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Geostatistical inversing for large-contrast transmissivity fields

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Abstract

The estimation of field parameters, such as transmissivity, is an important part of groundwater modeling. This work deals with the quasilinear geostatistical inverse approach to the estimation of the transmissivity fields from hydraulic head measurements. The standard quasilinear approach is an iterative method consisting of successive linearizations. We examine a synthetic case to evaluate the basic methodology and some modifications and extensions. The first objective is to evaluate the performance of the quasilinear approach when applied to strongly heterogeneous (or “high-contrast”) transmissivity fields and, when needed, to propose improvements that allow the solution of such problems. For large-contrast cases, the standard quasilinear method often fails to converge. However, by introducing a derivative-free line search as a polishing step after each Gauss–Newton iteration, we have found that convergence can be practically assured. Another issue is that the quasilinear procedure, which uses linearization about the best estimate to evaluate estimation variances, may lead to inaccurate estimation of the variance of the estimated variable. Our numerical results suggest that this may not be a particularly serious problem, though it is hard to say whether this conclusion will apply to other cases. Nevertheless, since the quasilinear approach is an approximation, we propose a potentially more accurate but computer-intensive Markov Chain Monte Carlo (MCMC) procedure based on conditional realizations generated through the quasilinear approach and accepted or rejected according to the Metropolis–Hastings algorithm. Six transmissivity fields with increasing contrast were generated and one thousand conditional realizations were computed for each studied case. The MCMC procedure proposed in this work gives an overall more accurate picture than the quasilinear approach but at a considerably higher computational cost.

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Abbreviations

m :

number of unknowns

n :

number of observations

p :

number of coefficients

s :

discretized unknown function (m × 1)

X :

known m × p matrix; represent the mean of the process

\({\varvec{\beta}}\) :

is the process mean (p × 1)

Q :

generalized prior covariance matrix of unknown function (m × m)

z :

vector of observation (n × 1)

h(s):

relationship between observation and unknown spatial process

v :

observation error (n × 1)

σ R :

measurement error

R :

error covariance matrix (n × n)

I :

identity matrix (n × n)

H :

n × m matrix it is the derivative of h

\({\varvec{\xi}}\) :

coefficients obtained by the solution of the system (3)

\(\overset{\wedge}{\bf s}\) :

estimated function (m × 1)

P yy :

n × n calculus matrix

P yb :

n × p calculus matrix

P bb :

p × p calculus matrix

V :

covariance matrix of estimation (m × m)

δ:

parameter of the line search

L :

objective function

G :

m × m calculus matrix

s u :

unconditional realization (m × 1)

s c :

conditional realization (m × 1)

ρ:

coefficient for stability of the MCMC chain, see Eq. (12)

α:

coefficient for stability of the MCMC chain, see Eq. (12)

u :

unconditional realization (m × 1)

l :

coefficient generated from the uniform distribution

\(\varsigma\) :

probability of acceptance

p′′(s c ):

posterior distribution

\(q\left( {\bf s}_{u,i}|{\bf s}_{u,i+1}\right)\) :

transition probability

x :

coordinates

y :

coordinates

d ij :

distance between the i and j point

T :

transmissivity

N :

Recharge rate

ϕ:

head

\({\varvec{ \theta}}\) :

vector of covariance function parameters

Q 1 :

statistics

Q 2 :

statistics

F :

Fisher information matrix

g :

gradient matrix of the objective function

λ:

Marquardt parameter

μ0 :

mean of the estimated log-transmissivity field

ν0 :

variance of the estimated log-transmissivity field

μ1 :

mean of the differences between the true field and the estimated one

ν1 :

variance of the differences between the true field and the estimated one

μ2 :

mean of the square differences between the true field and the estimated one

ν2 :

variance of the square differences between the true field and the estimated one

D 1 :

chi-square statistic

D 2 :

mean of the square differences between the true field and the one over the variance

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Authors and Affiliations

  1. Dipartimento di Ingegneria Civile, dell’Ambiente, del Territorio e Architettura, Università degli Studi di Parma, Viale G. P. Usberti 181/a, 43100, Parma, Italy

    Andrea Zanini

  2. Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, 94305-4020, USA

    Peter K. Kitanidis

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  1. Andrea Zanini
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Correspondence to Andrea Zanini.

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Zanini, A., Kitanidis, P.K. Geostatistical inversing for large-contrast transmissivity fields. Stoch Environ Res Risk Assess 23, 565–577 (2009). https://doi.org/10.1007/s00477-008-0241-7

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