Statistics for Spatial Data.

DOI 10.1007/s10040-0010176-2 Abstract Various techniques are available to quantify recharge; however, choosing appropriate techniques is often difficult. Important considerations in choosing a technique include space/time scales, range, and reliabili- ty of recharge estimates based on different techniques; other factors may limit the application of particular tech- niques. The goal of the recharge study is important be- cause it may dictate the required space/time scales of the recharge estimates. Typical study goals include water-re- source evaluation, which requires information on re- charge over large spatial scales and on decadal time scales; and evaluation of aquifer vulnerability to contam- ination, which requires detailed information on spatial variability and preferential flow. The range of recharge rates that can be estimated using different approaches should be matched to expected recharge rates at a site. The reliability of recharge estimates using different tech- niques is variable. Techniques based on surface-water and unsaturated-zone data provide estimates of potential recharge, whereas those based on groundwater data gen- erally provide estimates of actual recharge. Uncertainties in each approach to estimating recharge underscore the need for application of multiple techniques to increase

http://images.water.nv.gov/images/Misc/spring%20valley%20hearings/SNWA/568.pdf

Choosing appropriate techniques for quantifying groundwater recharge

The purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem. Parameter identification methods are classified under the error criterion used in the formulation of the inverse problem. The problem of ill-posedness in connection with the inverse problem is addressed. Typical inverse solution techniques are highlighted. The review also includes the evaluation of methods used for computing the sensitivity matrix. Statistics which can be used to estimate the parameter uncertainty are outlined. Attempts have been made to compare and contrast representative inverse procedures, and direction for future research is suggested.

Review of Parameter Identification Procedures in Groundwater Hydrology: The Inverse Problem

In this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions. The parameters include values and directions of principal hydraulic conductivities (or transmissivities) in anisotropic media, specific storage (or storativity), interior and boundary recharge or leakage rates, coefficients of head-dependent interior and boundary sources, and boundary heads. In transient situations, the initial head distribution can also be estimated if the system is originally at a steady state. Paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters. The inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters. Since not all the factors which contribute to the prior errors can be quantified statistically at the outset, the covariance matrices of these errors are expressed in terms of several parameters which, if unknown, can be estimated jointly with the hydraulic parameters by a stagewise optimization process. When transient head data are separated by a fixed time interval, the temporal structure of these data is approximated by a lag-one autoregressive model with a correlation coefficient that can be treated as another unknown parameter. Estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace. Paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature, all of which are based on the maximum likelihood concept, might be useful for selecting the best groundwater model (or the best method of parameterizing a particular model) among a number of given alternatives.

Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 1. Maximum Likelihood Method Incorporating Prior Information

Research on stochastic analysis of subsurface flow has developed rapidly in the last decade, but applications of this approach have been very limited. The purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large-scale behavior of naturally heterogeneous aquifers. Perturbation-based spectral theory, which presumes local statistical homogeneity, provides generic theoretical results for the head variance, effective conductivity tensor, and macrodispersivity tensor in a field. These results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field. Field information of variances and correlation scales of natural materials is summarized. The validity of some of the generic stochastic results is evaluated through comparisons with Monte Carlo simulations and field observations. A specific field application example is developed to illustrate how the stochastic results are used to estimate large-scale parameters and determine the reliability of three-dimensional numerical simulations. Using typical log conductivity covariance parameters, the effective hydraulic conductivity tensor, and the macrodispersivity tensor are estimated. The calculated head variance, based on the simulated mean hydraulic gradient, is used as a measure of the adequacy of the calculation of the steady state flow model. Discussion emphasizes limitations and extensions of this approach, and ongoing field evaluations of the results.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/WR022i09Sp0135S%4010.1002/%28ISSN%291944-7973.TRENDHYR1

Stochastic subsurface hydrology from theory to applications

The electrical behavior of thunderstorms triggered by local heating and sea-breeze convergence, a low pressure disturbance, and a weak frontal passage has been studied at the NASA Kennedy Space Center, Florida. A nonlinear least-squares minimization procedure has been developed to describe changes in the total electrostatic field produced by lightning in terms of point charge models for the cloud charge distributions. The results indicate that discharges to ground usually neutralize cloud charges in the range from -10 to -40 C. The computed charge altitudes for Florida are somewhat higher than for other geographical locations, 6 to 9.5 km, but the corresponding ambient air temperatures, -10 to -34 C, are similar. A large fraction of the discharges to ground show total field changes which are small or even reversed in polarity within 3 km of the discharges. It is suggested that ground discharges often neutralize a small positive charge, 0.5 to 4 C at altitudes of 1 to 3 km, in addition to the larger negative charge higher in the cloud.

Electrostatic Field Changes Produced by Florida Lightning

A method for obtaining pointwise or spatially averaged estimates of a nonintrinsic function is introduced based on residual kriging. The method relies on a stepwise iterative regression process for simultaneously estimating the global drift and residual semivariogram. Estimates of the function are then obtained by solving a modified set of simple kriging equations written for the residuals. The modification consists of replacing the true variogram in the kriging equations by the variogram of the residual estimates as obtained from the iterative regression process. The method is illustrated by considering groundwater levels in an Arizona aquifer. The results are compared with those obtained for the aquifer by the generalized covariance package BLUEPACK-3D.

Analysis of nonintrinsic spatial variability by residual kriging with application to regional groundwater levels

In part 1 of this paper a new, statistically based approach to the problem of estimating patially varying aquifer transmissivities on the basis of steady state water level data was presented. Part 2 describes a case study in which the new method is applied to actual field data from the Cortaro Basin in southern Arizona. The paper does not present a perfect example of how the new inverse method should be used but only a preliminary demonstration of some of its capabilities in dealing with realistic data. The estimated transmissivities are shown to compare favorably with those obtained earlier for the same basin by an ad hoc trial-and-error procedure. Both sets of transmissivity values have been used successfully in conjunction with a mixed explicit-implicit finite element model to reproduce 25 years of water level variations in the Cortaro Basin in response to pumpage during the period 1940–1965. The finite element model has revealed some important features of the local hydrogeological regime which have not been recognized previously.

A statistical approach to the inverse problem of aquifer hydrology: 2. Case study

Recent work has emphasized development of full-range water-retention functions that are applicable under both wet and dry soil conditions, but evaluation of such functions in numerical modeling has been limited. Here we show that simulations using the Rossi-Nimmo (RN) full-range function compared favorably with those using the common Brooks-Corey function and that the RN function can improve prediction of water potentials in near-surface soil, particularly under dry conditions. Simulations using the RN function also improved prediction of temperatures throughout the soil profile. Such improvements could be important for calculations of liquid and vapor flow in near-surface soils and in deep unsaturated zones of arid and semiarid regions.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000WR900193

Testing a full-range soil-water retention function in modeling water potential and temperature

Transmissivities determined by an inverse groundwater model are dependent on the prior hydraulic heads used to calibrate the model. It has been observed in a previous case study of the Avra Valley aquifer in southern Arizona that when gradients in the prior head distribution obtained by kriging do not match assumed no-flow boundary conditions, the inverse model produced low transmissivity values along these boundaries in an attempt by the model to reconcile inconsistencies in the head gradient and the boundary conditions. To alleviate this problem, an estimation method that includes constraints on the hydraulic head gradient across no-flow boundaries is presented and used to obtain prior heads estimated at node points for the Avra Valley aquifer. The method involves estimating the drift in the hydraulic head data and then kriging the hydraulic head residuals. The method is multiobjective in that the goal in the drift estimation is to fit simultaneously both the measured heads at well locations and the head gradient along the no-flow boundary conditions. The appropriate weighting of the no-flow boundary constraints is investigated by examining the tradeoff in fitting head measurements and fitting the no-flow boundary conditions. Changes in the semivariogram of the head residuals and the estimation error of the heads are also considered. It is found that boundary constraints can improve the fit of the estimated heads to the no-flow boundaries with little deterioration of the fit to the head measurements. Inverse modeling of the Avra Valley aquifer based on these estimated prior heads did not produce the low transmissivities along the no-flow boundaries. The constraints on the head gradient resulted in lower estimation errors on the prior heads and the subsequent lower estimation error of the transmissivities in regions of the aquifer in the vicinity of a no-flow boundary where few head measurements exist. Including boundary conditions in estimating prior heads for use in inverse modeling yielded more realistic transmissivities with lower estimation errors.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/98WR01762

Incorporation of constraints on hydraulic head gradients near no‐flow boundary conditions in the determination of spatial drift and their use in an inverse groundwater flow model

Elizabeth A. Jacobson

Papers

Electrostatic Field Changes Produced by Florida Lightning

Analysis of nonintrinsic spatial variability by residual kriging with application to regional groundwater levels

A statistical approach to the inverse problem of aquifer hydrology: 2. Case study

Testing a full-range soil-water retention function in modeling water potential and temperature

Incorporation of constraints on hydraulic head gradients near no‐flow boundary conditions in the determination of spatial drift and their use in an inverse groundwater flow model