

 14.5496 14.4265 14.2330 13.7231 13.4935 13.4726 

 4.78655 5.00656 5.43558 8.58150 11.4376 11.8188 

 3.78842 3.78562 3.73943 3.61346 3.49310 3.45992 

 2.72171 2.67882 2.62185 2.57641 2.58042 2.58755 

 1.77458 1.76269 1.75084 1.74006 1.73160 1.72757 

 0.873092 0.870080 0.866973 0.863704 0.860516 0.858411 

 14.8800 14.7800 14.8600 13.8700 13.0100 12.9700 

 5.61000 5.63000 5.94000 8.83000 11.0800 11.3500 

 3.70000 3.75000 3.52000 3.35000 3.42000 3.71000 

 2.37000 1.91000 1.76000 1.96000 1.82000 2.06000 

 1.54000 1.66000 1.78000 1.54000 1.70000 1.55000 

 0.940000 1.41000 1.22000 1.47000 1.08000 0.880000 

 0.13712154E-02 0.14478485E-02 0.14969611E-02 0.14298992E-02 0.13725139E-02 

0.13390466E-02 0.15545571E-02 0.15003525E-02 

 0.14538899E-02 0.13864241E-02 0.13934176E-02 0.13670539E-02 0.15608660E-02 

0.15425433E-02 0.16151958E-02 0.16452781E-02 

 0.15657784E-02 0.15886382E-02 0.18246844E-02 0.18830620E-02 0.18993714E-02 

0.18820695E-02 0.18301945E-02 0.18428763E-02 

 0.23594366E-02 0.23367761E-02 0.23298075E-02 0.23467569E-02 0.23140649E-02 

0.22475447E-02 0.65308255E-02 0.50689378E-02 

 0.50704796E-02 0.49490249E-02 0.47052181E-02 0.49441312E-02 

 -6.95528 -7.59430 

 -6.88616 -7.54031 

 -6.76804 -7.46494 

 -6.43567 -7.28740 

 -6.26939 -7.22403 

 -6.24739 -7.22516 

 -4.78414 -0.240649E-02 

 -4.91206 -0.944998E-01 

 -5.17005 -0.265524 

 -4.96155 -3.61991 

 -4.84161 -6.59591 

 -5.02165 -6.79714 

 -3.74236 -0.460639E-01 

 -3.72761 -0.580034E-01 

 -3.68348 -0.559502E-01 

 -3.59604 -0.174140E-01 

 -3.53574 0.426478E-01 

 -3.53499 0.750806E-01 

 -2.69891 -0.227933E-01 

123
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Input Files 

 -2.66038 -0.184378E-01 

 -2.61047 -0.113816E-01 

 -2.57549 -0.919775E-03 

 -2.59367 0.132558E-01 

 -2.61204 0.244984E-01 

 -1.76582 -0.875923E-02 

 -1.75638 -0.630512E-02 

 -1.74753 -0.331463E-02 

 -1.74041 0.349868E-03 

 -1.73644 0.484277E-02 

 -1.73592 0.835407E-02 

-0.870245 -0.284664E-02 

-0.868096 -0.198254E-02 

-0.865989 -0.983839E-03 

-0.863890 0.186636E-03 

-0.862079 0.156564E-02 

-0.861022 0.261434E-02 

 1 1 

 THE PARAMETER SETS FOLLOW 

 1.11920 0.125942 

 1.91444 0.785038E-01 

 1.78665 0.723296E-01 

 1.19926 0.136693 

_b2 file (example._b2): 

 1.11920 0.125942 

 14.9125 14.7902 14.5913 14.0530 13.7994 13.7717 

 6.25830 6.48017 6.91921 9.29229 11.4711 11.8419 

 4.92199 4.90968 4.85074 4.71361 4.59867 4.57772 

 3.54356 3.49045 3.42087 3.36873 3.38397 3.40104 

 2.31476 2.30107 2.28757 2.27605 2.26799 2.26511 

 1.13968 1.13657 1.13324 1.12992 1.12660 1.12444 

 1.91444 0.785038E-01 

 14.7982 14.6854 14.5117 14.0612 13.8643 13.8490 

 3.66131 3.86141 4.24862 8.33661 12.0036 12.3774 

 2.91287 2.91490 2.87901 2.76949 2.65662 2.61988 

 2.08898 2.05466 2.00876 1.97049 1.96871 1.97038 

 1.35974 1.34991 1.33980 1.33040 1.32244 1.31831 

 0.668514 0.665898 0.663172 0.660340 0.657408 0.655526 

 1.78665 0.723296E-01 

 15.9823 15.8606 15.6737 15.1897 14.9787 14.9621 

 3.92336 4.13889 4.55545 8.99430 12.9751 13.3778 

 3.12183 3.12423 3.08576 2.96817 2.84692 2.80706 

 2.23880 2.20211 2.15289 2.11173 2.10983 2.11143 

 1.45719 1.44666 1.43587 1.42568 1.41723 1.41261 

 0.716377 0.713626 0.710627 0.707445 0.704398 0.702631 

 1.19926 0.136693 

 13.8379 13.7248 13.5404 13.0412 12.8058 12.7800 

 5.84041 6.04568 6.45219 8.63549 10.6408 10.9837 

 4.59238 4.58067 4.52578 4.39841 4.29207 4.27328 

 3.30650 3.25695 3.19203 3.14359 3.15788 3.17407 

 2.15999 2.14697 2.13451 2.12386 2.11640 2.11393 

 1.06347 1.06053 1.05755 1.05446 1.05147 1.04941 

124
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Input Files 

_b3 file (example._b3): 

 8 6 0 2 1 0 

G1 G2 G3 G4 G5 G6 

 HK_1 HK_2 

 2.85654 11.9957 4.53921 2.14079 1.27045 3.20279 

 0.381022 -2.30827 

 0.96394832E-03 0.13820746E-02 0.16109805E-02 0.20488864E-02 0.30321409E-02 

0.16732761E-02 0.0000000 0.0000000 

 -2.84418 -0.123538E-01 

 -5.29811 -6.69752 

 -4.37059 -0.168617 

 -2.13973 -0.106125E-02 

 -1.27031 -0.137881E-03 

 -3.16943 -0.333658E-01 

 1.00000 0.00000 

 0.00000 1.00000 

 45.6380 15.9890 46.9660 45.4920 45.4770 

45.8750 

 45.4730 10.6680 

 THE PARAMETER SETS FOLLOW 

 1.82832 0.818010E-01 

 1.17192 0.120866 

 1.44237 0.111757 

 1.48551 0.884681E-01 

 1.83400 0.819924E-01 

 1.16830 0.120583 

 1.82767 0.817822E-01 

 1.17234 0.120893 

 1.82760 0.817804E-01 

 1.17238 0.120896 

 1.82936 0.818321E-01 

 1.17126 0.120820 

 1.17239 0.120897 

 1.82759 0.817798E-01 

 1.58544 0.875319E-01 

 1.35145 0.112952 

_b4 file (example._b4): 

 1.82832 0.818010E-01 

 14.7031 14.5883 14.4106 13.9490 13.7463 13.7301 

 3.83359 4.03804 4.43404 8.34384 11.8579 12.2337 

 3.04753 3.04903 3.01155 2.89900 2.78433 2.74775 

 2.18610 2.15054 2.10289 2.06339 2.06235 2.06468 

 1.42344 1.41320 1.40288 1.39314 1.38505 1.38092 

 0.699981 0.697227 0.694390 0.691413 0.688442 0.686534 

 2.29123 

 1.17192 0.120866 

 14.8142 14.6905 14.4912 13.9546 13.7038 13.6772 

 5.97744 6.20061 6.64103 9.13822 11.4268 11.8017 

 4.70650 4.69607 4.63942 4.50362 4.38650 4.36281 

 3.38693 3.33566 3.26840 3.21743 3.23023 3.24517 

 2.21175 2.19841 2.18516 2.17367 2.16539 2.16223 

125
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Input Files 

 1.08887 1.08568 1.08239 1.07904 1.07577 1.07355 

 3.56090 

 1.44237 0.111757 

 13.8048 13.6878 13.5018 13.0087 12.7843 12.7628 

 4.85767 5.06823 5.48016 8.25290 10.7782 11.1389 

 3.83863 3.83402 3.78730 3.66483 3.55091 3.52183 

 2.75907 2.71604 2.65919 2.61444 2.62048 2.62922 

 1.79967 1.78799 1.77638 1.76594 1.75790 1.75430 

 0.885609 0.882716 0.879694 0.876626 0.873547 0.871495 

 11.3327 

 1.48551 0.884681E-01 

 15.3788 15.2506 15.0503 14.5253 14.2911 14.2707 

 4.71698 4.94540 5.38994 8.95448 12.1801 12.5825 

 3.73883 3.73768 3.69200 3.56306 3.43687 3.39986 

 2.68481 2.64204 2.58503 2.53892 2.54111 2.54671 

 1.74967 1.73767 1.72564 1.71458 1.70570 1.70139 

 0.860568 0.857560 0.854368 0.851076 0.847743 0.845575 

 12.7448 

 1.83400 0.819924E-01 

 14.6641 14.5494 14.3722 13.9116 13.7094 13.6933 

 3.82153 4.02530 4.42007 8.32080 11.8267 12.2014 

 3.03798 3.03949 3.00215 2.88999 2.77559 2.73910 

 2.17938 2.14389 2.09638 2.05699 2.05588 2.05819 

 1.41909 1.40891 1.39851 1.38883 1.38071 1.37654 

 0.697785 0.695042 0.692217 0.689299 0.686325 0.684411 

 3.67501 

 1.16830 0.120583 

 14.8549 14.7309 14.5310 13.9929 13.7415 13.7148 

 5.99579 6.21957 6.66124 9.16414 11.4580 11.8340 

 4.72094 4.71058 4.65378 4.51760 4.40014 4.37634 

 3.39745 3.34600 3.27849 3.22733 3.24017 3.25521 

 2.21855 2.20513 2.19184 2.18036 2.17210 2.16895 

 1.09215 1.08902 1.08574 1.08239 1.07911 1.07688 

 5.59279 

 1.82767 0.817822E-01 

 14.7071 14.5921 14.4143 13.9522 13.7493 13.7331 

 3.83480 4.03922 4.43524 8.34582 11.8606 12.2365 

 3.04848 3.04998 3.01250 2.89997 2.78520 2.74860 

 2.18690 2.15128 2.10361 2.06408 2.06299 2.06532 

 1.42397 1.41377 1.40334 1.39364 1.38550 1.38132 

 0.700185 0.697445 0.694615 0.691690 0.688705 0.686784 

 1.71507 

 1.17234 0.120893 

 14.8097 14.6861 14.4868 13.9503 13.6996 13.6731 

 5.97510 6.19822 6.63859 9.13533 11.4235 11.7984 

 4.70472 4.69442 4.63782 4.50206 4.38493 4.36117 

 3.38579 3.33450 3.26720 3.21620 3.22897 3.24395 

 2.21092 2.19753 2.18428 2.17283 2.16460 2.16146 

 1.08839 1.08527 1.08200 1.07866 1.07538 1.07316 

 2.67229 

 1.82760 0.817804E-01 

 14.7076 14.5926 14.4148 13.9527 13.7498 13.7336 

 3.83486 4.03932 4.43540 8.34612 11.8609 12.2369 

 3.04855 3.05003 3.01255 2.89998 2.78521 2.74862 

126

 1.42387 1.41370 1.40331 1.39364 1.38554 1.38138 

 2.18688 2.15123 2.10356 2.06405 2.06301 2.06537 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Input Files 

 0.700138 0.697450 0.694631 0.691702 0.688741 0.686813 

 1.01752 

 1.17238 0.120896 

 14.8094 14.6857 14.4864 13.9499 13.6993 13.6727 

 5.97499 6.19809 6.63841 9.13506 11.4232 11.7980 

 4.70460 4.69424 4.63764 4.50189 4.38479 4.36106 

 3.38564 3.33438 3.26712 3.21614 3.22892 3.24387 

 2.21088 2.19752 2.18426 2.17279 2.16453 2.16138 

 1.08840 1.08524 1.08196 1.07862 1.07534 1.07313 

 1.58623 

 1.82936 0.818321E-01 

 14.6962 14.5812 14.4036 13.9419 13.7392 13.7231 

 3.83115 4.03541 4.43112 8.33939 11.8521 12.2277 

 3.04560 3.04711 3.00969 2.89727 2.78262 2.74605 

 2.18484 2.14928 2.10167 2.06218 2.06110 2.06341 

 1.42266 1.41247 1.40205 1.39235 1.38422 1.38004 

 0.699548 0.696803 0.693973 0.691048 0.688069 0.686150 

 2.57305 

 1.17126 0.120820 

 14.8212 14.6975 14.4980 13.9611 13.7102 13.6837 

 5.98058 6.20385 6.64455 9.14273 11.4323 11.8074 

 4.70899 4.69870 4.64208 4.50623 4.38903 4.36527 

 3.38891 3.33760 3.27026 3.21921 3.23200 3.24697 

 2.21301 2.19960 2.18633 2.17485 2.16660 2.16346 

 1.08942 1.08627 1.08300 1.07965 1.07638 1.07415 

 3.98442 

 1.17239 0.120897 

 14.8092 14.6856 14.4863 13.9498 13.6992 13.6726 

 5.97494 6.19805 6.63837 9.13499 11.4231 11.7979 

 4.70457 4.69421 4.63759 4.50184 4.38474 4.36101 

 3.38560 3.33434 3.26708 3.21610 3.22889 3.24384 

 2.21085 2.19750 2.18424 2.17277 2.16452 2.16136 

 1.08839 1.08523 1.08195 1.07861 1.07534 1.07312 

 0.159044 

 1.82759 0.817798E-01 

 14.7081 14.5931 14.4153 13.9533 13.7504 13.7341 

 3.83506 4.03960 4.43576 8.34654 11.8614 12.2374 

 3.04889 3.05034 3.01284 2.90027 2.78548 2.74884 

 2.18712 2.15151 2.10380 2.06422 2.06315 2.06546 

 1.42406 1.41382 1.40341 1.39367 1.38556 1.38139 

 0.700221 0.697469 0.694638 0.691677 0.688748 0.686813 

 0.602998 

 1.58544 0.875319E-01 

 15.0249 14.9012 14.7088 14.2056 13.9821 13.9632 

 4.42082 4.64083 5.06867 8.68673 11.9547 12.3465 

 3.50652 3.50644 3.46367 3.34027 3.21826 3.18160 

 2.51721 2.47698 2.42324 2.37953 2.38072 2.38536 

 1.64010 1.62869 1.61732 1.60674 1.59826 1.59412 

 0.806684 0.803738 0.800670 0.797407 0.794164 0.792197 

 -2.43575 

 1.35145 0.112952 

 14.1905 14.0700 13.8780 13.3672 13.1334 13.1104 

 5.18382 5.40041 5.82518 8.55177 11.0393 11.4081 

 4.09258 4.08664 4.03706 3.90959 3.79305 3.76468 

127

 1.92002 1.90771 1.89553 1.88469 1.87651 1.87290 

 2.94271 2.89710 2.83698 2.79010 2.79766 2.80790 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Input Files 

 0.944902 0.941932 0.938851 0.935610 0.932412 0.930263 

 -2.18079 

128
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

BEALE2-2k Output File 

#be Output File (example.#be): 

 NLBEALE 

 MODFLOW-2000 POST-PROCESSING PROGRAM TO CALCULATE 

 MEASURES OF TOTAL, INTRINSIC, AND COMBINED INTRINSIC NONLINEARITY 

 Version 1.0.0 31/10/2003 

 NUMBER OF ESTIMATED PARAMETERS...................: 2 

 NUMBER OF HEAD OBSERVATIONS......................: 36 

 NUMBER OF ALL OTHER OBSERVATIONS.................: 0 

 TOTAL NUMBER OF OBSERVATIONS.....................: 36 

 NUMBER OF PRIOR-INFORMATION EQUATIONS............: 0 

 NUMBER OF PRIOR INFORMATION W/ FULL WEIGHT MATRIX: 0 

 NUMBER OF DATA SETS USED FOR BEALES MEASURE......: 4 

 CALCULATED ERROR VARIANCE........................: 0.41296E-03 

 CORRECTION FACTOR FOR CONFIDENCE REGION..........: 33.243 

 OPTIMUM PARAMETERS 

 NO. NAME BOPT NO. NAME BOPT 

 1 HK_1 1.4638 2 HK_2 0.99433E-01 

 DEPENDENT VARIABLES COMPUTED WITH OPTIMUM PARAMETERS 

 NO. OBSERVATION FOPT NO. OBSERVATION FOPT 

 1 F1 14.550 19 F19 2.7217 

 2 F2 14.427 20 F20 2.6788 

 3 F3 14.233 21 F21 2.6219 

 4 F4 13.723 22 F22 2.5764 

 5 F5 13.493 23 F23 2.5804 

 6 F6 13.473 24 F24 2.5875 

 7 F7 4.7866 25 F25 1.7746 

 8 F8 5.0066 26 F26 1.7627 

 9 F9 5.4356 27 F27 1.7508 

 10 F10 8.5815 28 F28 1.7401 

 11 F11 11.438 29 F29 1.7316 

 12 F12 11.819 30 F30 1.7276 

 13 F13 3.7884 31 F31 0.87309 

 14 F14 3.7856 32 F32 0.87008 

 15 F15 3.7394 33 F33 0.86697 

 16 F16 3.6135 34 F34 0.86370 

 17 F17 3.4931 35 F35 0.86052 

 18 F18 3.4599 36 F36 0.85841 

 OBSERVED VALUES OF THE DEPENDENT VARIABLES 

 NO. OBSERVATION FOBS NO. OBSERVATION FOBS 

 1 F1 14.880 19 F19 2.3700 

 2 F2 14.780 20 F20 1.9100 

129

 3 F3 14.860 21 F21 1.7600 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 4 F4 13.870 22 F22 1.9600 

 5 F5 13.010 23 F23 1.8200 

 6 F6 12.970 24 F24 2.0600 

 7 F7 5.6100 25 F25 1.5400 

 8 F8 5.6300 26 F26 1.6600 

 9 F9 5.9400 27 F27 1.7800 

 10 F10 8.8300 28 F28 1.5400 

 11 F11 11.080 29 F29 1.7000 

 12 F12 11.350 30 F30 1.5500 

 13 F13 3.7000 31 F31 0.94000 

 14 F14 3.7500 32 F32 1.4100 

 15 F15 3.5200 33 F33 1.2200 

 16 F16 3.3500 34 F34 1.4700 

 17 F17 3.4200 35 F35 1.0800 

 18 F18 3.7100 36 F36 0.88000 

 RELIABILITY WEIGHTS FOR SAMPLE INFORMATION 

 NO. OBSERVATION W NO. OBSERVATION W 

 HEADS: 

 1 F1 0.13712E-02 19 F19 0.18247E-02 

 2 F2 0.14478E-02 20 F20 0.18831E-02 

 3 F3 0.14970E-02 21 F21 0.18994E-02 

 4 F4 0.14299E-02 22 F22 0.18821E-02 

 5 F5 0.13725E-02 23 F23 0.18302E-02 

 6 F6 0.13390E-02 24 F24 0.18429E-02 

 7 F7 0.15546E-02 25 F25 0.23594E-02 

 8 F8 0.15004E-02 26 F26 0.23368E-02 

 9 F9 0.14539E-02 27 F27 0.23298E-02 

 10 F10 0.13864E-02 28 F28 0.23468E-02 

 11 F11 0.13934E-02 29 F29 0.23141E-02 

 12 F12 0.13671E-02 30 F30 0.22475E-02 

 13 F13 0.15609E-02 31 F31 0.65308E-02 

 14 F14 0.15425E-02 32 F32 0.50689E-02 

 15 F15 0.16152E-02 33 F33 0.50705E-02 

 16 F16 0.16453E-02 34 F34 0.49490E-02 

 17 F17 0.15658E-02 35 F35 0.47052E-02 

 18 F18 0.15886E-02 36 F36 0.49441E-02 

 SENSITIVITIES FOR OPTIMUM PARAMETERS 

 PARAMETER F1 F2 F3 F4 F5 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -6.9553 -6.8862 -6.7680 -6.4357 -6.2694 

 HK_2 -7.5943 -7.5403 -7.4649 -7.2874 -7.2240 

 PARAMETER F6 F7 F8 F9 F10 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -6.2474 -4.7841 -4.9121 -5.1701 -4.9616 

 HK_2 -7.2252 -0.24065E-02 -0.94500E-01 -0.26552 -3.6199 

 PARAMETER F11 F12 F13 F14 F15 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -4.8416 -5.0216 -3.7424 -3.7276 -3.6835 

 HK_2 -6.5959 -6.7971 -0.46064E-01 -0.58003E-01 -0.55950E-01 

130

 PARAMETER F16 F17 F18 F19 F20 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -3.5960 -3.5357 -3.5350 -2.6989 -2.6604 

 HK_2 -0.17414E-01 0.42648E-01 0.75081E-01 -0.22793E-01 -0.18438E-01 

 PARAMETER F21 F22 F23 F24 F25 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -2.6105 -2.5755 -2.5937 -2.6120 -1.7658 

 HK_2 -0.11382E-01 -0.91977E-03 0.13256E-01 0.24498E-01 -0.87592E-02 

 PARAMETER F26 F27 F28 F29 F30 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -1.7564 -1.7475 -1.7404 -1.7364 -1.7359 

 HK_2 -0.63051E-02 -0.33146E-02 0.34987E-03 0.48428E-02 0.83541E-02 

 PARAMETER F31 F32 F33 F34 F35 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -0.87024 -0.86810 -0.86599 -0.86389 -0.86208 

 HK_2 -0.28466E-02 -0.19825E-02 -0.98384E-03 0.18664E-03 0.15656E-02 

 PARAMETER F36 

 ---------- ------------ 

 HK_1 -0.86102 

 HK_2 0.26143E-02 

 PARAMETER 

 NO. NAME LN 

 1 HK_1 1 

 2 HK_2 1 

 PARAMETERS FOR SAMPLE NO. 1 

 NO. NAME B NO. NAME B 

 1 HK_1 1.1192 2 HK_2 0.12594 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 1 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.913 19 F19 3.5436 

 2 F2 14.790 20 F20 3.4904 

 3 F3 14.591 21 F21 3.4209 

 4 F4 14.053 22 F22 3.3687 

 5 F5 13.799 23 F23 3.3840 

 6 F6 13.772 24 F24 3.4010 

 7 F7 6.2583 25 F25 2.3148 

 8 F8 6.4802 26 F26 2.3011 

 9 F9 6.9192 27 F27 2.2876 

 10 F10 9.2923 28 F28 2.2761 

 11 F11 11.471 29 F29 2.2680 

 12 F12 11.842 30 F30 2.2651 

 13 F13 4.9220 31 F31 1.1397 

 14 F14 4.9097 32 F32 1.1366 

 15 F15 4.8507 33 F33 1.1332 

 16 F16 4.7136 34 F34 1.1299 

 17 F17 4.5987 35 F35 1.1266 

 18 F18 4.5777 36 F36 1.1244 

131

 NO. NAME B NO. NAME B 

 PARAMETERS FOR SAMPLE NO. 2 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 1 HK_1 1.9144 2 HK_2 0.78504E-01 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 2 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.798 19 F19 2.0890 

 2 F2 14.685 20 F20 2.0547 

 3 F3 14.512 21 F21 2.0088 

 4 F4 14.061 22 F22 1.9705 

 5 F5 13.864 23 F23 1.9687 

 6 F6 13.849 24 F24 1.9704 

 7 F7 3.6613 25 F25 1.3597 

 8 F8 3.8614 26 F26 1.3499 

 9 F9 4.2486 27 F27 1.3398 

 10 F10 8.3366 28 F28 1.3304 

 11 F11 12.004 29 F29 1.3224 

 12 F12 12.377 30 F30 1.3183 

 13 F13 2.9129 31 F31 0.66851 

 14 F14 2.9149 32 F32 0.66590 

 15 F15 2.8790 33 F33 0.66317 

 16 F16 2.7695 34 F34 0.66034 

 17 F17 2.6566 35 F35 0.65741 

 18 F18 2.6199 36 F36 0.65553 

 PARAMETERS FOR SAMPLE NO. 3 

 NO. NAME B NO. NAME B 

 1 HK_1 1.7866 2 HK_2 0.72330E-01 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 3 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 15.982 19 F19 2.2388 

 2 F2 15.861 20 F20 2.2021 

 3 F3 15.674 21 F21 2.1529 

 4 F4 15.190 22 F22 2.1117 

 5 F5 14.979 23 F23 2.1098 

 6 F6 14.962 24 F24 2.1114 

 7 F7 3.9234 25 F25 1.4572 

 8 F8 4.1389 26 F26 1.4467 

 9 F9 4.5554 27 F27 1.4359 

 10 F10 8.9943 28 F28 1.4257 

 11 F11 12.975 29 F29 1.4172 

 12 F12 13.378 30 F30 1.4126 

 13 F13 3.1218 31 F31 0.71638 

 14 F14 3.1242 32 F32 0.71363 

 15 F15 3.0858 33 F33 0.71063 

 16 F16 2.9682 34 F34 0.70745 

 17 F17 2.8469 35 F35 0.70440 

 18 F18 2.8071 36 F36 0.70263 

 PARAMETERS FOR SAMPLE NO. 4 

 NO. NAME B NO. NAME B 

 1 HK_1 1.1993 2 HK_2 0.13669 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 4 

 NO. OBSERVATION FC NO. OBSERVATION FC 

132

 2 F2 13.725 20 F20 3.2569 

 1 F1 13.838 19 F19 3.3065 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 3 F3 13.540 21 F21 3.1920 

 4 F4 13.041 22 F22 3.1436 

 5 F5 12.806 23 F23 3.1579 

 6 F6 12.780 24 F24 3.1741 

 7 F7 5.8404 25 F25 2.1600 

 8 F8 6.0457 26 F26 2.1470 

 9 F9 6.4522 27 F27 2.1345 

 10 F10 8.6355 28 F28 2.1239 

 11 F11 10.641 29 F29 2.1164 

 12 F12 10.984 30 F30 2.1139 

 13 F13 4.5924 31 F31 1.0635 

 14 F14 4.5807 32 F32 1.0605 

 15 F15 4.5258 33 F33 1.0575 

 16 F16 4.3984 34 F34 1.0545 

 17 F17 4.2921 35 F35 1.0515 

 18 F18 4.2733 36 F36 1.0494 

 TOTAL NONLINEARITY (BNT).......... = 0.63930E-01 

 INTRINSIC NONLINEARITY (BNI)...... = 0.45443E-03 

 ****** NONLINEARITY MEASURES FOR CONFIDENCE INTERVALS 

 TOTAL NUMBER OF INTERVALS (NOINT)......................: 8 

 NUMBER OF INTERVALS FOR HEAD (NHI).....................: 6 

 NUMBER OF INTERVALS FOR HEAD DEPENDENT FLOWS (NQI).....: 0 

 NUMBER OF INTERVALS FOR PARAMETERS (NPI)...............: 2 

 INTERVAL TYPE (1=CONFIDENCE,2=PREDICTION)..............: 1 

 HEAD AND FLOW INTERVALS ON DIFFERENCES (0,1)=(NO,YES)..: 0 

 INTERVAL VARIABLES COMPUTED WITH OPTIMUM PARAMETERS 

 NO. INTERVAL GOPT NO. INTERVAL GOPT 

 1 G1 2.8565 5 G5 1.2704 

 2 G2 11.996 6 G6 3.2028 

 3 G3 4.5392 7 HK_1 0.38102 

 4 G4 2.1408 8 HK_2 -2.3083 

 RELIABILITY WEIGHTS FOR INTERVAL SAMPLE 

 NO. INTERVAL W NO. INTERVAL W 

 1 G1 0.0000 5 G5 0.0000 

 2 G2 0.0000 6 G6 0.0000 

 3 G3 0.0000 7 HK_1 0.0000 

 4 G4 0.0000 8 HK_2 0.0000 

 SENSITIVITIES FOR OPTIMUM PARAMETERS 

 PARAMETER G1 G2 G3 G4 G5 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -2.8442 -5.2981 -4.3706 -2.1397 -1.2703 

 HK_2 -0.12354E-01 -6.6975 -0.16862 -0.10612E-02 -0.13788E-03 

133
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 PARAMETER G6 HK_1 HK_2 

 ---------- ------------ ------------ ------------ 

 HK_1 -3.1694 1.0000 0.0000 

 HK_2 -0.33366E-01 0.0000 1.0000 

 CORRECTION FACTORS 

 1 G1 45.638 5 G5 45.477 

 2 G2 15.989 6 G6 45.875 

 3 G3 46.966 7 HK_1 45.473 

 4 G4 45.492 8 HK_2 10.668 

 *** INTERVAL NO. 1 

 PARAMETERS FOR SAMPLE NO. 1 

 NO. NAME B NO. NAME B 

 1 HK_1 1.8283 2 HK_2 0.81801E-01 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 1 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.703 19 F19 2.1861 

 2 F2 14.588 20 F20 2.1505 

 3 F3 14.411 21 F21 2.1029 

 4 F4 13.949 22 F22 2.0634 

 5 F5 13.746 23 F23 2.0624 

 6 F6 13.730 24 F24 2.0647 

 7 F7 3.8336 25 F25 1.4234 

 8 F8 4.0380 26 F26 1.4132 

 9 F9 4.4340 27 F27 1.4029 

 10 F10 8.3438 28 F28 1.3931 

 11 F11 11.858 29 F29 1.3851 

 12 F12 12.234 30 F30 1.3809 

 13 F13 3.0475 31 F31 0.69998 

 14 F14 3.0490 32 F32 0.69723 

 15 F15 3.0115 33 F33 0.69439 

 16 F16 2.8990 34 F34 0.69141 

 17 F17 2.7843 35 F35 0.68844 

 18 F18 2.7478 36 F36 0.68653 

 PREDICTED VALUE, GP= 2.2912 

 PARAMETERS FOR SAMPLE NO. 2 

 NO. NAME B NO. NAME B 

 1 HK_1 1.1719 2 HK_2 0.12087 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 2 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.814 19 F19 3.3869 

 2 F2 14.691 20 F20 3.3357 

 3 F3 14.491 21 F21 3.2684 

 4 F4 13.955 22 F22 3.2174 

 5 F5 13.704 23 F23 3.2302 

 6 F6 13.677 24 F24 3.2452 

134

 8 F8 6.2006 26 F26 2.1984 

 7 F7 5.9774 25 F25 2.2118 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 9 F9 6.6410 27 F27 2.1852 

 10 F10 9.1382 28 F28 2.1737 

 11 F11 11.427 29 F29 2.1654 

 12 F12 11.802 30 F30 2.1622 

 13 F13 4.7065 31 F31 1.0889 

 14 F14 4.6961 32 F32 1.0857 

 15 F15 4.6394 33 F33 1.0824 

 16 F16 4.5036 34 F34 1.0790 

 17 F17 4.3865 35 F35 1.0758 

 18 F18 4.3628 36 F36 1.0735 

 PREDICTED VALUE, GP= 3.5609 

 *** INTERVAL NO. 2 

 PARAMETERS FOR SAMPLE NO. 1 

 NO. NAME B NO. NAME B 

 1 HK_1 1.4424 2 HK_2 0.11176 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 1 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 13.805 19 F19 2.7591 

 2 F2 13.688 20 F20 2.7160 

 3 F3 13.502 21 F21 2.6592 

 4 F4 13.009 22 F22 2.6144 

 5 F5 12.784 23 F23 2.6205 

 6 F6 12.763 24 F24 2.6292 

 7 F7 4.8577 25 F25 1.7997 

 8 F8 5.0682 26 F26 1.7880 

 9 F9 5.4802 27 F27 1.7764 

 10 F10 8.2529 28 F28 1.7659 

 11 F11 10.778 29 F29 1.7579 

 12 F12 11.139 30 F30 1.7543 

 13 F13 3.8386 31 F31 0.88561 

 14 F14 3.8340 32 F32 0.88272 

 15 F15 3.7873 33 F33 0.87969 

 16 F16 3.6648 34 F34 0.87663 

 17 F17 3.5509 35 F35 0.87355 

 18 F18 3.5218 36 F36 0.87150 

 PREDICTED VALUE, GP= 11.333 

 PARAMETERS FOR SAMPLE NO. 2 

 NO. NAME B NO. NAME B 

 1 HK_1 1.4855 2 HK_2 0.88468E-01 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 2 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 15.379 19 F19 2.6848 

 2 F2 15.251 20 F20 2.6420 

 3 F3 15.050 21 F21 2.5850 

135

 5 F5 14.291 23 F23 2.5411 

 4 F4 14.525 22 F22 2.5389 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 6 F6 14.271 24 F24 2.5467 

 7 F7 4.7170 25 F25 1.7497 

 8 F8 4.9454 26 F26 1.7377 

 9 F9 5.3899 27 F27 1.7256 

 10 F10 8.9545 28 F28 1.7146 

 11 F11 12.180 29 F29 1.7057 

 12 F12 12.583 30 F30 1.7014 

 13 F13 3.7388 31 F31 0.86057 

 14 F14 3.7377 32 F32 0.85756 

 15 F15 3.6920 33 F33 0.85437 

 16 F16 3.5631 34 F34 0.85108 

 17 F17 3.4369 35 F35 0.84774 

 18 F18 3.3999 36 F36 0.84557 

 PREDICTED VALUE, GP= 12.745 

 *** INTERVAL NO. 3 

 PARAMETERS FOR SAMPLE NO. 1 

 NO. NAME B NO. NAME B 

 1 HK_1 1.8340 2 HK_2 0.81992E-01 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 1 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.664 19 F19 2.1794 

 2 F2 14.549 20 F20 2.1439 

 3 F3 14.372 21 F21 2.0964 

 4 F4 13.912 22 F22 2.0570 

 5 F5 13.709 23 F23 2.0559 

 6 F6 13.693 24 F24 2.0582 

 7 F7 3.8215 25 F25 1.4191 

 8 F8 4.0253 26 F26 1.4089 

 9 F9 4.4201 27 F27 1.3985 

 10 F10 8.3208 28 F28 1.3888 

 11 F11 11.827 29 F29 1.3807 

 12 F12 12.201 30 F30 1.3765 

 13 F13 3.0380 31 F31 0.69779 

 14 F14 3.0395 32 F32 0.69504 

 15 F15 3.0022 33 F33 0.69222 

 16 F16 2.8900 34 F34 0.68930 

 17 F17 2.7756 35 F35 0.68633 

 18 F18 2.7391 36 F36 0.68441 

 PREDICTED VALUE, GP= 3.6750 

 PARAMETERS FOR SAMPLE NO. 2 

 NO. NAME B NO. NAME B 

 1 HK_1 1.1683 2 HK_2 0.12058 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 2 

 NO. OBSERVATION FC NO. OBSERVATION FC 

136

 2 F2 14.731 20 F20 3.3460 

 1 F1 14.855 19 F19 3.3974 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 3 F3 14.531 21 F21 3.2785 

 4 F4 13.993 22 F22 3.2273 

 5 F5 13.741 23 F23 3.2402 

 6 F6 13.715 24 F24 3.2552 

 7 F7 5.9958 25 F25 2.2185 

 8 F8 6.2196 26 F26 2.2051 

 9 F9 6.6612 27 F27 2.1918 

 10 F10 9.1641 28 F28 2.1804 

 11 F11 11.458 29 F29 2.1721 

 12 F12 11.834 30 F30 2.1690 

 13 F13 4.7209 31 F31 1.0921 

 14 F14 4.7106 32 F32 1.0890 

 15 F15 4.6538 33 F33 1.0857 

 16 F16 4.5176 34 F34 1.0824 

 17 F17 4.4001 35 F35 1.0791 

 18 F18 4.3763 36 F36 1.0769 

 PREDICTED VALUE, GP= 5.5928 

 *** INTERVAL NO. 4 

 PARAMETERS FOR SAMPLE NO. 1 

 NO. NAME B NO. NAME B 

 1 HK_1 1.8277 2 HK_2 0.81782E-01 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 1 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.707 19 F19 2.1869 

 2 F2 14.592 20 F20 2.1513 

 3 F3 14.414 21 F21 2.1036 

 4 F4 13.952 22 F22 2.0641 

 5 F5 13.749 23 F23 2.0630 

 6 F6 13.733 24 F24 2.0653 

 7 F7 3.8348 25 F25 1.4240 

 8 F8 4.0392 26 F26 1.4138 

 9 F9 4.4352 27 F27 1.4033 

 10 F10 8.3458 28 F28 1.3936 

 11 F11 11.861 29 F29 1.3855 

 12 F12 12.236 30 F30 1.3813 

 13 F13 3.0485 31 F31 0.70019 

 14 F14 3.0500 32 F32 0.69744 

 15 F15 3.0125 33 F33 0.69462 

 16 F16 2.9000 34 F34 0.69169 

 17 F17 2.7852 35 F35 0.68871 

 18 F18 2.7486 36 F36 0.68678 

 PREDICTED VALUE, GP= 1.7151 

 PARAMETERS FOR SAMPLE NO. 2 

 NO. NAME B NO. NAME B 

 1 HK_1 1.1723 2 HK_2 0.12089 

137

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 2 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.810 19 F19 3.3858 

 2 F2 14.686 20 F20 3.3345 

 3 F3 14.487 21 F21 3.2672 

 4 F4 13.950 22 F22 3.2162 

 5 F5 13.700 23 F23 3.2290 

 6 F6 13.673 24 F24 3.2439 

 7 F7 5.9751 25 F25 2.2109 

 8 F8 6.1982 26 F26 2.1975 

 9 F9 6.6386 27 F27 2.1843 

 10 F10 9.1353 28 F28 2.1728 

 11 F11 11.424 29 F29 2.1646 

 12 F12 11.798 30 F30 2.1615 

 13 F13 4.7047 31 F31 1.0884 

 14 F14 4.6944 32 F32 1.0853 

 15 F15 4.6378 33 F33 1.0820 

 16 F16 4.5021 34 F34 1.0787 

 17 F17 4.3849 35 F35 1.0754 

 18 F18 4.3612 36 F36 1.0732 

 PREDICTED VALUE, GP= 2.6723 

 *** INTERVAL NO. 5 

 PARAMETERS FOR SAMPLE NO. 1 

 NO. NAME B NO. NAME B 

 1 HK_1 1.8276 2 HK_2 0.81780E-01 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 1 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.708 19 F19 2.1869 

 2 F2 14.593 20 F20 2.1512 

 3 F3 14.415 21 F21 2.1036 

 4 F4 13.953 22 F22 2.0640 

 5 F5 13.750 23 F23 2.0630 

 6 F6 13.734 24 F24 2.0654 

 7 F7 3.8349 25 F25 1.4239 

 8 F8 4.0393 26 F26 1.4137 

 9 F9 4.4354 27 F27 1.4033 

 10 F10 8.3461 28 F28 1.3936 

 11 F11 11.861 29 F29 1.3855 

 12 F12 12.237 30 F30 1.3814 

 13 F13 3.0485 31 F31 0.70014 

 14 F14 3.0500 32 F32 0.69745 

 15 F15 3.0126 33 F33 0.69463 

 16 F16 2.9000 34 F34 0.69170 

 17 F17 2.7852 35 F35 0.68874 

 18 F18 2.7486 36 F36 0.68681 

 PREDICTED VALUE, GP= 1.0175 

138

 NO. NAME B NO. NAME B 

 PARAMETERS FOR SAMPLE NO. 2 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 1 HK_1 1.1724 2 HK_2 0.12090 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 2 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.809 19 F19 3.3856 

 2 F2 14.686 20 F20 3.3344 

 3 F3 14.486 21 F21 3.2671 

 4 F4 13.950 22 F22 3.2161 

 5 F5 13.699 23 F23 3.2289 

 6 F6 13.673 24 F24 3.2439 

 7 F7 5.9750 25 F25 2.2109 

 8 F8 6.1981 26 F26 2.1975 

 9 F9 6.6384 27 F27 2.1843 

 10 F10 9.1351 28 F28 2.1728 

 11 F11 11.423 29 F29 2.1645 

 12 F12 11.798 30 F30 2.1614 

 13 F13 4.7046 31 F31 1.0884 

 14 F14 4.6942 32 F32 1.0852 

 15 F15 4.6376 33 F33 1.0820 

 16 F16 4.5019 34 F34 1.0786 

 17 F17 4.3848 35 F35 1.0753 

 18 F18 4.3611 36 F36 1.0731 

 PREDICTED VALUE, GP= 1.5862 

 *** INTERVAL NO. 6 

 PARAMETERS FOR SAMPLE NO. 1 

 NO. NAME B NO. NAME B 

 1 HK_1 1.8294 2 HK_2 0.81832E-01 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 1 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.696 19 F19 2.1848 

 2 F2 14.581 20 F20 2.1493 

 3 F3 14.404 21 F21 2.1017 

 4 F4 13.942 22 F22 2.0622 

 5 F5 13.739 23 F23 2.0611 

 6 F6 13.723 24 F24 2.0634 

 7 F7 3.8312 25 F25 1.4227 

 8 F8 4.0354 26 F26 1.4125 

 9 F9 4.4311 27 F27 1.4021 

 10 F10 8.3394 28 F28 1.3923 

 11 F11 11.852 29 F29 1.3842 

 12 F12 12.228 30 F30 1.3800 

 13 F13 3.0456 31 F31 0.69955 

 14 F14 3.0471 32 F32 0.69680 

 15 F15 3.0097 33 F33 0.69397 

 16 F16 2.8973 34 F34 0.69105 

 17 F17 2.7826 35 F35 0.68807 

 18 F18 2.7460 36 F36 0.68615 

139

 PREDICTED VALUE, GP= 2.5731 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 PARAMETERS FOR SAMPLE NO. 2 

 NO. NAME B NO. NAME B 

 1 HK_1 1.1713 2 HK_2 0.12082 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 2 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.821 19 F19 3.3889 

 2 F2 14.698 20 F20 3.3376 

 3 F3 14.498 21 F21 3.2703 

 4 F4 13.961 22 F22 3.2192 

 5 F5 13.710 23 F23 3.2320 

 6 F6 13.684 24 F24 3.2470 

 7 F7 5.9806 25 F25 2.2130 

 8 F8 6.2038 26 F26 2.1996 

 9 F9 6.6445 27 F27 2.1863 

 10 F10 9.1427 28 F28 2.1748 

 11 F11 11.432 29 F29 2.1666 

 12 F12 11.807 30 F30 2.1635 

 13 F13 4.7090 31 F31 1.0894 

 14 F14 4.6987 32 F32 1.0863 

 15 F15 4.6421 33 F33 1.0830 

 16 F16 4.5062 34 F34 1.0797 

 17 F17 4.3890 35 F35 1.0764 

 18 F18 4.3653 36 F36 1.0741 

 PREDICTED VALUE, GP= 3.9844 

 *** INTERVAL NO. 7 

 PARAMETERS FOR SAMPLE NO. 1 

 NO. NAME B NO. NAME B 

 1 HK_1 1.1724 2 HK_2 0.12090 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 1 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.809 19 F19 3.3856 

 2 F2 14.686 20 F20 3.3343 

 3 F3 14.486 21 F21 3.2671 

 4 F4 13.950 22 F22 3.2161 

 5 F5 13.699 23 F23 3.2289 

 6 F6 13.673 24 F24 3.2438 

 7 F7 5.9749 25 F25 2.2109 

 8 F8 6.1981 26 F26 2.1975 

 9 F9 6.6384 27 F27 2.1842 

 10 F10 9.1350 28 F28 2.1728 

 11 F11 11.423 29 F29 2.1645 

 12 F12 11.798 30 F30 2.1614 

 13 F13 4.7046 31 F31 1.0884 

 14 F14 4.6942 32 F32 1.0852 

 15 F15 4.6376 33 F33 1.0819 

 16 F16 4.5018 34 F34 1.0786 

140

 18 F18 4.3610 36 F36 1.0731 

 17 F17 4.3847 35 F35 1.0753 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 PREDICTED VALUE, GP= 0.15904 

 PARAMETERS FOR SAMPLE NO. 2 

 NO. NAME B NO. NAME B 

 1 HK_1 1.8276 2 HK_2 0.81780E-01 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 2 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.708 19 F19 2.1871 

 2 F2 14.593 20 F20 2.1515 

 3 F3 14.415 21 F21 2.1038 

 4 F4 13.953 22 F22 2.0642 

 5 F5 13.750 23 F23 2.0631 

 6 F6 13.734 24 F24 2.0655 

 7 F7 3.8351 25 F25 1.4241 

 8 F8 4.0396 26 F26 1.4138 

 9 F9 4.4358 27 F27 1.4034 

 10 F10 8.3465 28 F28 1.3937 

 11 F11 11.861 29 F29 1.3856 

 12 F12 12.237 30 F30 1.3814 

 13 F13 3.0489 31 F31 0.70022 

 14 F14 3.0503 32 F32 0.69747 

 15 F15 3.0128 33 F33 0.69464 

 16 F16 2.9003 34 F34 0.69168 

 17 F17 2.7855 35 F35 0.68875 

 18 F18 2.7488 36 F36 0.68681 

 PREDICTED VALUE, GP= 0.60300 

 *** INTERVAL NO. 8 

 PARAMETERS FOR SAMPLE NO. 1 

 NO. NAME B NO. NAME B 

 1 HK_1 1.5854 2 HK_2 0.87532E-01 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 1 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 15.025 19 F19 2.5172 

 2 F2 14.901 20 F20 2.4770 

 3 F3 14.709 21 F21 2.4232 

 4 F4 14.206 22 F22 2.3795 

 5 F5 13.982 23 F23 2.3807 

 6 F6 13.963 24 F24 2.3854 

 7 F7 4.4208 25 F25 1.6401 

 8 F8 4.6408 26 F26 1.6287 

 9 F9 5.0687 27 F27 1.6173 

 10 F10 8.6867 28 F28 1.6067 

 11 F11 11.955 29 F29 1.5983 

 12 F12 12.347 30 F30 1.5941 

 13 F13 3.5065 31 F31 0.80668 

141

 15 F15 3.4637 33 F33 0.80067 

 14 F14 3.5064 32 F32 0.80374 
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 16 F16 3.3403 34 F34 0.79741 

 17 F17 3.2183 35 F35 0.79416 

 18 F18 3.1816 36 F36 0.79220 

 PREDICTED VALUE, GP= -2.4358 

 PARAMETERS FOR SAMPLE NO. 2 

 NO. NAME B NO. NAME B 

 1 HK_1 1.3514 2 HK_2 0.11295 

 DEPENDENT VARIABLES COMPUTED FOR SAMPLE NO. 2 

 NO. OBSERVATION FC NO. OBSERVATION FC 

 1 F1 14.191 19 F19 2.9427 

 2 F2 14.070 20 F20 2.8971 

 3 F3 13.878 21 F21 2.8370 

 4 F4 13.367 22 F22 2.7901 

 5 F5 13.133 23 F23 2.7977 

 6 F6 13.110 24 F24 2.8079 

 7 F7 5.1838 25 F25 1.9200 

 8 F8 5.4004 26 F26 1.9077 

 9 F9 5.8252 27 F27 1.8955 

 10 F10 8.5518 28 F28 1.8847 

 11 F11 11.039 29 F29 1.8765 

 12 F12 11.408 30 F30 1.8729 

 13 F13 4.0926 31 F31 0.94490 

 14 F14 4.0866 32 F32 0.94193 

 15 F15 4.0371 33 F33 0.93885 

 16 F16 3.9096 34 F34 0.93561 

 17 F17 3.7931 35 F35 0.93241 

 18 F18 3.7647 36 F36 0.93026 

 PREDICTED VALUE, GP= -2.1808 

 COMBINED INTRINSIC NONLINEARITY 

 INTERVAL NO. BMI INTERVAL NO. BMI 

 1 G1 0.91136E-01 5 G5 0.92165E-01 

 2 G2 0.13976E-04 6 G6 0.89142E-01 

 3 G3 0.80532E-01 7 HK_1 0.37137E-01 

 4 G4 0.92049E-01 8 HK_2 0.22188E-01 

 COMBINED INTRINSIC NONLINEARITY AS IF F WERE LINEAR 

 INTERVAL NO. BMF0 INTERVAL NO. BMF0 

 1 G1 0.12232E-01 5 G5 0.12491E-01 

 2 G2 0.37404E-02 6 G6 0.11832E-01 

 3 G3 0.99148E-02 7 HK_1 0.58584E-12 

 4 G4 0.12468E-01 8 HK_2 0.59462E-10 

 COMBINED INTRINSIC NONLINEARITY AS IF G WERE LINEAR 

 INTERVAL NO. BMG0 INTERVAL NO. BMG0 

 1 G1 0.36826E-01 5 G5 0.37051E-01 

 2 G2 0.40559E-02 6 G6 0.36261E-01 

 3 G3 0.34126E-01 7 HK_1 0.37137E-01 

 4 G4 0.37003E-01 8 HK_2 0.22186E-01 

142
APPENDIX C. EXAMPLE SIMULATIONS – BEALE2-2k Output File 

 COMBINED INTRINSIC NONLINEARITY - MAX. SUM 

 INTERVAL NO. BMIMAX INTERVAL NO. BMIMAX 

 1 G1 0.11560 5 G5 0.11715 

 2 G2 0.80979E-02 6 G6 0.11280 

 3 G3 0.10036 7 HK_1 0.37137E-01 

 4 G4 0.11699 8 HK_2 0.22188E-01 

143
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Input Files 

RESAN2-2k Input Files 

Modified _rs file (example._rs): 

 2 36 36 0 0 0 1000 10059 1.2028920299866620D-02 

.01 0 0 1 1 

HK_1 HK_2 

 1.5984446317565970D-02 -9.3897459008995630D-02 

 1.2171445371320360D+00 

 3.70299257E-02 3.80506031E-02 3.86905819E-02 3.78140099E-02 3.70474569E-02 

3.65929864E-02 3.94278727E-02 3.87343839E-02 3.81299071E-02 3.72347161E-02 

3.73285078E-02 3.69736925E-02 3.95077951E-02 3.92752253E-02 4.01894972E-02 

4.05620262E-02 

 3.95699181E-02 3.98577228E-02 4.27163243E-02 4.33942601E-02 4.35817763E-02 

4.33828235E-02 4.27807719E-02 4.29287367E-02 4.85740304E-02 4.83402126E-02 

4.82680798E-02 4.84433360E-02 4.81047295E-02 4.74082753E-02 8.08135197E-02 

7.11964741E-02 

 7.12072998E-02 7.03493059E-02 6.85945898E-02 7.03145191E-02 

-7.04659367E+00-1.50004828E+00 

-7.06339741E+00-1.48310077E+00 

-7.10597181E+00-1.43996620E+00 

-7.27100706E+00-1.27518404E+00 

-7.40044165E+00-1.14634621E+00 

-7.43134212E+00-1.11610472E+00 

-6.84805489E+00-8.78273845E-02 

-6.79571819E+00-1.33685842E-01 

-6.68890238E+00-2.35920116E-01 

-6.33737326E+00-5.86289763E-01 

-6.05908632E+00-8.68575454E-01 

-5.98694611E+00-9.47160363E-01 

-5.29238510E+00-6.16868287E-02 

-5.25664139E+00-5.80927134E-02 

-5.23037148E+00-3.93029377E-02 

-5.26140881E+00 1.06110601E-02 

-5.35310459E+00 5.58182150E-02 

-5.40344524E+00 6.14485182E-02 

-3.86938190E+00-2.47615427E-02 

-3.83271551E+00-1.77744813E-02 

-3.78756404E+00-8.75341985E-03 

-3.77330446E+00 2.30506901E-03 

-3.84462976E+00 1.46126971E-02 

-3.90307307E+00 2.25643124E-02 

-2.55457449E+00-8.90955608E-03 

-2.54985857E+00-6.12230739E-03 

-2.54753423E+00-2.92208139E-03 

-2.54963517E+00 7.52098567E-04 

-2.55824447E+00 4.94222483E-03 

-2.56872201E+00 8.00079107E-03 

-1.26404727E+00-2.83111562E-03 

-1.26398706E+00-1.93543814E-03 

-1.26431274E+00-9.20786173E-04 

-1.26529419E+00 2.35229818E-04 

-1.26728976E+00 1.55798520E-03 

144
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Input Files 

-1.26906574E+00 2.53848336E-03 

 0.66151 0.57163 0.57791 0.65444 0.70896 0.78005 0.59267 0.58384 

 0.54952 0.43816 0.37298 0.38888 0.47908 0.43970 0.40578 0.41702 

 0.43077 0.42120 0.32116 0.30408 0.29091 0.27985 0.28804 0.26147 

 0.12901 0.15028 0.15210 0.16090 0.16435 0.17098 -.048154 -.018498 

-.0080944 .0071884 .033326 .055387 

729.28 667.57 615.68 586.33 557.59 534.25 584.85 598.77 

591.88 569.46 549.34 541.11 477.64 485.80 477.81 474.03 

490.18 486.31 326.73 325.31 323.54 332.79 344.61 339.91 

208.67 221.75 226.29 225.34 240.21 245.61 68.870 101.52 

103.49 110.91 118.31 98.075 

667.57 690.68 654.63 620.72 586.71 561.51 589.59 601.63 

595.99 577.41 559.76 549.42 478.84 483.95 474.92 467.99 

483.29 478.43 321.55 318.49 315.94 322.14 333.98 331.21 

214.14 224.01 227.44 222.64 233.67 236.83 72.997 106.27 

108.19 114.37 119.57 96.373 

615.68 654.63 668.02 651.43 618.29 587.79 577.61 590.09 

588.78 576.27 558.71 553.40 467.96 471.16 460.22 452.06 

465.32 460.42 308.79 303.50 300.50 304.75 314.61 312.42 

202.47 210.94 215.14 210.02 220.51 221.90 72.489 104.27 

105.26 110.19 114.24 92.439 

586.33 620.72 651.43 699.35 686.13 649.67 562.03 574.93 

574.79 566.42 550.59 553.37 452.96 452.53 440.17 430.63 

441.05 439.34 293.55 290.00 287.19 289.52 297.87 293.66 

193.48 201.20 204.83 198.65 206.20 206.93 69.506 99.734 

99.477 103.72 107.11 87.286 

557.59 586.71 618.29 686.13 728.59 707.08 543.25 559.94 

561.06 555.06 538.05 538.75 443.24 442.05 428.49 417.27 

424.38 421.98 288.46 285.79 281.38 283.14 288.82 282.64 

192.57 200.60 204.67 197.13 203.52 203.09 69.115 99.001 

98.459 104.13 107.79 87.132 

534.25 561.51 587.79 649.67 707.08 746.80 525.38 543.55 

547.06 545.23 529.14 527.88 437.16 436.68 423.31 411.63 

418.82 415.31 289.42 286.38 280.51 281.62 286.86 279.57 

191.49 200.05 202.37 193.86 198.61 197.71 65.800 95.801 

95.834 100.16 103.22 82.084 

584.85 589.59 577.61 562.03 543.25 525.38 643.27 632.46 

614.20 600.57 577.59 562.10 507.35 507.86 491.01 477.72 

483.99 476.30 331.83 326.31 325.01 329.89 338.91 334.53 

216.78 225.84 232.23 227.82 238.01 242.82 71.272 107.29 

108.79 114.71 123.51 104.54 

598.77 601.63 590.09 574.93 559.94 543.55 632.46 666.51 

657.18 644.11 618.41 599.20 524.83 525.59 509.64 499.58 

510.39 501.63 346.45 338.31 333.05 337.30 349.23 345.68 

221.67 230.89 237.21 232.60 242.72 247.01 73.206 109.94 

110.31 115.26 124.50 105.37 

591.88 595.99 588.78 574.79 561.06 547.06 614.20 657.18 

687.81 686.21 659.29 637.88 537.68 541.81 526.18 517.05 

531.07 522.43 362.45 354.14 347.07 349.82 361.59 358.58 

232.67 240.53 246.07 242.48 250.68 254.24 78.717 114.03 

114.30 119.04 127.26 106.76 

569.46 577.41 576.27 566.42 555.06 545.23 600.57 644.11 

686.21 721.28 701.42 676.61 554.47 562.74 551.08 543.76 

558.91 552.38 379.87 372.12 363.43 363.69 375.96 372.75 

145

120.19 124.31 131.48 110.02 

242.19 249.08 254.99 254.22 261.06 263.36 84.298 120.19 
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Input Files 

549.34 559.76 558.71 550.59 538.05 529.14 577.59 618.41 

659.29 701.42 717.66 702.17 553.51 566.83 558.53 551.51 

566.99 560.49 382.08 375.64 366.97 367.21 378.36 374.44 

245.30 252.08 257.29 255.75 261.15 262.41 88.611 121.30 

120.89 124.28 131.52 112.32 

541.11 549.42 553.40 553.37 538.75 527.88 562.10 599.20 

637.88 676.61 702.17 731.50 549.20 564.90 560.42 557.96 

574.98 566.97 382.48 378.58 371.67 371.59 381.56 374.77 

248.91 256.43 261.51 258.12 261.74 261.68 91.934 123.40 

123.28 127.11 134.16 113.32 

477.64 478.84 467.96 452.96 443.24 437.16 507.35 524.83 

537.68 554.47 553.51 549.20 640.67 626.22 591.51 563.61 

555.08 535.82 453.45 447.14 440.73 438.31 441.10 423.05 

295.30 305.66 312.49 311.97 316.94 320.88 105.64 144.94 

148.93 153.49 165.66 142.97 

485.80 483.95 471.16 452.53 442.05 436.68 507.86 525.59 

541.81 562.74 566.83 564.90 626.22 648.28 618.24 588.02 

580.79 558.88 460.96 454.52 448.96 446.67 449.88 432.98 

303.50 313.53 320.28 319.66 325.02 328.34 111.31 149.19 

151.82 155.30 165.42 142.83 

477.81 474.92 460.22 440.17 428.49 423.31 491.01 509.64 

526.18 551.08 558.53 560.42 591.51 618.24 619.12 597.36 

588.69 566.62 452.11 449.96 444.33 443.98 453.68 439.92 

300.25 308.91 315.03 313.82 319.76 324.34 108.06 145.56 

148.64 152.55 161.93 139.60 

474.03 467.99 452.06 430.63 417.27 411.63 477.72 499.58 

517.05 543.76 551.51 557.96 563.61 588.02 597.36 607.80 

599.81 577.83 445.39 443.84 438.99 440.63 456.72 446.56 

296.70 305.68 312.57 310.46 316.16 321.64 106.61 144.37 

146.88 150.81 159.16 137.11 

490.18 483.29 465.32 441.05 424.38 418.82 483.99 510.39 

531.07 558.91 566.99 574.98 555.08 580.79 588.69 599.81 

638.66 617.67 438.33 438.82 437.99 440.62 458.55 449.80 

292.93 303.57 309.94 308.99 313.48 315.66 105.79 143.49 

145.72 148.31 155.68 131.83 

486.31 478.43 460.42 439.34 421.98 415.31 476.30 501.63 

522.43 552.38 560.49 566.97 535.82 558.88 566.62 577.83 

617.67 629.47 428.62 428.57 430.59 437.01 456.18 447.76 

288.83 299.85 307.41 307.76 311.41 313.08 105.58 143.63 

146.56 150.52 156.68 132.63 

326.73 321.55 308.79 293.55 288.46 289.42 331.83 346.45 

362.45 379.87 382.08 382.48 453.45 460.96 452.11 445.39 

438.33 428.62 548.04 522.35 500.04 484.78 471.74 454.53 

379.15 390.62 392.18 383.88 378.94 379.17 137.98 179.29 

184.21 190.21 201.84 174.92 

325.31 318.49 303.50 290.00 285.79 286.38 326.31 338.31 

354.14 372.12 375.64 378.58 447.14 454.52 449.96 443.84 

438.82 428.57 522.35 531.05 515.27 499.46 486.02 465.12 

377.99 389.36 391.95 384.08 380.66 379.82 139.14 177.36 

182.01 187.59 198.06 171.04 

323.54 315.94 300.50 287.19 281.38 280.51 325.01 333.05 

347.07 363.43 366.97 371.67 440.73 448.96 444.33 438.99 

437.99 430.59 500.04 515.27 526.49 517.52 502.76 477.30 

381.07 395.46 399.39 394.44 392.14 390.23 145.44 183.03 

146

332.79 322.14 304.75 289.52 283.14 281.62 329.89 337.30 

188.11 193.42 204.07 177.28 
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Input Files 

349.82 363.69 367.21 371.59 438.31 446.67 443.98 440.63 

440.62 437.01 484.78 499.46 517.52 531.33 522.51 497.03 

371.13 389.59 396.26 394.86 398.34 402.69 143.86 182.63 

188.43 193.53 205.17 179.10 

344.61 333.98 314.61 297.87 288.82 286.86 338.91 349.23 

361.59 375.96 378.36 381.56 441.10 449.88 453.68 456.72 

458.55 456.18 471.74 486.02 502.76 522.51 546.39 526.83 

360.58 381.27 390.93 393.82 401.50 409.36 141.58 181.57 

187.82 192.83 203.97 177.63 

339.91 331.21 312.42 293.66 282.64 279.57 334.53 345.68 

358.58 372.75 374.44 374.77 423.05 432.98 439.92 446.56 

449.80 447.76 454.53 465.12 477.30 497.03 526.83 542.63 

346.27 367.35 378.05 383.54 394.49 404.22 134.42 174.09 

180.86 186.84 198.00 171.15 

208.67 214.14 202.47 193.48 192.57 191.49 216.78 221.67 

232.67 242.19 245.30 248.91 295.30 303.50 300.25 296.70 

292.93 288.83 379.15 377.99 381.07 371.13 360.58 346.27 

423.83 409.60 393.71 366.08 338.05 324.23 170.89 210.96 

207.57 202.51 198.52 162.88 

221.75 224.01 210.94 201.20 200.60 200.05 225.84 230.89 

240.53 249.08 252.08 256.43 305.66 313.53 308.91 305.68 

303.57 299.85 390.62 389.36 395.46 389.59 381.27 367.35 

409.60 427.94 417.14 392.68 365.92 350.74 173.99 216.39 

216.19 214.18 214.04 179.48 

226.29 227.44 215.14 204.83 204.67 202.37 232.23 237.21 

246.07 254.99 257.29 261.51 312.49 320.28 315.03 312.57 

309.94 307.41 392.18 391.95 399.39 396.26 390.93 378.05 

393.71 417.14 429.22 411.90 388.36 372.38 172.20 216.59 

217.49 217.96 221.48 188.33 

225.34 222.64 210.02 198.65 197.13 193.86 227.82 232.60 

242.48 254.22 255.75 258.12 311.97 319.66 313.82 310.46 

308.99 307.76 383.88 384.08 394.44 394.86 393.82 383.54 

366.08 392.68 411.90 426.12 411.10 395.28 165.65 211.41 

215.82 218.80 226.66 197.82 

240.21 233.67 220.51 206.20 203.52 198.61 238.01 242.72 

250.68 261.06 261.15 261.74 316.94 325.02 319.76 316.16 

313.48 311.41 378.94 380.66 392.14 398.34 401.50 394.49 

338.05 365.92 388.36 411.10 432.14 425.39 160.29 207.36 

214.39 219.04 231.88 204.96 

245.61 236.83 221.90 206.93 203.09 197.71 242.82 247.01 

254.24 263.36 262.41 261.68 320.88 328.34 324.34 321.64 

315.66 313.08 379.17 379.82 390.23 402.69 409.36 404.22 

324.23 350.74 372.38 395.28 425.39 444.93 153.77 201.63 

210.36 217.26 232.57 206.41 

68.870 72.997 72.489 69.506 69.115 65.800 71.272 73.206 

78.717 84.298 88.611 91.934 105.64 111.31 108.06 106.61 

105.79 105.58 137.98 139.14 145.44 143.86 141.58 134.42 

170.89 173.99 172.20 165.65 160.29 153.77 153.12 155.86 

140.83 128.13 111.47 93.066 

101.52 106.27 104.27 99.734 99.001 95.801 107.29 109.94 

114.03 120.19 121.30 123.40 144.94 149.19 145.56 144.37 

143.49 143.63 179.29 177.36 183.03 182.63 181.57 174.09 

210.96 216.39 216.59 211.41 207.36 201.63 155.86 197.28 

185.59 173.70 156.95 128.51 

147

114.30 120.19 120.89 123.28 148.93 151.82 148.64 146.88 

103.49 108.19 105.26 99.477 98.459 95.834 108.79 110.31 
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Input Files 

145.72 146.56 184.21 182.01 188.11 188.43 187.82 180.86 

207.57 216.19 217.49 215.82 214.39 210.36 140.83 185.59 

197.22 190.73 175.32 147.00 

110.91 114.37 110.19 103.72 104.13 100.16 114.71 115.26 

119.04 124.31 124.28 127.11 153.49 155.30 152.55 150.81 

148.31 150.52 190.21 187.59 193.42 193.53 192.83 186.84 

202.51 214.18 217.96 218.80 219.04 217.26 128.13 173.70 

190.73 202.06 192.15 162.93 

118.31 119.57 114.24 107.11 107.79 103.22 123.51 124.50 

127.26 131.48 131.52 134.16 165.66 165.42 161.93 159.16 

155.68 156.68 201.84 198.06 204.07 205.17 203.97 198.00 

198.52 214.04 221.48 226.66 231.88 232.57 111.47 156.95 

175.32 192.15 212.53 188.12 

98.075 96.373 92.439 87.286 87.132 82.084 104.54 105.37 

106.76 110.02 112.32 113.32 142.97 142.83 139.60 137.11 

131.83 132.63 174.92 171.04 177.28 179.10 177.63 171.15 

162.88 179.48 188.33 197.82 204.96 206.41 93.066 128.51 

147.00 162.93 188.12 202.26 

_ws file (example._ws): 

 0.5387840 0.1222136E-01 1 F1 

 0.5489494 0.1343849E-01 1 F2 

 0.5506946 0.2424740E-01 1 F3 

 0.5189314 0.5548934E-02 1 F4 

 0.4998959 -0.1790851E-01 1 F5 

 0.4930008 -0.1838977E-01 1 F6 

 0.1887295 0.3246087E-01 1 F7 

 0.1939302 0.2414436E-01 1 F8 

 0.2072599 0.1923179E-01 1 F9 

 0.3195286 0.9253923E-02 1 F10 

 0.4269593 -0.1335947E-01 1 F11 

 0.4369872 -0.1733579E-01 1 F12 

 0.1496757 -0.3496893E-02 1 F13 

 0.1486864 -0.1404292E-02 1 F14 

 0.1502898 -0.8822737E-02 1 F15 

 0.1465700 -0.1068726E-01 1 F16 

 0.1382196 -0.2890517E-02 1 F17 

 0.1379032 0.9968983E-02 1 F18 

 0.1162644 -0.1502673E-01 1 F19 

 0.1162447 -0.3336162E-01 1 F20 

 0.1142654 -0.3756152E-01 1 F21 

 0.1117733 -0.2674299E-01 1 F22 

 0.1103901 -0.3252914E-01 1 F23 

 0.1110804 -0.2264722E-01 1 F24 

 0.8620078E-01 -0.1139677E-01 1 F25 

 0.8520906E-01 -0.4964315E-02 1 F26 

 0.8450945E-01 0.1407734E-02 1 F27 

 0.8429423E-01 -0.9691494E-02 1 F28 

 0.8329535E-01 -0.1517311E-02 1 F29 

 0.8189733E-01 -0.8414512E-02 1 F30 

 0.7055598E-01 0.5408723E-02 1 F31 

 0.6194711E-01 0.3843991E-01 1 F32 

 0.6173658E-01 0.2513633E-01 1 F33 

148
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Input Files 

 0.6075818E-01 0.4265530E-01 1 F34 

 0.5902952E-01 0.1505264E-01 1 F35 

 0.6035101E-01 0.1525764E-02 1 F36 

149
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Output Files 

RESAN2-2k Output Files 

#nr file (example.#nr): 

 RESAN2-2k 

 MODFLOW-2000 POST-PROCESSING PROGRAM 

 TO PERFORM ANALYSIS OF NON-LINEARITY AND RESIDUALS 

 Version 1.0.0 25/06/2003 

 WEIGHTED RESIDUALS WILL BE READ FROM FILE Example._ws 

 NUMBER OF ESTIMATED PARAMETERS.............. 2 

 NUMBER OF OBSERVATIONS ..................... 36 

 NUMBER OF HEADS ............................ 36 

 NUMBER OF OBSERVATIONS OTHER THAN HEADS .... 0 

 NUMBER OF PRIOR EQUATIONS................... 0 

 NUMBER OF PRIOR WITH FULL WEIGHT MATRIX..... 0 

 NUMBER OF SETS OF RANDOM DEVIATES .......... 1000 

 NUMBER FOR RANDOM NUMBER GENERATOR ......... -10059 

 CALCULATED ERROR VARIANCE .................. 0.120289E-01 

 THEORETICAL ERROR VARIANCE ................. 0.100000E-01 

 PRINT (I-R) MATRIX.......................... 0 

 PRINT SIMULATED RESIDUALS................... 0 

 READ MEAN TRUE ERRORS....................... 1 

 READ OBSERVATION ERROR COVARIANCE MATRIX.... 1 

 COVARIANCE MATRIX FOR ESTIMATED PARAMETERS 

 HK_1 HK_2 

 .................................... 

 HK_1 1.59844E-02 -9.38975E-02 

 HK_2 -9.38975E-02 1.2171 

 RELIABILITY WEIGHTS FOR SAMPLE INFORMATION 

 OBSERVATION OBSERVATION 

 OBS# NAME W**.5 OBS# NAME W**.5 

 1 F1 0.37030E-01 19 F19 0.42716E-01 

 2 F2 0.38051E-01 20 F20 0.43394E-01 

 3 F3 0.38691E-01 21 F21 0.43582E-01 

 4 F4 0.37814E-01 22 F22 0.43383E-01 

 5 F5 0.37047E-01 23 F23 0.42781E-01 

 6 F6 0.36593E-01 24 F24 0.42929E-01 

 7 F7 0.39428E-01 25 F25 0.48574E-01 

 8 F8 0.38734E-01 26 F26 0.48340E-01 

 9 F9 0.38130E-01 27 F27 0.48268E-01 

 10 F10 0.37235E-01 28 F28 0.48443E-01 

 11 F11 0.37329E-01 29 F29 0.48105E-01 

 12 F12 0.36974E-01 30 F30 0.47408E-01 

 13 F13 0.39508E-01 31 F31 0.80814E-01 

150

 14 F14 0.39275E-01 32 F32 0.71196E-01 
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Output Files 

 15 F15 0.40189E-01 33 F33 0.71207E-01 

 16 F16 0.40562E-01 34 F34 0.70349E-01 

 17 F17 0.39570E-01 35 F35 0.68595E-01 

 18 F18 0.39858E-01 36 F36 0.70315E-01 

 SENSITIVITIES FOR OPTIMUM PARAMETERS 

 PARAMETER F1 F2 F3 F4 F5 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -7.0466 -7.0634 -7.1060 -7.2710 -7.4004 

 HK_2 -1.5000 -1.4831 -1.4400 -1.2752 -1.1463 

 PARAMETER F6 F7 F8 F9 F10 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -7.4313 -6.8481 -6.7957 -6.6889 -6.3374 

 HK_2 -1.1161 -0.87827E-01 -0.13369 -0.23592 -0.58629 

 PARAMETER F11 F12 F13 F14 F15 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -6.0591 -5.9869 -5.2924 -5.2566 -5.2304 

 HK_2 -0.86858 -0.94716 -0.61687E-01 -0.58093E-01 -0.39303E-01 

 PARAMETER F16 F17 F18 F19 F20 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -5.2614 -5.3531 -5.4034 -3.8694 -3.8327 

 HK_2 0.10611E-01 0.55818E-01 0.61449E-01 -0.24762E-01 -0.17774E-01 

 PARAMETER F21 F22 F23 F24 F25 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -3.7876 -3.7733 -3.8446 -3.9031 -2.5546 

 HK_2 -0.87534E-02 0.23051E-02 0.14613E-01 0.22564E-01 -0.89096E-02 

 PARAMETER F26 F27 F28 F29 F30 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -2.5499 -2.5475 -2.5496 -2.5582 -2.5687 

 HK_2 -0.61223E-02 -0.29221E-02 0.75210E-03 0.49422E-02 0.80008E-02 

 PARAMETER F31 F32 F33 F34 F35 

 ---------- ------------ ------------ ------------ ------------ ------------ 

 HK_1 -1.2640 -1.2640 -1.2643 -1.2653 -1.2673 

 HK_2 -0.28311E-02 -0.19354E-02 -0.92079E-03 0.23523E-03 0.15580E-02 

 PARAMETER F36 

 ---------- ------------ 

 HK_1 -1.2691 

 HK_2 0.25385E-02 

MEAN WEIGHTED RESIDUAL (EM) --------- = -.50018E-03 

SLOPE OF WEIGHTED RESIDUAL PLOT (SLP) = 0.35753E-02 

INTRINSIC NONLINEARITY MEASURE (QINT) = 0.19544E-03 (SHOULD BE <<0.14039E-01) 

 ORDERED, WEIGHTED RESIDUALS 

 OBSERVATION 

 NO. NAME E FRQ. (%) 

151

 2 F20 -.33362E-01 4.1667 

 1 F21 -.37562E-01 1.3889 
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Output Files 

 3 F23 -.32529E-01 6.9444 

 4 F22 -.26743E-01 9.7222 

 5 F24 -.22647E-01 12.500 

 6 F6 -.18390E-01 15.278 

 7 F5 -.17909E-01 18.056 

 8 F12 -.17336E-01 20.833 

 9 F19 -.15027E-01 23.611 

 10 F11 -.13359E-01 26.389 

 11 F25 -.11397E-01 29.167 

 12 F16 -.10687E-01 31.944 

 13 F28 -.96915E-02 34.722 

 14 F15 -.88227E-02 37.500 

 15 F30 -.84145E-02 40.278 

 16 F26 -.49643E-02 43.056 

 17 F13 -.34969E-02 45.833 

 18 F17 -.28905E-02 48.611 

 19 F29 -.15173E-02 51.389 

 20 F14 -.14043E-02 54.167 

 21 F27 0.14077E-02 56.944 

 22 F36 0.15258E-02 59.722 

 23 F31 0.54087E-02 62.500 

 24 F4 0.55489E-02 65.278 

 25 F10 0.92539E-02 68.056 

 26 F18 0.99690E-02 70.833 

 27 F1 0.12221E-01 73.611 

 28 F2 0.13438E-01 76.389 

 29 F35 0.15053E-01 79.167 

 30 F9 0.19232E-01 81.944 

 31 F8 0.24144E-01 84.722 

 32 F3 0.24247E-01 87.500 

 33 F33 0.25136E-01 90.278 

 34 F7 0.32461E-01 93.056 

 35 F32 0.38440E-01 95.833 

 36 F34 0.42655E-01 98.611 

 MEAN ORDERED, SIMULATED WEIGHTED RESIDUALS 

 OBSERVATION 

 NO. NAME D FRQ. (%) 

 1 F21 -.10938 1.3889 

 2 F20 -.92706E-01 4.1667 

 3 F23 -.81207E-01 6.9444 

 4 F22 -.71524E-01 9.7222 

 5 F24 -.63615E-01 12.500 

 6 F6 -.57005E-01 15.278 

 7 F5 -.50915E-01 18.056 

 8 F12 -.45545E-01 20.833 

 9 F19 -.40679E-01 23.611 

 10 F11 -.35812E-01 26.389 

 11 F25 -.31282E-01 29.167 

 12 F16 -.26841E-01 31.944 

 13 F28 -.22280E-01 34.722 

 14 F15 -.18282E-01 37.500 

 15 F30 -.14528E-01 40.278 

 16 F26 -.10723E-01 43.056 

152

 18 F17 -.30485E-02 48.611 

 17 F13 -.67828E-02 45.833 
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Output Files 

 19 F29 0.91987E-03 51.389 

 20 F14 0.45571E-02 54.167 

 21 F27 0.85930E-02 56.944 

 22 F36 0.12630E-01 59.722 

 23 F31 0.16606E-01 62.500 

 24 F4 0.20888E-01 65.278 

 25 F10 0.25444E-01 68.056 

 26 F18 0.30108E-01 70.833 

 27 F1 0.34861E-01 73.611 

 28 F2 0.39605E-01 76.389 

 29 F35 0.44706E-01 79.167 

 30 F9 0.50015E-01 81.944 

 31 F8 0.56072E-01 84.722 

 32 F3 0.62882E-01 87.500 

 33 F33 0.70577E-01 90.278 

 34 F7 0.80583E-01 93.056 

 35 F32 0.92808E-01 95.833 

 36 F34 0.10747 98.611 

 STD. DVS. OF ORDERED, SIMULATED WEIGHTED RESIDUALS 

 OBSERVATION 

 NO. NAME D FRQ. (%) 

 1 F21 0.37500E-01 1.3889 

 2 F20 0.33337E-01 4.1667 

 3 F23 0.30846E-01 6.9444 

 4 F22 0.27941E-01 9.7222 

 5 F24 0.25728E-01 12.500 

 6 F6 0.24604E-01 15.278 

 7 F5 0.23011E-01 18.056 

 8 F12 0.22052E-01 20.833 

 9 F19 0.21242E-01 23.611 

 10 F11 0.20091E-01 26.389 

 11 F25 0.19207E-01 29.167 

 12 F16 0.17644E-01 31.944 

 13 F28 0.16257E-01 34.722 

 14 F15 0.15356E-01 37.500 

 15 F30 0.14826E-01 40.278 

 16 F26 0.14434E-01 43.056 

 17 F13 0.14270E-01 45.833 

 18 F17 0.14161E-01 48.611 

 19 F29 0.14106E-01 51.389 

 20 F14 0.14000E-01 54.167 

 21 F27 0.14133E-01 56.944 

 22 F36 0.14548E-01 59.722 

 23 F31 0.14758E-01 62.500 

 24 F4 0.14946E-01 65.278 

 25 F10 0.15909E-01 68.056 

 26 F18 0.16967E-01 70.833 

 27 F1 0.17980E-01 73.611 

 28 F2 0.19061E-01 76.389 

 29 F35 0.20546E-01 79.167 

 30 F9 0.21745E-01 81.944 

 31 F8 0.23228E-01 84.722 

 32 F3 0.24758E-01 87.500 

153

 34 F7 0.30245E-01 93.056 

 33 F33 0.26490E-01 90.278 
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Output Files 

 35 F32 0.34256E-01 95.833 

 36 F34 0.37263E-01 98.611 

CORRELATION (CED) --------------- = 0.99559 

PROBABILITY OF CORRELATION (PROB) = 0.98800 

99% CONFIDENCE LIMIT (CL99) ----- = 0.99571 

95% CONFIDENCE LIMIT (CL95) ----- = 0.99380 

90% CONFIDENCE LIMIT (CL90) ----- = 0.99229 

_md file (example._md): 

 "ORD.WEIGHT.RES." "ORD.SIM.WT.RES." "STDVS.O.S.W.RS." "2*STDV.O.S.W.R." "FREQUENCY/PROB." 

"PROB.PLOT.POSI." "OBSERVATION" "PLOT-SYMBOL" 

 -0.37562E-01 -0.10938 0.37500E-01 0.75001E-01 1.3889 

-2.2003 F21 1 

 -0.33362E-01 -0.92706E-01 0.33337E-01 0.66674E-01 4.1667 

-1.7322 F20 1 

 -0.32529E-01 -0.81207E-01 0.30846E-01 0.61693E-01 6.9444 

-1.4804 F23 1 

 -0.26743E-01 -0.71524E-01 0.27941E-01 0.55881E-01 9.7222 

-1.2976 F22 1 

 -0.22647E-01 -0.63615E-01 0.25728E-01 0.51455E-01 12.500 

-1.1505 F24 1 

 -0.18390E-01 -0.57005E-01 0.24604E-01 0.49208E-01 15.278 

-1.0251 F6 1 

 -0.17909E-01 -0.50915E-01 0.23011E-01 0.46023E-01 18.056 

-0.91363 F5 1 

 -0.17336E-01 -0.45545E-01 0.22052E-01 0.44103E-01 20.833 

-0.81256 F12 1 

 -0.15027E-01 -0.40679E-01 0.21242E-01 0.42484E-01 23.611 

-0.71912 F19 1 

 -0.13359E-01 -0.35812E-01 0.20091E-01 0.40183E-01 26.389 

-0.63155 F11 1 

 -0.11397E-01 -0.31282E-01 0.19207E-01 0.38414E-01 29.167 

-0.54865 F25 1 

 -0.10687E-01 -0.26841E-01 0.17644E-01 0.35289E-01 31.944 

-0.46943 F16 1 

 -0.96915E-02 -0.22280E-01 0.16257E-01 0.32514E-01 34.722 

-0.39295 F28 1 

 -0.88227E-02 -0.18282E-01 0.15356E-01 0.30712E-01 37.500 

-0.31878 F15 1 

 -0.84145E-02 -0.14528E-01 0.14826E-01 0.29652E-01 40.278 

-0.24619 F30 1 

 -0.49643E-02 -0.10723E-01 0.14434E-01 0.28868E-01 43.056 

-0.17499 F26 1 

 -0.34969E-02 -0.67828E-02 0.14270E-01 0.28540E-01 45.833 

-0.10487 F13 1 

 -0.28905E-02 -0.30485E-02 0.14161E-01 0.28322E-01 48.611 

-0.34955E-01 F17 1 

 -0.15173E-02 0.91987E-03 0.14106E-01 0.28213E-01 51.389 

0.34955E-01 F29 1 

 -0.14043E-02 0.45571E-02 0.14000E-01 0.28000E-01 54.167 

0.10487 F14 1 

154
APPENDIX C. EXAMPLE SIMULATIONS – RESAN2-2k Output Files 

 0.14077E-02 0.85930E-02 0.14133E-01 0.28266E-01 56.944 

0.17499 F27 1 

 0.15258E-02 0.12630E-01 0.14548E-01 0.29096E-01 59.722 

0.24619 F36 1 

 0.54087E-02 0.16606E-01 0.14758E-01 0.29516E-01 62.500 

0.31878 F31 1 

 0.55489E-02 0.20888E-01 0.14946E-01 0.29892E-01 65.278 

0.39295 F4 1 

 0.92539E-02 0.25444E-01 0.15909E-01 0.31818E-01 68.056 

0.46943 F10 1 

 0.99690E-02 0.30108E-01 0.16967E-01 0.33933E-01 70.833 

0.54865 F18 1 

 0.12221E-01 0.34861E-01 0.17980E-01 0.35960E-01 73.611 

0.63155 F1 1 

 0.13438E-01 0.39605E-01 0.19061E-01 0.38123E-01 76.389 

0.71912 F2 1 

 0.15053E-01 0.44706E-01 0.20546E-01 0.41093E-01 79.167 

0.81256 F35 1 

 0.19232E-01 0.50015E-01 0.21745E-01 0.43490E-01 81.944 

0.91363 F9 1 

 0.24144E-01 0.56072E-01 0.23228E-01 0.46457E-01 84.722 

1.0251 F8 1 

 0.24247E-01 0.62882E-01 0.24758E-01 0.49517E-01 87.500 

1.1505 F3 1 

 0.25136E-01 0.70577E-01 0.26490E-01 0.52980E-01 90.278 

1.2976 F33 1 

 0.32461E-01 0.80583E-01 0.30245E-01 0.60491E-01 93.056 

1.4804 F7 1 

 0.38440E-01 0.92808E-01 0.34256E-01 0.68513E-01 95.833 

1.7322 F32 1 

 0.42655E-01 0.10747 0.37263E-01 0.74526E-01 98.611 

2.2003 F34 1

155
APPENDIX C. EXAMPLE SIMULATIONS – MODFLOW-2000 Input Files For Confidence Interval 

Calculation 

MODFLOW-2000 Input Files For Confidence Interval Calculation 

Name File (example.nam): 

GLOBAL 4 mf2k.global_listing 

LIST 3 output.dat 

DIS 95 discret.dat 

BAS6 1 bas6.dat 

ZONE 93 zone.dat 

MULT 94 multiple.dat 

LPF 31 lpf6.dat 

RCH 11 rech6.dat 

WEL 12 wel6.dat 

PCG 23 pcg2.dat 

OC 22 oc.dat 

DATA 10 inithead.dat 

OBS 40 obs.dat 

HOB 41 hob.dat 

PES 47 pes.dat 

SEN 46 sen.dat 

UNC 60 unc.dat 

DATAGLO 61 confint.dat 

DATAGLO 62 weight-res.dat

UNC File (unc.dat): 

 0 2 0 00 1 0 61 1 1 0 62 0 

.001 .0 .0 

 .0919 .0413 .0941 .0916 .0916 .0923 .0916 .0323 

HK_1 .0 

HK_2 .0

SEN File (sen.dat): 

# SEN file for MODFLOW-2000, UNC1NLI1 Report Example 

# 

 2 0 0 2 Item 1 

 1 0 0 0 Item 2 

HK_1 1 1 1.46378 .01 100 1E-10 Item 3 

HK_2 1 1 .099433 .01 100 1E-10 Item 3

HOB File (hob.dat): 

# HOBS file for MODFLOW-2000, UNC1NLI1 Report Example 

 42 0 0 6 

 1 1 

F1 1 37 13 1 0 0 0 14.88 729.28 0 1 

F2 1 30 13 1 0 0 0 14.78 690.68 0 1 

F3 1 25 13 1 0 0 0 14.86 668.02 0 1 

156
APPENDIX C. EXAMPLE SIMULATIONS – MODFLOW-2000 Input Files For Confidence Interval 

Calculation 

F4 1 20 13 1 0 0 0 13.87 699.35 0 1 

F5 1 14 13 1 0 0 0 13.01 728.59 0 1 

F6 1 8 13 1 0 0 0 12.97 746.80 0 1 

F7 1 37 26 1 0 0 0 5.61 643.27 0 1 

F8 1 30 26 1 0 0 0 5.63 666.51 0 1 

F9 1 25 26 1 0 0 0 5.94 687.81 0 1 

F10 1 20 26 1 0 0 0 8.83 721.28 0 1 

F11 1 14 26 1 0 0 0 11.08 717.66 0 1 

F12 1 8 26 1 0 0 0 11.35 731.50 0 1 

F13 1 37 39 1 0 0 0 3.7 640.67 0 1 

F14 1 30 39 1 0 0 0 3.75 648.28 0 1 

F15 1 25 39 1 0 0 0 3.52 619.12 0 1 

F16 1 20 39 1 0 0 0 3.35 607.80 0 1 

F17 1 14 39 1 0 0 0 3.42 638.66 0 1 

F18 1 8 39 1 0 0 0 3.71 629.47 0 1 

F19 1 37 52 1 0 0 0 2.37 548.04 0 1 

F20 1 30 52 1 0 0 0 1.91 531.05 0 1 

F21 1 25 52 1 0 0 0 1.76 526.49 0 1 

F22 1 20 52 1 0 0 0 1.96 531.33 0 1 

F23 1 14 52 1 0 0 0 1.82 546.39 0 1 

F24 1 8 52 1 0 0 0 2.06 542.63 0 1 

F25 1 37 65 1 0 0 0 1.54 423.83 0 1 

F26 1 30 65 1 0 0 0 1.66 427.94 0 1 

F27 1 25 65 1 0 0 0 1.78 429.22 0 1 

F28 1 20 65 1 0 0 0 1.54 426.12 0 1 

F29 1 14 65 1 0 0 0 1.7 432.14 0 1 

F30 1 8 65 1 0 0 0 1.55 444.93 0 1 

F31 1 37 78 1 0 0 0 .94 153.12 0 1 

F32 1 30 78 1 0 0 0 1.41 197.28 0 1 

F33 1 25 78 1 0 0 0 1.22 197.22 0 1 

F34 1 20 78 1 0 0 0 1.47 202.06 0 1 

F35 1 14 78 1 0 0 0 1.08 212.53 0 1 

F36 1 8 78 1 0 0 0 .88 202.26 0 1 

G1 1 22 46 1 0 0 0 0 1037.4 0 1 

G2 1 21 20 1 0 0 0 0 723.55 0 1 

G3 1 21 33 1 0 0 0 0 620.74 0 1 

G4 1 21 59 1 0 0 0 0 488.07 0 1 

G5 1 21 72 1 0 0 0 0 329.80 0 1 

G6 1 40 46 1 0 0 0 0 597.63 0 1

157
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

GLOBAL Output File For Confidence Interval Calculation 

GLOBAL Output File (mf2k.global_listing): 

 U.S. GEOLOGICAL SURVEY MODULAR FINITE-DIFFERENCE GROUND-WATER FLOW MODEL 

 VERSION 1.6SC1  ̈25/02/2003 

 This model run produced both GLOBAL and LIST files. This is the GLOBAL file. 

 GLOBAL LISTING FILE: mf2k.global_listing 

 UNIT 4 

 OPENING output.dat 

 FILE TYPE:LIST UNIT 3 STATUS:REPLACE 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING discret.dat 

 FILE TYPE:DIS UNIT 95 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING bas6.dat 

 FILE TYPE:BAS6 UNIT 1 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING zone.dat 

 FILE TYPE:ZONE UNIT 93 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING multiple.dat 

 FILE TYPE:MULT UNIT 94 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING lpf6.dat 

 FILE TYPE:LPF UNIT 31 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING rech6.dat 

 FILE TYPE:RCH UNIT 11 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wel6.dat 

 FILE TYPE:WEL UNIT 12 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING pcg2.dat 

 FILE TYPE:PCG UNIT 23 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING oc.dat 

 FILE TYPE:OC UNIT 22 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

158
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 OPENING inithead.dat 

 FILE TYPE:DATA UNIT 10 STATUS:UNKNOWN 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING obs.dat 

 FILE TYPE:OBS UNIT 40 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING hob.dat 

 FILE TYPE:HOB UNIT 41 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING pes.dat 

 FILE TYPE:PES UNIT 47 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING sen.dat 

 FILE TYPE:SEN UNIT 46 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING unc.dat 

 FILE TYPE:UNC UNIT 60 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING confint.dat 

 FILE TYPE:DATAGLO UNIT 61 STATUS:UNKNOWN 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING weight-res.dat 

 FILE TYPE:DATAGLO UNIT 62 STATUS:UNKNOWN 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 THE FREE FORMAT OPTION HAS BEEN SELECTED 

 DISCRETIZATION INPUT DATA READ FROM UNIT 95 

 # Discretization file for MODFLOW-2000, UNC1NLI1 Report Example 

 1 LAYERS 41 ROWS 91 COLUMNS 

 1 STRESS PERIOD(S) IN SIMULATION 

 MODEL TIME UNIT IS SECONDS 

 MODEL LENGTH UNIT IS UNDEFINED 

 THE GROUND-WATER TRANSPORT PROCESS IS INACTIVE 

 THE OBSERVATION PROCESS IS ACTIVE 

 THE SENSITIVITY PROCESS IS ACTIVE 

 THE PARAMETER-ESTIMATION PROCESS IS ACTIVE 

 MODE: PARAMETER ESTIMATION 

 ZONE OPTION, INPUT READ FROM UNIT 93 

 # Zone file for MODFLOW-2000, UNC1NLI1 Report Example 

 2 ZONE ARRAYS 

 MULTIPLIER OPTION, INPUT READ FROM UNIT 94 

 # Multiplier file for MODFLOW-2000, UNC1NLI1 Report Example 

 2 MULTIPLIER ARRAYS 

159

 Confining bed flag for each layer: 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 0 

 41173 ELEMENTS OF GX ARRAY USED OUT OF 41173 

 3731 ELEMENTS OF GZ ARRAY USED OUT OF 3731 

 11193 ELEMENTS OF IG ARRAY USED OUT OF 11193 

 DELR 

 READING ON UNIT 95 WITH FORMAT: (16G5.0) 

 DELC 

 READING ON UNIT 95 WITH FORMAT: (16G5.0) 

 TOP ELEVATION OF LAYER 1 = 1.00000 

 MODEL LAYER BOTTOM EL. = 0.00000 FOR LAYER 1 

 STRESS PERIOD LENGTH TIME STEPS MULTIPLIER FOR DELT SS FLAG 

 ---------------------------------------------------------------------------- 

 1 1.000000 1 1.000 SS 

 STEADY-STATE SIMULATION 

 MULT. ARRAY: HK1 = 1.00000 

 MULT. ARRAY: VK1 = 1.00000 

 ZONE ARRAY: ZHK1 

 READING ON UNIT 93 WITH FORMAT: (91I1) 

 ZONE ARRAY: ZVK1 = 999 

 LPF1 -- LAYER PROPERTY FLOW PACKAGE, VERSION 1, 1/11/2000 

 INPUT READ FROM UNIT 31 

 # Layer-Property Flow file for MODFLOW-2000, UNC1NLI1 Report Example 

 HEAD AT CELLS THAT CONVERT TO DRY= 999.00 

 3 Named Parameters 

 LAYER FLAGS: 

 LAYER LAYTYP LAYAVG CHANI LAYVKA LAYWET 

 --------------------------------------------------------------------------- 

 1 0 0 1.000E+00 0 0 

 INTERPRETATION OF LAYER FLAGS: 

 INTERBLOCK HORIZONTAL DATA IN 

 LAYER TYPE TRANSMISSIVITY ANISOTROPY ARRAY VKA WETTABILITY 

160

 LAYER (LAYTYP) (LAYAVG) (CHANI) (LAYVKA) (LAYWET) 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 --------------------------------------------------------------------------- 

 1 CONFINED HARMONIC 1.000E+00 VERTICAL K NON-WETTABLE 

 7462 ELEMENTS IN X ARRAY ARE USED BY LPF 

 6 ELEMENTS IN IX ARRAY ARE USED BY LPF 

 PCG2 -- CONJUGATE GRADIENT SOLUTION PACKAGE, VERSION 2.4, 12/29/98 

 MAXIMUM OF 50 CALLS OF SOLUTION ROUTINE 

 MAXIMUM OF 30 INTERNAL ITERATIONS PER CALL TO SOLUTION ROUTINE 

 MATRIX PRECONDITIONING TYPE : 1 

 14193 ELEMENTS IN X ARRAY ARE USED BY PCG 

 10500 ELEMENTS IN IX ARRAY ARE USED BY PCG 

 7462 ELEMENTS IN Z ARRAY ARE USED BY PCG 

 SEN1BAS6 -- SENSITIVITY PROCESS, VERSION 1.0, 10/15/98 

 INPUT READ FROM UNIT 46 

 # SEN file for MODFLOW-2000, UNC1NLI1 Report Example 

 # 

 NUMBER OF PARAMETER VALUES TO BE READ FROM SEN FILE: 2 

 ISENALL............................................: 0 

 SENSITIVITIES WILL BE STORED IN MEMORY 

 FOR UP TO 2 PARAMETERS 

 3745 ELEMENTS IN X ARRAY ARE USED FOR SENSITIVITIES 

 3731 ELEMENTS IN Z ARRAY ARE USED FOR SENSITIVITIES 

 4 ELEMENTS IN IX ARRAY ARE USED FOR SENSITIVITIES 

 PES1BAS6 -- PARAMETER-ESTIMATION PROCESS, VERSION 1.0, 07/22/99 

 INPUT READ FROM UNIT 47 

 # PES file for MODFLOW-2000, UNC1NLI1 Report Example 

 # 

 MAXIMUM NUMBER OF PARAMETER-ESTIMATION ITERATIONS (MAX-ITER) = 50 

 MAXIMUM PARAMETER CORRECTION (MAX-CHANGE) ------------------- = 2.0000 

 CLOSURE CRITERION (TOL) ------------------------------------- = 0.10000E-01 

 SUM OF SQUARES CLOSURE CRITERION (SOSC) --------------------- = 0.0000 

 FLAG TO GENERATE INPUT NEEDED BY BEALE-2000 (IBEFLG) -------- = 0 

 FLAG TO GENERATE INPUT NEEDED BY YCINT-2000 (IYCFLG) -------- = 0 

 OMIT PRINTING TO SCREEN (IF = 1) (IOSTAR) ------------------- = 0 

 ADJUST GAUSS-NEWTON MATRIX WITH NEWTON UPDATES (IF = 1)(NOPT) = 0 

 NUMBER OF FLETCHER-REEVES ITERATIONS (NFIT) ----------------- = 0 

 CRITERION FOR ADDING MATRIX R (SOSR) ------------------------ = 0.0000 

 VALUE USED TO INCREMENT MARQUARDT PARAMETER (RMAR) ---------- = 0.10000E-02 

 MARQUARDT PARAMETER MULTIPLIER (RMARM) ---------------------- = 1.5000 

 APPLY MAX-CHANGE IN REGRESSION SPACE (IF = 1) (IAP) --------- = 0 

 FORMAT CODE FOR COVARIANCE AND CORRELATION MATRICES (IPRCOV) = 1 

 PRINT PARAMETER-ESTIMATION STATISTICS 

 EACH ITERATION (IF > 0) (IPRINT) ----------------------- = 0 

 PRINT EIGENVALUES AND EIGENVECTORS OF 

 COVARIANCE MATRIX (IF > 0) (LPRINT) -------------------- = 0 

 SEARCH DIRECTION ADJUSTMENT PARAMETER (CSA) ----------------- = 0.80000E-01 

161

 MODIFY CONVERGENCE CRITERIA (IF > 0) (FCONV) ---------------- = 0.0000 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 CALCULATE SENSITIVITIES USING FINAL 

 PARAMETER ESTIMATES (IF > 0) (LASTX) -------------------- = 1 

 NUMBER OF USUALLY POS. PARAMETERS THAT MAY BE NEGATIVE (NPNG) = 0 

 NUMBER OF PARAMETERS WITH CORRELATED PRIOR INFORMATION (IPR) = 0 

 NUMBER OF PRIOR-INFORMATION EQUATIONS (MPR) ----------------- = 0 

 479 ELEMENTS IN X ARRAY ARE USED FOR PARAMETER ESTIMATION 

 81 ELEMENTS IN Z ARRAY ARE USED FOR PARAMETER ESTIMATION 

 52 ELEMENTS IN IX ARRAY ARE USED FOR PARAMETER ESTIMATION 

 OBS1BAS6 -- OBSERVATION PROCESS, VERSION 1.0, 4/27/99 

 INPUT READ FROM UNIT 40 

 # OBS file for MODFLOW-2000, UNC1NLI1 Report Example 

 OBSERVATION GRAPH-DATA OUTPUT FILES 

 WILL BE PRINTED AND NAMED USING THE BASE: Example 

 DIMENSIONLESS SCALED OBSERVATION SENSITIVITIES WILL BE PRINTED 

 HEAD OBSERVATIONS -- INPUT READ FROM UNIT 41 

 # HOBS file for MODFLOW-2000, UNC1NLI1 Report Example 

 NUMBER OF HEADS....................................: 42 

 NUMBER OF MULTILAYER HEADS.......................: 0 

 MAXIMUM NUMBER OF LAYERS FOR MULTILAYER HEADS....: 0 

 NUMBER OF HEADS USED FOR INTERVAL CALCULATION......: 6 

 826 ELEMENTS IN X ARRAY ARE USED FOR OBSERVATIONS 

 2 ELEMENTS IN Z ARRAY ARE USED FOR OBSERVATIONS 

 423 ELEMENTS IN IX ARRAY ARE USED FOR OBSERVATIONS 

 COMMON ERROR VARIANCE FOR ALL OBSERVATIONS SET TO: 1.000 

 UNC1NLI1 -- UNCERTAINTY-ESTIMATION PROCESS, VERSION 1.0, 03/08/02 

 INPUT READ FROM UNIT 60 

 IACT=0: COMPUTE CONFIDENCE OR PREDICTION INTERVALS 

 NUMBER OF HEAD INTERVALS...............................: 6 

 NUMBER OF PARAMETER INTERVALS..........................: 2 

 TOTAL NUMBER OF INTERVALS..............................: 8 

 HEAD AND FLOW INTERVALS ON DIFFERENCES (0,1)=(NO,YES)..: 0 

 INTERVAL LIMITS? (1=UPPER,-1=LOWER,O=UPPER & LOWER)....: 0 

 INTERVAL TYPE (1=CONFIDENCE,2=PREDICTION)..............: 1 

 NUMBER OF INITIAL PARAMETER SETS (0=USE DATASET 8).....: 0 

 RESULT SUMMARY IS WRITTEN TO UNIT NO (1=NOT WRITTEN)...: 61 

 CRITICAL VALUE CONST./VAR. (IDSQ=0/1)..................: 1 

 PRINT ITERATION LOG (1=YES,ELSE=NO)....................: 1 

 USE APPROXIMATE INCREMENTAL ADJUSTMENT? (1=YES,0=NO)...: 0 

 WEIGHTED RESIDUALS ARE WRITTEN TO UNIT NO (O=NOT WRIT.): 62 

 SCALING VALUES BS (1=READ VALUES, 0=COPY START. VAL.)..: 0 

 TOLP (CONVERGENCE CRITERIA FOR PARAMETER CHANGE).......: 0.100000E-02 

 TOLS (CONV. CRIT. FOR CHANGE OF OBJECTIVE FUNCTION)....: 0.00000 

 TOLY (CONV. CRIT. FOR CHANGE OF COMPUTED LIMIT VALUE)..: 0.00000 

 24 ELEMENTS IN X ARRAY ARE USED FOR PARAMETER ESTIMATION 

 6 ELEMENTS IN Z ARRAY ARE USED FOR PARAMETER ESTIMATION 

 48 ELEMENTS IN IX ARRAY ARE USED FOR PARAMETER ESTIMATION 

162
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 26729 ELEMENTS OF X ARRAY USED OUT OF 26729 

 11282 ELEMENTS OF Z ARRAY USED OUT OF 11282 

 11033 ELEMENTS OF IX ARRAY USED OUT OF 11033 

 7462 ELEMENTS OF XHS ARRAY USED OUT OF 7462 

 INFORMATION ON PARAMETERS LISTED IN SEN FILE 

 LOWER UPPER ALTERNATE 

 VALUE IN SEN REASONABLE REASONABLE SCALING 

 NAME ISENS LN INPUT FILE LIMIT LIMIT FACTOR 

 ---------- ----- -- ------------ ------------ ------------ ------------ 

 HK_1 1 1 1.4638 0.10000E-01 100.00 0.10000E-09 

 HK_2 1 1 0.99433E-01 0.10000E-01 100.00 0.10000E-09 

 ----------------------------------------------------------------------------- 

 FOR THE PARAMETERS LISTED IN THE TABLE ABOVE, PARAMETER VALUES IN INDIVIDUAL 

 PACKAGE INPUT FILES ARE REPLACED BY THE VALUES FROM THE SEN INPUT FILE. THE 

 ALTERNATE SCALING FACTOR IS USED TO SCALE SENSITIVITIES IF IT IS LARGER THAN 

 THE PARAMETER VALUE IN ABSOLUTE VALUE AND THE PARAMETER IS NOT LOG-TRANSFORMED. 

 ISENS IS GREATER THAN ZERO FOR 2 PARAMETERS 

 HEAD OBSERVATION VARIANCES ARE MULTIPLIED BY: 1.000 

 OBSERVED HEAD DATA -- TIME OFFSETS ARE MULTIPLIED BY: 1.0000 

 REFER. 

 OBSERVATION STRESS TIME STATISTIC PLOT 

 OBS# NAME PERIOD OFFSET OBSERVATION STATISTIC TYPE SYM. 

 1 F1 1 0.000 14.88 729.3 VARIANCE 1 

 2 F2 1 0.000 14.78 690.7 VARIANCE 1 

 3 F3 1 0.000 14.86 668.0 VARIANCE 1 

 4 F4 1 0.000 13.87 699.3 VARIANCE 1 

 5 F5 1 0.000 13.01 728.6 VARIANCE 1 

 6 F6 1 0.000 12.97 746.8 VARIANCE 1 

 7 F7 1 0.000 5.610 643.3 VARIANCE 1 

 8 F8 1 0.000 5.630 666.5 VARIANCE 1 

 9 F9 1 0.000 5.940 687.8 VARIANCE 1 

 10 F10 1 0.000 8.830 721.3 VARIANCE 1 

 11 F11 1 0.000 11.08 717.7 VARIANCE 1 

 12 F12 1 0.000 11.35 731.5 VARIANCE 1 

 13 F13 1 0.000 3.700 640.7 VARIANCE 1 

 14 F14 1 0.000 3.750 648.3 VARIANCE 1 

 15 F15 1 0.000 3.520 619.1 VARIANCE 1 

 16 F16 1 0.000 3.350 607.8 VARIANCE 1 

 17 F17 1 0.000 3.420 638.7 VARIANCE 1 

 18 F18 1 0.000 3.710 629.5 VARIANCE 1 

 19 F19 1 0.000 2.370 548.0 VARIANCE 1 

 20 F20 1 0.000 1.910 531.0 VARIANCE 1 

 21 F21 1 0.000 1.760 526.5 VARIANCE 1 

 22 F22 1 0.000 1.960 531.3 VARIANCE 1 

 23 F23 1 0.000 1.820 546.4 VARIANCE 1 

 24 F24 1 0.000 2.060 542.6 VARIANCE 1 

 25 F25 1 0.000 1.540 423.8 VARIANCE 1 

 26 F26 1 0.000 1.660 427.9 VARIANCE 1 

 27 F27 1 0.000 1.780 429.2 VARIANCE 1 

 28 F28 1 0.000 1.540 426.1 VARIANCE 1 

163

 29 F29 1 0.000 1.700 432.1 VARIANCE 1 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 30 F30 1 0.000 1.550 444.9 VARIANCE 1 

 31 F31 1 0.000 0.9400 153.1 VARIANCE 1 

 32 F32 1 0.000 1.410 197.3 VARIANCE 1 

 33 F33 1 0.000 1.220 197.2 VARIANCE 1 

 34 F34 1 0.000 1.470 202.1 VARIANCE 1 

 35 F35 1 0.000 1.080 212.5 VARIANCE 1 

 36 F36 1 0.000 0.8800 202.3 VARIANCE 1 

 37 G1 1 0.000 0.000 1037. VARIANCE 1 

 38 G2 1 0.000 0.000 723.5 VARIANCE 1 

 39 G3 1 0.000 0.000 620.7 VARIANCE 1 

 40 G4 1 0.000 0.000 488.1 VARIANCE 1 

 41 G5 1 0.000 0.000 329.8 VARIANCE 1 

 42 G6 1 0.000 0.000 597.6 VARIANCE 1 

 HEAD CHANGE 

 REFERENCE 

 OBSERVATION ROW COL OBSERVATION 

 OBS# NAME LAY ROW COL OFFSET OFFSET (IF > 0) 

 1 F1 1 37 13 0.000 0.000 0 

 2 F2 1 30 13 0.000 0.000 0 

 3 F3 1 25 13 0.000 0.000 0 

 4 F4 1 20 13 0.000 0.000 0 

 5 F5 1 14 13 0.000 0.000 0 

 6 F6 1 8 13 0.000 0.000 0 

 7 F7 1 37 26 0.000 0.000 0 

 8 F8 1 30 26 0.000 0.000 0 

 9 F9 1 25 26 0.000 0.000 0 

 10 F10 1 20 26 0.000 0.000 0 

 11 F11 1 14 26 0.000 0.000 0 

 12 F12 1 8 26 0.000 0.000 0 

 13 F13 1 37 39 0.000 0.000 0 

 14 F14 1 30 39 0.000 0.000 0 

 15 F15 1 25 39 0.000 0.000 0 

 16 F16 1 20 39 0.000 0.000 0 

 17 F17 1 14 39 0.000 0.000 0 

 18 F18 1 8 39 0.000 0.000 0 

 19 F19 1 37 52 0.000 0.000 0 

 20 F20 1 30 52 0.000 0.000 0 

 21 F21 1 25 52 0.000 0.000 0 

 22 F22 1 20 52 0.000 0.000 0 

 23 F23 1 14 52 0.000 0.000 0 

 24 F24 1 8 52 0.000 0.000 0 

 25 F25 1 37 65 0.000 0.000 0 

 26 F26 1 30 65 0.000 0.000 0 

 27 F27 1 25 65 0.000 0.000 0 

 28 F28 1 20 65 0.000 0.000 0 

 29 F29 1 14 65 0.000 0.000 0 

 30 F30 1 8 65 0.000 0.000 0 

 31 F31 1 37 78 0.000 0.000 0 

 32 F32 1 30 78 0.000 0.000 0 

 33 F33 1 25 78 0.000 0.000 0 

 34 F34 1 20 78 0.000 0.000 0 

 35 F35 1 14 78 0.000 0.000 0 

 36 F36 1 8 78 0.000 0.000 0 

 37 G1 1 22 46 0.000 0.000 0 

164

 38 G2 1 21 20 0.000 0.000 0 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 39 G3 1 21 33 0.000 0.000 0 

 40 G4 1 21 59 0.000 0.000 0 

 41 G5 1 21 72 0.000 0.000 0 

 42 G6 1 40 46 0.000 0.000 0 

 CRITICAL VALUE (DSQ) FOR INTERVALS 

 0.0919 0.0413 0.0941 0.0916 0.0916 0.0923 0.0916 0.0323 

 CONFIDENCE INTERVAL TO BE CALCULATED FOR THE FOLLOWING PARAMETERS 

 PAR. NO 

 HK_1 

 HK_2 

 SCALING VALUES FOR DAMPING PARAMETER CHANGES: 

 1.4638 0.99433E-01 

 SOLUTION BY THE CONJUGATE-GRADIENT METHOD 

 ------------------------------------------- 

 MAXIMUM NUMBER OF CALLS TO PCG ROUTINE = 50 

 MAXIMUM ITERATIONS PER CALL TO PCG = 30 

 MATRIX PRECONDITIONING TYPE = 1 

 RELAXATION FACTOR (ONLY USED WITH PRECOND. TYPE 1) = 0.10000E+01 

 PARAMETER OF POLYNOMIAL PRECOND. = 2 (2) OR IS CALCULATED : 1 

 HEAD CHANGE CRITERION FOR CLOSURE = 0.10000E-02 

 RESIDUAL CHANGE CRITERION FOR CLOSURE = 0.10000E-02 

 PCG HEAD AND RESIDUAL CHANGE PRINTOUT INTERVAL = 1 

 PRINTING FROM SOLVER IS LIMITED(1) OR SUPPRESSED (>1) = 0 

 DAMPING PARAMETER = 0.10000E+01 

 CONVERGENCE CRITERIA FOR SENSITIVITIES 

 PARAMETER HCLOSE RCLOSE 

 ---------- ------------ ------------ 

 HK_1 0.68316E-05 0.68316E-05 

 HK_2 0.10057E-03 0.10057E-03 

 -------------------------------------- 

 WETTING CAPABILITY IS NOT ACTIVE IN ANY LAYER 

 PARAMETERS DEFINED IN THE LPF PACKAGE 

 PARAMETER NAME:HK_1 TYPE:HK CLUSTERS: 1 

 Parameter value from package file is: 1.0000 

 This value has been changed to: 1.4638 , as read from 

 the Sensitivity Process file 

 LAYER: 1 MULTIPLIER ARRAY: HK1 ZONE ARRAY: ZHK1 

 ZONE VALUES: 1 

 PARAMETER NAME:HK_2 TYPE:HK CLUSTERS: 1 

 Parameter value from package file is: 1.0000 

 This value has been changed to: 9.94330E-02, as read from 

 the Sensitivity Process file 

 LAYER: 1 MULTIPLIER ARRAY: HK1 ZONE ARRAY: ZHK1 

165

 ZONE VALUES: 2 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 PARAMETER NAME:VK_0 TYPE:VK CLUSTERS: 1 

 Parameter value from package file is: 1.0000 

 LAYER: 1 MULTIPLIER ARRAY: VK1 ZONE ARRAY: ZVK1 

 ZONE VALUES: 999 

 HYD. COND. ALONG ROWS FOR LAYER 1 WILL BE DEFINED BY PARAMETERS 

 (PRINT FLAG= 31) 

 VERTICAL HYD. COND. FOR LAYER 1 WILL BE DEFINED BY PARAMETERS 

 (PRINT FLAG= 31) 

 ******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. 1 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G1 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

 WELL NO. LAYER ROW COL STRESS FACTOR 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL 1: FL= 4.1361 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. 2 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G2 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

166

 WELL NO. LAYER ROW COL STRESS FACTOR 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL 2: FL= 13.439 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. 3 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G3 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

 WELL NO. LAYER ROW COL STRESS FACTOR 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL 3: FL= 6.4088 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. 4 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G4 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

167

 WELL NO. LAYER ROW COL STRESS FACTOR 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL 4: FL= 3.1090 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. 5 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G5 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

 WELL NO. LAYER ROW COL STRESS FACTOR 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL 5: FL= 1.8458 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. 6 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G6 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

168

 WELL NO. LAYER ROW COL STRESS FACTOR 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL 6: FL= 4.6167 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. 7 

 STARTING PARAMETER SET NO. 1 

 PARNAM: HK_1 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

 WELL NO. LAYER ROW COL STRESS FACTOR 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL 7: FL= 0.42100 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. 8 

 STARTING PARAMETER SET NO. 1 

 PARNAM: HK_2 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

169

 WELL NO. LAYER ROW COL STRESS FACTOR 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL 8: FL= -0.87902 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. -1 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G1 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

 WELL NO. LAYER ROW COL STRESS FACTOR 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL -1: FL= 1.5895 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. -2 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G2 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

170

 WELL NO. LAYER ROW COL STRESS FACTOR 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL -2: FL= 10.581 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. -3 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G3 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

 WELL NO. LAYER ROW COL STRESS FACTOR 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL -3: FL= 2.5599 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. -4 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G4 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

171

 WELL NO. LAYER ROW COL STRESS FACTOR 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL -4: FL= 1.1892 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. -5 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G5 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

 WELL NO. LAYER ROW COL STRESS FACTOR 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL -5: FL= 0.70517 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. -6 

 STARTING PARAMETER SET NO. 1 

 OBSNAM: G6 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

172

 WELL NO. LAYER ROW COL STRESS FACTOR 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL -6: FL= 1.7864 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. -7 

 STARTING PARAMETER SET NO. 1 

 PARNAM: HK_1 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

 WELL NO. LAYER ROW COL STRESS FACTOR 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL -7: FL= 0.32539E-02 

******************************************************************************** 

 CALCULATION OF CONFIDENCE INTERVAL NO. -8 

 STARTING PARAMETER SET NO. 1 

 PARNAM: HK_2 

 1 Well parameters 

 PARAMETER NAME:WEL_0_1 TYPE:Q 

 Parameter value from package file is: 1.0000 

 NUMBER OF ENTRIES: 1 

173

 WELL NO. LAYER ROW COL STRESS FACTOR 
APPENDIX C. EXAMPLE SIMULATIONS – GLOBAL Output File For Confidence Interval Calculation 

 ---------------------------------------------- 

 1 1 21 46 -1.000 

 0 Recharge parameters 

 4 PARAMETERS HAVE BEEN DEFINED IN ALL PACKAGES. 

 (SPACE IS ALLOCATED FOR 500 PARAMETERS.) 

 C.I. LIMIT FOR INTERVAL -8: FL= -1.1052 

******************************************************************************** 

 PROCESS STARTED: 20040305 124557.200 

 PROCESS STOPPED: 20040305 124657.840

174
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

Summary And Weighted Residuals Output Files 

Summary Output File (confint.dat): 

C.I. 1: FL= 4.1361 ; SSE=0.91894E-01; ITER= 4; PAR: 1.007 0.1538 

C.I. 2: FL= 13.439 ; SSE=0.41351E-01; ITER= 3; PAR: 1.519 0.7981E-01 

C.I. 3: FL= 6.4088 ; SSE=0.94127E-01; ITER= 4; PAR: 1.003 0.1496 

C.I. 4: FL= 3.1090 ; SSE=0.91639E-01; ITER= 4; PAR: 1.008 0.1541 

C.I. 5: FL= 1.8458 ; SSE=0.91618E-01; ITER= 4; PAR: 1.007 0.1542 

C.I. 6: FL= 4.6167 ; SSE=0.92338E-01; ITER= 4; PAR: 1.006 0.1531 

C.I. 7: FL= 0.42100 ; SSE=0.91608E-01; ITER= 4; PAR: 2.636 0.6966E-01 

C.I. 8: FL= -0.87902 ; SSE=0.32298E-01; ITER= 4; PAR: 1.264 0.1321 

C.I. -1: FL= 1.5895 ; SSE=0.91861E-01; ITER= 4; PAR: 2.641 0.6964E-01 

C.I. -2: FL= 10.581 ; SSE=0.41295E-01; ITER= 4; PAR: 1.436 0.1282 

C.I. -3: FL= 2.5599 ; SSE=0.94083E-01; ITER= 4; PAR: 2.671 0.6945E-01 

C.I. -4: FL= 1.1892 ; SSE=0.91594E-01; ITER= 4; PAR: 2.636 0.6967E-01 

C.I. -5: FL= 0.70517 ; SSE=0.91630E-01; ITER= 4; PAR: 2.636 0.6967E-01 

C.I. -6: FL= 1.7864 ; SSE=0.92292E-01; ITER= 4; PAR: 2.646 0.6961E-01 

C.I. -7: FL= 0.32539E-02; SSE=0.91624E-01; ITER= 4; PAR: 1.008 0.1542 

C.I. -8: FL= -1.1052 ; SSE=0.32343E-01; ITER= 3; PAR: 1.750 0.7849E-01

Weighted Residuals Output File (weight-res.dat): 

WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL 1; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.167E-01 

 2 F2 0.171E-01 

 3 F3 0.266E-01 

 4 F4 0.460E-02 

 5 F5 -0.201E-01 

 6 F6 -0.206E-01 

 7 F7 0.316E-01 

 8 F8 0.250E-01 

 9 F9 0.230E-01 

 10 F10 0.808E-02 

 11 F11 -0.178E-01 

 12 F12 -0.198E-01 

 13 F13 -0.336E-02 

 14 F14 -0.103E-02 

 15 F15 -0.847E-02 

 16 F16 -0.110E-01 

 17 F17 -0.428E-02 

 18 F18 0.805E-02 

 19 F19 -0.151E-01 

 20 F20 -0.335E-01 

 21 F21 -0.378E-01 

 22 F22 -0.272E-01 

 23 F23 -0.333E-01 

 24 F24 -0.236E-01 

 25 F25 -0.116E-01 

 26 F26 -0.521E-02 

175

 27 F27 0.110E-02 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 28 F28 -0.101E-01 

 29 F29 -0.198E-02 

 30 F30 -0.895E-02 

 31 F31 0.521E-02 

 32 F32 0.382E-01 

 33 F33 0.249E-01 

 34 F34 0.423E-01 

 35 F35 0.147E-01 

 36 F36 0.117E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL 2; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.129E-01 

 2 F2 0.140E-01 

 3 F3 0.247E-01 

 4 F4 0.564E-02 

 5 F5 -0.180E-01 

 6 F6 -0.184E-01 

 7 F7 0.337E-01 

 8 F8 0.255E-01 

 9 F9 0.209E-01 

 10 F10 0.968E-02 

 11 F11 -0.138E-01 

 12 F12 -0.176E-01 

 13 F13 -0.248E-02 

 14 F14 -0.381E-03 

 15 F15 -0.780E-02 

 16 F16 -0.974E-02 

 17 F17 -0.208E-02 

 18 F18 0.108E-01 

 19 F19 -0.143E-01 

 20 F20 -0.326E-01 

 21 F21 -0.368E-01 

 22 F22 -0.260E-01 

 23 F23 -0.319E-01 

 24 F24 -0.220E-01 

 25 F25 -0.108E-01 

 26 F26 -0.441E-02 

 27 F27 0.195E-02 

 28 F28 -0.916E-02 

 29 F29 -0.100E-02 

 30 F30 -0.791E-02 

 31 F31 0.587E-02 

 32 F32 0.388E-01 

 33 F33 0.255E-01 

 34 F34 0.430E-01 

 35 F35 0.154E-01 

 36 F36 0.189E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL 3; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.124E-01 

 2 F2 0.128E-01 

176

 3 F3 0.223E-01 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 4 F4 0.618E-03 

 5 F5 -0.239E-01 

 6 F6 -0.243E-01 

 7 F7 0.318E-01 

 8 F8 0.250E-01 

 9 F9 0.228E-01 

 10 F10 0.622E-02 

 11 F11 -0.211E-01 

 12 F12 -0.234E-01 

 13 F13 -0.329E-02 

 14 F14 -0.987E-03 

 15 F15 -0.843E-02 

 16 F16 -0.110E-01 

 17 F17 -0.412E-02 

 18 F18 0.826E-02 

 19 F19 -0.151E-01 

 20 F20 -0.335E-01 

 21 F21 -0.378E-01 

 22 F22 -0.271E-01 

 23 F23 -0.332E-01 

 24 F24 -0.235E-01 

 25 F25 -0.115E-01 

 26 F26 -0.516E-02 

 27 F27 0.116E-02 

 28 F28 -0.100E-01 

 29 F29 -0.191E-02 

 30 F30 -0.887E-02 

 31 F31 0.525E-02 

 32 F32 0.383E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

 35 F35 0.148E-01 

 36 F36 0.123E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL 4; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.170E-01 

 2 F2 0.175E-01 

 3 F3 0.270E-01 

 4 F4 0.490E-02 

 5 F5 -0.198E-01 

 6 F6 -0.203E-01 

 7 F7 0.317E-01 

 8 F8 0.250E-01 

 9 F9 0.230E-01 

 10 F10 0.824E-02 

 11 F11 -0.175E-01 

 12 F12 -0.195E-01 

 13 F13 -0.333E-02 

 14 F14 -0.101E-02 

 15 F15 -0.847E-02 

 16 F16 -0.111E-01 

 17 F17 -0.431E-02 

 18 F18 0.802E-02 

177

 19 F19 -0.151E-01 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 20 F20 -0.335E-01 

 21 F21 -0.378E-01 

 22 F22 -0.272E-01 

 23 F23 -0.333E-01 

 24 F24 -0.236E-01 

 25 F25 -0.116E-01 

 26 F26 -0.522E-02 

 27 F27 0.109E-02 

 28 F28 -0.101E-01 

 29 F29 -0.199E-02 

 30 F30 -0.895E-02 

 31 F31 0.520E-02 

 32 F32 0.382E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

 35 F35 0.147E-01 

 36 F36 0.117E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL 5; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.171E-01 

 2 F2 0.175E-01 

 3 F3 0.270E-01 

 4 F4 0.499E-02 

 5 F5 -0.197E-01 

 6 F6 -0.202E-01 

 7 F7 0.317E-01 

 8 F8 0.250E-01 

 9 F9 0.231E-01 

 10 F10 0.829E-02 

 11 F11 -0.174E-01 

 12 F12 -0.194E-01 

 13 F13 -0.333E-02 

 14 F14 -0.101E-02 

 15 F15 -0.846E-02 

 16 F16 -0.111E-01 

 17 F17 -0.431E-02 

 18 F18 0.802E-02 

 19 F19 -0.151E-01 

 20 F20 -0.335E-01 

 21 F21 -0.379E-01 

 22 F22 -0.272E-01 

 23 F23 -0.333E-01 

 24 F24 -0.236E-01 

 25 F25 -0.116E-01 

 26 F26 -0.523E-02 

 27 F27 0.108E-02 

 28 F28 -0.101E-01 

 29 F29 -0.199E-02 

 30 F30 -0.895E-02 

 31 F31 0.519E-02 

 32 F32 0.382E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

178

 35 F35 0.147E-01 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 36 F36 0.117E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL 6; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.159E-01 

 2 F2 0.163E-01 

 3 F3 0.259E-01 

 4 F4 0.387E-02 

 5 F5 -0.208E-01 

 6 F6 -0.213E-01 

 7 F7 0.317E-01 

 8 F8 0.250E-01 

 9 F9 0.230E-01 

 10 F10 0.774E-02 

 11 F11 -0.184E-01 

 12 F12 -0.204E-01 

 13 F13 -0.334E-02 

 14 F14 -0.103E-02 

 15 F15 -0.848E-02 

 16 F16 -0.111E-01 

 17 F17 -0.428E-02 

 18 F18 0.806E-02 

 19 F19 -0.151E-01 

 20 F20 -0.335E-01 

 21 F21 -0.378E-01 

 22 F22 -0.272E-01 

 23 F23 -0.333E-01 

 24 F24 -0.236E-01 

 25 F25 -0.116E-01 

 26 F26 -0.521E-02 

 27 F27 0.110E-02 

 28 F28 -0.101E-01 

 29 F29 -0.198E-02 

 30 F30 -0.894E-02 

 31 F31 0.521E-02 

 32 F32 0.382E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

 35 F35 0.147E-01 

 36 F36 0.118E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL 7; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.172E-01 

 2 F2 0.177E-01 

 3 F3 0.273E-01 

 4 F4 0.573E-02 

 5 F5 -0.187E-01 

 6 F6 -0.192E-01 

 7 F7 0.317E-01 

 8 F8 0.249E-01 

 9 F9 0.227E-01 

 10 F10 0.885E-02 

179

 11 F11 -0.161E-01 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 12 F12 -0.183E-01 

 13 F13 -0.332E-02 

 14 F14 -0.103E-02 

 15 F15 -0.847E-02 

 16 F16 -0.110E-01 

 17 F17 -0.418E-02 

 18 F18 0.815E-02 

 19 F19 -0.151E-01 

 20 F20 -0.335E-01 

 21 F21 -0.378E-01 

 22 F22 -0.272E-01 

 23 F23 -0.332E-01 

 24 F24 -0.235E-01 

 25 F25 -0.116E-01 

 26 F26 -0.517E-02 

 27 F27 0.114E-02 

 28 F28 -0.100E-01 

 29 F29 -0.195E-02 

 30 F30 -0.892E-02 

 31 F31 0.523E-02 

 32 F32 0.383E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

 35 F35 0.148E-01 

 36 F36 0.120E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL 8; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.146E-01 

 2 F2 0.156E-01 

 3 F3 0.261E-01 

 4 F4 0.647E-02 

 5 F5 -0.173E-01 

 6 F6 -0.178E-01 

 7 F7 0.331E-01 

 8 F8 0.253E-01 

 9 F9 0.211E-01 

 10 F10 0.985E-02 

 11 F11 -0.136E-01 

 12 F12 -0.171E-01 

 13 F13 -0.276E-02 

 14 F14 -0.615E-03 

 15 F15 -0.803E-02 

 16 F16 -0.101E-01 

 17 F17 -0.259E-02 

 18 F18 0.101E-01 

 19 F19 -0.145E-01 

 20 F20 -0.329E-01 

 21 F21 -0.371E-01 

 22 F22 -0.263E-01 

 23 F23 -0.322E-01 

 24 F24 -0.224E-01 

 25 F25 -0.110E-01 

 26 F26 -0.463E-02 

180

 27 F27 0.172E-02 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 28 F28 -0.939E-02 

 29 F29 -0.124E-02 

 30 F30 -0.816E-02 

 31 F31 0.569E-02 

 32 F32 0.387E-01 

 33 F33 0.254E-01 

 34 F34 0.429E-01 

 35 F35 0.152E-01 

 36 F36 0.172E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL -1; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.168E-01 

 2 F2 0.173E-01 

 3 F3 0.269E-01 

 4 F4 0.533E-02 

 5 F5 -0.191E-01 

 6 F6 -0.196E-01 

 7 F7 0.317E-01 

 8 F8 0.249E-01 

 9 F9 0.227E-01 

 10 F10 0.864E-02 

 11 F11 -0.165E-01 

 12 F12 -0.187E-01 

 13 F13 -0.333E-02 

 14 F14 -0.104E-02 

 15 F15 -0.848E-02 

 16 F16 -0.110E-01 

 17 F17 -0.418E-02 

 18 F18 0.814E-02 

 19 F19 -0.151E-01 

 20 F20 -0.335E-01 

 21 F21 -0.378E-01 

 22 F22 -0.272E-01 

 23 F23 -0.332E-01 

 24 F24 -0.235E-01 

 25 F25 -0.116E-01 

 26 F26 -0.517E-02 

 27 F27 0.115E-02 

 28 F28 -0.100E-01 

 29 F29 -0.195E-02 

 30 F30 -0.891E-02 

 31 F31 0.522E-02 

 32 F32 0.383E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

 35 F35 0.148E-01 

 36 F36 0.120E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL -2; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.131E-01 

 2 F2 0.142E-01 

181

 3 F3 0.249E-01 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 4 F4 0.575E-02 

 5 F5 -0.179E-01 

 6 F6 -0.184E-01 

 7 F7 0.345E-01 

 8 F8 0.263E-01 

 9 F9 0.217E-01 

 10 F10 0.100E-01 

 11 F11 -0.139E-01 

 12 F12 -0.177E-01 

 13 F13 -0.185E-02 

 14 F14 0.258E-03 

 15 F15 -0.715E-02 

 16 F16 -0.909E-02 

 17 F17 -0.145E-02 

 18 F18 0.114E-01 

 19 F19 -0.137E-01 

 20 F20 -0.321E-01 

 21 F21 -0.363E-01 

 22 F22 -0.255E-01 

 23 F23 -0.314E-01 

 24 F24 -0.215E-01 

 25 F25 -0.105E-01 

 26 F26 -0.404E-02 

 27 F27 0.232E-02 

 28 F28 -0.879E-02 

 29 F29 -0.624E-03 

 30 F30 -0.754E-02 

 31 F31 0.617E-02 

 32 F32 0.391E-01 

 33 F33 0.258E-01 

 34 F34 0.433E-01 

 35 F35 0.157E-01 

 36 F36 0.217E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL -3; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.148E-01 

 2 F2 0.152E-01 

 3 F3 0.248E-01 

 4 F4 0.326E-02 

 5 F5 -0.212E-01 

 6 F6 -0.216E-01 

 7 F7 0.318E-01 

 8 F8 0.249E-01 

 9 F9 0.227E-01 

 10 F10 0.763E-02 

 11 F11 -0.184E-01 

 12 F12 -0.206E-01 

 13 F13 -0.331E-02 

 14 F14 -0.102E-02 

 15 F15 -0.846E-02 

 16 F16 -0.110E-01 

 17 F17 -0.416E-02 

 18 F18 0.816E-02 

182

 19 F19 -0.151E-01 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 20 F20 -0.335E-01 

 21 F21 -0.378E-01 

 22 F22 -0.272E-01 

 23 F23 -0.332E-01 

 24 F24 -0.235E-01 

 25 F25 -0.115E-01 

 26 F26 -0.516E-02 

 27 F27 0.116E-02 

 28 F28 -0.100E-01 

 29 F29 -0.194E-02 

 30 F30 -0.890E-02 

 31 F31 0.523E-02 

 32 F32 0.383E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

 35 F35 0.148E-01 

 36 F36 0.121E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL -4; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.171E-01 

 2 F2 0.176E-01 

 3 F3 0.273E-01 

 4 F4 0.566E-02 

 5 F5 -0.188E-01 

 6 F6 -0.193E-01 

 7 F7 0.317E-01 

 8 F8 0.249E-01 

 9 F9 0.227E-01 

 10 F10 0.880E-02 

 11 F11 -0.162E-01 

 12 F12 -0.184E-01 

 13 F13 -0.332E-02 

 14 F14 -0.103E-02 

 15 F15 -0.848E-02 

 16 F16 -0.110E-01 

 17 F17 -0.418E-02 

 18 F18 0.814E-02 

 19 F19 -0.151E-01 

 20 F20 -0.335E-01 

 21 F21 -0.378E-01 

 22 F22 -0.272E-01 

 23 F23 -0.332E-01 

 24 F24 -0.235E-01 

 25 F25 -0.116E-01 

 26 F26 -0.517E-02 

 27 F27 0.114E-02 

 28 F28 -0.100E-01 

 29 F29 -0.195E-02 

 30 F30 -0.892E-02 

 31 F31 0.522E-02 

 32 F32 0.383E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

183

 35 F35 0.148E-01 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 36 F36 0.120E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL -5; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.171E-01 

 2 F2 0.176E-01 

 3 F3 0.273E-01 

 4 F4 0.567E-02 

 5 F5 -0.188E-01 

 6 F6 -0.193E-01 

 7 F7 0.317E-01 

 8 F8 0.249E-01 

 9 F9 0.227E-01 

 10 F10 0.881E-02 

 11 F11 -0.162E-01 

 12 F12 -0.184E-01 

 13 F13 -0.332E-02 

 14 F14 -0.103E-02 

 15 F15 -0.847E-02 

 16 F16 -0.110E-01 

 17 F17 -0.418E-02 

 18 F18 0.814E-02 

 19 F19 -0.151E-01 

 20 F20 -0.335E-01 

 21 F21 -0.378E-01 

 22 F22 -0.272E-01 

 23 F23 -0.332E-01 

 24 F24 -0.236E-01 

 25 F25 -0.115E-01 

 26 F26 -0.516E-02 

 27 F27 0.115E-02 

 28 F28 -0.100E-01 

 29 F29 -0.195E-02 

 30 F30 -0.892E-02 

 31 F31 0.522E-02 

 32 F32 0.383E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

 35 F35 0.148E-01 

 36 F36 0.120E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL -6; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.165E-01 

 2 F2 0.170E-01 

 3 F3 0.267E-01 

 4 F4 0.504E-02 

 5 F5 -0.194E-01 

 6 F6 -0.199E-01 

 7 F7 0.317E-01 

 8 F8 0.249E-01 

 9 F9 0.227E-01 

 10 F10 0.849E-02 

184

 11 F11 -0.167E-01 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 12 F12 -0.189E-01 

 13 F13 -0.334E-02 

 14 F14 -0.104E-02 

 15 F15 -0.848E-02 

 16 F16 -0.110E-01 

 17 F17 -0.418E-02 

 18 F18 0.814E-02 

 19 F19 -0.151E-01 

 20 F20 -0.335E-01 

 21 F21 -0.378E-01 

 22 F22 -0.272E-01 

 23 F23 -0.332E-01 

 24 F24 -0.235E-01 

 25 F25 -0.115E-01 

 26 F26 -0.516E-02 

 27 F27 0.115E-02 

 28 F28 -0.100E-01 

 29 F29 -0.195E-02 

 30 F30 -0.892E-02 

 31 F31 0.523E-02 

 32 F32 0.383E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

 35 F35 0.148E-01 

 36 F36 0.120E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL -7; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.171E-01 

 2 F2 0.175E-01 

 3 F3 0.271E-01 

 4 F4 0.499E-02 

 5 F5 -0.197E-01 

 6 F6 -0.202E-01 

 7 F7 0.317E-01 

 8 F8 0.250E-01 

 9 F9 0.231E-01 

 10 F10 0.829E-02 

 11 F11 -0.174E-01 

 12 F12 -0.194E-01 

 13 F13 -0.333E-02 

 14 F14 -0.101E-02 

 15 F15 -0.846E-02 

 16 F16 -0.111E-01 

 17 F17 -0.431E-02 

 18 F18 0.802E-02 

 19 F19 -0.151E-01 

 20 F20 -0.335E-01 

 21 F21 -0.379E-01 

 22 F22 -0.272E-01 

 23 F23 -0.333E-01 

 24 F24 -0.236E-01 

 25 F25 -0.116E-01 

 26 F26 -0.524E-02 

185

 27 F27 0.108E-02 
APPENDIX C. EXAMPLE SIMULATIONS SIMULATIONS – Summary And Weighted Residuals Output 

Files 

 28 F28 -0.101E-01 

 29 F29 -0.199E-02 

 30 F30 -0.895E-02 

 31 F31 0.519E-02 

 32 F32 0.382E-01 

 33 F33 0.249E-01 

 34 F34 0.424E-01 

 35 F35 0.147E-01 

 36 F36 0.118E-02 

 WEIGHTED RESIDUALS AT LIMIT FOR INTERVAL -8; STARTING VALUES 1 

 OBSNO OBS. NAME WEIGHT. RES. 

 1 F1 0.140E-01 

 2 F2 0.150E-01 

 3 F3 0.255E-01 

 4 F4 0.603E-02 

 5 F5 -0.177E-01 

 6 F6 -0.182E-01 

 7 F7 0.328E-01 

 8 F8 0.249E-01 

 9 F9 0.206E-01 

 10 F10 0.954E-02 

 11 F11 -0.138E-01 

 12 F12 -0.173E-01 

 13 F13 -0.303E-02 

 14 F14 -0.895E-03 

 15 F15 -0.832E-02 

 16 F16 -0.104E-01 

 17 F17 -0.283E-02 

 18 F18 0.990E-02 

 19 F19 -0.147E-01 

 20 F20 -0.331E-01 

 21 F21 -0.373E-01 

 22 F22 -0.265E-01 

 23 F23 -0.324E-01 

 24 F24 -0.226E-01 

 25 F25 -0.112E-01 

 26 F26 -0.477E-02 

 27 F27 0.158E-02 

 28 F28 -0.954E-02 

 29 F29 -0.139E-02 

 30 F30 -0.832E-02 

 31 F31 0.556E-02 

 32 F32 0.386E-01 

 33 F33 0.253E-01 

 34 F34 0.428E-01 

 35 F35 0.152E-01 

 36 F36 0.161E-02

186