Parameter estimation for MODFLOW


MODFLOWP is the USGS "Computer Program for Estimating Parameters of a Transient or Steady-State, Three-Dimensional, Ground-Water Flow Model Using Nonlinear Regression."


Data used to estimate parameters in MODFLOWP can include existing independent estimates of parameter values, observed hydraulic heads or temporal changes in hydraulic heads, and observed gains and losses along head-dependent boundaries (such as streams). Model output includes statistics for analyzing the parameter estimates and the model. These statistics can be used to quantify the reliability of the resulting model, suggest changes in model construction and compare results of models constructed in different ways. Code has been added to the original MODFLOWP program to facilitate the input of file names and the opening of appropriate files. Five new computer programs are included in MODFLOWP for testing weighted residuals and calculating linear confidence and prediction intervals on MODFLOWP results. MODFLOWP utilizes extended memory.

MODFLOWP Parameter Data
MODFLOWP Calibration
MODFLOWP Supplemental Programs
MODFLOWP Requirements


MODFLOWP is nearly identical to MODFLOW when the parameter-estimation package is not used. Parameters are estimated by minimizing a weighted least-squares objective function by the modified Gauss-Newton method or by a conjugate-direction method. Parameters used to calculate the following MODFLOW model inputs can be estimated:

  • Transmissivity and storage coefficient of confined layers
  • Hydraulic conductivity and specific yield of unconfined layers
  • Vertical leakance
  • Vertical anisotropy (used to calculate vertical leakance)
  • Horizontal anisotropy
  • Hydraulic conductance of the River, Streamflow-Routing, General-Head Boundary, and Drain Packages
  • Areal recharge rates
  • Maximum evapotranspiration
  • Pumpage rates, and
  • Hydraulic head at constant-head boundaries


Most numerical models of ground-water flow systems need to be calibrated; that is, the model needs to be made to match the physical system being modeled. The model and the physical system are compared based on calibration criteria that are defined by the user. For example, typical calibration criteria are that model parameter values are to be consistent with independent estimates of associated field parameters and that simulated hydraulic-head values are to be similar to observed values.

Numerical models of ground-water flow systems can be calibrated by trial-and-error in which simulated aspects of the physical system are repeatedly, manually changed until the model satisfactorily matches the physical system as measured using the defined calibration criteria. Although trial-and-error calibration is conceptually simple, it has three limitations. First, there is no way to know if the estimated parameter values satisfy the calibration criteria better than some untested set of parameter values. This lack of knowing makes it difficult to test hypotheses about a ground-water flow system because a model constructed using one hypothesis might produce better results because of the parameter values used, and not because that hypothesis is better than another. (The process of comparing different hypotheses is called model discrimination.) Second, it is difficult to determine if estimated parameters are highly correlated- that is, that coordinated changes in model parameters would produce identical results in terms of the calibration criteria. When high correlations are present, it is impossible to uniquely estimate the parameter values. Third, the reliability of parameter estimates and simulated results can only be assessed by the tedious process of manually perturbing parameter values to perform a sensitivity analysis. The process also is inexact because results depend on how much the parameter values are perturbed, and the appropriate value is generally unknown. The lack of precision makes it difficult to evaluate whether the calibrated model is accurate enough to be used to make conclusions about the aquifer system or to predict aquifer response.

Alternatively, numerical models of ground-water flow systems can be calibrated by nonlinear regression in which the model itself is used to determine changes in parameter values. Nonlinear regression is accomplished in the following steps:

l. Using the calibration criteria, define an objective function which is a measure of how closely the model matches the physical system.

2. Determine the parameter values that produce the smallest value of the objective function. This is called minimization or optimization of the objective function, and, using the Parameter-Estimation Package of MODFLOWP, can be accomplished with either the modified Gauss-Newton method or a conjugate-direction method. Because the ground-water flow equation is nonlinear with respect to many of the parameters that are most commonly estimated, the optimization methods are iterative-that is, the same procedure is repeated to update parameter values until the optimal parameter values are reached.

3. Calculate statistics by which model discrimination and assessment of model reliability can be accomplished easily and objectively.

Although MODFLOWP uses nonlinear regression to effectively estimate parameter values of ground-water flow systems, each regression must be evaluated to determine whether the regression is valid. MODFLOWP prints some statistics needed in this evaluation but does not support two commonly-used graphical methods of evaluating weighted residuals, linear confidence and prediction intervals.


The MODFLOWP package also includes five additional computer programs for testing weighted residuals and calculating linear confidence and prediction intervals on results from MODFLOWP. The five programs, which utilize extended memory, are: YR, NORM, BCINT, YCINT and BEALEP. These programs use results from MODFLOWP to:

  • Produce data sets that can be used to create two graphs commonly used to test weighted residuals
  • Calculate linear confidence and prediction intervals, and
  • Test the validity of one of the assumptions of the method used to calculate the linear confidence and prediction intervals

The programs YR and NORM in MODFLOWP are used to produce data sets for two types of graphs commonly used to evaluate weighted residuals. The first graph, using data from YR, plots weighted residuals against the weighted simulation values. For a regression to be valid, the weighted residuals need to be randomly distributed above and below a weighted residual value of zero for all weighted simulated values.

The second graph using data from NORM in MODFLOWP is a normal probability graph of the weighted residuals. The weighted residuals approximately fall on a straight line if they are independent and normally distributed.

If the weighted residuals are independent and normally distributed, confidence intervals on estimated parameter values and confidence and prediction intervals on simulated values may be calculated using a normal probability distribution (programs BCINT and YCINT in MODFLOWP). Confidence intervals are used to quantify the precision with which parameter values or simulated values are determined with the available data. BCINT produces data that can be used to produce graphs of estimated parameter values and associated linear confidence intervals. MODFLOWP's YCINT calculates confidence or prediction intervals on simulated hydraulic heads and flows along head-dependent boundaries. The results are presented in tabular form in MODFLOWP.

The program BEALEP in MODFLOWP is used to determine whether a ground-water flow model is roughly linear for parameter values close to the optimized parameter values. A model must be roughly linear for the linear confidence intervals produced by BCINT and YCINT to be accurate.

A residual analysis, RESAN, (Cooley and Naff, 1990, USGS TWRI Bk 3 Ch B4), is also included in MODFLOWP.

MODFLOWP includes the latest source code, executable version, user's guide, and technical support. The MODFLOWP Report/Doc is not included but is available for purchase at a nominal fee.

MODFLOWP Requirements: PC 386/486 with 2 MB RAM and math coprocessor

Incredible NetworksΤελευταία Ενημέρωση 27 Ιουλίου 2004 - Last Revised on July 27th 2004
Προηγούμενη σελίδα
Copyright 1998-2005 MP & Associates - Φορμίωνος 119-121 - ΑΘΗΝΑ 16121 - Τηλ: (210) 7600955 - Fax: (210) 7600956