# EFFECTIVE GROUNDWATER MODEL CALIBRATION: With Analysis of Data, Sensitivities, and Uncertainty 

# *by* Mary C. Hill and Claire R. Tiedeman

# Computer Instructions for the Exercises, using the US Geological Survey computer programs 

# MODFLOW-2005 and MODPATH; 

# UCODE_2005, Residual_Analysis, Model_Linearity, Corfac_Plus, and Model_Linearity_Adv; and

**MFI2005, GW_Chart, and ModelViewer**

This document first describes the software set up, and then describes
how to conduct the computer runs required by some of the exercises of
Hill and Tiedeman (2007) using MODFLOW-2005 (Harbaugh, 2005) and MODPATH
(Pollock, 1994) to simulate the system involved, and UCODE_2005 (Poeter
and others, 2005) to perform sensitivity analysis and estimate parameter
values. The input files are constructed using MFI2005 (Harbaugh, 2007,
written commun.), which is a basic graphical interface that supports
MODFLOW-2005 and UCODE_2005. This document provides instructions for the
exercises in which input files for these programs are prepared, the
programs are executed, and directions for locating where the information
shown in the book is presented in the model output files. For other
aspects of the exercises and for all references cited in this document,
readers are referred to Hill and Tiedeman (2007).

Refer to Hill and Tiedeman (2007, p. 21-24) for description and
illustration of the flow system and management problem involved in these
exercises. There it says that MODFLOW-2000 and its ADV Package (Anderman
and Hill, 2001) are used to simulate the flow system for calibration and
prediction condition; here, we use MODFLOW-2005 and MODPATH.

The documentation for all programs used is included in the
"documentation" directory of the directories distributed for class.

The setup of the files distributed for doing the exercises is described
in the remainder of this section.

The executables needed to perform the exercises are in the bin and the
exer directories (see section "File Setup". You could also do one of the
following: (1) Copy the executables to a directory listed in the PATH
environment variable (type "path" at the DOS prompt to display the value
of PATH); or (2) Place the files in a different directory and modify the
PATH variable on your computer to add the directory to the PATH. To keep
the class files self-contained and to avoid changing the PATH on
computers used temporarily for class, we just place them in the
directory students use to perform the exercises.

## File setup

The directories used in the exercises are listed in Table 1.

Table 1. The directories used in the exercises.

+---------------------+------------------------------------------------+
| **Directory**       | **Contents**                                   |
+---------------------+------------------------------------------------+
| **First level**     |                                                |
+---------------------+------------------------------------------------+
| bin-plotting\\      | See table below under "Software setup"         |
+---------------------+------------------------------------------------+
| documentation\\     | This document.                                 |
|                     |                                                |
|                     | MODFLOW-MODPATH\\ UCODE-2005\\ OPR-PPR\\ MMA\\ |
+---------------------+------------------------------------------------+
| Exe                 | answers\\ exer\\ initial\\                     |
| rcises-UCODE_2005\\ |                                                |
+---------------------+------------------------------------------------+
| **Second level**    |                                                |
+---------------------+------------------------------------------------+
| answers\\           | Final files for most exercises. This is what   |
|                     | exer\\ would contain if all the exercises were |
|                     | conducted as described in these instructions.  |
|                     |                                                |
|                     | Subdirectories ex5.1a\\ ex5.1b\\ ex8           |
+---------------------+------------------------------------------------+
| bin\\               | Most of the executables needed to run the      |
|                     | exercises.                                     |
+---------------------+------------------------------------------------+
| exer\\              | The mfi2005 executable and selected files      |
|                     | needed to run most exercises.                  |
|                     |                                                |
|                     | All files and folders that you create when     |
|                     | doing the exercises.                           |
+---------------------+------------------------------------------------+
| initial\\           | Files needed to conduct exercises that can not |
|                     | be created easily or at all by MFI2005.        |
|                     |                                                |
|                     | Subdirectories ex3.2.i\\ ex5.1a.i\\ ex5.1b.i\\ |
|                     | ex8.i\\ ex9.4i\\                               |
+---------------------+------------------------------------------------+
| **Third level**     |                                                |
+---------------------+------------------------------------------------+
| ex3.2i\\ ex5.1a.i\\ | Files that contain data needed for exercises   |
| ex5.1b.i\\ ex8.i\\  | 3.2, 5.1a, 5.1b, 8, and 9.4.                   |
| ex9.4.i\\           |                                                |
|                     | Distributed in directory initial\\.            |
+---------------------+------------------------------------------------+
| ex5.1a\\ ex5.1b\\   | Final files for exercises 5.1a, 5.1b and all   |
| ex8\\               | parts of exercise 8.                           |
|                     |                                                |
|                     | Distributed in directory answers\\..           |
+---------------------+------------------------------------------------+

## Software setup

The executables for the software needed to do the exercises.

Table 2. The executables distributed for the exercises.

  --------------------- ------------------------------ -------------------------------
  **Software program**  **Executable file**            **Comment**

  **The following are                                  
  in the bin\\                                         
  directory**                                          

  MODFLOW-2005          mf2005.exe                     Ground-water flow model

  UCODE_2005            ucode_2005.exe \*\*\*get new   Inverse modeling and
                        version                        sensitivity analysis

  Model_Linearity       model_linearity.exe            Calculates the modified Beale's
                                                       measure of model linearity

  Model_Linearity_Adv   model_linearity_adv.exe        Calculates the total,
                                                       intrinsic, and combined
                                                       measures of model linearity

  Corfac_Plus           Corfac_Plus.exe                Calculates correction factors
                                                       used to adjust confidence and
                                                       prediction intervals in some
                                                       circumstances. See Poeter and
                                                       others (2007, p. 204) and
                                                       references cited therein.

  Residual_Analysis     residual_analysis.exe          Calculates statistics for
                                                       residual analysis

  Linear_Uncertainty    linear_uncertainty             Calculates linear intervals on
                                                       predictions

  MODPATH               Mpathr4_3.exe                  Post-processor for MODFLOW that
                                                       calculates advective transport
                                                       paths

  PathLineRead          Pathlineread.exe               Used to convert MODPATH output
                                                       to more convenient form.

                        Mf2k.exe                       \*\*\*remove if not needed

  **The following are                                  
  in the exer                                          
  directory**                                          

  MFI2005               00-mfi2005.exe                 Simple preprocessor for
                                                       MODFLOW-2005

  **The following are                                  
  in the                                               
  bin\\bin-plotting\\                                  
  directory**                                          

  GW_Chart              GW_Chart\\gw_chart.exe         Graphs output from UCODE_2005

  ModelViewer           ModelViewer\\modelviewer.exe   Visualizes results from
                                                       MODFLOW-2005
  --------------------- ------------------------------ -------------------------------

To make it easier to access the main programs used most in class, create
shortcuts on the Windows desktop for (1) the MFI2005 executable in
*exer* (named *00-mfi2005.exe* there so it can be listed first
alphabetically to make it easier to find), (2) the gw_chart.exe
executable in bin-plotting\\GW_Chart, and (3) the ModelViewer.exe
executable in bin-plotting\\ModelViewer. To do this, in Windows File
Explorer, **right** click and drag each of these two programs from
Explorer to the desktop and select "Create Shortcuts Here". From then
on, these programs can be started by double-clicking on the shortcut
icon.

In addition to the MFI2005 executable, the exer\\ directory includes the
following files.

Table 3. Files included in the exer\\ directory in addition to the
MFI2005 executable.

  -------------- -------------------------------- -----------------------------------
  **Exercise**   **Batch or input file**          **Comment**

  **Exercises                                     
  6.2 and 7.3 do                                  
  analyses for                                    
  the regression                                  
  completed in                                    
  exercise                                        
  5.2c**                                          

  6.2e           ex6.2e-residual_analysis.bat     Analyzes model fit to observations.

  7.3            ex7.3-model_linearity.bat        Calculates modified Beale's measure
                                                  of model linearity.

                 ex7.3-corfac_plus.bat            Defines that confidence intervals
                                                  (instead of prediction intervals)
                                                  are to be calculated.

                 ex7.3_ucode.corfac               Input file needed by Corfac_Plus
                                                  when it is run using
                                                  ex7.3-corfac_plus.bat.

                 ex7.3-model_linearity_adv.bat    Calculates total and intrinsic
                                                  measures of model linearity.

  **Exercise                                      
  9.11 does the                                   
  analyses                                        
  indicated for                                   
  the regression                                  
  completed in                                    
  exercise                                        
  9.7.**                                          

  9.11           ex9.11-model_linearity.bat       Calculates modified Beale's measure
                                                  of model linearity.

                 ex9.11-corfac_plus.bat           Input file needed by Corfac_Plus
                                                  when it is run using
                                                  ex9.11-corfac_plus.bat

                 ex9.11_ucode.corfac              Input file needed by Corfac_Plus
                                                  when it is run using
                                                  ex9.11-corfac_plus.bat.

                 ex9.11-model_linearity_adv.bat   Calculates total and intrinsic
                                                  measures of model linearity.
  -------------- -------------------------------- -----------------------------------

### MFI2005: Toubleshooting problems getting going

When using MFI2005, the user initiates a MODFLOW-2005 run by using the
"Run MF2005" item on the FILE menu. When you select "Run MF2005",
MFI2005 tries to invoke a batch file named MFI2005RUN.BAT; this file is
in the *exer* directory. If, when you select "Run MF2005", a black
window appears and immediately disappears, it means that the operating
system cannot find MFI2005RUN.BAT. This will occur if MFI2005RUN.BAT is
neither in a directory listed in the PATH, nor in the directory where
the MODFLOW data set resides. If this happens, copy MFI2005RUN.BAT into
a suitable directory. Similarly, when MFI2005RUN.BAT is executed, it
invokes mf2005.exe, which must be in a directory in PATH or in the
current directory.

Similarly, when usingMFI2005, the user initiates a UCODE_2005 run by
using the "Run UCODE" item on the FILE menu. When you select "Run
UCODE", MFI2005 tries to invoke a batch file named MFIUCODERUN.BAT; this
file is in the *exer* directory. If, when you select "Run UCODE", a
black window appears and immediately disappears, it means that the
operating system cannot find MFIUCODERUN.BAT. This will occur if
MFIUCODERUN.BAT is neither in a directory listed in the PATH, nor in the
directory where the MODFLOW data set resides. If this happens, copy
MFIUCODERUN.BAT into a suitable directory. Similarly, when
MFIUCODERUN.BAT is executed, it invokes UCODE_2005.exe, which needs to
be in a directory in PATH or in the current directory.

The paths of the MODFLOW-2005 and UCODE_2005 executables are defined in
the files MFI2005RUN.BAT and MFIUCODERUN.BAT.

### GW_Chart and ModelViewer

GW_Chart (Winston, 2000)is a plotting program for output from
MODFLOW-2005, and UCODE_2005, and other programs.

ModelViewer (Hsieh and Winston, 2002) enables three-dimensional
visualization of spatially variable model input and output; both
programs can be downloaded from the book website.

### Software versions

It is advantageous to use the executables distributed in the bin and
exer directory in class, because these versions of the programs are
consistent with the exercise instructions in this document. The latest
versions of the programs can be downloaded at
http://water.usgs.gov/nrp/gwsoftware/modflow.html.

## Exercise 3.1: Steady-state model setup using parameters

This exercise corresponds to exercise 3.1 in Hill and Tiedeman (p. 36).

[Objective]{.underline}: Prepare all input for the steady-state
simulation used in the exercises and run the simulation. Use the
preprocessing program MFI2005 to prepare the input files. Use parameters
to define system properties and the parameter values listed in Table 4.
These values will be the starting values when nonlinear regression is
performed.

Table 4: Parameter names and starting values for properties of the
steady-state flow-system for which parameters are estimated in the
exercises.

\[m/s, meters per second; cm/yr, centimeters per year. This is the same
as Table 3.1 of Hill and Tiedeman, 2007, p. 38)\]

+---------------------------------------------+----------+------------+
| **Flow-system property**                    | **P      | **Starting |
|                                             | arameter | value**    |
|                                             | name**   |            |
+---------------------------------------------+----------+------------+
| Horizontal hydraulic conductivity of layer  | HK_1     | 3.0×10^-4^ |
| 1, in m/s                                   |          |            |
+---------------------------------------------+----------+------------+
| Hydraulic conductivity of the riverbed, in  | K_RB     | 1.2×10^-3^ |
| m/s                                         |          |            |
+---------------------------------------------+----------+------------+
| Vertical hydraulic conductivity of          | VK_CB    | 1.0×10^-7^ |
| confining bed, in m/s                       |          |            |
+---------------------------------------------+----------+------------+
| Horizontal hydraulic conductivity of layer  | HK_2     | 4.0×10^-5^ |
| 2 in columns 1 and 2, in m/s                |          |            |
+---------------------------------------------+----------+------------+
| Recharge in recharge zone 1, in cm/yr       | RCH_1    | > 63.072   |
+---------------------------------------------+----------+------------+
| Recharge in recharge zone 2, in cm/yr       | RCH_2    | > 31.536   |
+---------------------------------------------+----------+------------+

#### Getting started

-   Start MFI2005 by double clicking the executable in the *exer*
    directory.

-   Enter *ex3.1.nam* when prompted for the name file. This creates a
    new MODFLOW-2005 dataset named *ex3.1* in directory *exer*.

-   Specify 18 rows, 18 columns, 2 model layers, and free format.

#### Discretization (DIS) input

The Discretization (DIS) input is described by Harbaugh (2005, p. 4-3 to
4-5, 8-11)**.**

-   Click on the "DIS" menu item. In the "DIS Menu" dialog box, click on
    the buttons below "Check" and enter the following data.

```{=html}
<!-- -->
```
-   Set model time and length units to seconds and meters, respectively.

-   Set DELR = DELC = 1000 m.

-   In LAYCBD, specify that a confining bed below layer 1 is to be
    simulated.

-   For the top elevation of layer 1, use 100 m.

-   Bottom elevations of layer 1, the confining bed, and layer 2 can be
    determined from fig. 2-1 in Hill and Tiedeman (p. 22).

-   The default in MFI2005 is to simulate one steady-state stress period
    which is what is needed here. Thus, no change is needed for the
    "Stress Periods" button.

-   Click "Close"

```{=html}
<!-- -->
```
-   BAS input (Harbaugh, 2005, Ch. 4 and p. 8-10 to 8-11): Click on the
    "BAS" menu item. Do the following for layers 1 and 2.

```{=html}
<!-- -->
```
-   Specify IBOUND=1 for all cells, which means that all cells are
    active. Though a real number is in the box, the program uses it as
    an integer.

-   Set initial head = 200 m.

-   Check to be sure all are specified for both layers.

-   Click "Close"

### Exercise 3.1a: Parameters that define list data:

List data and using parameters to define list data are discussed by
Harbaugh (2005, p. 8-1 to 8-3). For this problem, the River and
General-Head Boundary Packages use list data. In the MODFLOW-2005
documentation, these are classified as stress packages; however,
generally they define Cauchy boundary conditions for the system.

In the River Package a parameter is defined to calculate the riverbed
conductance. In the General-Head Boundary Package no parameters are
defined.

#### River( RIV) Package input 

The River Package is described by Harbaugh (2005, 6-6 to 6-12 and p.
8-34 to 8-36)**,** and is used here to represent the river.

-   Click on the "STRESS" menu item, then click on the RIV button. See
    that this menu allows rivers to be defined using parameters on the
    left and without parameters on the right. We will be defining the
    river using a parameter.

-   Click on "New" and define one parameter named K_RB, which will
    represent hydraulic conductivity of the riverbed. The initial value
    of 1.2×10^-3^ is given in Table 4. Close this window.

-   To specify the cells of the river reach and data associated with
    them, click "Modify" in the lower left corner of the "List Data"
    window. In the window that comes up, click "Internal editing".

```{=html}
<!-- -->
```
-   Specify that the river reach is in layer 1, rows 1-18, and column 1.

-   Stage in the river is 100 m, and the elevation of the riverbed
    bottom is 90 m.

-   The RIV Package requires riverbed conductance (K×L×W/M) as input,
    which is calculated in MODFLOW-2005 as the product of the parameter
    and Condfact. Here, the parameter is defined as the hydraulic
    conductivity of the confining unit, which is K in the equation, so
    that Condfact = (L×W/M) = 1000 m for each river cell. Lesser values
    would be entered if, for example, the stream gages were located such
    that only part of a cell was to be included in the measured reach,
    the width or length of stream in each cell varied, to include an
    independently determined variation of K along the length of the
    stream, or to support additive parameters used, for example, to
    represent linear variation of streambed hydraulic conductivity
    between values at each end of a reach.

```{=html}
<!-- -->
```
-   Click the OK.

-   Now you are back in the "List Parameter Data" window. Click "OK".

-   Finally, in the "List Data" window, click the "Activate for this
    stress period" button to activate K_RB for stress period 1. Click
    "OK".

#### General-Head Boundary (GHB) input 

General-Head Boundary (GHB) Package is described by Harbaugh (2005, p.
6-4 to 6-5, 8-49 to 8-51), and is used here to represent flow from the
highlands bordering the area on the side opposite the river.

-   In the Stress Dialog box, click on GHB. Under "Non parameter data",
    click Package, and "Internal editing".

```{=html}
<!-- -->
```
-   Define 36 GHB cells in rows 1-18 of column 18, in both layers 1 and
    2.

-   The external head (Bhead) is 350 m.

-   The hydraulic conductance between the cell and the external
    source/sink is 1.0×10^-7^ m^2^/s (you can enter "1e-7").

```{=html}
<!-- -->
```
-   Click "OK" or "Close" in all the windows.

-   Save the dataset (Under the "File" menu).

-   Execute MODFLOW-2005 from the FILE menu. The output files will be
    produced in directory exer\\. Because a flow package (here it will
    be LPF) has not been activated, an error message is printed in the
    DOS window. However, the list file (*ex3.lst*) should show 18 river
    reaches with stage = 100, conductance = 1.2, and bottom
    elevation = 90. There also should be 36 GHB cells listed with
    stage = 350 and conductance = 1.0E-7.

-   Click on the black window where MODFLOW was running and press the
    space bar before retuning to the MFI window.

### Exercise 3.1b: Multiplier and Zone Arrays

Two multiplication arrays and one zone array are needed to define the
parameters of this problem. They are all defined here, and are then
referred to when the parameters are defined in subsequent exercises.

#### Multiplier arrays 

Multiplier arrays are described by Harbaugh (2005, p.4-6, 8-3 to 8-7,
8-10, and 8-15 to 8-16). Here we use one multiplication array that is a
constant value for all cells to convert recharge calculated using the
recharge parameters, which are in centimeters per year, to the units of
the model, which are meters and seconds. The second multiplication array
represents the spacial distribution of the hydraulic conductivity of the
lower model layer.

-   Click on the "Zone-Mult" menu item and choose "Multiplier Arrays".

-   Click on "ADD" to define a multiplier array named "RCH_CONV" and set
    it to the constant 3.170979E-10; this is used to convert recharge
    parameter values provided in cm/yr to the model units of m/s.

-   Click "ADD" again to define a multiplier array named "HK_2_MULT",
    which is used to simulate the variability of hydraulic conductivity
    of layer 2. Click "OK".

-   Click "Package" and "Internal editing".

-   Using the 18 by 18 array, approximate a linear variation from 1.0 to
    9.0 by taking nine steps. Do this by placing the values 1.0 in all
    rows of columns 1 and 2, 2.0 in all rows of columns 3 and 4, 3.0 in
    columns 5 and 6, and so on, ending with values of 9.0 in columns 17
    and 18. Click "OK".

-   Set multiplier = 1.0 in the "Define Multiplier Arrays" dialog box.
    Click "OK".

#### Zone array 

Zone arrays are discussed by Harbaugh (2005, p.4-6, 8-4, 8-10, 8016).
Here, a zone array is needed to represent the two areas of recharge
defined for this problem by Hill and Tiedeman (2007, p. 22).

-   Click on the "Zone-Mult" menu item and choose "Zone Arrays".

-   Click "ADD" to define a zone array named "RCH_ZONE". Click
    "Package", then "Internal editing" and define a zone array that has
    a value of 1 in columns 1-9 and a value of 2 in columns 10-18. Click
    "OK", set multiplier = 1.0, and click "OK".

    **Note:** If you run MODFLOW-2005 at this point, an error message is
    produced because not all needed files have been produced. However,
    the MODFLOW_2005 output file can be checked to be sure the files
    produced so far are being read correctly.

### Exercise 3.1c: Parameters that define properties for the top of the system

Properties that can be defined over the top of the system include, for
example, areal recharge and evapotranspiration. This problem has areal
recharge, which is simulated using the Recharge Package of MODFLOW-2005
(Harbaugh, p. 6-3 to 6-4 and 8-37 to 8-39):

-   Click the "STRESS" menu item and click on the RCH button.

-   Under "Parameter Data", click "New" and the "Array Parameter
    Definition" dialog box appears. Define parameter name RCH_1, which
    controls recharge in recharge zone 1. Use the parameter value from
    Table 4 above, 63.072 cm/yr.

-   In the first line of the table in this dialog box, specify that zone
    array RCH_ZONE and multiplier array RCH_CONV apply to this parameter
    by typing in the array names. Set Z1 to 1 to indicate that the
    parameter applies to zone number 1 in the zone array. Click "OK".

-   Click "New" again, and follow the same steps to define parameter
    RCH_2. Use the value from Table 1 above, 31.536 cm/yr. For this
    parameter, set Z1 to 2. Click "OK".

-   Specify that recharge applies to the top model layer, and activate
    both parameters for stress period 1. Click "OK" and close the STRESS
    Dialog box.

### Exercise 3.d: Parameters that define flow properties within the system

MFI2005 supports two of the three internal flow packages available with
MODFLOW-2005: Block-Centered Flow (BCF) and Layer-Property Flow (LPF).
It does not support Hydrogeologic Unit Flow (HUF). Here, use LPF, which
is discussed by Harbaugh (2005, p. chapter 5 and 8-28 to 8-31).

-   In MFI2005, click the "Internal Flow" menu item and choose LPF.

-   Click "LAYTYP", to specify model layer flags. Set LAYVKA to "Hor. To
    Vert. Ani." for both layers; this makes it so parameters are used to
    calculate horizontal to vertical anisotropy for each model layer
    instead of vertical hydraulic conductivity. Keep the default values
    for the other items (LAYTYP=confined, LAYAVG=Harmonic, horizontal
    anisotropy=1.0), and click "OK".

-   Question: Both layers are being simulated as confined. Why would one
    want to do that in a system with a free surface top boundary?!

-   Click "Hydraulic" to enter parameter data. On the left under
    "Parameter Data" there is a list of three types of parameters that
    can be defined for the LPF Package. HK refers to hydraulic
    conductivity along rows, VANI stands for vertical anisotropy of
    model layers, and VKCB stands for vertical hydraulic conductivity of
    implicit confining beds. VANI would be replaced by VK if the other
    option for LAYVKA had been chosen in the previous step. Parameters
    can not be defined for horizontal anisotropy in the interface
    MFI2005, though they can be defined in MODFLOW-2005. See exercise
    3.1h for more on vertical anaisotropy.

-   Make sure "HK" is chosen. Click "New" and see the "Array Parameter
    Definition" dialog box.

-   Follow steps similar to the RCH definition to define parameter HK_1
    to control hydraulic conductivity (HK) of layer 1. Use the parameter
    value in Table 4 above, 3.0×10^-4^ m/s.

-   Enter a "1" for "Layer". This parameter has no associated multiplier
    or zone arrays; the default "ALL" listed under "Zone Array" means
    that the parameter value applies to all cells in layer 1. Click "OK"

-   Follow similar steps to define HK_2 to control HK of layer 2, using
    the parameter value from Table 1 of 4.0×10^-5^ m/s and multiplier
    array HK_2_MULT. You can check the name of the multiplier array by
    clicking the box labeled "Edit Multiplier Arrays". Click "OK".

-   Change the parameter type to VKCB. Follow the steps for defining
    HK_1 to define a VKCB parameter called VK_CB; use the parameter
    value in Table 4 above, 1.0×10^-7^ m/s.

-   Finally, define VKA (which here refers to horizontal to vertical
    anisotropy) using non-parameter input, and specify a constant value
    of 1.0 for both layers.

-   Click "OK". Click "CLOSE".

### Exercise 3.1e: Solver

Solvers are needed to calculate heads for all cells of the
finite-difference grid based on the entered information.

[Instructions]{.underline}:

-   In MFI2005, choose the "Solver" option. Four solvers are available
    through MFI2005: PCG2, SIP, DE4, and LMG (LMG is freely available to
    USGS users only). Any solver can be selected for this problem. The
    completed set of files distributed in class in the directory
    "answers" uses the PCG2 solver.

-   If PCG2 is selected, change the value of RCLOSE to match HCLOSE. The
    value needed for HCLOSE and RCLOSE depend on the units used, as
    discussed by Hill (1990, p. 12-13). Here, the use of seconds for
    time means that the smaller value of RCLOSE is needed.

-   If LMG is selected, BCLOSE may need to be changed. If the global
    budget error reported in the MODFLOW output file *ex3.1.lst* is
    larger than 0.01 (1 percent), reduce the value of BCLOSE by a factor
    of 2 or more. If the global budget error is 0.00, it may be possible
    to increase BCLOSE to obtain a faster solution, though for this
    small test case the short execution time is not a problem. (LMG is
    freely available to USGS users only).

### Exercise 3.1f: Output Control

Output Control is described by Harbaugh (2005, p. 3-8, 8-17 to 8-21).

-   Click the "OUTPUT" Menu item, and then click "Output file names and
    formats". This automatically results in some results, such as heads,
    being saved to default file names and formats. Change the Head Print
    format to 4, which will provide two digits to the right of the
    decimal point instead of one.

-   Use default file name for a binary output file to which heads are to
    be saved, and click "OK".

-   Click "Times for Saving Head" and select option to save heads every
    time step.

-   Click "Times for Printing Budget" and select option to print budgets
    every time step.

-   Click "OK" and close the "Output Control Data" dialog box.

-   Click on the "OPTIONS" menu item and choose "Show Files". Note that
    the LIST output file is *EX3.1.LST*. Also note on the main screen
    what packages are being activated, and that they are all from
    MODFLOW-2005's Ground-Water Flow Process (OBS is not listed).

### Exercise 3.1g: Final check and first execution

-   Click on the "OPTIONS" menu item and choose "Check Complete" to
    verify that the data entry for all packages is complete.

-   Save the dataset.

-   Execute MODFLOW-2005 from the FILE menu. The output files will be
    produced in directory exer\\.

-   In your ex3.1.lst file, the heads in layer 1, column 1 (beneath the
    steam) should be 100.23. for all rows. The head in column 18 should
    all be 190.83m for all rows. Why are the heads the same for all rows
    in both model layers? Check the global budget at the end of the List
    file to be sure the percent discrepancy is less than 1.00.

-   If different heads are listed, make sure that vertical anisotropy is
    applied to both model layers, that the right parameter values are
    entered, and that the recharge and head-dependent boundary
    conditions have been activated for the stress period. If problems
    persist, check the other model inputs.

### Exercise 3.1h (Optional): Define horizontal and vertical anisotropy parameters

MODFLOW-2005 supports definition of parameters governing horizontal and
vertical anisotropy; this exercise considers definition of parameters
for vertical anisotropy. The vertical anisotropy or vertical hydraulic
conductivity for each model layer can be defined as parameters with
parameter type VANI or VK in the Layer-Property Flow Package (Harbaugh,
2005, p. chapter 5 and 8-28 to 8-31). These parameter types affect
simulation of the vertical flow of water between the model layers, and
are most important when there is no intervening confining unit.

[Instructions]{.underline}:

-   In MFI2005, use the Save As option on the FILE menu to save the
    current dataset (ex3.1) as a new dataset called ex3.1h and use the
    ex3.1h dataset for this exercise.

-   Set CHANI = 0 for both layers in LPF input.

-   Define horizontal anisotropy of layers 1 and 2 as parameter HANI_1&2
    in LPF input. Set the horizontal anisotropy to 1.0 to produce output
    consistent with the previous results. (This part of the exercise can
    not be done with MFI2005. Either skip this or complete the exercise
    by modifying the files directly.)

-   Define the vertical anisotropy of layer 1 as parameter VANI_1 in LPF
    input, and define the vertical anisotropy of layer 2 as parameter
    VANI_2. No multiplication or zone arrays are needed for these
    parameter definitions because in this exercise vertical anisotropy
    is constant throughout layers 1 and 2. Set vertical anisotropy to
    1.0 to produce output consistent with the previous results.

-   Execute MODFLOW-2005:

-   Run MODFLOW-2005, and compare the output file *ex3.1e.lst* to the
    output file *ex3.1.lst* from exercise 1.

-   Make sure that the simulated hydraulic heads and global volumetric
    rate budgets are identical. If they are not identical, check your
    definition of parameters HANI_1&2, VANI_1, and VANI_2.

### Exercise 3.1i (Optional): Define additive parameters for interpolation methods.

MODFLOW-2005 allows model inputs to be calculated as the sum of
contributions from more than one parameter (Harbaugh, 2005, p. 8-2, 8-5
to 8-7). This is important when hydraulic conductivity, recharge, and
other quantities vary gradually over an area or volume, and are not well
represented using zones of constant value. Although this capability is
used to represent a simple linear variation in this exercise, it can be
used to represent complex interpolation such as kriging, and can be used
to add stochastic components to large-scale trends. One deficiency of
the present version of MODFLOW-2005 is that the interpolation can not be
accomplished using log-transformed parameter values and the exponential
taken of the resulting distribution, as often is needed for kriging.
However, this capabilitiy is expected in the near future. In this
exercise, you will apply additive parmeters to represent the linear
variation in the hydraulic conductivity of layer 2 of the test case.

The goal is to maintain the same hydraulic-conductivity distribution in
layer 2, but to allow the variation to be more flexible. Here, we will
maintain the assumption that hydraulic conductivity varies in a
step-wise linear fashion in layer 2, but we will allow the parameter
values to represent the increase in hydraulic conductivity within the
layer. Setting up the parameters in this manner facilitates imposing
different values of hydraulic conductivity at the two ends of the layer,
and would allow estimating these values by regression. If the estimated
parameter values differed from the startig values, the increase in
hydraulic conductivity might be other than a nine-fold increas and(or)
the value under the stream might be different.

[Instructions]{.underline}:

In MFI2005, use the Save As option on the FILE menu to save the current
dataset as a new dataset called ex3.1i and use the ex3.1i dataset for
this exercise.

Replace the definition of parameter HK_2 with new parameters HK_2a and
HK_2b. Both new parameters will extend throughout layer 2. Make one
parameter value equal the hydraulic conductivity in grid columns 1 and 2
(the value for HK_2 in Table 4 is 4.0×10^-5^ m/s); make the other equal
the hydraulic conductivity in grid columns 17 and 18 (this will be 9
times the first value, or 36.0×10^-5^). Use multiplier array names MULTa
and MULTb. The arrays need to be defined such that HK_2a × MULTa(i,j) +
HK_2b × MULTb(i,j) equals the hydraulic conductivity that occurred at
cell i,j before.

-   Execute MODFLOW-2005

-   Compare the output file ex3.1h.lst to the output file ex3.1.lst from
    exercise 1.

-   Make sure that the simulated hydraulic heads and global volumetric
    rate budgets are identical. If they are not identical, check your
    definition of parameters HK_2a and HK_2b and of multiplier arrays
    MULTa and MULTb.

## Exercise 3.2: Observations for the steady-state problem

This exercise corresponds to exercise 3.2 in Hill and Tiedeman (2007, p.
38).

Defining observations for the test case with MODFLOW-2005 requires one
input file for the hydraulic-head observations and one for the flow
observations. Observed hydraulic heads for the steady-state model are in
table 3-2 of Hill and Tiedeman (2007, p. 51).

The MODFLOW-2005 input instructions for observations are described in
OBS.PDF, which is provided in the documentation directory districtured
with these exercises.

-   In MFI2005, under the "File" menu item, save the *ex3.1* dataset as
    dataset *ex3.2* and use the *ex3.2* dataset in this exercise.

### Exercise 3.2a: Hydraulic-head observations.

-   In MFI2005, click Observations. Select the "Single Time Head" option
    on the Observations Dialog window.

-   Review the 12 columns for which data need to be provided to define
    head observations. Columns 1 through 9 contain data used to
    construct the MODFLOW head observation input file. Columns 1, 7 and
    10 through 12 are used to define observation in the UCODE_2005 input
    files. Does it make sense that the data provided in columns 2 though
    6 and 8 and 9 are not needed by UCODE_2005? Does it make sense that
    the data listed in columns 10 through 12 are not needed by MODFLOW?
    Why would the data in columns 1 and 7 be used by both codes?

-   Enter data for the 10 steady-state head-observations listed in Table
    3.2 of Tiedeman and Hill (2007, p. 39). This can be accomplished in
    any of three ways. In all cases, the observation wells are assumed
    to be located at the centers of model cells, so ROFF and COFF equal
    0.0. Here you can use any one of the methods. In practice the chosen
    method will depend on what software is available to assist in
    constructing the head observation file.

(1) Type the data in directly from Table 3.2.

(2) Use the file head-data.txt in the initial/ex3.2i directory. If you
    have a text editor such as Textpad that allows single columns of
    data to be copied, the copied columns can be inserted into the
    MFI2005 window by placing the cursor in the top box to be copied
    into and typing ctrl-v.

(3) Use the already constructed head observation file ex3.2.obh in the
    initial/ex3.2i directory. To accomplish this, enter a fake data
    point in the "Single-Tine Head Observations" window, specify output
    file ex3.2.\_os, and click "OK" and then "Close". Save the files and
    close MFI2005. In the exer directory, replace the file ex3.2.obh
    with the file from the initial/ex3.2i directory. Restart MFI2005.
    Clock "Observations" and "Single-Time Head Observations" to check
    that the observations are being read.

-   Close the Observations Dialog window and save the MFI2005 files.

### Exercise 3.2b: Define flow observations

The flow observations are defined as head-dependent flows represented
using the River Package, as described in OBS.PDF. For the steady-state
simulation, one flow observation, a gain to the river of 4.4 m^3^/s, is
available for the 18,000-ft reach of the river simulated in the model.
The coefficient of variation for the observation is 10%.

A gain to the river is a loss to the ground-water system. The MODFLOW
convention is that flows out of the ground-water system are negative.
Thus, the observation needs to be entered as a negative value; that is,
-4.4 m^3^/s.

-   In MFI2005, click Observations. Select the "RIV observations" option
    on the Observations Dialog window.

-   Click "ADD Cell Group" and then "EDIT Observation Times". Name the
    observation "flow01.ss" and enter the observed value (-4.4) and
    coefficient of variation used for the statistic value (0.10). Assign
    a plot symbol of 2. Click "OK".

-   Click "EDIT Cell Locations". The cell group will include all 18 cell
    locations defined for the river in exercise 1. FACTOR = 1.0 for all
    cells in the cell group. These values can be obtained from the file
    *ex3.2.riv* of the *exer* directory using a text editor that allows
    single columns of numbers to be copied, as described above in input
    option (2) for heads.

-   For the output file, enter ex3.2.\_os.

-   Close all dialog boxes and save the files.

### Exercise 3.2c: Check head and flow observations and their simulated equivalents

-   Execute MODFLOW-2005.

-   In ex3.2.lst, examine the listing of the observation input and the
    table of observed and simulated heads and flows. Check simulated
    values against those in figure 3.2 of Hill and Tiedeman (p. 39). The
    flow observation and its simulated equivalent are printed with more
    significant digits in *ex3.2.\_os*. This will be important when
    using UCODE_2005.

-   Address the question posed for this exercise in Hill and Tiedeman
    (2007, p. 39).

-   Use ModelViewer to visualize the hydraulic-conductivity distribution
    and the simulated heads over the model domain.

ModelViewer Instructions (also see the instruction file in the
bin-plotting directory and your notebooks):

-   To start: Execute ModelViewer. Under the File menu item, select New;
    then in the Model Selection dialog box, select Modflow 2000 (which
    works for MODFLOW-2005). Click "OK". In the Modflow 2000 Data files
    dialog box, specify the name file by browsing to directory *exer*,
    and selecting ex3.2.nam. Click "OK".

-   In the Data Selection dialog box, keep the defaults.

-   When the model domain appears, under the Show menu item, select
    "Solid" and "Color Bar".. Hydraulic heads are shown. Under the
    Toolbox menu item, select Geometry and set z to 10 to specify 10x
    vertical exaggeration. Click on a corner of the figure and move
    around to see the head distribution throughout the volume.

-   To visualize hydraulic conductivity, under the Toolbox menu item
    select Data, and choose K along rows. Under the Toolbox menu item,
    select Solid, and then select Blocky.

-   To show the location of the boundary conditions simulated using the
    River and General-Head Boundary Packages, under the Show menu select
    Model Features.

Additional ModelViewer capabilities and options are described in the
online help system.

### Exercise 3.2d: Calculate weights on hydraulic-head and flow observations 

MODFLOW allows observations to be specified, but to define weights the
capabiities of UCODE_2005 are needed. MFI2005 has already used the
information entered about observations to produce the files needed by
UCODE_2005. Look at the file *ex3.2_ucode.obs*. This is a typical
UCODE_2005 input block. Here, STATFLAG is used to identify whether the
STATISTIC provided to calculate the weight is a variance or coefficient
of variation. Other options are described by Poeter and others (2005, p.
83).

Follow the instructions for exercise 3.2d from Hill and Tiedeman (2007,
p. 39-40). Check your calculations against the weights printed in Figure
3.2. The weights also are listed in the UCODE_2005 main output file
produced in exercise 3.3 in a table entitled "FIT OF SIMULATED
EQUIVALENTS TO OBSERVATIONS".

## Exercise 3.3: Evaluate model fit using the starting parameter values

If the evaluation from exercise 3.2 indicates that all observations,
equivalent simulated values, and defined parameters are specified
correctly, the model fit resulting from the starting parameter values
can be evaluated. This exercise corresponds to exercise 3.3 in Hill and
Tiedeman (p. 40). Do the first part of the exercise as described in the
book. In addition to using figure 3.2, locate the information in file
*ex3.2.lst*.

### Exercise 3.3a: Produce the Parameter Value File

In the exercises above, the parameter values were defined in the package
input files. There are many circumstances in which it is useful to have
one file that can be used to alter any of the parameter values. This is
provided by MODFLOW-2005 through the Parameter-Value File described by
Harbaugh (2005, p. 4-6, 8-10, 8-23 to 8-24). Here we use MFI2005 to
produce that file.

[Instructions]{.underline}:

-   In MFI2005, save the ex3.2 dataset as dataset ex3.3 and use the
    ex3.3 dataset in this rest of this exercise.

-   In MFI2005, choose the "Parameter Values" option. Click the button
    on the top left and choose to have all defined parameters included
    in the PVAL file.

-   The parameter values from the package input files are listed in the
    third column, which is labeled "GWF File Value". This column can not
    be modified in this window, which is indicated by the shading.

-   The values in the second column, which is labeled "PVAL File Value
    (B)", are set to the values of the third column, but can be changed.
    The other columns define values used by UCODE_2005, and default
    values derived from the package parameter values have automatically
    been entered by MFI2005. Keep the default values for now.

-   Click OK and save the files.

-   Execute MODFLOW-2005. Check to see that the same results are
    obtained as in exercise 3.2. If not, click "Parameter Values" and
    check to be sure the parameter values are the same as the values in
    the package files.

-   Use the parameter values ability of MFI2005 to do the final part of
    exercise 3.3 described by Hill and Tiedeman (2007, p. 40).

-   When done, return all parameter values to their starting values,
    which are equivalent to the values from the packages listed in the
    third column of the parameter values dialog box.

## Exercise 4.1: Sensitivity analysis for the initial model

These instructions apply to the parts of exercise 4.1 in Hill and
Tiedeman (p. 60-66) for which computer runs are needed.

### Exercise 4.1a: Calculate sensitivities for the steady-state flow system using UCODE_2005

Here the sensitivites are calculated with UCODE_2005 by perturbing the
parameter values, one at a time, and the sensitivities are used to
calculate model evaluation statistics. This mode is listed as
"Sensitivity Analysis" mode in the UCODE_2005 documetation (Poeter and
others, 2005, table 3, p. 30).

-   In MFI2005, save the ex3.3 dataset as dataset *ex4.1* and use the
    *ex4.1* dataset in this exercise.

-   In MFI2005, click "UCODE" to get the menu used to select UCODE_2005
    capabilities. At the very top left is a window with the Verbose
    option, which controls the level of input repetition and error and
    warning messages produced by UCODE_2005. Below on the left, the
    UCODE_2005 modes are listed. Output options are listed further to
    the right but still under the "UCODE"Conrol_Data" banner. Further to
    the right are the options used to control nonlinear regression. At
    he bottom is the input used to control prior information, which is
    used to represent knowledge about the parameter values that is not
    represented in the observations used in the regression.

-   Click "Sensitivities-Central". Click OK and then "File" and "Save".

-   Use a text editor to look at the files with \_ucode in their
    filename. These are:

```{=html}
<!-- -->
```
-   *ex4.1_ucode.in*, the main UCODE input file;

-   *ex4.1_ucode.obs*, the Observation_Data input block;

-   *ex4.1_ucode.par*, the Parameter_Data input block;

-   *ex4.1_ucode.tpl*, a template file used to construct the
    MODFLOW-2005 input file *ucode.pvl*, and

-   *ex4.1_ucode.inst*, instructions for reading simulated equivalents
    from the MODFLOW-2005 *ex4.1.\_os* file.

```{=html}
<!-- -->
```
-   Use the UCODE_2005 documentation to answer the following questions.

```{=html}
<!-- -->
```
-   In the main input file, *ex4.1_ucode.in*, what do the pairs of
    statements starting with BEGIN and END define? (See chapter 5). What
    are keywords?

-   What are the three options for organizing data within the input
    blocks?

-   In the first input block, called "Options", what does the "3" mean?
    What other options are there?

-   In the UCODE_Control_Data input block, what keywords control what is
    called the mode in Table 3 of the UCODE_2005 documentation (p. 30)?

-   In the Model_Command_Lines input block, what is the command used to
    run MODFLOW-2005? Here, MODFLOW-2005 is run directly, but often it
    is run through a batch file. Batch files would, for example, allow a
    sequence of models to be executed, such as MODFLOW-2005 and MODPATH
    or MT3DMS.

-   The data for the Parameter_Data input block and the Observation_Data
    input block are read from files *ex4.1_ucode.par* and
    *ex4.1_ucode.obs*, respectively. Look in *ex4.1_ucode.par* and find
    the parameter values you specified using the MFI2005 Parameter
    Values option. Look in *ex4.1_ucode.obs* and find the observed
    values.

-   The last two input blocks in *ex4.1_ucode.in* define how UCODE_2005
    interacts with the process model (here, MODFLOW-2005) input and
    output files.

-   Identify the name of the MODFLOW-2005 input file and look at the
    template file. How big are the fields into which UCODE_2005 can
    substitute numbers? Why might this be important?

-   Identify the name of the MODFLOW-2005 output file and look at the
    instructions used to read it.

-   The input blocks in the UCODE_2005 main input file need to be listed
    in the order in which they are presented in Table 4 of the
    documentation (p. 43), though all input blocks need not be present.
    List which input blocks are not included in this UCODE_2005 main
    input file.

-   Most keywords in the UCODE_2005 input files have default values that
    are used if the keyword does not appear. Thus, UCODE_2005 input
    files can be very short in many circumstances. Here, it is easier
    for an interface such as MFI2005 to always include the same set of
    keywords. Go through the UCODE_Control_Data input block and
    determine which keywords could have been omitted because the
    defaults are specified.

```{=html}
<!-- -->
```
-   Execute UCODE_2005 by clicking "File" and then "Run UCODE".

-   Make sure that the simulated hydraulic heads and global volumetric
    rate budgets in *ex4.1.lst* are the same as those in *ex3.3.lst*. If
    the hydraulic heads and volumetric rates in the two files are not
    the same, check the parameter entries in *ex4.1.pvl*. In
    *ex4.1.pvl*, the columns after the first two are not read by
    MODFLOW-2005; they are used by MFI2005.

-   Using GW_Chart (make sure you specify "UCODE_2005", plot the
    dimensionless scaled sensitivities (listed in file
    *ex4.1-ucode.\_sd*), composite scaled sensitivities
    (*ex4.1-ucode.\_sc*), and leverage statistics and
    (*ex4.1-ucode.\_so*). Inspect the parameter correlation coefficients
    (*ex4.1-ucode.\_pcc*).

### Exercises 4.1b, c, d, and e

Please see Hill and Tiedeman (2007, p. 60-63).

Table 5. The statistics used in these exercises and filename extensions
of the files where they are listed are as follows. The file name prefix
is always ex4.1.

+---------+----------------------------+-------------------------------+
| E       | Statistic name             | Filename extension^1^         |
| xercise |                            |                               |
+---------+----------------------------+-------------------------------+
| 4.1b    | dimensionless and          | \_sd                          |
|         |                            |                               |
|         | composite scaled           | \_sc                          |
|         | sensitivities              |                               |
+---------+----------------------------+-------------------------------+
| 4.1c    | parameter correlation      | \_pcc: values larger than or  |
|         | coefficients               | equal to 0.95                 |
|         |                            |                               |
|         |                            | \_mc: the entire parameter    |
|         |                            | correlation matrix            |
+---------+----------------------------+-------------------------------+
| 4.1d    | one-percent scaled         | \_s1: values related to       |
|         | sensitivities              | individual observations:      |
+---------+----------------------------+-------------------------------+
| 4.1e    | leverage                   | \_so                          |
+---------+----------------------------+-------------------------------+

^1^ Files with scaled sensitivities and leverage have extensions that
start with \_s. Files that contain matrices have extensions that start
with \_m.

## Exercise 5.1: Modified Gauss-Newton and Application to a Two-Parameter Problem

Read the description of this exercise in Hill and Tiedeman (2007, p.
80-81). This document provides instructions for obtaining results such
as those shown in Figure 5.4 and Tables 5.1 to 5.3 using UCODE_2005.
MFI2005 does not support the Derived_Parameters input block of
UCODE_2005 needed to combine the parameters or the Observation_Groups
input block that can be used to make omission of selecgted observations
easier. As a result, we will be working with many files produced
previously by MFI2005, and some that will need to be modified in a text
editor. The runs will be accomplished using batch files executed outside
of MFI2005.

### Exercise 5.1a: Assess relation of objective-function surfaces to parameter correlation coefficients

-   Copy the directory initial\\.ex5.1a.i to exer\\ex5.1a. In this
    directory, most of the files to be inspected, changed, or run have
    names beginning with "00-" so that they are listed at the top of the
    directory when the "name" header bar is clicked in the Windows file
    manager. The instructions below refer to the ex5.1a directory unless
    specified.

-   There are two main UCODE_2005 input files: 00-ex5.1_ucode.in and
    00-ex5.1_ucode.noflow.in. Inspect the UCODE_Control_Data input block
    and find the keyword used to determine what this run will do.
    Consider the instructions on Page 55 to 56 of the UCODE_2005 input
    instructions (Poeter and others, 2005).

-   In the Parameter_Data in put block, identify keywords
    LowerConstraint, UpperConstraint, and SOSincrement, which are
    discussed by Poeter and others (2007, p. 70 and 72). How many runs
    will be executed to create the data set for Figure 5.4a or b?

-   Inspect the Observation_Groups input block in either of the main
    input files and note the keyword "UseFlag". The Observation \_Data
    input block identifies files 00-EX5.1_ucode.head.obs and
    00-EX5.1_ucode.flow.obs where the observation data are listed.

-   Produce data needed for the graphs of Figures 5.4a and b by
    executing the batch files in this directory. For Figure 5.4c, the
    file 00-ex5.1_ucode.flow.obs change the coefficient of variation
    used for the flow observation.

### Exercise 5.1b: Examine the performance of the modified Gauss-Newton method

-   Copy the directory initial\\.ex5.1b.i to exer\\ex5.1b.

-   To perform the first set of four regression runs listed on page 83
    of the text book for the model with observations only, modify
    00-ex5.1_ucode.noflow.in and run UCODE_2005 by double-clicking the
    batch file 00-ex5.1_ucoderun.noflow.bat. The runs require changing
    MaxChange and the parameter values. MaxChange is in the
    Reg_GN_Controls input block (the third input block from the top).

-   Plot the sets of parameter values calculated by the regression on
    the full contour plots provided with the course notes. Small
    versions of the plots are shown in Figure 5.4a and 5.4b on p. 82 of
    Hill and Tiedeman. The parameter values for KMult and RchMult are in
    the Parameter_Data input block (the fifth input block from the top).
    The parameter values from each iteration needed in the exercise will
    be printed in file ex5.1_ucode.\_pa and file ex5.1_ucode.\_summary
    (the last is new and is not mentioned in the UCODE_2005
    documentation.

-   Model fit to the observations as the regression proceeds is measured
    using the weighted least-squares objective function values listed in
    ex5.1_ucode.\_ss. Plot these values using GW_CHART. Start GW_Chart.
    Choose UCODE_2005. From the "File" menu item, choose "Open file" and
    navigate to the desired directory. Close GW_CHART before proceeding;
    MFI2005 sometimes can not proceed correctly when GW_Chart is open.

-   You are next asked to repeat runs 1 through 4 above with the river
    observation weighted using a coefficient of variation of 10 percent
    and then 1 percent. Use the 00‑ex5.1_ucode.in and
    00-ex5.1_ucoderun.bat files. The coefficient of variation is already
    10 percent. To make it 1 percent, change the value in the file
    00‑ex5.1_ucode.flow.obs.

## Exercise 5.2: Nonlinear regression

These instructions apply to the parts of exercise 5.2 in Hill and
Tiedeman (p. 87) for which computer runs are needed.

-   In MFI2005, save the *ex4.1* dataset as dataset *ex5.2* and use the
    *ex5.2* dataset in this exercise.

### Exercise 5.2a: Define range of reasonable parameter values

-   [In MFI2005, click "Parameter Values" MFI has inserted values for
    "LowerValue" and "UpperValue" that are higher and lower than the
    starting value by two orders of magnitude. These are
    non-constraining values used to identify unreasonable parameter
    estimates, which can be useful indicators of model error (see Hill
    and Tiedeman, 2007, p.140-142). Do the values specified make sense
    for these parameters? Change values as needed to reflect the ranges
    you think would be apply to typical field sites. To approximate the
    results discussed in exercise 7.1g, use the following values. These
    imply that a range defined by multiplying and dividing by a factor
    or 0.4 for the hydraulic conductivity parameters forms a reasonable
    range. Comment on this assumption.]{.underline}

  ----------------- -------------------------- --------------------------
  Parameter name    Lower reasonable value     Upper reasonable value

  HK_1              1.20E-04                   7.50E-04

  HK_2              1.60E-05                   1.00E-04

  VK_CB             4.00E-08                   2.50E-07

  K_RB              6.00E-04                   3.00E-03

  RCH_1             32                         126

  RCH_2             16                         63
  ----------------- -------------------------- --------------------------

-   Make all the parameters Adjustable (check in box) and do not log
    transform any parameters (uncheck the Log Trans box).

### Exercise 5.2b: First attempts at estimating parameters by nonlinear regression

-   In mfi2005, click "UCODE". Activate "Optimize". See that
    MAX-ITER=10, MaxChange=2.0, TolPar=0.01, and TrustRegion="No". These
    are the regression related variables that are changed most often.

-   Click "Parameter Values" and place constraints on the four
    hydraulic-conductivity parameter values so that negative values for
    these parameters are not allowed. Do this by turning on "Constrain"
    and assigning lower constraints equal to the starting value times
    0.01. For this problem the upper constraints are less critical; use
    the starting value times 100. Negative values could also be avoided
    by log-transforming the parameter values; this is considered later.

-   Save files and perform nonlinear regression by executing UCODE. Look
    at the iteration information in the file ex5.2_ucode.#uout. In
    iteration 1, the regression tries to assign parameter K_RB a
    negative value, and in iteration 2, the regression tries to assign
    parameter HK_2 a negative value. This indicates that the solution is
    performing poorly.

-   Use GW_CHART to look at ex5.2_ucode.\_pa (click "yes" when asked to
    divide the values by their initial values). Note that in file
    ex5.2_ucode.\_pa, the values of the K parameters are positive. This
    results from constraining the parameter values and permits
    successive parameter estimation iterations. Look at any other files
    you wish, and then close GW_Chart.

-   In ex5.2_ucode.\_pasub, the parameter values are in a format that
    can be used in the Parameter_Values input block of UCODE_2005. If
    desired, parameter estimation could be restarted from any
    intermediate iteration by copying the parameter values from
    ex5.2_ucode.\_pasub into a Parameter_Values input block.

-   Save file ex5.2_ucode.#uout as ex5.2_ucode.#uout-nolog.

-   To prevent the regression from calculating negative values for
    parameters K_RB and HK_2 and to hopefully obtain a better posed
    regression problem, the log transform of these parameters can be
    estimated instead of the native value. To specify that the log
    transform is to be estimated, in MFI2005 click "Parameter Values"
    and click to obtain check marks in the "LogTrans." boxes for the
    four hydraulic-conductivity parameters. Remove the constraints on
    the parameter values by removing the check under "Constrian".

-   Execute UCODE_2005 to again perform parameter estimation.

-   Save file *ex5.2_ucode.#uout* as *ex5.2_ucode.#uout-log*. If
    constraints were not removed as suggested, this run regression
    converges with the value of parameter HK_2 on its constraint.
    Without constraints the run does not converge.

-   Results are in a table at the bottom of *ex5.2_ucode.#uout.log*. Use
    gw_chart to examine *ex5.2_ucode.\_pa* and *ex5.2_ucode.\_ss*.
    Performance is similar to that shown in Table 5.4; differences are
    caused by differences in the accuracy of the sensitivities and in
    differences in the regression routines.

-   Do the Problem on p. 88-89 of Hill and Tiedeman.

### Exercise 5.2c: Assign prior information on parameters.

-   Save *ex5.2* as *ex5.2c* and use *ex5.2c* for this exercise

-   In "Parameter Values", remove all checks under "logTrans" so that
    native parameter values are estimated. Click OK.

-   To define prior information on parameters, click UCODE.

-   For K_RB, the prior equation name (PriorName on the menu) is
    "K_RB_PR". Use the starting values of the parameters as the prior
    estimates (1.2×10^-3^). The equation is the parameter identifier,
    K_RB.

-   For VK_CB, the prior equation name is "VK_CB_PR". Use the starting
    value (1.0×10^-7^). The equation is the parameter identifiers,
    VK_CB.

-   Use a coefficient of variation of 0.3 to define the weights of both
    parameters. Specify a plot symbol of 3. General instructions for
    constructing prior information equations in UCODE_2005 are provided
    by (Poeter and others (2005, p. 93-97).

-   Execute UCODE_2005. If it does not converge with TrustRegion=no, try
    one of the other options. Differences occur depending on whether the
    constraints on th parameter values are defined in the "Parameter
    Values" menu of MFI2005.

-   Do the Problem on p. 91 of Hill and Tiedeman.

-   Use ModelViewer to visualize the hydraulic heads and hydraulic
    conductivities (see instructions in Exercise 3.2b).

## Exercise 6.1

### Exercise 6.1a, c, and d

See Hill and Tiedeman (2007, p. 114). The values are located in file
ex5.2c.#uout toward the bottom. You can search for "CALCULATED ERROR
VARIANCE".

## Exercise 6.2

### Exercise 6.2a, b, and d

The graphs mentioned can be produced with GW_Chart and files produced in
exercise 5.2c.

For Figure 6.7a with unweighted residuals on the horizontal axis, use
*ex5.2c_ucode.\_ws*.

For Figure 6.7b, use file *ex5.2c_ucode.\_ww*.

For Figure 6.7c, use *ex5.2c_ucode.\_os.*

For Figure 6.11, use file *ex5.2c_ucode.\_nm*.

### Exercise 6.2e: Determine acceptable deviations from independent normal weighted residuals

Figures 6.13 and 6.14 are plotted using output files from the program
RESIDUAL_ANALYSIS, which is not supported by MFI2005.

-   In most circumstances, the input needed for the RESIDUAL_ANALYSIS
    program is produced by UCODE_2005 upon completion of a successful
    regression run. This requires that DataExchange=yes in the
    UCODE_Control_Data input block by designation or default. For this
    exercise, RESIDUAL_ANALYSIS needs to be executed in the directory
    with your output files from exercise 5.2c, and specify *ex5.2c* on
    the command line. This can be accomplished using the batch file
    *ex6.2e-residual_analysis.bat* located in the *exer* directory.
    Double click on this file.

-   The output files from RESIDUAL_ANALYSIS are *ex5.2c_ucode.#resan*,
    which contains all the generated random numbers and other
    information, *ex5.2c_ucode.\_rd*, which contains the independent
    random numbers, and *ex5.2c_ucode.\_rg*, which contains the
    correlated random numbers. The latter two files can be used to make
    normal probability plots such as those shown in figures 6-14 using
    GW_Chart .

-   The other files produced by RESIDUAL_ANALYSIS are
    *ex5.2c_ucode.\_rb* and *ex5.2c_ucode.\_rc* ,which contain values of
    the influence statistics DFBETAS and Cook's D, respectively. These
    statistics are the subject of exercise 7.1c.

-   Do the [Problem]{.underline} on p. 123 of Hill and Tiedeman.

## Exercise 7.1: Parameter Statistics

### Exercise 7.1a and b

The composite scaled sensitivities and leverage statistics needed for
these exercise are in files *ex5.2c_ucode.\_sc* and *ex5.2c_ucode.\_so*,
respectively*.*

### Exercise 7.1c

The DFBETAS and Cook's D statistics needed for this exercise were
calculated as part of exercise 6.2e and are in files *ex5.2c_ucode.\_rb*
and *ex5.2c_ucode.\_rc*.

### Exercise 7.1e: Evaluate the uniqueness of the parameter estimates using parameter correlation coefficients

#### Part 1

This exercise requires that the flow observation be removed. With
MFI2005 this can be accomplished as follows.

-   Click "Options"

-   Click "Deactivate"

-   Click "RVOB"

-   Click "OK"

-   When an observation is deactivated the \_os filename is removed for
    all remaining observations. To reestablish it for the other
    observations, click "Observations", then "Sngl. Time Heads". Define
    the output file as "ex7.1e.\_os".

-   Run UCODE

Proceed with the exercise questions listed in Hill and Tiedeman (2007,
p. 150-151).

#### Part 2

Part 2 of this exercise starts on page 151. To start the regression from
different initial values, do the following.

-   Close MFI2005, then re-execute the program and open dataset
    *ex5.2c*. Save the *ex5.2c* dataset as dataset *ex7.1e2* and use the
    *ex7.1e2* dataset for this part of exercise 7.1e.

-   Click "Parameter Values" and change the starting parameter values to
    the values in set 1 of table 7-7 (p. 151), perform nonlinear
    regression. For the first set of parameter values, the regression
    does not converge with TrustRegion=no, but it does with
    TrustRegion=dogleg. Rename the file *ex7.1e2_ucode.#out* to
    *ex7.1e2_ucode.#out.1.* Then, change the values to those in set 2 of
    table 7-4, perform nonlinear regression and rename the *#uout* from
    this run to *.1e2_ucode.#out.2*.

-   Do the [Problem]{.underline} at the top of p. 151 of Hill and
    Tiedeman.

## Exercise 7.1f

The parameter standard deviations, linear confidence intervals, and
coefficients of variation are listed in the main output file
ex5.2c_ucode.#uout and the data-exchange file ex5.2c_ucode.\_pc.

## Exercise 7.1g

GW_Chart can use the parameter estimates, reasonable range upper and
lower values, and confidence interval upper and lower limits listed in
the ex5.2c_ucode.\_pc file. The resulting graph will be similar to the
graph shown by Hill and Tiedeman (2007, p. 153, fig. 7.7) if the
following values are used for the reasonable ranges.

+---------------+--------------------+------------+--------------------+
| Parameter     | Lower reasonable   | Starting   | Upper reasonable   |
| name          | limit              | value      | limit              |
+---------------+--------------------+------------+--------------------+
| HK_1          | > 1.2×10^-4^       | >          | > 7.5×10^-4^       |
|               |                    | 3.0×10^-4^ |                    |
+---------------+--------------------+------------+--------------------+
| HK_2          | > 1.6×10^-5^       | >          | > 1.0×10^-4^       |
|               |                    | 4.0×10^-5^ |                    |
+---------------+--------------------+------------+--------------------+
| VK_CB         | > 4.0×10^-8^       | >          | > 2.5×10^-7^       |
|               |                    | 1.0×10^-6^ |                    |
+---------------+--------------------+------------+--------------------+
| K_RB          | > 6.0×10^-4^       | >          | > 3.0×10^-3^       |
|               |                    | 1.2×10^-3^ |                    |
+---------------+--------------------+------------+--------------------+
| RCH_1         | > 32               | > 63.072   | > 126              |
+---------------+--------------------+------------+--------------------+
| RCH_2         | > 16               | > 31.536   | > 63               |
+---------------+--------------------+------------+--------------------+

In GW_CHART, accept the option of dividing by the parameter values to
produce a graph similar to figure 7.7, except that the values are the
fraction of the estimated value instead of the percent. That is, the
values in figure 7.7 are multiplied by 100 while those plotted by
GW_CHART are not.

## Exercise 7.2

This exercise has no computer runs.

## Exercise 7.3: Test for Linearity

-   To begin, with MFI2005, copy model ex5.2c to ex7.3.

To test linearity, sets of model parameters are generated, as discussed
by Hill and Tiedeman (2007, p.143, eq. 7.11). For the hydraulic
conductivity parameters to remain positive in the generated parameter
sets, it is necessary in this problem (indeed, it is often necessary) to
log-transform them. When such a change is made, it is also necessary to
repeat the regression using the changed status of the parameters so that
the variance-covariance matrix of the parameters of eq. 7.11 is correct.
Another complication in this problem is that some of the parameters that
are to be log-transformed have prior information. This requires a change
in the prior information equation used, and a change in the statistic
using Hill and Tiedeman (2007, p. 130, eq. 7.6), however UCODE_2005 now
automatically makes these cahnges for you. The needed runs are
accomplished through the following steps.

-   Click Parameter Values. Click to get checks in the LogTrans column
    for the four hydraulic-conductivity parameters (HK_1, HK_1, VK_CB,
    K_RB). The starting parameter values do not need to be modified;
    execution time would be saved by substituting in the previously
    estimated values, but that is not important for the model considered
    here. Click OK.

-   The following is now done for you by UCODE_2005. The prior
    information equations are changed to "log10(K_RB)" and
    "log10(VK_CB)". Based on, equation 7.6 of Hill and Tiedeman
    (2007, p. 130), [standard deviations]{.underline} on the
    log-transformed values equivalent to the coefficients of variation
    on the native values are determined to be 0.123 for both
    log-transformed K_RB and VK_CB. Check the weights for th prior
    information on the parameters to show that the weights are
    consistent with this standard deviation.

-   "Optimize" should still be selected on the upper left side of the
    window. Click OK.

-   Save the files and run UCODE to repeat the regression with the
    changed status of the parameters.

-   Click UCODE. Click to select "Linearity" on the upper left side of
    the window. Click OK.

-   Save the files and run UCODE to produce the sets of parameter values
    and do the forward runs needed to evaluate model linearity, as
    described by Hill and Tiedeman (2007, p.142-144). The main output
    file has the filename extension #umodlin.

-   Produce modified Beale's measure and other indications of model
    linearity by executing the computer program MODEL_LINEARITY using
    the batch file *ex7.3-model_linearity.bat*. The results found in
    Figure 7.9 of Hill and Tiedeman (2007, p. 156) can be found a little
    up from the bottom of the file *ex7.2_ucode.#modlin*. Change the
    name of this file to *ex7.2_ucode.#modlin.cv-.3*.

Before changing the statistic for the prior information, it is easier to
proceed to ex7.3b to calculate the total and intrinsic model
nonlinearity for this weighting of the prior information. This requires
the following steps.

-   Execute CORFAC_PLUS using the batch file *ex7.3-corfac_plus.bat*.
    For this problem, the main purpose of this run is to provide a list
    of quantities for which uncertainty intervals are to be calculated.
    While we are only interested in the total and intrinsic model
    nonlinearity in this exercise, we will also end up with the combined
    intrinsic model nonlinearity which will indicate how well linear
    intervals are likely to perform for these quantities. Here, the
    quantities are defined as the six parameter estimates. The input
    file *ex7.3_ucode.corfac* is constructed using the input
    instructions of Poeter and others (2005, p. 203-210). There is no
    need to look at the output from this run.

-   In MFI2005, click UCODE and choose LinearityAdv to be "Conf" to
    indicate that confidence intervals are to be calculated. This is
    used for a nonlinearity measure important to intervals calculated
    for predictions; which are not being calculated here. Click OK and
    run UCODE. The main output file has the filename extension
    #umodlinadv_conf.

-   Execute MODEL_LINEARITY_ADV using the batch file
    *ex7.3-model_linearity_adv.bat.* The total and intrinsic model
    nonlinearity are listed about half-way through the output file
    *ex7.3_ucode.#modlinadv*. Rename this file to
    *ex7.3_ucode.#modlinadv.cv-.3*

Now the sequence of runs is repeated using different values of the
statistics for the prior information.

-   The coefficient of variation of 0.3 used for the prior information
    reflects greater confidence in the parameter values than one might
    reasonably expect in practice. Repeat the analysis using a
    coefficient of variation of 1.0, which UCODE_2005 uses to produce a
    standard deviation of 0.31 for the log-transformed parameter values.
    The measures calculated are slightly different than those shown in
    the book because a slightly different statistic was used to weight
    the prior information.

Briefly, the sequence of runs is as follows.

1.  With the Optimize option in the UCODE menu, run UCODE_2005 through
    MFI2005.

2.  With the Linearity option in the UCODE menu, run UCODE_2005 through
    MFI2005.

3.  Execute *ex7.3-model_linearity.bat* to get the modified Beale's
    measure in *ex7.3_ucode.#modlin*. Change the name of this file to
    *ex7.3_ucode.#modlin.cv-1.0*.

4.  Execute *ex7.3_ucode.corfac*

5.  With the ModelLinearity Adv=Conf option in the UCODE menu, run
    UCODE_2005 through MFI2005

6.  Execute *ex7.3-model_linearity_adv.bat* to get measures of total and
    intrinsic model nonlinearity in *ex7.3_ucode.#modlinadv_conf*.
    Rename this file to *ex7.3_ucode.#modlinadv_conf.cv-1.0*.

-   Address the last part of exercise 7.3a on page 156 of Hill and
    Tiedeman (2007).

-   Address the last part of exercise 7.3b on page 157 of Hill and
    Tiedeman (2007).

## Exercise 8

Many of the statistics for analysis of predictions are not yet supported
by GW_Chart. Therefore, you will be shown where the data are located but
in class we will just use the figures in the book to show you what the
associated graphs look like.

## Exercises 8.1 and 8.2

Exercises 8.1 and 8.2 involve predictions, which are not yet supported
by MFI2005. For these exercises, the necessary input files have already
been constructed, and you will mainly be executing batch files and
examining the results. The file structure for these exercises is under
folder *initial/ex8.i/*, as follows:

+---------------------+------------------------------------------------+
| **Directory**       | **Contents**                                   |
+---------------------+------------------------------------------------+
| First level         |                                                |
+---------------------+------------------------------------------------+
| ex8\\               | mf-mp-runs\\ ucode-opr-ppr-runs\\              |
+---------------------+------------------------------------------------+
| Second level        |                                                |
+---------------------+------------------------------------------------+
| mf-mp-runs\\        | data-ss\\ data-preds\\ data-new-obs\\          |
|                     | data-grid-sensitivities\\                      |
+---------------------+------------------------------------------------+
| u                   | ex8.1a\\ ex8.1b\\ ex8.1b-obs-pred\\ ex8.1c\\   |
| code-opr-ppr-runs\\ | ex8.1c-predgroups\\ ex8.1d\\ ex8.1e\\ ex8.1f\\ |
|                     | ex8.1f+grid\\ ex8.2a\\ ex8.2b\\ files-ss\\     |
|                     | files-preds\\ files-new-obs\\                  |
+---------------------+------------------------------------------------+
| Third level         |                                                |
+---------------------+------------------------------------------------+
| data-ss\\           | Files for running MODFLOW-2005 and MODPATH     |
| data-preds\\\       |                                                |
| data-new-obs\\      |                                                |
|                     |                                                |
| data-g              |                                                |
| rid-sensitivities\\ |                                                |
+---------------------+------------------------------------------------+
| ex8.1a\\ ex8.1b\\   | Batch files and input files for exercises 8.1a |
|                     | to 8.1f, 8.2a, and ex8.2b                      |
| ex8.1b-obs-pred\\   |                                                |
| ex8.1c\\            |                                                |
| ex8.1c-predgroups\\ |                                                |
| ex8.1d\\ ex8.1e\\   |                                                |
| ex8.1f\\            |                                                |
| ex8.1f+grid\\       |                                                |
| ex8.2a\\ ex8.2b     |                                                |
+---------------------+------------------------------------------------+
| files-ss\\          | Batch files and input files used by the runs   |
| files-preds\\\      | conducted in folders ex8.1a\* and ex8.2a       |
| files-new-obs\\     |                                                |
+---------------------+------------------------------------------------+

In exercises 8.1 and 8.2 for this class, advective transport is
simulated using MODPATH, rather than the Advective Transport Observation
Package of MODFLOW-2000, which was used to produce the output for
prediction analyses presented in exercises 8.1 and 8.2 of Hill and
Tiedeman (2007). ADV will no longer be supported; MODPATH is being
modified to include selected capabilities previously available only in
ADV.

As distributed in answers, these directories to not include the output
files because these can be easily created by double-clicking the batch
files. Once executed, the output files can be removed using the
00-clean.bat file located in each directory.

## Exercise 8.1. Predict advective transport and perform sensitivity analysis

-   In *exer*\\, create folder *ex8*\\ and copy all folders in *ex8.i*\\
    into *exer*\\*ex8*\\.

### 8.1a. Predict advective transport

The input files for this exercise, which uses the model as calibrated in
Exercise 5.2c, are in folder *ex8.1a*. Most of the files for the
MODFLOW-2005 and MODPATH runs are in folder *files-preds*. The
MODFLOW-2005 files in this folder have the root name *ex8-preds*, and
are based on those used in Exercise 5.2c. The files for Exercise 8.1a
have been modified as follows.

The Well Package input file, *ex8-preds.wel*, has been constructed and
its name added to the name file *ex8-preds.nam* and
*ex8-preds.nam.ucode* using file type WEL. (This was done in MFI2005
using the Stress menu item and clicking on "WEL"). Other changes to the
files include (1) the estimated parameter values from *ex5.2c.\_paropt*
have been copied into *ex8-preds.pvl*, and (2) the head and flow
observations have been omitted.

For the MODPATH run, the following files have been constructed: the Main
input file *ex8-preds.mpath.mai*, the Name file *ex8-preds.mpath.nam*,
and the response file *ex8.1a.mpath.rsp*. The first two files reside in
folder *files-preds*, and the latter in folder *ex8.1a*. In the MODPATH
run, one particle is placed on the upper face of the center of the cell
in row 2, column 16, layer 1, in the northeast part of the model (Hill
and Tiedeman, fig. 8.7, p. 197). The file *prediction.times* lists the
10 year, 50 year, and 100 year times at which predictions are of
interest. This file is read by MODPATH, and in the *pathline* output
file, coordinates of the particle at these times are listed in addition
to the standard pathline file output, which lists a particle's position
for times when it moves from one cell to another.

-   Copy the two *folders* "mf-mp-runs" and "ucode-opr-ppr-runs" under
    ex8.i into the new directory exer\\ex8.

-   In *ex8\\ex8.1a*, execute batch file *01-ex8.1a.bat*, which runs
    MODFLOW-2005 and MODPATH.

-   Use the output in the *pathline* file to answer the Problem in
    Exercise 8.1a (Hill and Tiedeman, p. 195).

In the pathline file, there are 10 columns, as labeled in the following
example.

@ \[ MODPATH Version 4.00 (V4, Release 3, 7-2003) (TREF= 0.000000E+00 )
\]

1 1.55000E+04 1.65000E+04 9.99900E-01 9.99950E+01 0.00000E+00 16 2 1 1

1 1.51565E+04 1.63903E+04 7.87957E-01 8.93978E+01 -3.15000E+08 16 2 1 1

1 1.51565E+04 1.63903E+04 7.87957E-01 8.93978E+01 3.15000E+08 16 2 1 1

1 1.50000E+04 1.63422E+04 7.10433E-01 8.55216E+01 4.46763E+08 15 2 1 1

1 2 3 4 5 6 7 8 9 10

The columns contain the following:

+-------+------------------------------------+-------+-----------------+
| C     | Contents                           | C     | Contents        |
| olumn |                                    | olumn |                 |
+-------+------------------------------------+-------+-----------------+
| 1     | Particle number                    | 6     | Time. In        |
|       |                                    |       | seconds.        |
|       |                                    |       |                 |
|       |                                    |       | "-" identifies  |
|       |                                    |       | observation     |
|       |                                    |       | time.           |
+-------+------------------------------------+-------+-----------------+
| 2     | Distance parallel to rows,         | 7     | Column          |
|       | measured from outside edge of      |       |                 |
|       | column 1                           |       |                 |
+-------+------------------------------------+-------+-----------------+
| 3     | Distance along columns, measured   | 8     | Row             |
|       | from the outside edge of the last  |       |                 |
|       | row. (ADV starts from the outside  |       |                 |
|       | edge of the first row.)            |       |                 |
+-------+------------------------------------+-------+-----------------+
| 4     | Vertical location measured in      | 9     | Layer           |
|       | local model layer coordinates.     |       |                 |
|       | Negative values locate particles   |       |                 |
|       | within implicit confining layers   |       |                 |
+-------+------------------------------------+-------+-----------------+
| 5     | Vertical location measured from    | 10    | Particle number |
|       | the datum consistent with the      |       |                 |
|       | Discretization file.               |       |                 |
+-------+------------------------------------+-------+-----------------+

#### Plotting the pathline

The pathline can be plotted on the pages provided showing the model
grid, or in three dimensions using ModelViewer. The instructions for
using ModelViewer are as follows:

(1) Modpath and Modflow files need to be in same directory. To obtain
    this, copy all files from

> Ex8\\mf-mp-runs\\data-preds
>
> to
>
> Ex8.1a

(2) Start ModelViewer.

> File\> New
>
> MODFLOW2000
>
> In the next menu, browse to fine
>
> Name file: ex8-preds.nam
>
> Pathline file: pathline

(3) Show: pathlines

### 8.1b. Determine the parameters that are important to the predictions using prediction scaled sensitivities and parameter correlation coefficients

The input files for Exercise 8.1b have been constructed and are in
directories *ex8.1b*\\ and *ex8.1b-pcc*\\.

In folder *ex8.1b*\\, batch file *01-ucode-obs-sensitivity-noprior.bat*
runs UCODE-2005 in Sensitivity Analysis mode with the prior information
on parameters K_RB and VK_CB removed, to calculate the *pcc* with only
the calibration observations. The process model is a MODFLOW-2005 run of
the steady-state model without pumping; files for this simulation are
located in folder *data-ss*. Examine file
*01-ucode-obs-sensitivity-noprior.in* and verify that the input is
correct for performing a run in Sensitivity Analysis mode with no prior
information.

Batch file *02-ucode-prediction.bat* calculates the prediction scaled
sensitivities (*pss*) for the advective transport predictions, using
UCODE input file *02-ucode-prediction.in*. The process models are the
MODFLOW-2005 run of the steady-state model with pumping and the MODPATH
run; files for these simulations are located in folder *data-preds*.
Examine the UCODE input file *02-ucode-prediction.in* and note that (1)
predictions=yes in the UCODE Control_Data block; (2) an extra parameter
for the predictive run, the effective porosity of aquifers 1 and 2
(POR_1&2) has been defined in the PARAMETER_DATA_FOR_PREDICTION block;
and (3) nine predictions have been defined, corresponding to transport
in the x, y, and z directions at 10, 50, and 100 years; and (4) prior
information has been defined for the effective porosity parameter. The
weights defined for the prior information are discussed under Exercise
8.1c on p. 200 of Hill and Tiedeman.

To obtain parameter correlation coefficients with the calibration
observations and the predictions, the predictions need to be defined as
observations and included with the calibration observations in a
Sensitivity Analysis mode run. In folder *ex8.1b-pcc*, this has been
done in input file *01-ucode-obs-pred-sensitivity.in*. Examine the
contents of this file.

-   In folder *ex8.1b*\\, run batch file
    01-ucode-obs-sensitivity-noprior.bat to calculate the pcc without
    prior information on VK_CB and K_RB. The pcc matrix is in the main
    UCODE output file; pcc \> 0.90 are in listed *ex8.1b.\_pcc*.

-   In folder ex8.1b\\, run batch file 02-ucode-prediction.bat to
    calculate the pss. Of the four different pss files produced by
    UCODE-2005, data-exchange file \_*sppr* contains the scaling used to
    produce Figure 8.8 of Hill and Tiedeman (p. 198).

-   In folder *ex8.1bobs-pred*, run batch file
    *01-ucode-obs-pred-sensitivity.bat* to calculate the pcc with both
    the calibration observations and the predictions.

-   Verify that the output from these runs is similar to that shown in
    Figure 8.8 and Tables 8.4 and 8.5.

### Overview of OPR-PPR input files for Exercises 8.1c, 8.1d, and 8.1f

The *opr* and *ppr* statistics are calculated using the computer
software OPR-PPR (Tonkin et al., in press). OPR-PPR is designed to use
as input up to seven data-exchange files that are produced as output by
UCODE_2005 calibration and prediction simulations. These include files
containing general model data (\_*dm*, \_*dmp*), sensitivities (\_*su*,
\_*spu*, \_*supri*, \_*suprip*), and weighting ( \_*wt*, \_*wtpri*,
\_*wtprip*) files. The only additional, user prepared, file that is
needed by OPR-PPR is its main input file, which specifies the type of
analysis to be conducted and various *opr* and *ppr* calculation
options. Input instructions for the OPR-PPR main input file are in the
pdf version of the OPR-PPR documentation in your *Documentation+*
folder.

The UCODE data-exchange files from the sensitivity analysis and
prediction runs of Exercise 8.1b that are used in the OPR-PPR
calculations of Exercises 8.1c, 8.1d, and 8.1f are:

ex8.1b.\_dm model data file

ex8.1b.\_dmp model data file from prediction run

ex8.1b.\_su existing observations sensitivity file

ex8.1b.\_wt existing observations weight file

ex8.1b.\_spu prediction sensitivities file

ex8.1b.\_suprip sensitivities file for prior on prediction-only
parameters

ex8.1b.\_wtprip weights file for prior on prediction-only parameters

### Exercises 8.1c, d, f. Using GW_Chart

GW_Chart can now be used to plot the ppr and opr statistics. Navigate to
the appropriate directory from GW_Chart. All the ppr and opr values or
average values for each parameter or observation (averaged over the
predictions) can be plotted.

### 8.1c. Determine the parameters that are important to the predictions using the parameter-prediction statistic

In folder *ex8.1c*, the OPR-PPR main input files for the *ppr* runs of
Exercise 8.1c are *ppr-pargroups-no.in* and *ppr-pargroups-yes.in*. Both
these files are in folder *ex8.1c* and folder *ex8.1c-predgroups*. In
the input files for the latter folder, a prediction group is defined
that includes all 9 of the predictions, so the values listed in the
*\_ppr* output file for individual parameters (*pargroups-no*) are the
average *ppr* statistics for all predictions, as shown in Figure 8.9a of
Hill and Tiedeman (p. 201).

-   Run batch files *01a-PPR-PARGROUPS-NO.BAT* and
    *01b-PPR-PARGROUPS-YES.BAT* in both folders *ex8.1c* and folder
    *ex8.1c-predgroups*, to produce the *ppr* statistics for individual
    and groups of parameters.

### 8.1d. Assess the importance of existing observations to the predictions using the observation-prediction (opr) statistic

In folder *ex8.1d*, the OPR-PPR main input file for the *opr* run of
Exercise 8.1d is *opromit.in*.

-   Run batch file *01-opromit.bat* to produce the *opr* statistics
    shown in Figure 8.10a of Hill and Tiedeman. The *\_OPR_ABSCHG* file
    contains the values plotted in Figure 8.10b.

### 8.1e Assess the likely importance of potential new observations to the predictions using dimensionless and composite scaled sensitivities and parameter correlation coefficients

In folder *ex8.1e*\\, batch file *01-ucode-obs-newobs-sensitivity.bat*
runs UCODE-2005 in Sensitivity Analysis mode with the prior information
on parameters K_RB and VK_CB removed, and with the calibration
observations and the potential new observations. The process model for
simulating the new observations is a MODFLOW-2005 run of the
steady-state model with pumping, and with the potential head and flow as
observations (in files *ex8.obh* and *ex8.obr*); files for this
simulation are located in folder *data-new-obs*. Examine file
*01-ucode-obs-newobs-sensitivity.in* and verify that the input is
correct for performing this UCODE run.

-   In folder *ex8.1e*\\, run batch file
    *01-ucode-obs-newobs-sensitivity.bat* to calculate the *pcc* with
    the potential new observations and the calibration observations. The
    pcc produced should be similar to those in Table 8.7(c) of Hill and
    Tiedeman. The pcc matrix is in the main UCODE output file; pcc \>
    0.90 are listed in *ex8.1e.\_pcc*.

-   To produce the pcc matrix shown in Table 8.7(a) of Hill and
    Tiedeman, open file *01-ucode-obs-newobs-sensitivity.in* and in the
    OBSERVATION_GROUPS block, change the USEFLAG for observation
    'newhead' to 'no'. To produce the *pcc* matrix shown in Table 8.7(b)
    of Hill and Tiedeman, change the USEFLAG for observation 'newhead'
    back to 'yes', and change the USEFLAG for observation 'newflow' to
    'no'.

### 8.1f Assess the likely importance of potential new observations to the predictions using the observation-prediction (opr) statistic

Files for the first part of this analysis are in folder *ex8.1f*.

To produce the *opr* statistics for the two potential new observations,
first a UCODE sensitivity analysis run is needed to obtain the
sensitivities of the simulated equivalents of the potential new
observations, and then OPR-PPR is run for an OPRADD analysis.

For the UCODE-2005 run, with main input file *01-ucode-new-obs.in*, the
process model is that in folder *data-new-obs*, described above. This
UCODE-2005 run produces the sensitivity file *ex8.1f-new.\_su*, which is
listed in the main OPR-PPR input file for this exercise, *opradd.in*, so
it can be read by OPR-PPR when conducting the analysis of adding the
potential new observations. Weights on the potential observations are
required for the OPRADD analysis; for this example the weights on the
potential head and flow are the same as those on the head and flow
calibration observations, respectively.

This run produces an error message from UCODE_2005 because it tries to
calculate a variance-covariance matrix using the two potential
observations. Given the six parameters defined, this produces a singular
matrix and the program fails trying to take the inverse. All we want are
the sensitivities, which are produced. So, while annoying, the error is
not problematic for our purposes.

-   First run batch file *01-ucode-new-obs.bat*. Then run batch file
    *01-opradd.bat* to produce the *opr* statistics shown in Figure 8.12
    of Hill and Tiedeman.

Files for the second part of this analysis are in folder *ex8.1f+grid*.
The capability in OPR-PPR of calculating *opr* statistics for a
potential new head observation at every node of the model requires a
'grid sensitivity' file containing the sensitivity of head to all model
parameters, for every model node. This file can easily be produced using
the Sensitivity Process of MODFLOW-2000, but not yet MODFLOW-2005.
Obtaining it using UCODE_2005 is tedious. For this exercise, we use a
file produced in a MODFLOW-2000 run, under pumping conditions (file
*withpumping.\_grid-sensitivities*). The grid sensitivities file can be
produced using any process model, as long as it is formatted so that it
can be read by OPR-PPR. This run also requires weights for the potential
new observations at every model node; these are the same as those for
the calibration observations of head and are specified in file
*grid.\_wt*. These grid sensitivity and grid weights files are listed in
the main OPR-PPR input file for this analysis.

-   Run batch file *01-opraddnode.bat* to produce the grid of *opr*
    statistics used to produce Figure 8.13 of Hill and Tiedeman. Values
    in file opraddnode.\_OPCNOD are used to produce Figure 8.14.

## Exercise 8.2. Prediction Uncertainty Measured Using Inferential Statistics

### 8.2a. Calculate linear confidence intervals on the components of advective transport

The post processing program LINEAR_UNCERTAINTY is used to calculate
linear confidence intervals on predictions. This run requires input from
a sensitivity analysis or regression run, and from a prediction run. In
folder *ex8.2a*, batch file *01-ucode-obs-sensitivity.bat* produces the
sensitivities for the calibration observations, calculated at the
optimal parameter values, and batch file *02-ucode-prediction.bat*
produces the prediction sensitivities. Consider these batch files in the
context of the flow chart for calculating linear intervals presented by
Poeter and others (2005, p. 158) and repeated here.

![](media/image1.wmf)

-   Run batch files *01-ucode-obs-*sensitivity.bat and
    02-ucode-prediction.bat in sequence

-   Run batch file 03-linunc.bat to produce the linear confidence
    intervals. The LINEAR_UNCERTAINTY program has no main input file;
    all input comes from the data-exchange files produced in the
    sensitivity analysis and prediction runs. The main output file is
    *ex8.2a.#linunc*, and the data-exchange file containing the
    intervals is *ex8.2a.\_linp*.

    The program LINEAR_UNCERTAINTY always calculates confidence and
    prediction intervals. Find them in the output file ending in \_linp.

### 8.2b. Calculate nonlinear confidence intervals on the components of advective transport

Files for calculating nonlinear confidence intervals are provided in
directory initial\\ex8.i\\ucode-opr-ppr-runs\\ex8.2b. These files are
constructed usingthe steps listed by Poeter and others (2005, p.
216-218). Look at the following aspects of these files.

-   Inspect the batch files that are used to accomplish the runs noted
    in the following flow chart, which is modified slightly from Poeter
    and others (2005, p. 193). They are named
    *01-ucode-sensitivity.bat*, *01-zcreate-init.bat*,
    *02-ucode-prediction.bat*, *03-corfac.bat*, and
    *04-ucode-nonlin-uncert.bat*. Consider these batch files in the
    context of the flow chart for calculating nonlinear intervals
    presented by Poeter and others (2005, p. 193). A modified version is
    repeated here.

![](media/image2.wmf)

\* is replaced by "conf" when confidence intervals are calculated and
"pred" when prediction intervals are calculated. (see Poeter and others,
2005, p. 189-190).

\*\* is replaced as described by dm, mv, su, and, when prior information
is defined, supri. See Poeter and others (2005, p. 189-191).

-   Inspect the UCODE_2005 main input files to determine how the
    04-ucode-nonlin-uncert.bat accounts for the calibration conditions
    and observations, and predictive conditions and predictions, which
    are accounted for separately in other input files. In particular,
    identify the following.

1.  The model command lines in the different files.

2.  The parameter definition in the different files. Note that the
    porosity parameter is included for all runs, even the sensitivity
    runs for which is has no affect on the simulated values. Future
    versions of UCODE_2005 may allow parameters that apply only to
    predictions to be omitted from the runs to which they do not apply.

3.  The observations and predictions in the different files.

4.  The prior information in the different files (note that it is the
    same for all runs in this set of files).

5.  The model input files and template files defined for each run.

6.  The model output files and instruction files defined for each run.

-   Inspect how the limits calculated are controlled by the input block
    Reg_GN_NonLinInt, keyword WhichLimits, and the input blocks
    Prediction_Groups and Prediction_Data.

-   Run the batch files *01-ucode-sensitivity.bat*,
    *01-zcreate-init.bat*, *02-ucode-prediction.bat*, *03-corfac.bat*,
    and *04-ucode-nonlin-uncert.bat*

-   Inspect the output file *ex8.2b.\_intconf*. Identify the interval
    limit, the sum-of-squared weighted residuals at the limit, and the
    objective function goal. Note how the identifier in the first column
    is negative for the lower limit and positive for the upper limit.

-   Inspect the parameter values used to calculate each of the limits
    are listed in the output file *ex8.2b.\_intconfpar*.

-   Inspect the summary of the regressions for each limit in the output
    file *ex8.2b.\_intconfsum*.

-   Inspect the weighted residuals calculated using the parameter values
    for each limit in the output file *ex8.2b.\_intconfwr*.

## Exercise 9.1 and 9.2: Simulate Transient Hydraulic Heads and Perform Preparatory Steps

Prepare input for transient simulation using parameters, and run the
simulation in forward mode. The parameter values listed in table 2
should be used in the input files. These will be the starting values
when nonlinear regression is performed.

Table 2: Parameter names and starting values for properties of the
transient system for which parameters are estimated in the exercises.

\[Same as Table 9.2 of Hill and Tiedeman (2007, p. 231)\]

  ------------------------------------------ --------------- -------------
  Flow-system property                       Parameter name  Starting
                                                             value

  Specific Storage of layer 1 (m^-1^)        SS_1            2.65e-5

  Specific Storage of layer 2 (m^-1^)        SS_2            4.0E-6

  Pumping rate in each model layer 1 and 2   Q_1&2           -1.1
  (m^3^/s)                                                   
  ------------------------------------------ --------------- -------------

[Instructions]{.underline}:

-   In MFI2005, save the *ex4.1* dataset as dataset *ex9.1* and use the
    *ex9.1* dataset in this exercise.

-   DIS input: Click "Stress Periods". Specify that the simulation will
    include a total of 5 stress periods. The first stress period will be
    steady-state, and the others will be transient. Stress-period
    length, number of time steps, and time-step multiplier are shown in
    the following table. (Harbaugh, 2005, p. 4-4 to 4-4, 8-11 to 8-14)

  ---------- ---------------- -------------- -------------- --------------
  Stress     Stress-period    Number of time Time-step      SS or TR
  period     length (seconds) steps          multiplier     

  1          0                1              1.0            SS

  2          87,162           1              1.0            TR

  3          261,486          1              1.0            TR

  4          522,972          1              1.0            TR

  5          23,567,451       9              1.2            TR
  ---------- ---------------- -------------- -------------- --------------

-   Internal Flow input: LPF. Click "Hydraulic".

    Two SS parameters (see table 2 for parameter values):

    Define parameter SS_1 to control specific storage (SS) of layer 1.

    Define parameter SS_2 to control SS of layer 2.

```{=html}
<!-- -->
```
-   Stress input: Click "RCH". Activate both RCH parameters for all
    stress periods.

-   Click "RIV". Activate parameter K_RB for all stress periods.

-   Click "GHB". Activate all GHB cells for all stress periods..
    Non-parameter data can be reused by selecting the "Previous" option.

The General-Head boundary Package is used to represent the boundary
condition on the side of the region farthest from the stream. It is not
defined using a parameter, and its activation for additional stress
periods is different than for other aspects of the system. For GHB,
activate for additional stress periods by clicking on the stress period
and clicking on "Previous". This makes the definition of the input the
same as the previous stress period. Start with stress period 2. Here,
stress period 5 has been selected and "Previous" is ready to be
selected.

![](media/image3.png){width="5.9944444444444445in"
height="3.134027777777778in"}

-   Click "WEL". Define one parameter called Q_1&2 to control two well
    cells, in layers 1 and 2, row 9, column 10. See table 2 above for
    parameter value. Activate the parameter for stress periods 2-5.
    (Harbaugh, 2005, p. 8-40 to 8-42)

-   Click "Parameter Values". Click button at upper left "Select
    Parameter Values..." and select the three new parameters.

-   Save the dataset and execute MODFLOW-2005 from the File menu. Output
    files are produced in directory *exer*.

-   Evaluate the resulting heads as follows.

The MFI2005 default is to print the final heads of the simulation in the
LST output file. A good check when running a transient model that starts
with a steady-state stress period is to compare the heads from the
initial steady-state stress period with the heads from the model that
included only the steady-state stress period. To print the heads from
the steady-state stress period, do the following.

-   Click "Output"

-   Choose "Times for Printing Heads"

-   Select list

-   List two print times: stress period 1, timestep 1, and stress period
    5, timestep 9.

-   Rerun MODFLOW-2005.

-   Heads will be printed to output file ex9.1.lst.

Compare the steady-state heads with the values in ex4.1.lst.

Evaluate the heads at later time steps by looking at them in the output
file or by plotting them using ModelViewer. Presently, only the final
heads are saved. Additional heads can be saved using the MFI2005 Output
menu mentioned above.

Also look at the global budgets in the output file. Is the system at
steady-state at the end of the simulation? That is, is the change in
storage small?

## Exercise 9.3: Transient Parameter Definition

Define the three parameters listed in Table 9.2 of Hill and Tiedeman
(2007, p. 231). Double check that the proper layers and stress periods
are used. For example, pumpage applies only to stress periods 2 through
5. Also, under Parameter Values, use the "Select parameters to be in
PVAL file" button.

## Exercise 9.4: Observations for the Transient Problem

Transient observations are listed in tables 9.3 and 9.4 of Hill and
Tiedeman, p. 232-234. They are also provided in files in the
initial\\ex9.4.i\\ directory. Input instructions for observations for
MODFLOW-2005 are provided in OBS.PDF. Input instructions for
observations for UCODE_2005 are provided in Poeter and others (2005).

-   In MFI2005, save the *ex9.1* dataset as dataset *ex9.4* and use the
    *ex9.4* dataset in this exercise.

-   To make use of a partially complete file already prepared, exit
    MFI2005 and copy the file *ex9.4.hob* from the
    *exer\\initial.mfi2005\\ex9.4.i* directory to the *exer* directory,
    overwriting the existing *ex9.4.hob* file. Restart MFI2005.

-   Use the "Multi-Time Head" button to add observations for well 1
    (hd01.ss, dd01.1, dd01.283; here we use slightly different
    observation names than those defined by Hill and Tiedeman, 2007).

```{=html}
<!-- -->
```
-   Set TOMULTH to 86400, which allows the time offset (TOFFSET) from
    the beginning of the reference stress period (IREFSP) to be in units
    of days, although the model time unit is seconds.

-   For the header name, use well1 and well2, respectively.

-   When transient data are defined, MODFLOW-2005 allows the regression
    to use either the measured hydraulic heads or the initial hydraulic
    head and subsequent drawdowns as observations. Specify that
    drawdowns are to be calculated by MODFLOW-2005 and used in the
    objective function (applies to observations following the first for
    each well). In both cases, HOBS needs to be head; the program will
    take the difference to produce drawdowns. The statistic for all but
    the initial value depends on whether heads or drawdowns are used at
    these times. This is because subtracting subsequent heads from the
    initial heads results in some types of errors being omitted from the
    resulting drawdowns. See Hill and Tiedeman (2007, p. 231).

```{=html}
<!-- -->
```
-   Use the "RIV Observations" button to add the two gains observed in
    river flow presented in table 9.4 of Hill and Tiedeman (2007, p.
    234).

```{=html}
<!-- -->
```
-   Set the time offset multiplier to 86400 so that observation-time
    offsets can be entered in days.

-   Flow is out of the ground-water system. MODFLOW sign conventions
    require observed river gains to be negative.

-   Assign plot symbol as 2.

```{=html}
<!-- -->
```
-   Use MFI to run MODFLOW-2005, and examine the tables of simulated and
    observed heads and flows.

## Exercise 9.5: Evaluate Transient Model Fit Using Starting Parameter Values

Address the problem in Hill and Tiedeman (p. 235) using the table of
simulated equivalents of the observations, the residuals, and weighted
residuals located at the end of the file *ex9.4.lst*.

## Exercise 9.6: Sensitivity Analysis for the Initial Model

Producing the results needed to conduct a sensitivity analysis of the
initial model can be accomplished by the following steps. Exercises 9.6
b and c (Hill and Tiedeman, 2007, p. 240 and 243) are addressed using
results in the *ex9.6.\_sc* and *ex9.6.\_pcc* files, respectively.

-   Save the *ex9.4* model to *ex9.6*.

-   Click UCODE. Click the button next to "Sensitivities - Central".

-   As before, we will begin without prior information to obtain a clear
    understanding of what information the observations provide on the
    parameter. Remove the prior information by deleting the PriorNames.
    Click OK. Click UCODE again to see that, indeed, the prior
    information has been removed.

-   Use MFI to run UCODE.

-   Use GW_Chart to look at the results in the files *ex9.6.\_sc* and
    *ex9.6.\_pcc.*

## Exercise 9.7: Estimate Parameters for the Transient System by Nonlinear Regression

Regression can be performed by the following steps.

-   Save the ex9.6 model to ex9.7.

-   Click "Parameter Values" and log-transform the six parameters
    related the hydraulic conductivity and storage characteristics.
    Click OK.

-   Click UCODE. Click the button next to "Optimize". Click OK.

-   Use MFI to save the files and run UCODE.

-   The results presented in Hill and Tiedeman (2007, p. 245 to 246) for
    this problem were produced by MODFLOW-2000 using
    sensitivity-equation sensitivities. Here, we are using less accurate
    forward-difference sensitivities. For this problem, the method used
    to solve the regression matters when forward difference
    sensitivities are used. In the UCODE menu on the right side, the
    input for the Reg_GN_Controls input block of UCODE_2005 are
    presented. If the only change made is to change "Trustregion" to
    "Dogleg", regression converges in eight iterations. If
    Trustregion=no or HookStep, regression does not converge in 10
    iterations; if Generally, the HookStep method is the most robust,
    but for this problem the DogLeg method performs better.

-   The results shown in Hill and Tiedeman (2007, figure 9.12a, p. 245)
    are listed in two tables located toward the end of file
    *ex9.7.#uout*. All of the methods achieve nearly the same model fit
    and estimated parameters, despite two being unable to satisfy a
    TolPar convergence criterion of 0.01.

-   Produce the graph shown in Hill and Tiedeman (2007, figure
    9.12b, p. 245) using GW_Chart and file *ex9.7.\_pa*.

## Exercise 9.8: Evaluate Measures of Model Fit

A modified version of the fragment of output shown in the lower part of
Figure 9.13 of Hill and Tiedeman (2007, p. 246) occur toward the end of
the output file *ex9.7_ucode.#uout*. The new versions provide confidence
intervals on the calculated error variance and the standard error of the
regression, to make it easier to determine whether the values are
significantly different than 1.0. Here, we see that they are not, which
is consistent with our knowledge that the model is correct and the
weights correctly reflect the observation erros.

LEAST-SQUARES OBJ FUNC (OBS. ONLY)\-\-\-\-- = 23.992

NUMBER OF INCLUDED OBSERVATIONS\-\-\-\-\-\-\-- = 35 OF 35

NUMBER OF PRIOR ESTIMATES\-\-\-\-\-\-\-\-\-\-\-\-\-- = 0

NUMBER OF ESTIMATED PARAMETERS\-\-\-\-\-\-\-\-- = 9

CALCULATED ERROR VARIANCE (CEV)\-\-\-\-\-\-\-- = 0.92275

95% CONFIDENCE INTERVAL ON CEV\-\-\-\-\-\-\-\-- = 0.57232 1.7335

STANDARD ERROR \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-- = 0.96060

95% CONFIDENCE INTERVAL ON STD ERR\-\-\-\-- = 0.75652 1.3166

CORRELATION COEFFICIENT\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-- = 0.99999

ITERATIONS\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-- = 8

STATISTICS FOR EVALUATING ALTERNATIVE MODELS:

No prior used in this parameter estimation

MAX LIKE OBJ FUNC OBSERVATIONS ONLY (MLOFO)\-\-\-\-\-\-\-\-\-\-- =
-13.218

LN DETERMINANT of Fisher Information Matrix (OBS ONLY) = 118.51

MODEL EVALUATION MEASURES:

OBSERVATIONS ONLY (no prior used in this parameter estimation)

AIC STATISTIC\-\-\-\-\-\-\-\-\-\-\-\-- = 6.7823

AICc STATISTIC\-\-\-\-\-\-\-\-\-\-\-- = 15.949

BIC STATISTIC\-\-\-\-\-\-\-\-\-\-\-\-- = 22.336

KASHYAP STATISTIC\-\-\-\-\-\-\-\-- = 77.473

## Exercise 9.9: Perform Graphical Analyses of Model Fit and Evaluate Related Statistics.

The graphs shown in Figure 9.14 of Hill and Tiedeman (2007, p.247) can
be produced using GW_Chart and files (a) *ex9.7_ucode.\_ws*, (b)
*ex9.7_ucode .\_ww*, and(c) *ex9.7_ucode .\_os*. The graph of Figure
9.17a can be produced using GW_Chart and the file *ex9.7_ucode.\_nm*.

## Exercise 9.10: Evaluate Estimated Parameter Values

Figure 9.18 of Hill andTiedeman (2007, p.251) can be constructed using
GW_Chart and the file ex9.7_ucode.\_sc.

A graph similar to the one shown in Figure 9.19 can be constructed using
GW_Chart and file ex9.7_ucode.\_pc if the reasonable lower and upper
limits are defined consistently.

The large parameter correlation coefficients of Table 9.7 are listed in
file *ex9.7_ucode.\_pcc*.

## Exercise 9.11: Test for Linearity

Model nonlinearity can be evaluated as for exercise 7.3 using the
following sequence of runs. There are no complicating factors such as
the need to log-transform parameters (they are already transformed). The
sequence of steps, briefly, is as follows.

-   Begin by using MFI2005 to copy the *ex9.7* files to *ex9.11*.

-   Use MFI2005 and the Optimize option in the UCODE menu.

-   Use MFI2005 and the Linearity option in the UCODE menu.

-   Execute ex9.11-model_linearity.bat

-   Execute ex9.11_ucode.corfac

-   Use MFI2005 with the ModelLinearity Adv=Conf in the UCODE menu.

-   Execute *ex9.11-model_linearity_adv.bat*

## Exercises 9.12: Predictions

Files are not provided for this exercise. Please perform the exercises
using results provided in Hill and Tiedeman (2007).