PRECIPITATION

ET INTERCEPTION

RECHARGE

PERCOLATION

INFILTRATION

RUNOFF

UNSATURATED

ZONE

REJECTED

RECHARGE ET

STREAM

CHANNEL

VERTICAL BASE FLOW

LEAKAGE

HSPF

MODFLOW

WATER TABLE

HORIZONTAL FLOW

HORIZONTAL FLOW

SATURATED

ZONE

SAS

ICU

UFA VERTICAL

LEAKAGE

Application of Nonlinear Least-Squares

Regression to Ground-Water Flow

Modeling, West Central Florida

U.S. Department of the Interior

U.S. Geological Survey

Prepared in cooperation with

TAMPA BAY WATER and the

SOUTHWEST FLORIDA WATER MANAGEMENT DISTRICT

Water-Resources Investigations Report 00-4094
Cover photograph: Crystal Springs

(Manny Lopez, Southwest Florida Water Management District)
U.S. GEOLOGICAL SURVEY

Water-Resources Investigations Report 00-4094 

Tallahassee, Florida

2000

Application of Nonlinear Least-Squares 

Regression to Ground-Water Flow Modeling, 

West-Central Florida

Prepared in cooperation with

TAMPA BAY WATER and the

SOUTHWEST FLORIDA WATER MANAGEMENT DISTRICT

By Dann K. Yobbi
U.S. DEPARTMENT OF THE INTERIOR

BRUCE BABBITT, Secretary

U.S. GEOLOGICAL SURVEY

Charles G. Groat, Director

Copies of this report can be 

purchased from:

U.S. Geological Survey

Branch of Information Services

Box 25286

Denver, CO 80225-0286

888-ASK-USGS

The use of firm, trade, and brand names in this report is for identification purposes only and 

does not constitute endorsement by the U.S. Geological Survey.

For additional information

write to:

District Chief

U.S. Geological Survey

Suite 3015

227 N. Bronough Street

Tallahassee, FL 32301

Additional information about water resources in Florida is available on the

World Wide Web at http://fl.water.usgs.gov
Contents iii

CONTENTS

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

Introduction ........................................................................................................................................................................... 1

Purpose and Scope....................................................................................................................................................... 2

Description of the Study Area ..................................................................................................................................... 2

Description of the Existing CNTB Area Hydrologic Model ....................................................................................... 6

Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling....................................................... 12

Nonlinear Least-Squares Regression Method.............................................................................................................. 12

Simulation Model ........................................................................................................................................................ 13

Observation Data ............................................................................................................................................... 15

Parameter Structure............................................................................................................................................ 15

Model Analysis............................................................................................................................................................ 15

Initial Parameter Sensitivities and Correlations (phase 1)................................................................................. 21

Recharge................................................................................................................................................... 21

Hydraulic Conductivity of the Surficial Aquifer System......................................................................... 24

Leakance of the Intermediate Confining Unit.......................................................................................... 25

Transmissivity of the Upper Floridan aquifer.......................................................................................... 26

Anisotropy of the Upper Floridan aquifer................................................................................................ 28

Hydraulic–Head Sensitivity ..................................................................................................................... 28

Optimal Estimates of Parameter Values (phase 2) ............................................................................................ 29

Model Agreement .............................................................................................................................................. 44

Limitations of Model Analysis .......................................................................................................................... 51

Summary................................................................................................................................................................................ 52

Selected References............................................................................................................................................................... 52

Appendix—Initial and optimized parameter values, relative composite sensitivity for optimal estimates of 

input parameter values, and correlation between parameters ........................................................................................... 56

FIGURES

1-3. Maps showing:

1. Location of study area and well fields........................................................................................................... 3

2. Physiographic provinces................................................................................................................................ 4

3. Location of rivers and springs ....................................................................................................................... 5

4. Chart showing generalized stratigraphic and hydrogeologic section, west-central Florida.................................. 6

5-8. Maps showing:

5. Primary components of the surface-water system simulated by HSPF and the ground-water system 

simulated by MODFLOW ............................................................................................................................. 7

6. Model grid used in simulation of ground-water flow system........................................................................ 8

7. Model boundary conditions for the surficial aquifer system (layer 1) .......................................................... 9

8. Model boundary conditions for the Upper Floridan aquifer (layer 2)........................................................... 10

9. Diagram showing simulated annual average hydrologic budget for the central northern Tampa Bay

area hydrologic model 1971-93............................................................................................................................. 11

10-28. Maps showing:

10. Areal distribution of simulated rejected recharge from the surficial aquifer system, calendar year 1987 .... 13
iv Contents

11. Simulated percolation (recharge) rates from the surface-water system to the ground-water system, 

calender year 1987 ........................................................................................................................................ 14

12. Watershed subbasins within the study area................................................................................................... 16

13. Calibrated hydraulic conductivity of the surficial aquifer system for the existing central northern 

Tampa Bay area hydrologic model ............................................................................................................... 17

14. Calibrated leakance of the intermediate confining unit for the existing central northern Tampa Bay 

area hydrologic model................................................................................................................................... 18

15. Calibrated transmissivity of the Upper Floridan aquifer for the existing central northern Tampa Bay 

area hydrologic model................................................................................................................................... 19

16. Calibrated anisotropy of the Upper Floridan aquifer for the existing central northern Tampa Bay 

area hydrologic model................................................................................................................................... 20

17. Areas of the central northern Tampa Bay area hydrologic model that are insensitive to the estimation 

of recharge..................................................................................................................................................... 22

18. Simulated depth to water table using the initial central northern Tampa Bay area hydrologic model 

parameter values, calendar year 1987 ........................................................................................................... 24

19. Areas of the central northern Tampa Bay area hydrologic model that are insensitive to the estimation 

of leakance .................................................................................................................................................... 26

20. Areas of the central northern Tampa Bay area hydrologic model that are insensitive to the estimation 

of transmissivity of the Upper Floridan aquifer............................................................................................ 27

21A. Sensitivity of simulated water levels to a 2-percent increase in the initial value of recharge ...................... 30

21B. Sensitivity of simulated water levels to a 2-percent increase in the initial value of recharge ...................... 31

22A. Sensitivity of simulated water levels to a 2-percent increase in the initial value of hydraulic

conductivity of the surficial aquifer system.................................................................................................. 32

22B. Sensitivity of simulated water levels to a 2-percent increase in the initial value of hydraulic

conductivity of the surficial aquifer system.................................................................................................. 33

23A. Sensitivity of simulated water levels to a 2-percent increase in the initial value of leakance ...................... 34

23B. Sensitivity of simulated water levels to a 2-percent increase in the initial value of leakance ...................... 35

24A. Sensitivity of simulated water levels to a 2-percent increase in the initial value of Upper Floridan 

aquifer transmissivity.................................................................................................................................... 36

24B. Sensitivity of simulated water levels to a 2-percent increase in the initial value of Upper Floridan 

aquifer transmissivity.................................................................................................................................... 37

25A. Sensitivity of simulated water levels to a 2-percent increase in the initial value of Upper Floridan 

aquifer anisotropy ratio ................................................................................................................................. 38

25B. Sensitivity of simulated water levels to a 2-percent increase in the initial value of Upper Floridan 

aquifer anisotropy ratio ................................................................................................................................. 39

26. Simulated rate of leakage through the intermediate confining unit based on steady-state 1987 conditions 40

27. Delineation of Upper Floridan aquifer zones used for parameter estimation and simulated 

transmissivity-zone multiplier for the initial parameter values .................................................................... 41

28. Delineation of leakance zones of the intermediate confining unit used for parameter estimation and 

simulated leakance-zone multiplier for the initial parameter values ............................................................ 42

29-32. Graphs showing:

29. Relative composite sensitivity for optimal estimates of input parameter values.......................................... 43

30. Simulated parameter-zone multipliers derived from two separate sets of water-level observations ............ 44

31. Comparison of simulated to measured water levels for the parameter-estimation model ............................ 46

32. Comparison of simulated to measured water levels for the initial central northern Tampa Bay area 

hydrologic model .......................................................................................................................................... 47

33-34. Maps showing

33. Location of water-level measurements, value of water-level residuals, and simulated water table in 

the surficial aquifer system, calendar year 1987.......................................................................................... 48

34. Location of water-level measurements, value of water-level residuals, and simulated potentiometric 

surface of the Upper Floridan aquifer, calendar year 1987........................................................................... 49
Contents v

TABLES

1. Relative composite sensitivity, number of water-level measurements, and areal size of each recharge zone

for the initial parameter values ............................................................................................................................... 23

2. Relative composite sensitivity, number of water-level measurements, and areal size of each hydraulic 

conductivity zone of the surficial aquifer system for the initial parameter values................................................. 25

3. Relative composite sensitivity, number of water-level measurements, and areal size of each leakance zone 

of the intermediate confining unit for the initial parameter values ........................................................................ 25

4. Relative composite sensitivity, number of water-level measurements, and areal size of each transmissivity 

zone of the Upper Floridan aquifer for the initial parameter values....................................................................... 27

5. Relative composite sensitivity, number of water-level measurements, and areal size of each Upper Floridan 

aquifer anisotropy zone for the initial parameter values ........................................................................................ 28

6. Statistical summary of optimal parameter-zone multipliers derived from 119 and 549 water-level 

observations ............................................................................................................................................................ 45

7. Statistical summary of differences between simulated and measured water levels for the initial and 

optimized simulations............................................................................................................................................. 45

8. Comparison of model-simulated and estimated base flows, calendar year 1987 ................................................... 50

9. Comparison of model-simulated and measured spring flows, calendar year 1987 ............................................... 51

10. Simulated hydrologic budget of the aquifer system in the study area, calendar year 1987 ................................... 51
vi Contents
Introduction 1

Application of Nonlinear Least-Squares Regression to 

Ground-Water Flow Modeling, West-Central Florida

By Dann K. Yobbi

Abstract

A nonlinear least-squares regression technique for estimation of ground-water flow model 

parameters was applied to an existing model of the 

regional aquifer system underlying west-central 

Florida. The regression technique minimizes the 

differences between measured and simulated 

water levels. Regression statistics, including 

parameter sensitivities and correlations, were calculated for reported parameter values in the existing model. Optimal parameter values for selected 

hydrologic variables of interest are estimated by 

nonlinear regression. Optimal estimates of parameter values are about 140 times greater than and 

about 0.01 times less than reported values. Independently estimating all parameters by nonlinear 

regression was impossible, given the existing 

zonation structure and number of observations, 

because of parameter insensitivity and correlation. 

Although the model yields parameter values similar to those estimated by other methods and reproduces the measured water levels reasonably 

accurately, a simpler parameter structure should be 

considered. Some possible ways of improving 

model calibration are to: (1) modify the defined 

parameter-zonation structure by omitting and/or 

combining parameters to be estimated; (2) carefully eliminate observation data based on evidence 

that they are likely to be biased; (3) collect additional water-level data; (4) assign values to insensitive parameters, and (5) estimate the most 

sensitive parameters first, then, using the optimized values for these parameters, estimate the 

entire data set. 

INTRODUCTION

A variety of techniques have been used to analyze the water-supply problems in west-central Florida 

including the use of numerical ground-water flow models. A common use of the numerical models is to predict the response of an aquifer to planned stresses. 

While the mathematical and computational aspects of 

such response predictions are reasonably well developed, the question of confidence in parameter estimates 

has not been completely resolved. Model calibration is 

traditionally accomplished by manual trial-and-error 

approach during which the modeler iteratively selects 

parameter values to improve the model results using 

intuition about model response to changes in parameter 

values. A calibration obtained using a trial-and-error 

approach alone does not guarantee the statistically best 

solution. Consequently, there is no practical way to 

assess model uniqueness, if the observations support 

the level of model complexity, or if a simpler model 

would significantly improve model fit.

Model calibration can be facilitated using an 

inverse model (such as nonlinear least-squares regression), in which the parameter values are adjusted automatically to match field observations as closely as 

possible. Inverse modeling can improve the quality of 

ground-water models and yield results that are not 

readily available through trail-and-error calibration 

efforts (Poeter and Hill, 1996). Inverse modeling 

reveals data shortcomings and needs, lack of sensitivity 

of estimated parameters to calibration data, and 

extreme parameter correlation that can be easily overlooked during trial-and-error calibration. Using 

inverse modeling for model calibration quantifies the 

uncertainty in parameter estimates and statistically 

gives the most appropriate solution for the given input 

parameters. 
2 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

To facilitate the routine use of the nonlinear 

least-squares regression method of inverse modeling, 

the U.S. Geological Survey (USGS) in cooperation 

with Tampa Bay Water and the Southwest Florida 

Water Management District (SWFWMD), began a 

study in 1997 to apply this method to the existing central northern Tampa Bay (CNTB) area hydrologic flow 

model. Nonlinear least-squares regression was used to 

determine parameter sensitivities and correlations and 

to estimate parameter values of the existing model. 

Purpose and Scope 

The purpose of this report is to describe the 

application of nonlinear least-squares regression to the 

existing ground-water flow model of the CNTB area. 

The study was limited to the CNTB area hydrologic 

model defined by SDI Environmental Services, Inc. 

(SDI), (1997). The ground-water flow model utilized 

for this study is based on the data and information presented by SDI (1997) for the CNTB area hydrologic 

flow model. The hydrologic and geologic setting of the 

study area is briefly described in subsequent sections. 

A short description of the CNTB area hydrologic 

model and regression procedure follows. The remainder of the report is devoted to parameter-estimation 

analysis, as it was applied to the CNTB area hydrologic 

model. Discussions include descriptions of modeling 

procedures, evaluation of parameter-value regression 

statistics, including parameter sensitivities and correlations for parameter values reported by SDI (1997). The 

optimal set of parameter values and associated statistics determined by regression is presented last.

Description of the Study Area

Detailed descriptions of the study area are presented in SDI (1997) and SWFWMD (1993). The 

study area encompasses approximately 2,000 square 

miles (mi2

) that includes all of Pasco County; most of 

Hernando, Pinellas and Hillsborough Counties; and 

part of Polk County (fig. 1). Surface topography is 

characterized by relatively flat, marshy lowlands along 

the coast, rolling hills of intermediate relief throughout 

the central part of Pasco County, and sand terraces to 

the northeast. The most prominent topographic feature 

of the area is the Brooksville Ridge, which is located in 

central Hernando and eastern Pasco Counties (fig. 2). 

Land surface altitudes range from sea level near the 

coast to over 300 feet (ft) above sea level along the 

Brooksville Ridge.

Several rivers (six) and their tributaries, several 

small streams along the coast, and some internally 

drained and ephemeral streams that flow only during 

extreme rainfall events compose the surface-water system of the study area (fig. 3). The two largest riverine 

systems are the Hillsborough and Withlacoochee Rivers. There are hundreds of lakes, swampy plains, and 

intermittent ponds dispersed throughout the study area, 

ranging in size from less than 1/4 acre to more than 

2,500 acres. A large concentration of lakes exist in several areas, namely northwest Hillsborough County, 

central Pasco County, and along the Brooksville and

Lakeland Ridges.

Numerous springs (17) are located in the study 

area and are either found inland, flowing to adjacent 

rivers or along the coast, discharging directly to the 

Gulf of Mexico (fig. 3). The two most important 

springs are Weeki Wachee Spring, located in western 

Hernando County and Crystal Springs, located along 

the northern reaches of the Hillsborough River. Crystal 

Springs provides a large portion of the Hillsborough 

River’s base flow while Weeki Wachee Spring is the 

source of flow in the Weeki Wachee River. 

The ground-water flow system beneath the study 

area is a multilayered system consisting of a thick 

sequence of carbonate rocks overlain by clastic deposits (fig. 4). The surficial deposits and carbonate rocks 

are subdivided into a hydrogeologic framework that 

forms a sequence of two aquifers and one confining 

unit. This framework includes the unconfined surficial 

aquifer system and the confined Floridan aquifer system. A low permeability intermediate confining unit 

separates the aquifers. The Floridan aquifer system 

consists of the Upper and Lower Floridan aquifers that 

are separated by a middle confining unit. The middle 

confining unit contains saltwater in the study area, and 

the freshwater flow is limited to the Upper Floridan 

aquifer. The Upper Floridan aquifer is underlain by 

low-permeablility evaporitic limestone that forms the 

bottom of the fresh ground-water flow system.
Introduction 3

PASCO COUNTY

HERNANDO COUNTY

PASCO COUNTY

HILLSBOROUGH COUNTY

HILLSBOROUGH COUNTY

POLK COUNTY

PINELLAS COUNTY

STUDY AREA

North Pasco BOUNDARY

Well field

Brooksville

Starkey

Well field

Cross Bar

Ranch

Well field

Cypress

Creek

Well field

Eldridge

Wilde

Well field

Cypress

Bridge

Wellfield

South Pasco

Well field

Cosme

Odessa

Well field

Section 21

Well field

Morris

Bridge

Well field

Cone Ranch

Well field

(proposed)

South Central

Hillsborough

Regional Well field

Northwest

Hillsborough

Well field

Land O

Lakes

New

Port

Richey

Temple

Terrace

Tarpon

Springs

Clearwater

Tampa

Brandon

Plant City

Zephyrhills

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

LOCATION OF

STUDY AREA

  



  

	   

0 5 10 MILES

WELL FIELD 0 5 10 KILOMETERS

EXPLANATION

FLORIDA

Figure 1. Location of study area and well fields.
4 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

PASCO COUNTY

HILLSBOROUGH COUNTY

HILLSBOROUGH COUNTY

POLK COUNTY

PINELLAS COUNTY

STUDY AREA

BOUNDARY

Brooksville

Land O

Lakes

New

Port

Richey

Temple

Terrace

Tarpon

Springs

Clearwater

Tampa

Brandon

Plant City

Zephyrhills

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

GULF

COAST

LOWLANDS

BROOKSVILLE

RIDGE

ZEPHYRHILLS GAP

(Within WESTERN VALLEY)

POLK

UPLAND

LAKELAND

RIDGE

POLK

UPLAND

COASTAL SWAMPS

  



  

	   

0 5 10 MILES

0 5 10 KILOMETERS

EXPLANATION

WELL FIELD

PHYSIOGRAPHIC CONTACT

Figure 2. Physiographic provinces (modified from SDI Environmental Services, Inc., 1997).
Introduction 5

HERNANDO COUNTY

PASCO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Land O

Lakes

Temple

Terrace

Tarpon

Springs

Clearwater

Tampa

Brandon

Plant

City

Zephyrhills

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Mud

Spring

Salt

Spring Weeki

Wachee

Spring

Bobhill

Spring

Gator

Spring

Magnolia

Springs

Horseshoe

Spring

Hudson

Spring

Salt Spring

Shady

Spring

Health

Spring

Sulphur

Springs

Eureka

Springs

Lettuce

Lake

Spring

Buckhorn

Spring Lithia

Springs

Crystal

Springs

Weeki Wachee River River

Pithlachascotee

River

Anclote

River

Brooker

Creek

Flint

Creek

Creek

Creek

Rocky

Brushy

Sweetwater

Creek

Hillsborough

River

Hi l l sborouhg

Creek

Cypress

Creek

Trout

Blackwater

Creek

Withlacoochee

Alafia

River

  



  

	   

0 5 10 MILES

WELL FIELD 0 5 10 KILOMETERS

GENERAL LOCATION OF SPRING

EXPLANATION

Figure 3. Location of rivers and springs.
6 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

Description of the Existing CNTB Area 

Hydrologic Model

The existing CNTB area hydrologic model is a 

coupled surface-water and ground-water flow model 

developed by SDI (1997) for the CNTB area. The 

CNTB area hydrologic flow model links together the 

surface-water model HSPF (Johanson and others, 

1984) with the ground-water model MODFLOW 

(McDonald and Harbaugh, 1988) (fig. 5). HSPF simulates the basin water budget and the processes above 

the saturated ground-water system (precipitation, 

evapotranspiration, interception, surface-water withdrawal, infiltration, interflow, runoff, rejected 

recharge, and percolation to the ground-water system). 

HSPF is the tool for converting precipitation data into 

the quantity of water that reaches the water table as 

ground-water inflow (recharge) and the quantity of 

water that is compared directly to measured stream discharges. Recharge is defined as the amount of water 

that has infiltrated and percolated through the unsaturated soil zone. This water represents an outflow flux 

from HSPF and can be thought of as the water that 

reaches the water table and becomes part of the saturated flow system or ground water. MODFLOW simulates the saturated ground-water system and the 

processes of recharge, leakage, baseflow, and groundwater withdrawals. At times, ground water can rise 

above land surface during periods of high recharge. 

Water above land surface is either allowed to pond or is 

transferred from the ground-water system to the surface-water system as “rejected recharge” and is routed 

as basin runoff. 

The primary feature of the CNTB area hydrologic model is the integrating software that provides the 

linkage and exchange of water between the two component models, HSPF and MODFLOW. Integrating software provides linkage between the models by 

reformatting the output data from one hydrologic 

model (HSPF) for input into the other model (MODFLOW). Exchange of water between the two models 

occurs in several ways. Using hourly rainfall data, 

HSPF calculates the weekly percolation (recharge) 

from the surface-water system to the ground-water system and the stages in surface-water reaches of streams. 

MODFLOW is then run with recharge and stream 

stages calculated by HSPF. MODFLOW calculates 

aquifer water levels and base flow to and from surfacewater features. Base flow calculated by MODFLOW 

becomes input to HSPF and stages are adjusted. The 

System Series Stratigraphic unit Major

lithologic

unit

Hydrogeologic unit

Quaternary Holocene and

Pleistocene

Surficial sand,

terrace sand, phosphorite Sand Surficial aquifersystem

Pliocene

Miocene

Tertiary

Oligocene

Eocene

Paleocene

Intermediate confining unit

Undifferentiated

deposits

Peace River

Formation

Suwannee

Limestone

Ocala

Limestone

Oldsmar and

Cedar Key Formation

Avon Park

Formation

Arcadia

Formation

Tampa

Member

Hawthorn Group

Upper

confining

unit

“Water-bearing

units”

Upper

Floridan

aquifer

Lower

Floridan

aquifer

Lower

confining unit

Middle

confining unit

Limestone

Limestone

and

dolomite

D

imestone

olomite

and

l

Sand,

imestone

clay, and

l

Florida aquifer system

Figure 4. Generalized stratigraphic and hydrogeologic section, west-central Florida.
Introduction 7

corresponding subbasin and reach numbers from HSPF 

to individual cells in MODFLOW are linked in the 

CNTB model using a Geographic Information System 

(GIS). The integrating software translates HSPF 

results (hourly increments) into stress periods used by 

MODFLOW (weekly), and MODFLOW results are 

partitioned into appropriate periods for HSPF.

Areally, the CNTB area hydrologic model is 

divided into a grid of 131 rows by 121 columns, with 

0.25 mi2 cells in the center of the model and cells up to 

1 mi2

 elsewhere (fig. 6). The ground-water part of the 

CNTB area hydrologic model consists of two layers. 

The upper layer (layer 1) represents the surficial aquifer system as an unconfined layer. The lower layer 

(layer 2) represents the Upper Floridan aquifer as a 

confined/unconfined layer. Vertical leakage through 

the intermediate confining unit was simulated implicitly using a leakance array. Assigning a high leakance 

value (0.35 day-1) simulates the absence of the confining unit. When using this leakance value, simulated 

water levels in model layers 1 and 2 are equal. Rivers 

are modeled as river cells in layer 1 and, in those locations where rivers are believed to be in direct hydraulic 

connection with the Upper Floridan aquifer, in model 

layer 2 (figs. 7 and 8). Five lakes (Stemper Lake, Bell 

Lake, Big Fish Lake, Crews Lake, and King Lake East) 

were represented as individual reaches and were simulated directly in HSPF (fig. 7). Other lakes were considered to be “windows” in the surficial aquifer system 

through which the water table can be observed and 

were assumed to behave in the same way as the surficial aquifer system. Wetlands are modeled as river 

cells in layer 1. Springs are represented by one or more 

drain cells in model layer 2. Several boundary conditions were used to constrain the lateral extent of the 

simulated flow system. Based on regional groundwater flow, most of the lateral extent in layer 1 is a noflow boundary, except where the boundary coincides 

with the coastline of the Gulf of Mexico and Tampa 

Bay, which are represented in the model as specified 

heads. In layer 2, the southeastern and most of the 

northern boundaries are no flow boundaries representing flow lines in the Upper Floridan aquifer. A part of 

the northern boundary is represented by a general head 

boundary. The extreme eastern edge of the model is 

represented as a specified head boundary. The coastline is represented as a no-flow boundary. 

The CNTB area hydrologic model was developed as a numerical tool to assess hydrologic issues 

related to resource/well field management and wateruse permit renewal applications for the Cypress Bridge, 

Cypress Creek, and Cross Bar Ranch well fields 

(fig. 1). The simulation period is from 1971 to 1993 

(approximately 1,200 weeks). The calibration period 

was the 12-yr period from 1976 through 1987. Simulation of the 5-yr period from 1971 through 1975 prior 

to calibration was used to stabilize water levels and 

flows in the model. The 6-yr period from 1988 through 

1993, following the calibration period, was chosen as 

the model verification period.

PRECIPITATION

ET INTERCEPTION

RECHARGE

PERCOLATION

INFILTRATION

RUNOFF

UNSATURATED

ZONE

REJECTED

RECHARGE ET

STREAM

CHANNEL

VERTICAL BASE FLOW

LEAKAGE

HSPF

MODFLOW

WATER TABLE

HORIZONTAL FLOW

HORIZONTAL FLOW

SATURATED

ZONE

SAS

ICU

UFA

EXPLANATION

ET is evapotranspiration

VERTICAL

LEAKAGE

SAS is surficial aquifer system

ICU is Intermediate aquifer system

UFA is Upper Floridan aquifer

Figure 5. Primary components of the surface-water system simulated by HSPF and the ground-water system 

simulated by MODFLOW.
8 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

130

COLUMNS

120

110

100

90

80

70

60

50

40

30

20

10

1

1 10 20 30 40 50 60 70 80 90 100 110 120

ROWS

EXPLANATION

INACTIVE MODEL CELL

ACTIVE MODEL CELL

STUDY AREA BOUNDARY

Brooksville

Land O

Lakes

New

Port

Richey

Temple

Terrace

Tarpon

Springs

Clearwater

Tampa

Brandon

Plant City

Zephyrhills

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

  



  

	   

0 5 10 MILES

0 5 10 KILOMETERS

Figure 6. Model grid used in simulation of ground-water flow system. (Modified from SDI Environmental Services, Inc., 

1997.)
Introduction 9

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

Big Fish

Lake

Bell Lake

King Lake East

Crews Lake

Lake

Stemper

  



  

	   

0 5 10 MILES

0 5 10 KILOMETERS

EXPLANATION

SPECIFIED HEAD BLOCK

RIVER BLOCK

NO FLOW BOUNDARY

Figure 7. Model boundary conditions for the surficial aquifer system (layer 1). (Modified from SDI 

Environmental Services, Inc., 1997.)
10 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Land O

Lakes

New

Port

Richey

Temple

Terrace

Tarpon

Springs

Clearwater

Tampa

Brandon

Plant City

Zephyrhills

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO







	   

0 5 10 MILES

0 5 10 KILOMETERS

EXPLANATION

SPECIFIED HEAD BLOCK

SPRING BLOCK

GENERAL HEAD BOUNDARY

FLOW BLOCK (GHB)

SPRING AND RIVER BLOCK

RIVER AND GHB BLOCK

RIVER BLOCK

WELL FIELD

NO FLOW BOUNDARY

Figure 8. Model boundary conditions for the Upper Floridan aquifer (layer 2). (Modified from SDI Environmental 

Services, Inc., 1997.)
Introduction 11

A generalized conceptual model of the components of the CNTB area hydrologic model is shown 

schematically in figure 9. The surface-water system 

extends to the saturated zone of the ground-water system (water table in the surficial aquifer system) and the 

components include rainfall, runoff, infiltration, percolation, evapotranspiration (ET), and streamflow. The 

ground-water components include recharge, groundwater pumpage, stream baseflow, lateral ground-water 

flow, and leakage between the surficial aquifer system 

and the Upper Floridan aquifer. Rainfall enters the soil 

layer of the unsaturated zone (modeled by HSPF) 

through infiltration. Water enters the saturated zone 

(modeled by MODFLOW) from the surface-water 

component through percolation. Water above land surface that is not allowed to pond (rejected recharge) is 

treated as excess water and routed as basin runoff to lateral surface-water flow. Rainfall either runs off or percolates downward and recharges the surficial aquifer 

system. In the surficial aquifer system, water may 

move laterally to discharge where it intersects land surface, be lost as ET, or leak downward to the Upper 

Floridan aquifer. Water in the Upper Floridan aquifer 

moves laterally to lowland discharge areas such as the 

Gulf coast where it leaks upward.

Figure 9 summarizes the annual average hydrologic budget for the CNTB model area for the 1971-93 

simulation period. The simulated inflows and outflows 

illustrate the primary processes used to represent the 

surface-water system and the ground-water system in 

the CNTB area hydrologic model. In the simulated 

budget, about 75 percent of rainfall is lost to ET, 11 percent runs off, 5 percent is discharge to streams, 3 percent is discharge to springs, 2 percent leaves the area 

as lateral ground-water outflow, and 4 percent is 

pumped from the ground-water system.

Recharge to the ground-water system averages 

9.6 in/yr. Part of the rainfall (2.0 in/yr) is rejected as 

recharge and contributes to additional ET and surface 

runoff. Simulated net leakage between the surficial 

aquifer system and the Upper Floridan aquifer was 

6.8 in/yr downward, which represented 50 percent of 

the total flow in the surficial aquifer system. This is in 

agreement with high leakance characteristics of the 

intermediate confining unit. Consequently, hydrologic 

conditions in the surficial aquifer system can substantially effect conditions in the Upper Floridan aquifer. 

Likewise, hydrologic conditions in the Upper Floridan 

aquifer can substantially effect conditions in the surficial aquifer system.

Precipitation

52.0 in

SURFACE-WATER SYSTEM (sw)

SURFICIAL AQUIFER SYSTEM

UPPER FLORIDAN AQUIFER

Evapotranspiration

(ET) 38.5 in

Percolation

12.5 in

Recharge

9.6 in

Leakage

6.8 in

ET (gw)

2.9 in

ET (sw)

35.6 in

Runoff

5.9 in

Base flow

0.8 in

Base flow

1.6 in

Rejected recharge

2.0 in

Base flow

2.4 in

Spring flow

1.8 in

Pumpage

2.2 in

Net lateral outflow

1.2 in

GROUND-WATER SYSTEM (gw)

Figure 9. Simulated annual average hydrologic budget for the central northern Tampa Bay area 

hydrologic model 1971-93. (Modified from SDI Environmental Services, Inc., 1997).
12 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

APPLICATION OF NONLINEAR LEASTSQUARES REGRESSION TO

GROUND-WATER FLOW MODELING

 Fundamentally, the process of model calibration is the same using either inverse models or the 

trial-and-error approach: parameter values and other 

aspects of the model are adjusted until the dependent 

variables (water levels and flows) match field observations. Important advantages of using nonlinear 

least-squares regression, however, are the ability to 

determine parameter values that produce the best 

match to field observations and the ability to quantify 

the quality of model calibration using statistical measures. The statistical framework of this process can 

be used to determine strengths and weaknesses of the 

model, the likely accuracy of simulated results, and 

measures of parameter uncertainty. In addition, 

results can be used to help evaluate whether model 

parameter estimates are reliably calculated with available data and what additional data could be most useful in improving the model (Poeter and Hill, 1996). 

Consequently, inverse modeling can improve the 

quality of ground-water models and yield results that 

are not readily available through trial-and-error calibration efforts. Also, if the inverse model converges, 

the model is likely to be a unique set of parameter 

values.

Ground-water flow is simulated using the USGS 

MODFLOW model (McDonald and Harbaugh, 1988). 

To perform inverse modeling, the flow model is linked 

with a nonlinear least-squares regression routine 

(Halford, 1992). The combined ground-water flow 

model and nonlinear regression is called MODOPTIM. 

MODOPTIM was used to: (1) identify sensitivity of 

water-level data to the estimation of each parameter, 

(2) identify parameters that are highly correlated, and 

(3) indicate optimal parameter values for available 

water-level data.

Nonlinear Least-Squares Regression Method

The nonlinear weighted least-squares regression 

method minimizes the sum-of-squared residuals (SS) 

between measured and simulated quantities and is 

based on a modified Gauss-Newton method (Gill and 

others, 1981). The SS is defined as:

, (1)

 

 where

hjs is the jth simulated water level, in feet;

hjm is the jth measured water level, in feet;

wi is the weighting factor, and

n is the number of water-level comparisons.

Water levels were assigned a weighting factor of 

1.0 as there was little basis for differentiating among 

measurements. Stream discharge measurements were 

not formally used during regression analysis because of 

difficulties of accurately determining the ground-water 

component of total gaged stream discharge. Spring flow 

also was not formally used during regression analysis 

because the conductance term is poorly known and 

because many springs are assigned to multiple grid cells.

Although the SS serves as the objective function 

(measure of model fit), root-mean-square error 

(RMSE) is reported instead because RMSE is more 

directly comparable to actual values and serves as a 

composite of the average and the standard deviation of 

a set (Halford, 1997). RMSE is related to the SS by:

. (2)

The first step in the parameter-estimation process is to perform one execution of the model to establish the initial differences (residuals) between 

simulated and measured water levels. The residuals are 

squared and summed to produce the sum-of-squared 

residuals objective function (eq. 1), which is used by 

the regression to measure model fit to the observations. 

In the next step, the sensitivity coefficients (derivatives 

of simulated water-level change with respect to parameter change) are calculated by the influence coefficient 

method (Yeh, 1986) using the initial model results. 

After the residuals and the sensitivities are calculated, 

a single parameter-estimation iteration is performed. 

The current arrays of sensitivity coefficients and residuals are used by a quasi-Newton procedure (Gill and 

others, 1981. p. 137) to compute the parameter change 

that should improve the model. The model is updated 

to reflect the latest parameter estimates and a new set of 

residuals is calculated. The entire process of changing 

a parameter in the model, calculating new residuals, 

and computing a new value for the parameter is continued iteratively until model error or model-error change 

is reduced to a specified level or until a specified 

SS wi hjs hjm [ ] ( ) – 2

j = 1

n

= ∑

RMSE SS n ( ) ⁄ 0.5 =
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 13

number of iterations is made (Halford, 1992). Logs of 

the parameters are estimated because log-parameters 

are better behaved from a numerical perspective, and 

because logs of the parameters prevent the actual 

parameter values from becoming negative.

Simulation Model

The CNTB area hydrologic model developed by 

SDI (1997) was used in this study with three modifications. First, the CNTB area hydrologic model was decoupled to run without HSPF. Second, rejected recharge 

simulated in the CNTB area hydrologic model (fig. 10) is 

simulated in this model by drains, where the elevation of 

the drain is set at land surface. Third, steady-state model 

simulations were used for the parameter estimation part of 

this study. Steady-state analysis assumes that groundwater levels, hydraulic gradients, and the velocity distribution of ground-water flow do not change with time. 

Although ground-water levels fluctuate seasonally, the 

annual range fluctuates around long-term averages, and 

therefore, the state of the aquifer system approximates a 

dynamic equilibrium. Time-averaged hydrologic 

conditions for the 1987 calendar year were chosen for 

steady-state conditions and parameter estimation. The 

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

0

0 to 10

10 to 25

25 to 50

Greater than 50

EXPLANATION

REJECTED RECHARGE RATE, IN INCHES PER YEAR

Figure 10. Areal distribution of simulated rejected recharge from the surficial aquifer system, calendar year 1987.
14 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

average 1987 hydrologic conditions were considered suitable for several reasons.

1. The frequency of data collection was sufficient for 

calculation of representative annual average values.

2. Measured annual precipitation in 1987 was close to the 

long-term average value.

3. The small net change in water levels measured in wells 

indicates that the change in storage in the aquifer 

systems was small during 1987.

Data input were obtained directly or indirectly 

from the CNTB area hydrologic model (SDI, 1997) and 

included: starting water-level values, hydraulic conductivity, bottom altitude of the surficial aquifer system, riverbed conductance values, intermediate aquifer 

system leakance, transmissivity of the Upper Floridan 

aquifer, horizontal anisotropy values of the Upper 

Floridan aquifer, boundary heads and boundary conductance values for the Upper Floridan aquifer, pumping rates, and drain altitudes and conductances for the 

surficial aquifer system and the Upper Floridan aquifer. 

Input data arrays also included specified recharge and 

discharge rates to/from the surficial aquifer system 

(fig. 11). Net recharge rates are calculated by substracting ET from recharge in each model cell, thus in 

cells where ET exceeds recharge, a negative value of 

recharge is obtained.

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

EXPLANATION

-9 to 0

0

0 to 10

10 to 20

SIMULATED RECHARGE AND DISCHARGE, IN INCHES PER YEAR

20 to 30

30 to 40

40 to 50

SUBBASIN BOUNDARY

Figure 11. Simulated percolation (recharge) rates from the surface-water system to the ground-water system, 

calender year 1987. (Modified from SDI Environmental Services, Inc., 1997).
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 15

Observation Data

Water levels measured in 223 surficial aquifer 

system wells and 326 Upper Floridan aquifer wells in 

1987 are the observation data used for comparison during the simulation modeling; whereas only 119 observations were used in the SDI (1997) simulation 

modeling. Measured water levels were used by 

MODOPTIM during parameter estimation to provide 

values to define the objective function for the model 

simulation. Because all water levels from wells in 

wellfield areas were used, the SS may be slightly 

biased toward these areas. As previously indicated, 

base flow values were not matched by parameter estimation due to difficulties of accurately determining the 

base flow (ground-water discharge) component of total 

streamflow. Spring flow values also were not matched 

by the regression because the conductance term is 

poorly known, because many individual springs were 

simulated with multiple grid cells, and because of the 

lack of accurate altitude control on spring pool elevation (altitudes of most springs were determined from 

topographic maps). 

For each water-level observation, the following 

quantities were specified: measured water-level value, 

well location, and model layer number. Because measured water levels rarely coincide with the center of a 

cell, simulated water levels were interpolated laterally 

to points of measurements from the surrounding cells. 

Simulated water levels can be laterally interpolated 

because they are assumed to be part of a continuous 

distribution. Vertical interpolation of water levels was 

not performed because each aquifer was modeled as a 

single layer and the generally low permeability of the 

intermediate confining unit results in a discontinuity in 

vertical head distributions.

Parameter Structure

Simplification of the subsurface framework is 

inherent in the modeling process because of the 

required spatial discretization. Each finite-difference 

cell is assigned one value for each hydraulic parameter, 

which represents a spatially averaged uniform value. 

To minimize nonuniqueness problems caused by trying 

to estimate too many parameters, the spatial distribution of parameters is represented in MODOPTIM by 

dividing the model domain into zones with homogenous hydrologic properties (each zone characterized by 

one constant value). The zones and corresponding 

parameter values are defined as the “parameter structure” (figs. 12-16). 

The parameter structure for the existing CNTB 

area hydrologic model (SDI, 1997) consists of five 

parameters divided into 200 parameter zones. The 

parameters examined in this study are (1) recharge 

(Qre), (2) hydraulic conductivity (K) of the surficial 

aquifer system, (3) leakance (L) of the intermediate 

confining unit, (4) transmissivity (T) of the Upper 

Floridan aquifer, and (5) horizontal anisotropy ratio 

(Kx/Ky) of the Upper Floridan aquifer. 

Model Analysis

Analysis of the existing CNTB area hydrologic 

model consisted of two simulation phases. The goal of 

the first phase of model analysis was to statistically 

evaluate the parameter structure reported by SDI 

(1997). Parameter value regression statistics, including 

sensitivities and correlations, were calculated for the 

SDI (1997) calibration values by executing only one 

parameter-estimation iteration. The goal of the second 

phase of model analysis was to “optimize” or statistically select the “best fit” parameter values using nonlinear least-squares regression. The “best fit”

parameter values are those that yield the minimum 

value of SS (eq. 1) and are dependent on the given set 

of observations.

A total of 200 potential independent parameters 

were defined by SDI (1997), which were many more 

parameters than could be estimated with the available 

water-level measurements. Because most data sets 

only support the estimation of relatively few parameters, the number of parameter values estimated generally needs to be a fraction of the number of 

observations used to estimate them (Hill, 1992). 

Therefore, the entire parameter set was divided into 

five individual data sets; each data set corresponds to 

one of the five parameters of interest. Sensitivities and 

correlations were then run individually for each parameter zone of the five parameters of interest. The sensitivity for each parameter-zone value was used as a 

measure of the “reasonableness” of estimating a 

parameter and as a guide for deciding which parameters should be estimated. Insensitive parameters should 

not be estimated because there is no basis to do so as 

measured by the objective function (eq. 1). Although 

200 parameter zones were initially defined for the 

CNTB area hydrologic model, not all parameters were 

estimated. 
16 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Land O

Lakes

New

Port

Richey

Temple

Terrace

Tarpon

Springs

Clearwater

Tampa

Brandon

Plant City

Zephyrhills

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

4

5

6

80

16

46

3

2

72

9 8

7

1

23

88

91

20

34

57

59

62

12

85

10

68

91

15

33 29

77

31

90

55

76

21

65

91

91

24

13

79

53

60

83

67

81

30

64

58

50

36

75

69

86

18

89

14

73

61

78

47

82

84

70

25

49

87

39

71

22

27

43

28 32

74

38

66

40

19

37

52

35

44

26

11

56

57

54 47

42

35

63

17

51

91

45

48

23

41

51

41

33

36

91

91

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

WELL FIELD

SUBBASIN BOUNDARY—Number

identifies recharge zone

EXPLANATION

91

Figure 12. Watershed subbasins within the study area. (Modified from SDI Environmental Services, Inc., 1997).
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 17

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

15

10

1

0.1

EXPLANATION

HYDRAULIC CONDUCTIVITY, IN FEET PER DAY

4.5

3

Figure 13. Calibrated hydraulic conductivity of the surficial aquifer system for the existing central northern 

Tampa Bay area hydrologic model. (Modified from SDI Environmental Services, Inc., 1997).
18 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

3.5 10 to 1.0 10

1.0 10 to 1.0 10

1.0 10 to 1.0 10

x x

x x

x x

-1 -2

-2 -3

-3 -4

1.0 10 to 1.0 10

1.0 10

PARAMETER-ZONE IDENTIFICATION NUMBER

x x

x

-4 -5

-6

EXPLANATION

LEAKANCE OF THE INTERMEDIATE CONFINING UNIT, IN FEET PER DAY PER FOOT

2

24

8

1

14

16

13

12

62

20

9

35

60

69

55

15

22

71

34

11

6

48

41

65

23 5

10

26

33

63

54

31

39

3 29

66

27

7

47

50

70

40

63

49 32

38

18

64 58

28

17

56

30

42

61

57

44

67

51

46

19

21

45

52

68

4

59

53

37

25

36

14

26

43

Figure 14. Calibrated leakance of the intermediate confining unit for the existing central northern Tampa Bay 

area hydrologic model. (Modified from SDI Environmental Services, Inc., 1997).
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 19

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

7

6

19

18

4

5

8

22

10

14

9

15

17

13

12

2

21

1

16

3

23 20

20

11

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

4

10,000 to 35,000

35,000 to 70,000

EXPLANATION

TRANSMISSIVITY, IN FEET SQUARED PER DAY

70,000 to 130,000

130,000 to 300,000

300,000 to 500,000

PARAMETER-ZONE IDENTIFICATION NUMBER

Figure 15. Calibrated transmissivity of the Upper Floridan aquifer for the existing central northern Tampa Bay 

area hydrologic model. (Modified from SDI Environmental Services, Inc., 1997).
20 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

8

6

9

2

4

7

3

5

1

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

5

EXPLANATION

0.1

0.3

1.0

3.0

ANISOTROPY RATIO

4.0

10.0

PARAMETER-ZONE IDENTIFICATION NUMBER

Figure 16. Calibrated anisotropy of the Upper Floridan aquifer for the existing central northern Tampa Bay area 

hydrologic model. (Modified from SDI Environmental Services, Inc., 1997).
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 21

Initial Parameter Sensitivities 

and Correlations (phase 1)

The overall sensitivity of the parameters to the 

observations reflect how well the parameters are 

defined by the observations and indicate how well the 

parameters will be estimated. Composite sensitivity 

(CS) is the statistic that is used to measure this overall 

sensitivity, and indicates the cumulative amount of 

information that the measurements contain toward the 

estimation of that parameter. The magnitude of the 

main diagonal of the covariance matrix is a rough estimate of the sensitivity of the model to a parameter. The 

CS of the jth parameter was quantified by:

, (3)

where

CSi,j is the Jacobian matrix or sensitivity matrix of 

SS(x), and is the partial derivative of SS(x) 

(eq. 1) at all observations (o) with respect to 

parameter change (number of observations 

by number of parameters matrix), 

n is the number of water-level comparisons, 

and

 is the sensitivity coefficient of the ith

observation (o) with respect to the jth parameter estimated.

For a given model and objective function, the 

measure calculated by eq. 3 increases as parameter sensitivity increases. Parameter sensitivity was reported in 

terms of the relative composite sensitivity (RCS), 

which is the square root of the main diagonal value 

divided by the maximum main diagonal for each 

parameter. The RCS for the jth parameter was quantified by:

(4)

The most sensitive parameter has a RCS equal to 

1.00 and the RCS of all other estimated parameters is 

less than one. The larger the value of the RCS, the 

more sensitive the model is to that parameter, as a 

whole. Parameters with larger RCS values relative to 

those for other parameters are likely to be easily estimated by the regression; parameters with smaller RCS 

values may be more difficult to estimate. Parameters 

with smaller RCS values also tend to have higher 

parameter uncertainty and broader confidence intervals. For some parameters, the available measurements may not provide enough information for 

estimation. If the RCS value is less than 0.02, the optimization procedure has difficulty estimating the 

parameter. 

In considering model sensitivity to a particular 

parameter, it also is important to consider the areal size 

of the zone (relative to the total model area) and the 

number of water levels within the zone. This information along with RCS values should be considered when 

assessing, in a qualitative manner, the relative sensitivity of the model (either as a whole or locally) to each 

parameter.

Correlation between parameters indicates 

whether or not the parameter estimates are unique with 

the given model construction and observations. It is an 

indicator of the degree of linear dependency in the sensitivity matrix and reflects the redundancy of the problem. Correlation coefficients are calculated by 

inverting a matrix that is singular when correlations are 

-1.0 or +1.0 (Poeter and Hill, 1996). If two parameters 

are highly correlated, then changing the parameter values in a linearly coordinated way will result in a similar 

value of the objective function. Correlation coefficients greater than 0.95 usually indicate a pair of 

parameters are highly correlated (Hill, 1992). Parameters that are highly correlated are not desirable because 

they cannot be independently estimated. 

The general results from the analysis described 

in the following sections indicate that there is insufficient observation data to independently estimate all 

SDI (1996) parameter values given the present zonation structure. A simpler parameter-zonation structure 

should be considered given the lack of information 

contained in the data that are available for calibration. 

Some possible ways of improving model calibration 

are to: (1) modify the defined parameter-zonation 

structure by omitting and/or combining parameters to 

be estimated; (2) carefully eliminate observation data 

based on evidence that the data are likely to be biased; 

(3) collect additional water-level data; (4) assign values 

to insensitive parameters, and (5) estimate the most 

sensitive parameters first, then using the optimized values for these parameters, estimate the entire data set 

(Hill, 1992, appendix B; Yager, 1993). 

Recharge

Recharge to the surficial aquifer system was 

defined with 91 parameter zones (fig. 12); zones correspond to each of the 91 surface-water subbasins represented in the CNTB area hydrologic model. The 1987 

recharge rates were derived from the surface-water 

CSi j , xj ∂

∂

oi ( )    

i = 1

n

= ∑

xj ∂

∂

oi ( )

RCSj ( ) CS j j , max CS ( )j j , ⁄ ]0.5 = [
22 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

model component (HSPF) and averaged 13.6 in/yr. 

Sensitivity values for the parameter zones, number of 

water-level measurements within each zone, and the 

areal size of each zone are shown in table 1. Results 

show that there are insufficient water-level data to reliably estimate the defined parameter-zone values 

(fig. 17 and table 1). Sensitivity is less than 0.02 for 

most of the parameter zones, and the small RCS values 

indicate that these parameters are not well defined with 

the available observations and given model construction. Possible solutions to improve calibration are to 

set parameter-zone values to specified values, estimate 

fewer parameters by combining zones, or collect more 

water-level data that will uniquely define all parameter 

values. Generally, the low sensitivity areas coincide 

with areas where the simulated water table is constrained by land surface and water levels are within 5 ft 

of land surface (fig. 18). Many of the parameters also 

are highly correlated, which limits the ability of the 

model to uniquely determine the parameters.

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Land O

Lakes

New

Port

Richey

Temple

Terrace

Tarpon

Springs

Clearwater

Tampa

Brandon

Plant City

Zephyrhills

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

5

16

46

2

72

9

7

1

23

88

91

20

34

57

59

62

85

10

68

91

15

33 29

77

31

90

55

21

65

91

91

24

13

79

53

60

83

67

81

30

64

50

36

75

86

18

89

14

73

61

78

47

84

70

25

49

87

39

71

22

27

43

28 32

74

38

66

40

19

37

52

35

44

26

11

56

57

54 47

42

35

63

17

51

91

45

48

23

41

51

41

33

36

91

91

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

EXPLANATION

91 SUBBASIN BOUNDARY TO WHICH THE MODEL IS RELATIVELY INSENSITIVE

TOWARD THE ESTIMATION OF DISCHARGE—The relative composite sensitivity

is less than 0.02. Number is identifier found in table 1

Figure 17. Areas of the central northern Tampa Bay area hydrologic model that are insensitive to the estimation 

of recharge.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 23

Table 1. Relative composite sensitivity, number of water-level measurements, and areal size of each recharge zone for the 

initial parameter values

[RCS, relative composite sensitivity; SAS, surficial aquifer system; UFA, Upper Floridan aquifer; mi2, square miles; zone numbers are

shown in figure 12]

Zone 

number RCS

Number of water-level 

measurements Zone area 

(mi2)

Zone 

number RCS

Number of water-level 

measurements Zone area 

(mi2)

SAS UFA Total SAS UFA Total

6 1.0000 7 16 23 76.62 44 0.0032 1 2 3 2.94

4 0.9882 1 9 10 169.29 15 0.0031 7 13 20 22.53

3 0.9882 0 2 2 46.74 79 0.0031 3 11 14 14.63

82 0.0346 1 5 6 8.02 21 0.0031 0 2 2 17.50

80 0.0342 7 7 14 32.85 25 0.0030 1 2 3 7.13

58 0.0323 10 11 21 11.39 90 0.0030 3 6 9 18.58

69 0.0255 1 4 5 11.00 81 0.0029 1 2 3 11.98

76 0.0204 3 6 9 18.00 31 0.0026 1 0 1 19.33

12 0.0203 13 15 28 30.06 51 0.0026 0 0 0 1.50

8 0.0203 8 16 24 32.85 10 0.0023 1 1 2 27.16

72 0.0199 5 10 15 70.03 14 0.0023 2 3 5 8.75

46 0.0185 2 3 5 113.34 29 0.0023 1 1 2 21.55

87 0.0170 3 2 5 7.01 2 0.0022 0 3 3 37.71

34 0.0167 26 28 54 37.53 64 0.0022 2 0 2 11.47

85 0.0158 5 10 15 27.26 49 0.0020 0 1 1 7.13

5 0.0150 1 6 7 107.17 42 0.0020 2 5 7 2.00

7 0.0141 0 1 1 27.22 27 0.0019 0 1 1 5.75

67 0.0140 0 1 1 13.01 36 0.0017 0 0 0 11.20

50 0.0131 13 12 25 11.25 84 0.0011 0 0 0 7.99

57 0.0121 2 3 5 39.67 66 0.0011 3 2 5 4.06

65 0.0112 2 2 4 16.93 26 0.0011 0 0 0 2.94

59 0.0107 13 10 23 36.66 73 0.0010 0 4 4 8.74

60 0.0106 4 4 8 14.02 48 0.0010 2 2 4 0.57

54 0.0097 1 0 1 2.06 18 0.0008 0 0 0 9.56

56 0.0096 2 1 3 2.44 52 0.0006 1 0 1 3.25

68 0.0095 1 3 4 26.39 13 0.0006 0 4 4 15.02

77 0.0091 1 1 2 19.74 35 0.0004 0 0 0 4.63

16 0.0077 2 4 6 114.17 75 0.0004 0 0 0 11.13

71 0.0072 1 0 1 6.25 78 0.0004 1 0 1 8.52

40 0.0072 12 14 26 3.77 86 0.0003 0 2 2 9.62

23 0.0060 2 2 4 44.15 74 0.0003 0 1 1 4.87

88 0.0059 5 8 13 39.75 19 0.0003 0 0 0 3.72

47 0.0056 1 1 2 10.51 32 0.0003 0 0 0 5.00

30 0.0054 15 14 29 11.51 63 0.0003 0 0 0 1.69

39 0.0052 1 0 1 6.82 45 0.0002 0 0 0 0.76

61 0.0052 0 0 0 8.57 1 0.0002 0 0 0 25.03

20 0.0046 8 4 12 38.08 70 0.0002 0 0 0 7.50

9 0.0043 0 1 1 34.92 37 0.0002 0 1 1 3.26

33 0.0042 0 1 1 20.25 24 0.0002 0 1 1 15.27

28 0.0041 3 6 9 5.28 55 0.0002 0 0 0 18.26

89 0.0039 2 0 2 9.19 41 0.0002 0 0 0 0.91

43 0.0039 3 1 4 5.37 17 0.0002 0 0 0 1.36

53 0.0039 2 2 4 14.51 62 0.0001 0 1 1 35.86

38 0.0037 0 0 0 4.50 11 0.0001 0 0 0 2.56

83 0.0037 1 5 6 13.88 91 0.0001 1 2 3 109.62

22 0.0033 0 2 2 6.06
24 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

Hydraulic Conductivity of the Surficial Aquifer System 

Hydraulic conductivity (K) of the surficial aquifer 

system was defined with six parameter zones (fig. 13). 

Sensitivity values for the parameter zones, zone areas, 

and number of water-level measurements within each 

zone are shown in table 2. Regression results indicate 

that the available water-level data provide sufficient 

information to reliably estimate hydraulic conductivity 

of the surficial aquifer system in three of the five parameter zones (RCS > 0.02). Sensitivity is highest for the 

high K parameters (15.0, 10.0, and 4.5 ft/day) and is 

lowest for the low K parameters (0.1, 1.0, and 3.0 ft/d). 

For improved calibration, the parameter value for the 

low sensitivity zones should be set to the specified value, 

combined with adjacent zones, or more water-level measurements will need to be added to the regression. The 

low sensitivity for the 1.0, 0.1, and 3.0 ft/d parameter 

zones is primarily due to a lack of water-level measurements within these zones and the small parameter-zone 

area relative to total model area. The area of the zones is 

less than 5 percent of the total model area and only six 

water-level measurements are within the zones. The Kparameter zones also are not highly correlated to one 

another. The highest degree of correlation is between the 

1.0 and 0.1 ft/d hydraulic conductivity zones (r = 0.91).

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

5 and greater

OBSERVATION WELL AND SIMULATED

DEPTH TO WATER TABLE, IN FEET

BELOW LAND SURFACE

EXPLANATION

SIMULATED DEPTH TO WATER TABLE,

IN FEET BELOW LAND SURFACE

Less than 5 Less than 5 5 and greater

Figure 18. Simulated depth to water table using the initial central northern Tampa Bay area hydrologic model 

parameter values, calendar year 1987.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 25

Table 2. Relative composite sensitivity, number of waterlevel measurements, and areal size of each hydraulic 

conductivity zone of the surficial aquifer system for the initial 

parameter values

[K, hydraulic conductivity; RCS, relative composite sensitivity; SAS, surficial aquifer system; UFA, Upper Floridan aquifer; mi2, square miles; ft/d, 

foot per day; zones are shown in figure 13]

K-Parameter

Zone RCS

Number of waterlevel

 measurements

Zone area

(mi2)

SAS UFA

15.0 ft/d 1.000 77 119 610.71

10.0 ft/d 0.700 138 200 1,360.06

 4.5 ft/d 0.044 7 6 17.32

 3.0 ft/d 0.017 1 4 105.83

 1.0 ft/d 0.008 0 0 0.06

 0.1 ft/d 0.003 0 1 0.06

Table 3. Relative composite sensitivity, number of water-level measurements, and areal size of each leakance zone of the 

intermediate confining unit for the initial parameter values

[RCS, relative composite sensitivity; SAS, surficial aquifer system; UFA, Upper Floridan aquifer; mi2, square miles; zone numbers are shown in figure 14]

Zone 

number RCS

Number of water-level measurements

Zone area

(mi2) Zone 

number RCS

Number of water-level measurements

Zone area

(mi2)

SAS UFA Total SAS UFA Total

20 1.0000 9 18 27 74.78 30 0.0540 8 15 23 6.07

71 0.4400 26 27 53 32.60 7 0.0490 3 6 9 6.76

63 0.4300 18 13 31 35.71 10 0.0430 3 2 5 25.34

52 0.3600 4 4 8 3.01 32 0.0430 1 1 2 10.26

24 0.3400 5 7 12 213.3 48 0.0430 4 0 4 29.97

69 0.3200 2 4 6 50.73 4 0.0410 1 0 1 1.31

14 0.2600 20 24 44 136.63 42 0.0370 1 1 2 6.00

15 0.2300 2 6 8 40.21 31 0.0340 3 8 11 18.64

62 0.2300 10 12 22 79.06 44 0.0320 1 1 2 5.18

43 0.2200 7 6 13 1.01 64 0.0320 1 5 6 10.13

66 0.2100 6 8 14 14.63 70 0.0290 0 0 0 11.98

58 0.1800 4 1 5 9.38 2 0.0280 6 16 22 162.92

51 0.1700 5 1 10 4.24 12 0.0250 0 4 4 98.88

34 0.1500 4 4 8 31.78 61 0.0210 1 1 2 5.45

56 0.1400 1 1 2 6.26 18 0.0200 1 4 5 10.14

55 0.1300 1 4 5 38.65 27 0.0200 3 3 6 14.63

9 0.1100 3 3 6 33.29 11 0.0190 0 5 5 30.31

19 0.1100 2 0 2 3.74 36 0.0180 1 1 2 0.75

39 0.1100 0 1 1 17.51 53 0.0180 0 0 0 1.89

40 0.1100 0 1 1 15.95 46 0.0160 0 0 0 4.19

57 0.1100 7 6 13 5.24 49 0.0150 0 2 2 10.98

13 0.0950 2 4 6 101.86 17 0.0140 0 0 0 6.58

65 0.0930 1 3 4 31.79 50 0.0110 0 0 0 12.01

5 0.0910 5 8 13 14.19 28 0.0100 1 9 10 12.46

41 0.0890 8 13 21 29.46 37 0.0100 0 0 0 1.33

23 0.0860 0 4 4 28.74 21 0.0095 1 1 2 3.44

45 0.0780 1 3 4 3.43 33 0.0094 0 1 1 24.59

54 0.0770 9 8 17 19.42 47 0.0082 0 0 0 12.26

67 0.0730 4 5 9 4.39 29 0.0071 0 0 0 14.96

35 0.0700 5 7 12 58.21 59 0.0068 0 0 0 2.51

68 0.0680 0 3 3 2.75 26 0.0062 0 1 1 26.98

60 0.0670 0 3 3 57.27 22 0.0060 0 0 0 34.02

6 0.0610 2 4 6 15.15 1 0.0059 0 3 3 71.78

8 0.0610 1 6 7 106.24 25 0.0056 1 1 2 0.88

16 0.0610 0 2 2 59.94 38 0.0024 0 0 0 10.17

3 0.0540 8 7 15 7.55

Leakance of the Intermediate Confining Unit 

Leakance of the intermediate confining unit was 

defined by 71 parameter zones (fig. 14). Sensitivity 

values for the parameter zones, number of water-level 

measurements within each zone, and the areal size of 

each zone are shown in table 3. Regression results indicate that the available water-level data provide sufficient information to reliably estimate leakance of the 

intermediate confining unit for most parameter zones; 

however, estimating all leakance-zone values simultaneously may be impossible due to parameter insensitivity (fig. 19 and table 3). It also may not be reasonable 

to estimate leakance for several parameter zones given 

the lack of sensitivity of the parameter-zone values. 
26 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

Generally, sensitivity is highest for parameter zones that 

have an abundance of observation data within the zones. 

Leakance zones also are not strongly correlated to one 

another. The strongest correlation is between zones 

60 and 70 (r = 0.87). The second highest correlation is 

between zones 1 and 25 (r = 0.86).

Transmissivity of the Upper Floridan Aquifer

Transmissivity of the Upper Floridan aquifer 

was defined by 23 parameter zones (fig. 15). Sensitivity values for the parameter zones, number of waterlevel measurements within each zone, and the areal 

size of each zone are shown in table 4. Regression 

results indicate that the available water-level data provide sufficient information to reliably estimate transmissivity of the Upper Floridan aquifer in all but two 

isolated zones (20 and 21) (fig. 20), given the lack of 

sensitivity. Zone 21 includes specified head boundary 

cells while zone 20 includes river boundary cells 

(figs. 7 and 8). Sensitivity is low because these boundary conditions prevent the simulated water level from 

changing substantially for different parameter values. 

The low sensitivity also is due to limited water-level 

measurements within each of the zones. Transmissivity zones also are not strongly correlated. The strongest 

correlation is between zones 23 and 20 (r = 0.90). The 

second highest correlation is between zones 2 and 13 

(0.87).

1

22

11

26

33

29

47

50

49

38

28

17

46

21

59

53

37

25

36

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Land O

Lakes

New

Port

Richey

Temple

Terrace

Tarpon

Springs

Clearwater

Tampa

Brandon

Plant City

Zephyrhills

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

 	
  




 
 
 

 	
 
 



0 5 10 MILES

26 0 5 10 KILOMETERS

EXPLANATION

BOUNDARY OF ZONE TO WHICH THE MODEL IS RELATIVELY INSENSITIVE

TO LEAKANCE—The relative composite scaled sensitivity is less than 0.02. Number

is identifier found in Table 1

Figure 19. Areas of the central northern Tampa Bay area hydrologic model that are insensitive to the estimation 

of leakance.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 27

 

 

Table 4. Relative composite sensitivity, number of water-level measurements, and areal size of each transmissivity zone of 

the Upper Floridan aquifer for the initial parameter values

[RCS, relative composite sensitivity; SAS, surficial aquifer system; UFA, Upper Floridan aquifer; mi2, square miles; zone numbers are shown in figure 15]

Zone 

number RCS

Number of water-level measurements

Zone 

area 

(mi2)

Zone 

number RCS

Number of water-level measurements

Zone 

area 

(mi2 SAS UFA ) Total SAS UFA Total

1 1.000 1 5 6 162.02 19 0.176 3 6 9 127.10

9 0.571 92 109 201 365.50 23 0.143 1 6 7 65.92

2 0.473 1 3 4 62.90 3 0.091 0 2 2 54.35

10 0.470 47 50 97 156.92

18 0.290 3 8 11 121.91 8 0.088 1 0 1 18.01

5 0.078 4 11 15 58.81

17 0.268 20 29 49 171.26 11 0.078 0 2 2 24.01

16 0.226 2 8 10 121.51 22 0.047 3 2 5 11.36

14 0.216 14 21 35 127.59 15 0.045 4 3 7 40.50

6 0.214 2 3 5 206.97

4 0.202 15 31 46 115.97 12 0.044 1 3 4 0.88

21 0.019 0 4 4 7.12

7 0.200 5 16 21 34.46 20 0.012 0 0 0 23.24

13 0.188 4 4 8 15.07 19 0.176 3 6 9 127.10

20

21

20

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Land O

Lakes

New

Port

Richey

Temple

Terrace

Tarpon

Springs

Clearwater

Tampa

Brandon

Plant City

Zephyrhills

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

EXPLANATION

21 BOUNDARY OF ZONE TO WHICH THE MODEL IS RELATIVELY INSENSITIVE

TO UPPER FLORIDAN AQUIFER TRANSMISSIVITY—The relative composite

scaled sensitivity is less than 0.02. Number is identifier found in Table 1

Figure 20. Areas of the central northern Tampa Bay area hydrologic model that are insensitive to the estimation 

of transmissivity of the Upper Floridan aquifer.
28 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

Anisotropy of the Upper Floridan Aquifer 

Anisotropy of the Upper Floridan aquifer was 

defined by nine parameters zones (fig. 16). Sensitivity 

values for the parameter zones, number of water-level 

measurements within each zone, and the areal size of 

each zone are shown in table 5. Regression results indicate that available water-level data provide sufficient 

information to reliably estimate anisotropy of the 

Upper Floridan aquifer for most parameter-zone values. Generally, sensitivity is highest for parameter 

zones that have an abundance of observation measurements within the zones. Parameters also are not highly 

correlated to one another (r ≤ 0.25). In another simulation, the degree to which anisotropy and transmissivity 

zones of the Upper Floridan aquifer are correlated was 

tested. Regression results indicate a strong negative 

correlation (-0.98) between anisotropy of the Upper 

Floridan aquifer in zones 3 and 9 and transmissivity of 

the Upper Floridan aquifer in zones 8 and 23, respectively. The high correlation associated with these 

zones preclude independent estimation of them. The 

third highest correlation is between anisotropy zone 8 

and transmissivity zone 13 (-0.84). 

 

Hydraulic–Head Sensitivity 

A second sensitivity analysis was performed in 

addition to computing the relative composite sensitivities. The model response investigated in the sensitivity 

analysis was hydraulic head. The parameters selected 

for testing were recharge, hydraulic conductivity of the 

surficial aquifer system, leakance of the intermediate

confining unit, transmissivity of the Upper Floridan 

aquifer, and anisotropy of the Upper Floridan aquifer. 

Hydraulic-head sensitivities are a measure of the 

change in simulated water levels due to changes in 

parameter values. Results from this analysis can suggest which model inputs are likely to have improved 

parameter estimates, and can be used to identify areas 

where additional data are most likely to effect simulated water levels. If the analysis shows that the model 

is not sensitive to changes in certain parameters, efforts 

to improve parameter estimates in the modeled area 

would not improve the simulation capability of the 

model. Conversely, if the sensitivity analysis shows 

that the model is sensitive to changes in a particular 

parameter, additional data collection and analysis to 

better define or verify the parameter values in the 

model area could result in improved simulation capability. Collection of water-level data in areas of high 

sensitivity would be more valuable than obtaining 

water-level data in areas of low sensitivity.

Model sensitivity was described in terms of the 

amount that water levels would change with a 2-percent increase in the parameter value (figs. 21-25). 

Water-level change was calculated for the entire model 

and for each layer so that the relative sensitivities of the 

units could be compared.

The spatial patterns of the 2-percent sensitivities 

of simulated water levels in layers 1 and 2 to recharge 

are similar to each other (fig. 21) and are positive 

(water-level increase). The increase in recharge results 

in a corresponding increase in water levels due to 

increased flux through the flow system. To transmit 

this increased flux through the ground-water system, 

the hydraulic gradient steepens, which requires higher 

water levels. In the area southeast of Brooksville 

(fig. 21), water levels in layer 1 show little or no 

change. This is an area of the model where the surficial 

aquifer system is thin and discontinuous and only isolated, perched water-table conditions exist. Consequently, the Upper Floridan aquifer is unconfined, and 

recharge is applied to layer 2 and not to layer 1. Simulated water levels generally are sensitive to changes in 

recharge. The absolute mean water-level change was 

0.26 ft for the surficial aquifer system and 0.18 ft for 

the Upper Floridan aquifer.

The spatial patterns of the 2-percent sensitivities 

of simulated water levels in layers 1 and 2 to hydraulic 

conductivity of the surficial aquifer system are 

Table 5. Relative composite sensitivity, number of waterlevel measurements, and areal size of each Upper Floridan 

aquifer anisotropy zone for the initial parameter values

[Kx/Ky, horizontal anisotropy ratio; RCS, relative composite sensitivity; 

SAS, surficial aquifer system; UFA, Upper Floridan aquifer; mi2

, square 

miles; zone numbers are shown in figure 16]

Kx/Ky 

Parameter

zone

Kx/Ky RCS

Number of waterlevel

 measurements

Zone 

area

(mi2)

SAS UFA

3 3.0 1.000 1 6 39.96

2 0.3 0.821 24 25 15.77

9 10.0 0.801 2 0 21.02

6 4.0 0.759 5 10 71.06

1 0.1 0.710 4 7 11.46

5 3.0 0.573 2 3 25.00

7 4.0 0.361 0 1 71.06

4 3.0 0.161 1 1 23.45

8 10.0 0.071 1 0 4.49
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 29

different from each other and are mostly negative 

(water-level decrease) (fig 22). The negative values 

indicate that in response to an increase in hydraulic 

conductivity, water levels would decline, resulting in a 

flattening of the lateral hydraulic gradient through 

layer 1. Water levels in layer 2 decrease in response to 

a decrease in flux due to lower surficial aquifer system 

water levels. Simulated water levels are generally 

insensitive to changes in hydraulic conductivity of the 

surficial aquifer system. The absolute mean waterlevel change was 0.01 ft for the surficial aquifer system 

and 0.003 ft for the Upper Floridan aquifer.

The 2-percent sensitivities of simulated water 

levels in layers 1 and 2 to leakance of the intermediate 

confining unit show different patterns, and are mostly 

negative (water-level decrease) for layer 1 and positive 

(water-level increase) for layer 2 (fig. 23). These spatial patterns result because if leakance is increased, 

more of the specified recharge will flow through the 

intermediate confining unit, thus lowering water levels 

in layer 1 and raising water levels in layer 2. Simulated 

leakage rate through the intermediate confining unit 

provides a better indication of the quantity and spatial 

distribution of the flux between the surficial aquifer 

system and the Upper Floridan aquifer. Figure 26 illustrates that the majority of the study area is dominated 

by diffuse downward leakage to the Upper Floridan 

aquifer. Hence, an overall increase in leakance of the 

intermediate confining unit or transmissivity of the 

Upper Floridan aquifer will result in an overall decline 

in water levels. The negative water levels in layer 1 are 

larger than the positive water levels in layer 2, which is 

a reflection of permeability contrasts between the layers. Simulated water levels are generally sensitive to 

changes in leakance of the intermediate confining unit. 

The absolute mean water-level change was 0.04 ft for 

the surficial aquifer system and 0.04 ft for the Upper 

Floridan aquifer.

The spatial patterns of the 2-percent sensitivities 

of simulated water levels in layers 1 and 2 to transmissivity of the Upper Floridan aquifer are similar to each 

other and are mostly negative (water-level decrease) 

(fig. 24). Because recharge to the surficial aquifer system does not change when transmissivity of the Upper 

Floridan aquifer changes, the hydraulic gradient should 

decrease proportionately for the same amount of water 

to be conveyed laterally through the Upper Floridan 

aquifer. The water-level change in layer 1 is similar to 

the water-level change in layer 2 because the vertical 

leakance between layers 1 and 2 is relatively large. In 

addition, the spatial pattern of water-level change for 

transmissivity is similar to that for recharge (compare 

figs. 21 and 25) because the ground-water system is 

dominated by leakage to the Upper Floridan aquifer in 

the study area. Simulated water levels are generally 

sensitive to changes in transmissivity of the Upper 

Floridan aquifer. The absolute mean water-level 

change was 0.08 ft for the surficial aquifer system and 

0.14 ft for the Upper Floridan aquifer.

The spatial patterns of the 2-percent sensitivities 

of simulated water levels in layers 1 and 2 to anisotropy 

of the Upper Floridan aquifer are similar to each other 

and mostly negative (water-level decline) (fig. 25). 

Because flux in the system does not change substantially with a change in anisotropy of the Upper Floridan 

aquifer, the change in anisotropy causes the lateral 

hydraulic gradient to change proportionately for the 

same amount of water transmitted laterally. Waterlevel changes are greatest in model cells where anisotropy is not equal to one (compare figs. 16 and 26). 

Overall, simulated water levels are insensitive to 

changes in this parameter. The absolute mean waterlevel change was 0.01 ft for the surficial aquifer system 

and 0.02 ft for the Upper Floridan aquifer.

In summary, the simulated water levels are most 

sensitive to changes in recharge and transmissivity of 

the Upper Floridan aquifer. In third place of importance 

is leakance of the intermediate confining unit. The 

least important factors are hydraulic conductivity of the 

surficial aquifer system and anisotropy of the Upper 

Floridan aquifer.

Optimal Estimates of Parameter Values (phase 2)

A total of 96 parameter zones were selected for 

parameter estimation by nonlinear least-squares regression. Based on the RCS and correlation values and the 

consideration of water-level data availability, this set of 

parameters includes most of the important system characteristics. Sensitivities and correlation values, however, may be different for different parameter structures. 

Multipliers were used to modify the initial value of 

hydraulic conductivity of the surficial aquifer system, 

leakance of the intermediate confining unit, transmissivity of the Upper Floridan aquifer, or anisotropy of the 

Upper Floridan aquifer by a fixed amount in the zone 

assigned to each parameter. Initial values for all parameters were set equal to their calibrated values in the 

existing CNTB area hydrologic model (SDI, 1997).

Recharge was assumed to be known and was 

specified as simulated by SDI (1997). Recharge was 
30 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

SURFICIAL AQUIFER SYSTEM

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

0.00 to 0.05

0.05 to 0.25

Greater than 1.00

General area of model where the potentiometric surface of

the Upper Floridan aquifer is always below the top of the

aquifer (SDI Environmental Services, Inc., 1997, fig. 47)

EXPLANATION

WATER-LEVEL CHANGE, IN FEET

0.25 to 0.50

0.50 to 1.00

Figure 21A. Sensitivity of simulated water levels to a 2-percent increase in the initial value of recharge.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 31

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

UPPER FLORIDAN AQUIFER

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

0.00 to 0.05

0.05 to 0.25

EXPLANATION

WATER-LEVEL CHANGE, IN FEET

0.25 to 0.50

Greater than 0.50

Figure 21B. Sensitivity of simulated water levels to a 2-percent increase in the initial value of recharge.
32 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

SURFICIAL AQUIFER SYSTEM

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

-1.00 to -0.50

-0.50 to -0.25

0.05 to 0.25

EXPLANATION

WATER-LEVEL CHANGE, IN FEET

-0.25 to -0.05

-0.05 to 0.05

Figure 22A. Sensitivity of simulated water levels to a 2-percent increase in the initial value of hydraulic 

conductivity of the surficial aquifer system.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 33

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

UPPER FLORIDAN AQUIFER

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

EXPLANATION

-0.50 to -0.01

WATER-LEVEL CHANGE, IN FEET

-0.01 to 0.01 0.01 to 0.02

Figure 22B. Sensitivity of simulated water levels to a 2-percent increase in the initial value of hydraulic 

conductivity of the surficial aquifer system.
34 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

SURFICIAL AQUIFER SYSTEM

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

-1.00 to 0.50

-0.50 to -0.25

0.05 to 0.25

0.25 to 0.50

EXPLANATION

WATER-LEVEL CHANGE, IN FEET

-0.25 to -0.05

-0.05 to 0.05

Figure 23A. Sensitivity of simulated water levels to a 2-percent increase in the initial value of leakance.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 35

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

EXPLANATION

-0.10 to -0.05

WATER-LEVEL CHANGE, IN FEET

-0.05 to 0.05 0.05 to 0.25

UPPER FLORIDAN AQUIFER

Figure 23B. Sensitivity of simulated water levels to a 2-percent increase in the initial value of leakance.
36 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

SURFICIAL AQUIFER SYSTEM

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

-1.00 to 0.50

-0.50 to -0.25

0.05 to 0.25

0.25 to 0.50

EXPLANATION

WATER-LEVEL CHANGE, IN FEET

-0.25 to -0.05

-0.05 to 0.05

Figure 24A. Sensitivity of simulated water levels to a 2-percent increase in the initial value of Upper Floridan 

aquifer transmissivity.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 37

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

UPPER FLORIDAN AQUIFER

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

-1.00 to 0.50

-0.50 to -0.25

0.05 to 0.25

0.25 to 0.50

EXPLANATION

WATER-LEVEL CHANGE, IN FEET

-0.25 to -0.05

-0.05 to 0.05

Figure 24B. Sensitivity of simulated water levels to a 2-percent increase in the initial value of Upper Floridan 

aquifer transmissivity.
38 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

EXPLANATION

-0.25 to -0.05

WATER-LEVEL CHANGE, IN FEET

-0.05 to 0.05 0.05 to 0.25

SURFICIAL AQUIFER SYSTEM

Figure 25A. Sensitivity of simulated water levels to a 2-percent increase in the initial value of Upper Floridan 

aquifer anisotropy ratio.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 39

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

UPPER FLORIDAN AQUIFER

EXPLANATION

-0.25 to -0.05

WATER-LEVEL CHANGE, IN FEET

-0.05 to 0.05 0.05 to 0.25

Figure 25B. Sensitivity of simulated water levels to a 2-percent increase in the initial value of Upper Floridan 

aquifer anisotropy.
40 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

EXPLANATION

Less than 0

0 to 10

10 to 25

25 to 50

SIMULATED LEAKAGE RATE, IN INCHES PER YEAR

Figure 26. Simulated rate of leakage through the intermediate confining unit based on steady-state 1987 

conditions.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 41

not estimated because of low sensitivity of parameter 

zones and because high correlation was expected 

between recharge and transmissivity. High correlation 

was expected because the ground-water flow equation 

can be written in terms of the ratio (Q = KA(dh/dl)). 

Unless independent information on recharge or transmissivity is available, the regression cannot be used to 

distinguish each individual component of the ratio. An 

incorrectly specified recharge matrix could affect the 

values, to some degree, and add to the uncertainty in 

the estimated parameter values. 

A total of 23 parameter zones (SDI, 1997) were 

used to represent transmissivity of the Upper Floridan 

aquifer. Regression results indicate that the estimate 

of most transmissivity-zone values is consistent with 

hydrogeologic information. Generally, values of 

transmissivity are within 50 percent of the initial values (fig. 27). Optimal estimates of individual transmissivity parameters range from a factor of about 13.9 

above to a factor of about 0.1 below the initial values. 

Several of the values for parameter zones highlighted 

in figure 27 are probably outside the range of likely 

values estimated from field data. 

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Land O

Lakes

New

Port

Richey

Temple

Terrace

Tarpon

Springs

Clearwater

Tampa

Brandon

Plant City

Zephyrhills

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

0.83

(110,000)

0.63

(50,000)

1.01

(20,000)

0.53

(250,000)

1.63

(130,000)

1.17

(50,000)

1.13

(70,000)

0.07

(10,000)

13.86

(20,000)

1.27

(60,000)

1.28

(20,000)

10.65

(500,000)

11.52

(35,000)

13.66

(25,000)

1.22

(500,000)

1.41

(400,000)

2.71

(100,000)

0.66

(20,000)

0.85

(40,000)

3.60

(400,000)

1.47

(300,000)

0.71

(300,000)

0.19

(130,000)

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0.53 0 5 10 KILOMETERS

(250,000)

Less than 0.50

0.50 to 1.50

More than 1.50

PARAMETER-ZONE MULTIPLIER Black number is parameter-zone

multiplier. Red number is the initial

value of Upper Floridan aquifer

transmissivity, in feet squared per day

EXPLANATION

Figure 27. Delineation of Upper Florida aquifer zones used for parameter estimation and simulated 

transmissivity-zone multiplier for the initial parameter values.
42 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

A total of 71 parameter zones (SDI, 1997) were 

used to represent leakance of the intermediate confining 

unit. Regression results indicate that the estimate of 

most leakance parameters is within the expected range of 

likely values. Generally, values are within an order of 

magnitude of the initial values (fig. 28). Optimal estimates of individual leakance parameters range from a 

factor of about 143.8 above to a factor of about 0.01 

below initial values. Several of the values for parameterzones highlighted in figure 28 are probably outside the 

range of likely values estimated from field data.

One global parameter value served as a multiplier for estimating the spatially variable hydraulic 

conductivity of the surficial aquifer system. Regression results indicate that the initial estimate of hydraulic conductivity corresponds closely to estimates 

obtained from other independent sources and is within 

the range of likely values estimated from field data. 

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

0.12

1.0E-02

11.39

6.0E-04

1.46

3.5E-01

0.50

1.5E-03

0.85

3.0E-03

3.06

1.0E-03

4.13

1.8E-04

1.00

3.0E-03 4.04

1.0E-03 2.86

1.0E-02

1.48

3.0E-03

1.26

2.5E-03

0.66

4.0E-05

4.22

2.0E-03

0.12

1.0E-03

70.99

6.0E-04

1.68

3.0E-04

0.69

8.0E-04

1.46

8.0E-04

0.16

7.0E-04

1.35

5.0E-04

1.52 4.0E-04

0.53

1.0E-03

0.68

4.0E-04

3.82

7.0E-04

1.46

8.0E-04

0.59

1.0E-04

0.58

1.5E-03

4.52

8.0E-04

2.37

3.0E-04

0.03

1.0E-04

31.25

2.0E-03

0.91

2.0E-03 1.60

1.0E-03

0.77

8.0E-04

1.35

2.0E-04

1.31

6.0E-04

2.18

1.5E-03

0.34

6.0E-05

1.79

1.0E-04

0.27

4.0E-04 0.99

4.0E-04

0.19

6.0E-04

4.73

1.5E-03

0.23

4.0E-04

1.26

2.0E-04

0.50

4.0E-05

2.00

1.0E-06

7.018.0E-04

0.16

6.0E-03

2.31

4.0E-03

1.51

8.0E-04

8.01

5.0E-04

0.55

8.0E-04

0.78

6.0E-04

0.34

1.5E-03

2.14

2.0E-04

0.27

4.0E-04

0.27

4.0E-04

0.341.0E-02

1.10

5.0E-05

2.03

1.0E-05

0.01

1.0E-03

0.21

5.0E-05

0.04

1.0E-03

0.96

9.0E-05

0.78

8.0E-04

1.35

2.0E-04

0.95

3.0E-04

2.11

0.04 2.0E-04

3.5E-01

0.95 3.0E-04

1.02

1.0E-03

0.24

7.0E-05

143.80

7.0E-04

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0.04 0 5 10 KILOMETERS 3.5E-01

Less than 0.10

0.10 to 10

More than 10

LEAKANCE-ZONE MULTIPLIER Black number is leakance-zone

multiplier. Red number is the initial

leakance of intermediate confining

unit, in feet per day per foot

EXPLANATION

Figure 28. Leakance zones of the intermediate confining unit used for parameter estimation and simulated 

leakance-zone multiplier for the initial parameter values.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 43

LEAKANCE OF THE INTERMEDIATE CONFINING UNIT

RELATIVE COMPOSITE SENSITIVITY

1 234 5678 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 48 47 49 50

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

1.0

0.8

0.6

0.4

0.2

0

KX 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 KZ

PARAMETER-ZONE IDENTIFICATION SYMBOL

TRANSMISSIVITY OF THE UPPER FLORIDAN AQUIFER (1-23)

HYDRAULIC CONDUCTIVITY OF THE SURFICIAL AQUIFER SYSTEM (KX)

ANISOTROPY OF THE UPPER FLORIDAN AQUIFER (KZ)

RELATIVE COMPOSITE

SENSITIVITY

1.0

0.8

0.6

0.4

0.2

0

1.0

0.8

0.6

0.4

0.2

0

PARAMETER-ZONE IDENTIFICATION SYMBOL

1.0

0.8

0.6

0.4

0.2

0

Figure 29. Relative composite sensitivity for optimal estimates of input 

parameter values.

The optimal estimate of hydraulic conductivity is a factor of about 0.78 below the initial values. 

One global parameter value served 

as a multiplier for estimating the spatially 

variable anisotropy ratio of the Upper 

Floridan aquifer. Regression results produced an estimated value of anisotropy of 

the Upper Floridan aquifer within the range 

of likely values based on very limited field 

data. The optimal estimate of anisotropy of 

the Upper Floridan aquifer is a factor of 

about 0.25 below the initial values. 

RCS values for the ninety-six estimated parameter-zone values in the optimized final model are shown in figure 29. 

The final (optimized) RCS values changed 

somewhat, but were still quite similar to the 

initial SDI (1997) parameter values (determined at the first iteration of the optimized 

simulation). Generally, sensitivity is highest for Upper Floridan aquifer transmissivity zones and lowest for intermediate 

confining unit zones. As indicated previously, parameters with smaller RCS values 

relative to those for other parameters are 

likely to have higher parameter uncertainty; 

parameters with higher RCS values are 

likely to have lower parameter uncertainty. 

Most of the estimated parameters are 

not highly correlated to one another, as 

indicated by small correlation coefficients 

(most less than 0.50, appendix). The most 

highly correlated pair is adjacent leakance 

zones 47 and 25 (appendix) with a correlation coefficient of –0.99. The second highest correlated pair is adjacent leakance 

zones 13 and 22 with a correlation coefficient of –0.88.

The effects using a different set of 

water-level observations also were evaluated. Comparisons were made between an 

inverse model with the 119 calibration 

water-level observations used in the SDI 

(1997) model calibration and the present 

inverse model which uses 549 water-level 

observations. Simulated parameter-zone 

multipliers derived from the two data sets 

are presented in figure 30 and a statistical 
44 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

PARAMETER-ZONE IDENTIFICATION SYMBOL

LEAKANCE OF THE INTERMEDIATE CONFINING UNIT

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 48 47 49 50

TRANSMISSIVITY OF THE UPPER FLORIDAN AQUIFER (1-23)

HYDRAULIC CONDUCTIVITY OF THE SURFICIAL AQUIFER SYSTEM (KX)

ANISOTROPY OF THE UPPER FLORIDAN AQUIFER (KZ)

KX 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23KZ

PARAMETER-ZONE IDENTIFICATION SYMBOL

10.0

1.0

0.1

0.01

50.0

LEAKANCE-ZONE MULTIPLIER PARAMETER-ZONE MULTIPLIER

OPTIMIZED VALUES USING 119

WATER-LEVEL OBSERVATIONS

OPTIMIZED VALUES USING 549

WATER-LEVEL OBSERVATIONS

EXPLANATION

123 4 6 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

10.0

1.0

0.1

0.01

50.0

10.0

1.0

0.1

0.01

50.0

10.0

1.0

0.1

0.01

50.0

Figure 30. Simulated parameter-zone multipliers derived from two separate sets of water-level observations.

summary of the results is presented in 

table 6. Most of the statistics shown in 

table 6 and many of the optimized values 

shown in figure 30 are significantly different for the two data sets. These differences reflect the dependency of inverse 

model results on the parameter-zonation 

structure and the observation data set. 

Inverse model instability occurs when too 

many parameters are estimated based on 

a limited number of observations.

Model Agreement

Model fit of the parameter-estimation model was evaluated both objectively and subjectively. A statistical 

analysis of residuals (simulated values 

minus measured values) was used to 

objectively assess the overall goodness of 

model fit. Residuals are important indicators of model fit, but are dependent on 

both the quality of observation data and 

model accuracy. The quantities included 

in the evaluation are: (1) water levels; 

(2) river base flows; and (3) spring flows. 

Inspection of the spatial distribution of 

errors of water level, direction and magnitude of ground-water flow, and river 

and spring discharges were used to subjectively analyze model performance. 

Observation data available for 

comparison with simulated values 

included 549 measured water levels in 

wells completed in the surficial aquifer 

system and the Upper Floridan aquifer, 

15 estimated river base flows, and 

23 measured spring flows. However, 

only the 549 hydraulic water-level values 

were used in the inverse model. 
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 45

The parameter-estimation model generally produced simulated water levels in close agreement with 

measured water levels (table 7 and fig. 31). The waterlevel residuals were normally distributed; approximately 70 percent of the simulated water levels are 

within 2 ft of the measured water levels and approximately 90 percent are within 5 ft of measured levels. 

The residual statistics show that the overall model fit 

for both layers is improved when parameters are optimized using nonlinear regression. The RMSE was 

reduced from 7.35 to 5.63 ft and the average residual 

was reduced from 0.37 to 0.09 ft (fig. 32 and table 7). 

The small improvement relative to the overall error is 

primarily due to large surficial aquifer system residuals 

in both models and the bias of the SS to well field areas 

where dense clusters of wells are present. 

Simulated water-table altitudes for the surficial 

aquifer system are shown in figure 33. The flow model 

simulated average annual steady-state water levels for 

the period January through December 1987. Simulated 

water levels were slightly higher than measured water 

levels; the average difference between simulated and 

measured water levels at 223 measurement sites was 

0.37 ft. The general directions of simulated groundwater flow and the magnitude of hydraulic gradients 

within the surficial aquifer system were in close agreement with measured water levels. There are a few notable exceptions to this agreement where model results 

may be deficient. For example, in a small area of Hernando and Pasco County, south of Brooksville and 

north of Dade City, the surficial aquifer system cells go 

dry and the model does not simulate the water table 

accurately. This is in an area where the Upper Floridan 

aquifer is unconfined and where hydraulic separation 

between the surficial aquifer system and the Upper 

Floridan aquifer does not exist (SDI, 1997).

 In cells that go dry (Brooksville area), the altitude of the water-table surface in the surficial aquifer 

system is assumed to be equal to the altitude of the 

potentiometric surface in the Upper Floridan aquifer. 

Also in the Brandon and Clearwater areas, the model 

does not simulate water levels accurately (residuals 

Table 6. Statistical summary of optimal parameter-zone multipliers derived from 119 and 549 water-level 

observations

[UFA, Upper Florida aquifer; ICU, intermediate confining unit; K, hydraulic conductivity; Kx/Ky, horizontal anisotropy ratio; 

SAS, surficial aquifer system]

Statistic

Transmissivity

of the UFA

Leakance

 of the ICU

K

of the SAS

Kx/Ky

of the UFA

119 549 119 549 119 549 119 549

Number of zones 23 23 71 71 1 1 1 1

Minimum 0.01 0.07 0.02 0.01

Maximum 34.58 13.86 490.50 143.80

Mean 2.96 3.09 9.12 5.02 1.96 0.78 0.77 0.25

Standard deviation 7.29 4.49 58.15 19.03

Table 7. Statistical summary of differences between simulated and measured water levels 

for the initial and optimized simulations

Number of

 observations

Root mean

 square error

 (feet)

Average

 water-level

 residuals

 (feet)

Minimum waterlevel residuals 

(feet)

Maximum

water-level

residuals

(feet)

Surficial aquifer system

Initial values 223 8.15 -0.67 -55.36 15.02

Optimized values 223 5.89 0.37 -45.02 35.56

Upper Floridan aquifer

Initial values 326 4.06 1.39 -9.90 13.08

Optimized values 326 2.62 0.03 -9.94 9.01

Entire model

Initial values 549 7.35 0.37 -55.36 15.02

Optimized values 549 5.63 0.09 -45.02 35.56
46 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

150

100

50

0

0 50 100 150

MEASURED WATER LEVELS, IN FEET

SIMULATED WATER LEVEL

S, IN FEET

60

50

40

30

20

10

0

-50 -40 -30 -20 -10 0 10 20 30 40 50

NUMBER OF WATER-LEVEL

MEASUREMENTS

WATER-LEVEL RESIDUALS

Number of

water-level

measurements

Minimum

residual

(feet)

Maximum

residual

(feet)

Average

error

(feet)

Root mean

square error

(feet)

SURFICIAL AQUIFER SYSTEM

120

80

40

0

0 40 80 120

MEASURED WATER LEVELS, IN FEET

SIMULATED WATER LEVEL

S, IN FEET

50

40

30

20

10

0

-10 -5 0 5 10

WATER-LEVEL RESIDUALS

NUMBER OF WATER-LEVEL

MEASUREMENTS

Number of

water-level

measurements

Minimum

residual

(feet)

Maximum

residual

(feet)

Average

error

(feet)

Root mean

square error

(feet)

326 -9.94 9.01 0.03 2.62

UPPER FLORIDAN AQUIFER

223 -45.02 35.56 0.37 5.89

Figure 31. Comparison of simulated to measured water levels for the parameter-estimation 

model.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 47

150

100

50

0

0 50 100 150

MEASURED WATER LEVELS, IN FEET

SIMULATED WATER LEVELS, IN FEET

NUMBER OF WATER-LEVEL

MEASUREMENTS

WATER-LEVEL RESIDUALS

Number of

water-level

measurements

Minimum

residual

(feet)

Maximum

residual

(feet)

Average

error

(feet)

Root mean

square error

(feet)

223 -55.36 15.02 -0.68 8.15

SURFICIAL AQUIFER SYSTEM

-60 -40 -20 0 20 40 60

50

40

30

20

10

0

50

40

30

20

10

0

WATER-LEVEL RESIDUALS

NUMBER OF WATER-LEVEL

MEASUREMENTS

Number of

water-level

measurements

Minimum

residual

(feet)

Maximum

residual

(feet)

Average

error

(feet)

Root mean

square error

(feet)

326 -9.90 13.08 1.39 4.06

UPPER FLORIDAN AQUIFER

-15 -10 -5 0 5 10 15

120

80

40

0

0 40 80 120

MEASURED WATER LEVELS, IN FEET

SIMULATED WATER LEVELS, IN FEET

Figure 32. Comparison of simulated to measured water levels for the initial central northern 

Tampa Bay area hydrologic model.
48 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

greater than 10 ft below measured). In the Brandon and 

Clearwater areas, measured water levels may represent 

local or perched water-level conditions that cannot be 

simulated due to the localized nature of the perched 

system. The residuals also could indicate that either 

recharge rate in these areas was underestimated or the 

leakance of the intermediate confining unit between 

layers was overestimated. No spatial trends in the distribution of water-level residuals are apparent (fig. 33).

The simulated potentiometric surface of the 

Upper Floridan aquifer is shown in figure 34. Simulated water levels were slightly higher than measured 

water levels; the average difference between simulated 

and measured water levels at the 326 measurement 

sites was 0.03 ft. The magnitude and direction of simulated hydraulic gradients are similar to measured 

water levels. No spatial trends in the distribution of 

water-level residuals are apparent (fig. 34).

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

GENERALIZED BOUNDARY

OF DRY CELLS

+

+

+

+

+ + +

+

+

+ +

+

+

+ +

+ +

++++

+++ ++ + + + ++ ++

+ + + +

+

+

+

++++++++ +

+

+

+

+

+ +++

++++

+++++

+

+

+

+

+ +

++

+

+

+

+

+

+

+

+ + +

+

+

+

++ ++ +

+ +

+

+

+ +

+

+ + + +

+

+

+ + + +

+

+

+

++

+ +

+

+

+

+ ++

+ +

+ +

+

+ +

++ + +++

++ +++ + ++

+

+ + +

+

+ + +

+ +

+ +

++

+

+

+

+

+

	









 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

0 to 20

20 to 40

40 to 60

60 to 80

OBSERVATION WELL AND WATERLEVEL RESIDUAL—Measured value

minus simulated value, in feet

EXPLANATION

SIMULATED WATER TABLE IN THE

SURFICIAL AQUIFER SYSTEM, IN FEET.

DATUM IS SEA LEVEL

80 to 100

100 to 120

120 to 140

Greater than 140

0 to 5

5 to 10

+ Greater than 10

Less than -10 +

-5 to -10

0 to -5

Figure 33. Location of water-level measurements, value of water-level residuals, and simulated water table in 

the surficial aquifer system, calendar year 1987.
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 49

Simulated river base flow was compared at 

21 sites (table 8). The simulated annual average 

ground-water discharge to streams was simulated as 

84 ft3/s for the 1987 calendar year. Base flow, determined by hydrograph separation techniques using a 

computerized program (White and Sloto, 1991), was 

estimated to be 557 ft3/s. The simulated discharge 

compares poorly with the estimated discharge, probably due to the scale of the model. A greater level of 

detail in river-bed leakance, stage, and bottom elevation, which is consistent at all scales, should be incorporated into this model to more accurately simulate 

base flow rates. 

Simulated spring flow was compared at 13 sites 

(table 9). The simulated annual average ground-water 

discharge to springs was estimated to be 392 ft3/s for 

the 1987 calendar year. Total spring flow, determined 

from direct measurements and estimates was about 

0 to 20

20 to 40

40 to 60

60 to 80

OBSERVATION WELL AND WATERLEVEL RESIDUAL—Simulated value

minus observed value, in feet

EXPLANATION

SIMULATED POTENTIOMETRIC SURFACE

OF THE UPPER FLORIDAN AQUIFER, IN

FEET. DATUM IS SEA LEVEL

80 to 100

100 to 120

120 to 133

+

+

-9 to -5

-5 to 0

0 to 5

5 to 12

PASCO COUNTY

HERNANDO COUNTY

HILLSBOROUGH COUNTY

POLK

COUNTY

PINELLAS

COUNTY

STUDY AREA

BOUNDARY

Brooksville

Clearwater

Tampa

Lakeland

Dade

City

OLD

TAMPA

BAY

GULF OF MEXICO

Zephyrhills

 	
  




 
 
 

 	
 
 



0 5 10 MILES

0 5 10 KILOMETERS

+

+

+

+

+

+

+ +

+

+

+

+

+

+

+

+

+

+

++ +

++

+

+

+

+ ++ + +

+

+ + + +

+ ++

+ +

++ +

+ + + + + +

+

+ +

+ +

+

+

+

++ + +

+ + +

+

+

+

++

+++

++ + + +

++ +

+ + +

+ +

+

++

+++

+

+

+

+

+

++

++++++++++

+

+

+

+

+

+

+ + + + + +

+ + + +

+ +

+ + +

+ +

+ +

+ +

+ +

+

+ +

+ +

+

+ + + + +

+ + +

++

++

++

+ + +

+

+ ++

+

+

+ +

+ +

+

+ +

+

+

+ +

+

+

+

+ +

+++ + + + + ++++

+++ +++++++

+

+ + +

+ + +

+

+

++

++++++ +

+

++ + + ++

+

+ ++

++

+

++

+ ++++

+ +

+

+

+

+

+

+

+

+

+

+

++

+ + +

+ +++++ + +

+

+

+

Figure 34. Location of water-level measurements, value of water-level residuals, and simulated potentiometric 

surface of the Upper Floridan aquifer, calendar year 1987.
50 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

Table 8. Comparison of model-simulated and estimated base flows, calendar year 1987

[ft3/s; cubic foot per second]

Station name Station

 number

Estimated

base flow

(ft3/s)

Maximum

discharge

(ft3/s)

Minimum

discharge

(ft3/s)

Average

discharge

(ft3/s)

Optimized

base flow

(ft3/s)

Anclote River:

 South Branch near Odessa 02309848 1.8 184 0.01 5.33 0.5

 Near Odessa 02309980 22.7 1,840 0.85 78.6 4.1

 Near Elfers 02310000 26.5 82.9 3.60 82.9 4.6

Brooker Creek:

 At Van Dyke 02307200 2.0 97.0 0.01 5.45 0.9

 Near Lake Fern 02307323 2.5 153 0.01 9.05 0.8

 Near Lake Tarpon 02307359 9.2 434 0.42 21.8 0.8

Hillsborough River:

 Near Zephyrhills 02303000 60.11 3,980 83.0 220 24.2

 At Morris Bridge 02303330 105.71 3,160 92.0 270 7.6

 Near Tampa 02304500 151.82 3,020 0.26 359.06 10.6

Hillsborough River Tributaries:

 Blackwater Creek near Knights 02302500 17.0 1,760 4.50 67.0 0.2

 Cypress Creek

 Near San Antonio 02303400 8.6 996 0.12 25.46 0.2

 At Worthington Gardens 02303420 20.1 1,430 0.72 73.54 4.5

 Near Sulphur Springs 02303800 41.9 1,430 2.40 104 5.6

 Flint Creek near Thonotosassa 02303300 16.1 322 2.00 43.2 1.1

 Trout Creek near Sulphur Springs 02303350 3.5 30.8 0.00 30.8 2.6

Pithlachascotee River:

 Near Fivay Junction 02310280 4.5 130 0.41 11.7 4.6

 Near New Port Richey 02310300 10.7 879 0.91 33.5 6.0

Rocky Creek:

 At State Highway 587 02306774 7.5 195 0.85 15.0 0.1

 Near Sulphur Springs 02307000 32.1 724 4.50 66.8 2.3

Rocky Creek Tributaries:

 Brushy Creek near Tampa 02303910 10.0 259 2.90 21.2 0.3

 Sweetwater Creek near Tampa 02306647 3.1 310 1.40 25.2 2.4

 Totals 557.4 84.0

1

Crystal Springs discharge excluded

2

Includes surface withdrawals of 85 ft3/s for the City of Tampa
Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling 51

432 ft3

/s. Total simulated spring flow compares favorably with total measured spring flow; however, the simulated spring flow for individual springs compares poorly 

with the measured and estimated discharge. Matching 

individual spring flows is difficult and also is of questionable value because of discretization and data input problems. Many individual springs have been assigned to 

multiple grid cells instead of single grid cells; including 

Weeki Wachee, Lithia, Gator, Bobhill, Magnolia Springs, 

and Salt Springs. In addition, many of the spring altitudes 

were determined from topographic maps resulting in estimated spring-pool altitudes that affect simulated flow. 

Spring-pool altitudes assigned to the cell containing Salt 

and Mud Springs (7 ft above sea level), and the cells containing Gator, Bobhill, and Magnolia Springs (13 ft above 

sea level) are set too high, and therefore, flow from these 

springs is reduced.

Components of the simulated hydrologic budget 

of the modeled area for the 1987 calendar year are 

shown in table 10. Of the 15.3 in/yr of flow through the 

surficial aquifer system and the Upper Floridan aquifer 

(sum of inflows or outflows shown in bold in table 10), 

nearly 90 percent consists of recharge from rainfall to 

the surficial aquifer system. In addition, simulated net 

leakage to the Upper Floridan aquifer from the surficial 

aquifer system was 7.2 in/yr, which represents 46 and 67 

percent of the total flows in the surficial aquifer system 

and the Upper Floridan aquifer, respectively. 

Table 9. Comparison of model-simulated and measured 

spring flows, calendar year 1987 

[ft3/s, cubic foot per second]

Spring name

Measured or

 estimated flow

(ft3/s)

Optimized flow

(ft3/s)

Buckhorn 9.4 10.3

Crystal 54.7 76.8

Gator, Bobhill, Magnolia 6.8 0.0

Heath 6.5 24.5

Horeshoe 8.6 0.9

Hudson 30.0 1.4

Lettuce, Eureka 11.0 0.0

Lithia 44.5 23.9

Salt (Pasco) 9.5 10.3

Salt, Mud 27.2 0.0

Shady 0.4 1.6

Sulphur 38.2 31.9

Weeki Wachee 185 210.9

Totals 431.8 392.5

Limitations of Model Analysis 

This inverse model, or any other model, is limited 

by simplification of the conceptual model, discretization 

effects, difficulty in obtaining sufficient measurements to 

account for all of the spatial variation in hydraulic properties throughout the model area, and limitations in the 

accuracy of land surface altitude measurements. The 

inverse model simulates average annual conditions and 

does not account for seasonal changes in ground-water 

recharge and discharge or seasonal variability in values of 

hydraulic head. The model yields parameter-zone values 

similar to estimates from field data and produces simulated water levels in close agreement with measured 

water levels. This model, however, is not unique, and different ground-water flow model constructions with optimal parameter estimates may fit the available 

observations equally well. Results showed that there are 

not 

Table 10. Simulated hydrologic budget of the aquifer system 

in the study area, calendar year 1987

[All values are fluxes averaged over the model area. Sum of bold number 

represent total inflow or outflow through Upper Florida aquifer (UFA) and 

the surficial aquifer system (SAS)]

Budget

component

Inflow

(inches per 

year)

Outflow

(inches per 

year)

SURFICIAL AQUIFER SYSTEM

Recharge from rainfall 13.6

Upward Leakage from UFA 2.0

Downward leakage to UFA 9.2

Rejected recharge (drains) 3.8

Discharge to rivers 1.7

Discharge to coast 0.9

Pumpage from wells < 0.01

Total flow 15.6 15.6

UPPER FLORIDAN AQUIFER

Leakage from SAS 9.2

Direct recharge from rainfall 1.2

Specified head boundary 0.3

Leakage from rivers 0.2

Discharge to springs 3.3

Pumpage from wells 2.9

Upward leakage to SAS 2.0

Discharge to rivers 1.8

General head boundary 0.6

Discharge to Bear Sink and

 Round Sink

0.3

Total flow 10.9 10.9
52 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

sufficient water-level data to independently estimate all 

possible parameters. In addition, several parameter 

values estimated by the regression are probably not 

reasonable. The inverse model is designed not as a 

predictive tool, but as an interpretive one; it is intended 

to gain modeling insight given the proposed conceptualization. Due to the limited availability of observation 

data, a simpler parameter structure should be considered to produce a more unique solution. 

SUMMARY

This report presents the results of a study to 

describe application of nonlinear least-squares regression to the existing ground-water flow model of the 

central northern Tampa Bay (CNTB) area. The study 

has an area of approximately 2,000 mi2 that includes all 

of Pasco County, most of Hernando, Pinellas and 

Hillsborough Counties, and part of Polk County. Six 

rivers and their tributaries, several small streams along 

the coast, and some internally drained systems that 

flow only during extreme rainfall events, define the 

surface-water system of the study area. The two largest 

systems are the Hillsborough and Withlacoochee 

Rivers. There are hundreds of lakes, swampy plains, 

and intermittent ponds dispersed throughout the study 

area, ranging in size from less than 1/4 acre to more 

than 2,500 acres. In the study area, a total of 17 springs 

are either found inland flowing to adjacent rivers or 

along the coast discharging directly to the Gulf of Mexico. The ground-water flow system beneath the study 

area is a multilayered system consisting of a thick 

sequence of carbonate rock overlain by clastic deposits. 

The hydrogeologic framework includes the unconfined 

surficial aquifer system and the confined Upper 

Floridan aquifer. A low permeablility intermediate 

confining unit separates the aquifers. The Upper 

Florida aquifer is underlain by a low-permeablility 

evaporite limestone that forms the bottom of the fresh 

ground-water flow system.

Ground-water flow was simulated using MODFLOW, the USGS three-dimensional ground-water 

flow model. The flow model was linked with a nonlinear weighted least-squares regression routine for solution of the inverse problem. The ground-water flow 

model constructed for this study was based on existing 

data and information and assumed steady-state conditions. Regression statistics for the reported parameter 

values (SDI, 1997), including parameter sensitivities 

and correlation, were calculated. The analysis 

procedure consisted of two simulation phases. Phase 1 

was designed to calculate parameter sensitivities and 

correlations for the SDI (1997) parameter values and to 

assess model parameterization. Phase 2 was designed 

to determine the optimal parameter values for the 

hydrologic features of interest using nonlinear regression and to evaluate these optimal values.

A total of 96 parameter-zone values were estimated. Recharge was assumed to be known and specified as an unestimated parameter. Optimal estimates of 

individual transmissivity-zone values are generally 

within 50 percent of the initial values and range from a 

factor of about 13.9 above to a factor of about 

0.1 below than the initial values. Optimal estimates of 

individual leakance-zone values are generally within 

an order of magnitude of the initial values and range 

from a factor of about 143.8 above to a factor of about 

0.01 below than initial values. The optimal estimate of 

hydraulic conductivity of the surficial aquifer system is 

a factor of about 0.78 below than the initial. The optimal estimate of anisotropy of the Upper Floridan aquifer is a factor of about 0.25 below than the initial 

values. Several estimates of transmissivity of the 

Upper Floridan aquifer and leakance of the intermediate confining unit are probably outside the range of 

likely values estimated from field data.

The parameter-estimation model generally produced simulated water levels in close agreement with 

measured water levels. Approximately 70 percent of 

the simulated water levels are within 2 ft of the measured water levels and 90 percent are within 5 ft of 

measured levels. The general directions of groundwater flow and the magnitude of hydraulic gradients 

simulated in the surficial aquifer system and the Upper 

Floridan aquifer agree well with the regional groundwater flow system.

It was impossible to independently estimate all 

parameters given the present zonation structure and 

observation data sets due to parameter insensitivity and 

correlation. A simpler parameter structure should be 

considered. Possible solutions are to (1) collect more 

water-level data; (2) estimate fewer parameters by 

either combining zones or assigning values to insensitive parameters; and (3) estimate the most sensitive 

parameters first, then using the optimized values for the 

most sensitive parameters as initial values, estimate the 

entire data set. 
Selected References 53

SELECTED REFERENCES

Gill, T.E., Murray, W., and Wright, M.H., 1981, Practical 

optimization: Orlando, Fla., Academic Press, Inc., 

401 p.

Halford, K.J., 1992, Incorporating reservoir characteristics 

for automatic history matching: Baton Rouge, Louisiana State University, Ph.D. dissertation, 150 p.

———1997, Ground-water flow in the surficial aquifer 

system and potential movement of contaminants from 

Selected waste-disposal sites at Naval Station Mayport, 

Florida: U.S. Geological Survey Water-Resources 

Investigations Report 97-4262, 104 p.

Hill, M.C., 1992, A computer program (MODFLOWP) for 

estimating parameters of a transient, three-dimensional, 

ground-water flow model using non-linear regression: 

U.S. Geological Survey Open-File Report 91-484, 

358 p.

Johanson, R,C., Imhoff, J.C., Kittle, J.l., and Donigan. A.S., 

1984, Hydrological simulation program-FORTRAN 

(HSPF): User’s manual for release 8.0: U.S.Environmental Protection Agency, Athens, Georiga.

McDonald, M.G., and Harbaugh, A.W., 1988, A modular 

three-dimensional finite-difference ground-water flow 

model: U.S. Geological Survey Techniques of WaterResources Investigations, book 6, chap. A1, 576 p.

Poeter, E.P., and Hill, M.C., 1996, Inverse Models: A necessary next step in ground-water modeling: Ground 

Water, v. 35, no. 2, p. 250-260.

SDI Environmental Services, Inc., 1997, Water resources 

evaluation and integrated hydrologic model of the 

northern Tampa Bay region: Consultant’s report in the 

files of Tampa Bay Water, Clearwater, Florida, 146 p.

Southwest Florida Water Management District, 1993 (draft), 

Computer model of ground-water flow in northern 

Tampa Bay: Southwest Florida Water Management 

District: Brooksville, Florida, 120 p.

White, K.E., and Sloto, R.A., 1991, Base flow frequency 

characteristics of selected Pennsylvania streams: U.S. 

Geological Survey Water-Resources Investigations 

Report 90-4160, 66 p.

Yager, R.M., 1993, Simulated three-dimensional groundwater flow in the Lockport Group, a fractured dolomite 

aquifer near Niagara Falls, New York: U.S. WaterResources Investigations Report 92-4189, 43 p.

Yeh, W.W-G., 1986, Review of parameter identification procedures in groundwater hydrology: The inverse problem: Water Resources Research, v. 22, no. 2, p. 95-108.
54 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida
APPENDIX
56 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

Appendix. Initial and optimized parameter values, relative composite sensitivity for optimal

estimates of input parameter values, and correlation between parameters

[RCS, relative composite sensitivity; ft/d, foot per day; ft/d/ft, foot per day per foot; K, hydraulic

conductivity; L, leakance; T, transmissivity; Kx/Ky, horizontal anisotropy ratio]

Parameterzone

number

RCS Initial value Optimized

value

Correlated parameters

Most correlated 2nd most

correlated

Hydraulic conductivity of the surficial aquifer system1 (ft/d)

 K 0.320 21 20.78 L20 (-.56) L69(-.38)

Transmissivity of the Upper Floridan aquifer3 (ft2/d)

 T1 1.000 500,000 610,000 T2 (-.61) T3 (.52)

 T2 0.670 400,000 564,000 L17 (.63) T2(-.61)

 T3 0.110 300,000 441,000 L60 (.84) L23(-.70)

 T4 0.320 250,000 132,500 KxKy(-.43) T8 (.37)

 T5 0.090 130,000 24,700 V65(-.39) T18,T4 (.15)

 T6 0.240 110,000 91,300 T19 (.74) L25,L47(.36)

 T7 0.360 100,000 271,000 L17 (.67) T (.44)

 T8 0.065 70,000 9,100 KxKy(-.78) K13 (.52)

 T9 0.950 60,000 76,200 L66 (.63) T17 (0.40)

T10 0.720 50,000 58,500 L3 (.31) KxKy (-.27)

T11 0.130 40,000 34,000 L47(-.84) L25 (.82)

T12 0.073 35,000 403,200 L17 (.49) T7 (.39)

T13 0.180 25,000 341,500 KxKy(-.67) L17 (.53)

T14 0.350 20,000 25,600 L66(-.30) T12(-.28)

T15 0.016 10,000 700 L26(-.28) L66 (.19)

T16 0.410 20,000 13,200 KxKy (.42) L21 (.37)

T17 0.370 20,000 277,200 L54(-.76) L55(-.51)

T18 0.420 20,000 20,200 T4 (.37) L49 (.36)

T19 0.280 50,000 31,500 T6 (.74) L16(-.35)

T20 0.013 300,000 213,000 T23(-.56) L16(-.31)

T21 0.006 500,000 532,5000 L33 (.18) L18 (.12)

T22 0.078 130,000 211,900 T23(-.36) L16 (-.25)

T23 0.210 400,000 1,440,000 L16 (.60) T20 (-.56)

Anisotropy of the Upper Floridan aquifer4

Kx/Ky 0.310 21 20.25 T8(.78) T13(-.67)

Leakance of the intermediate confining unit5 (ft/d/ft)

 L1 0.001 3.50E-01 5.11E-01 L25 (.76) L47(-.75)

 L2 0.200 1.00E-02 1.20E-03 L12(-.18) L33(-.10)

 L3 0.280 6.00E-03 9.60E-04 T10 (.31) L45 (.13)

 L4 0.032 4.00E-03 9.24E-03 L71(-.32) L68(-.12)

 L5 0.130 3.00E-03 3.00E-03 L65(-.39) T18,T4 (.15)

 L6 0.051 2.50E-03 3.15E-03 L23(-.23) L60 (.13)

 L7 0.018 2.00E-03 8.44E-03 L37(-.53) K13(-.12)

 L8 0.026 1.50E-03 7.10E-03 T23(-.24) T20(-.24)

 L9 0.180 1.00E-03 5.30E-04 L61(-.28) L68(-.21)

L10 0.058 8.00E-04 1.17E-03 L59(-.25) L11(-.24)

L11 0.026 7.00E-04 1.12E-04 L61(-.31) L10(-.24)

L12 0.043 6.00E-04 6.83E-03 L21(-.45) L2(-.18)

L13 0.018 5.00E-04 4.01E-03 L22(-.88) T18(-.27)

L14 0.310 4.00E-04 1.08E-04 L71 (.17) L19(-.16)

L15 0.220 3.00E-04 2.85E-04 L26(-.17) L28(-.15)

L16 0.003 2.00E-04 2.52E-04 T23 (.60) L22(-.41)

L17 0.014 1.80E-04 7.43E-04 L23(-.79) T7 (.67)

L18 0.009 1.00E-04 5.90E-05 L21(-.45) L23(-.23)

L19 0.110 9.00E-05 8.64E-05 L14(-.16) L66(-.16)

L20 0.300 7.00E-05 1.68E-05 K (-.56) L20 (.23)
Appendix 57

Leakance of the intermediate confining unit5(ft/d/ft)

L21 0.001 6.00E-05 2.06E-05 L46 (-.40) T16(.37)

L22 0.090 5.00E-05 1.03E-05 L13 (-.88) L33(.31)

L23 0.090 4.00E-05 3.40E-03 L60 (-.83) L17 (-.79)

L24 0.338 1.00E-05 2.65E-05 L50 (-.30) L20 (.15)

L25 0.006 1.00E-06 1.99E-06 L47 (-.99) T11,L70 (.82)

L26 0.027 3.50E-01 1.40E-02 T15 (.28) L62(-.25)

L27 0.008 1.00E-02 2.86E-02 L33 (.31) T14(-.17)

L28 0.020 1.00E-02 3.40E-03 L64(-.17) L15(-.15)

L29 0.002 3.00E-03 4.44E-03 L31 (.33) L41(-.30)

L30 0.068 3.00E-03 2.55E-03 L11 (.17) L61 (.12)

L31 0.024 2.00E-03 6.25E-02 L29 (.33) L35(-.16)

L32 0.048 2.00E-03 1.82E-03 T17(-.13) L55(-.12)

L33 0.025 1.50E-03 7.50E-04 L27 (.31) L59(-.22)

L34 0.190 1.50E-03 8.70E-04 L34 (.13) L71 (.09)

L35 0.069 1.50E-03 3.27E-03 L48(-.28) L18 (.19)

L36 0.036 1.50E-03 5.10E-04 K8 (.19) KxKy(-.11)

L37 0.009 1.00E-03 4.04E-03 L7(-.53) K10 (.05)

L38 0.002 1.00E-03 3.06E-03 L55(-.24) L11 (.22)

L39 0.096 1.00E-03 1.02E-03 L40(-.80) L55(-.42)

L40 0.009 1.00E-03 4.00E-05 L39(-.80) L20 (.23)

L41 0.095 1.00E-03 1.00E-05 -- --6

L42 0.023 1.00E-03 1.20E-04 L17(-.46) L60(-.31)

L43 0.170 1.00E-03 1.60E-03 K17(-.13) L54 (.09)

L44 0.170 3.00E-04 5.04E-04 L60 (.24) L46(-.22)

L45 0.029 8.00E-04 3.62E-03 T10 (.13) L3 (.13)

L46 0.037 8.00E-04 5.52E-04 L60 (-.80) T3(-.66)

L47 0.001 8.00E-04 5.61E-03 L25 (-.99) T11,L70(-.84)

L48 0.002 8.00E-04 1.17E-03 L35 (-.28) T14(-.27)

L49 0.004 8.00E-04 1.21E-03 T18 (.36) L16(-.29)

L50 0.010 8.00E-04 4.40E-04 L24 (-.30) T19(.22)

L51 0.250 8.00E-04 6.16E-04 L51(-.34) L51(-.11)

L52 0.380 8.00E-04 6.24E-04 L52 (.16) T9 (.13)

L53 0.001 7.00E-04 2.67E-03 L51(-.34) T9(-.19)

L54 0.008 7.00E-04 1.01E-01 T17(-.76) L39(-.35)

L55 0.017 6.00E-04 4.26E-02 T17(-.51) L39(-.42)

L56 0.092 6.00E-04 7.86E-04 T14(-.08) T8 (.07)

L57 0.120 6.00E-04 4.68E-04 KxKy(.18) T14(-.15)

L58 0.091 6.00E-04 1.14E-04 T15(-.08) L26 (.05)

L59 0.004 5.00E-04 6.75E-04 L61(-.81) L10(-.25)

L60 0.170 4.00E-04 9.20E-05 T3 (.84) L23(-.83)

L61 0.021 4.00E-04 6.08E-04 L59(-.81) L9(-.28)

L62 0.190 4.00E-04 3.96E-04 L26(-.25) K(-.06)

L63 0.610 2.00E-04 2.70E-04 K(-.13) L20 (.07)

L64 0.039 2.00E-04 4.28E-04 L28(-.17) T22(-.07)

L65 0.038 2.00E-04 4.22E-04 T6(-.39) T4 (.13)

Appendix. Initial and optimized parameter values, relative composite sensitivity for optimal

estimates of input parameter values, and correlation between parameters

[RCS, relative composite sensitivity; ft/d, foot per day; ft/d/ft, foot per day per foot; K, hydraulic

conductivity; L, leakance; T, transmissivity; Kx/Ky, horizontal anisotropy ratio]

Parameterzone

number

RCS Initial value Optimized

value

Correlated parameters

Most correlated 2nd most

correlated
58 Application of Nonlinear Least-Squares Regression to Ground-Water Flow Modeling, West-Central Florida

Leakance of the intermediate confining unit (ft/d/ft)5

L66 0.006 1.00E-04 3.00E-06 T9 (.63) T17(.27)

L67 0.120 1.00E-04 1.79E-04 L35(-.14) T4(.08)L57(-.08)

L68 0.031 4.00E-04 2.72E-04 T17 (.25) L68,L66(-.21)

L69 0.260 5.00E-05 5.50E-05 K(-.38) L20(.21)

L70 0.026 4.00E-05 2.00E-05 L47(-.84) L25(.82)

L71 0.590 3.00E-04 7.11E-04 L4(-.32) T14(.17)

1expected reasonable range of optimized values 1 to 40 feet per day.

2value is a multiplier for the spatially variable parameter values.

3expected reasonable range of optimized values 10,000 to 1,000,000 feet squared per day.

4expected reasonable range of optimized values 0.1 to 20.

5expected reasonable range of optimized values 1.0E-06 to 1.0E-02.

6not calculated because L41 reached its lower limit of 0.01 times the initial value.

Appendix. Initial and optimized parameter values, relative composite sensitivity for optimal

estimates of input parameter values, and correlation between parameters

[RCS, relative composite sensitivity; ft/d, foot per day; ft/d/ft, foot per day per foot; K, hydraulic

conductivity; L, leakance; T, transmissivity; Kx/Ky, horizontal anisotropy ratio]

Parameterzone

number

RCS Initial value Optimized

value

Correlated parameters

Most correlated 2nd most

correlated