Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol24.Issue2.2000:

Sensitivity of temporal moments calculated by the adjoint-state

method and joint inversing of head and tracer data

Olaf A. Cirpka

*

_{, Peter K. Kitanidis}

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

Received 27 August 1999; received in revised form 7 January 2000; accepted 19 January 2000

Abstract

Including tracer data into geostatistically based methods of inverse modeling is computationally very costly when all concen-tration measurements are used and the sensitivities of many observations are calculated by the direct di€erentiation approach. Harvey and Gorelick (Water Resour Res 1995;31(7):1615±26) have suggested the use of the ®rst temporal moment instead of the complete concentration record at a point. We derive a computationally ecient adjoint-state method for the sensitivities of the temporal moments that require the solution of the steady-state ¯ow equation and two steady-state transport equations for the forward problem and the same number of equations for each ®rst-moment measurement. The eciency of the method makes it feasible to evaluate the sensitivity matrix many times in large domains. We incorporate our approach for the calculation of sen-sitivities in the quasi-linear geostatistical method of inversing (``iterative cokriging''). The application to an arti®cial example of a tracer introduced into an injection well shows good convergence behavior when both head and ®rst-moment data are used for

inversing, whereas inversing of arrival times alone is less stable. Ó _{2000 Elsevier Science Ltd. All rights reserved.}

Keywords:Inverse modeling; Sensitivity; Adjoint state; Temporal moments

1. Introduction

In the inverse problem of groundwater hydrology, we infer the hydraulic conductivity distribution within a particular aquifer from measured hydraulic heads and other information, such as tracer data collected at the site. Due to the non-uniqueness of the solution and measurement errors, the inverse problem is best solved in a statistical framework [22,29]. A common distinction is made between regression-like techniques [7,12] in which parameters, much fewer in number than the ob-servations, of a deterministic model are ®tted by mini-mizing a least-square criterion and methods based on a geostatistical description of the aquifer variability [10,15±17,19,25,26,30,32,36,37]. Requiring a known distribution of zones or trends, the regression techniques are more restrictive than the geostatistically based models in which the a priori knowledge is limited to the spatial correlation of the estimated property. The pa-rameters that describe the geostatistical structure may

themselves be inferred from the measurements [16,17,19].

The dependency of the heads and concentrations on the hydraulic conductivity is nonlinear. For small vari-ations of the conductivities, linearization about the mean conductivity (or its logarithm) is valid [10,15,19, 26,30,32]. If the (log-)conductivity varies more strongly, linearization may be done about the estimate itself. This approach is referred to as quasi-linear geostatistical in-versing [16] or iterative cokriging [36]. The present paper describes a quasi-linear method that is applicable to highly variable conductivity ®elds.

In the geostatistical method of inversing, calculating the sensitivities of the measurements with respect to the parameters is often the computationally most costly step. The sensitivity matrix, or Jacobian, multiplied with the covariance matrix of the parameters yields the cross-covariance matrix connecting the variations of the pa-rameters with those of the measurements. From the standpoint of code-development, the direct di€erentia-tion method is the simplest approach: the forward problem (¯ow and transport with given parameters) is repeated with a small variation of a single parameter in each run. Dividing the variations of all measured www.elsevier.com/locate/advwatres

*

Corresponding author. Fax: +1-650-725-9720.

E-mail address:ocirpka@stanford.edu (O.A. Cirpka).

quantities by the variation of each parameter yields the sensitivity matrix. Thus, for m parameters and n measurements,m‡1 problems must be solved. Since we like to approximate the log-conductivity on a nodal or element-wise basis, the number of parameters, in this case element-related log-conductivities, is usually larger by orders of magnitudes than the number of measure-ments. Nonetheless, the approach has been used by various authors since it is easy to implement and ap-plicable to complicated coupled problems [10,15].

The adjoint-state method for the determination of the sensitivity matrix requires the solution of only one for-ward problem andnadjoint problems which are similar to the forward problem. The adjoint-state method has become therefore the common approach for evaluating the sensitivity of head data with respect to the log-ductivity distribution [22,29,33,34]. For inversing con-centration data, however, the approach is more complicated because concentrations are not directly dependent on the hydraulic conductivity. It is also prone to numerical errors because of instabilities arising from advection domination. In the present study, we modify the adjoint-state method of Sun and Yeh [30] for con-centration data. Particularly, our modi®cation in the development of the adjoint-state equations leads to a more stable behavior, so that satisfactory sensitivity matrices can be evaluated under the application of linear Finite Element methods.

Most inverse methods in groundwater hydrology have been restricted to hydraulic-head measurements. Sun and Yeh [30,31] developed a continuous adjoint-state method and inversing procedure for concentration data. Skaggs and Barry [28] presented a discrete adjoint-state method for transient transport solved by the La-place transform Galerkin method. Mayer and Huang [21] presented an inversing approach for heads and tracer data based on a genetic algorithm, including the ®tting of geostatistical parameters. However, the geo-statistical parameters were used for kriging of log-con-ductivity measurements only, neglecting any direct spatial correlation of heads and concentrations with conductivities. Harvey and Gorelick [10] showed that including the ®rst temporal moment in inversing of head-data leads to a signi®cant improvement of the es-timate. In that study, they evaluated the moments by solving the transient transport equation and integrating the concentration history over time; then they calculated the sensitivities by the direct di€erentiation approach. The same authors [11] developed moment-generating equations for the direct solution of the temporal mo-ments without solving for the concentration history. James et al. [15] used these equations in order to eval-uate the distribution of retardation factors from con-servative and partitioning tracer data. They used the direct di€erentiation approach for the calculation of the sensitivity matrix. All these studies were strictly linear

and were therefore limited to cases of a small log-con-ductivity variance.

The advantages of using travel-time data rather than actual concentrations for the purpose of inversing have been shown by Ezzedine and Rubin [6]. The same authors analyzed the travel times of the Cape Cod data set [27]. Their aim, however, was to infer the structural parameters of the log-conductivity ®eld rather than to solve the complete inverse problem and obtain the actual distribution. Varni and Carrera [35] developed moment-generating equations for transient ¯ow ®elds. They demonstrated that neglecting dispersion in the evaluation of groundwater ages leads to discontinuous distributions causing instabilities in inversing.

In the present paper, we present a computationally ecient method for the determination of the log-conductivity distribution inferred from measurements of the hydraulic heads and tracer data. The method is ef-®cient because we use temporal moments rather than concentration data themselves and we apply an im-proved adjoint-state method for the evaluation of the sensitivity matrix. The focus of the paper is on the derivation of the adjoint-state method which is a mod-i®cation of the continuous approach developed by Sun and Yeh [30] and explained in detail by Sun [29]. As we will show, our modi®cation of the method leads to a signi®cant improvement of its numerical stability. We integrate the adjoint-state method into a quasi-linear approach of inversing based on iterative cokriging. To our knowledge, this is the ®rst application of the ad-joint-state method to temporal moments of tracer data. Due to the eciency of the method, we can incorporate tracer data into a non-linear geostatistical method of inversing that is not restricted to the case of a small variance of the log-conductivity. We complete the paper by applying the method to an example of a tracer in-troduced into an injection well.

2. Governing equations

Consider ¯ow and transport in an aquifer with spa-tially varying but locally isotropic hydraulic

conductiv-ity K. The steady-state groundwater ¯ow equation

without internal sinks and sources is

o

o_xi K

o_/

o_xi

ˆ0 1_†

subject to the boundary conditions:

/_ˆ/^ on C1; …2†

ni o_/

o_xiKˆq^ on C2 …3†

boundary Cand pointing outwards. /^ andq^are pre-scribed values of the head at the boundary and the speci®c-discharge component normal to it. The speci®c dischargeqiis given by Darcy's law

qiˆ ÿK o_/

o_xi: …4†

We consider transport of a conservative tracer, intro-duced via an in¯ow boundary, described by the well-known advection±dispersion equation

o_c

o_t‡ o

o_xi vic

ÿDij o_c

o_xj

ˆ0 5_†

in whichviis the seepage velocity andDij is the

disper-sion tensor given by:

Dijˆ

vivj v

k k…alÿat† ‡dij…atv‡Dm†; …6†

viˆ

qi

h; …7†

where al and at are the longitudinal and transverse

dispersivities, Dm the molecular di€usion coecient,dij

the Kronecker delta-function which equals unity if the indices are identical and zero otherwise, and h is the porosity. We apply the following boundary condi-tions:

niDij o_c

o_xjˆ0 on Cn Cin; …8†

ni vic

ÿDij o_c

o_xj

ˆnivi^c…xC;t† on Cin …9†

in which Cin is the in¯ow boundary, with nivi<0,

and Cn Cin denotes the part of the boundary that

does not belong to Cin; ^c…xC;t† is a prescribed

con-centration in the in¯ow. In the most general case,

^

c…xC;t† varies with time t and xC denoting the spatial

coordinates along the in¯ow boundary. In the fol-lowing analysis, we consider that the tracer is intro-duced as a pulse with a spatially homogeneous in¯ow concentration:

^

c…xC;t† ˆ^cd…tÿt0†: …10†

We may characterize the concentration historyc…x;t†at any point within the domain by its temporal moments mk…x†

mk…x† ˆ

Z 1

tˆ0

tkc…x;t†dt 11_†

in which the subscript k determines the order of the temporal moment. Multiplying Eq. (5) bytk_{, integrating}

over time and applying integration by parts leads to the moment-generating equations [11]

o

o_xi vimk

ÿDij o_mk

o_xj

ˆk mkÿ1 …12†

subject to the boundary conditions:

niDij o_mk

o_xj ˆ0 on CnCin; …13†

ni vimk

ÿDij o_mk

o_xj

ˆnivim^k…xC;t† on Cin …14†

in which m_^k…xC;t†is thekth moment of ^c…xC;t†.

In the case of the Dirac-pulse in¯ow concentration,

^

mk…xC;t†simpli®es to a constant for the zeroth moment

and to zero for all higher moments. Since the transport equation is linear with respect to c, the temporal de-velopment of the concentration distribution can be evaluated from the solution of a Dirac-pulse boundary-value problem by convolution. The moments of a function derived by convolution are the sums of mo-ments of the input function and the transfer function. Hence, for homogeneously distributed in¯ow concen-trations, we can determine the temporal moments re-lated to the Dirac-pulse in¯ow concentration by subtracting the moments of the in¯ow concentration from that of the measured breakthrough curve.

In the following, we will restrict our analysis to the zeroth and ®rst moment. Eq. (11) considers the resident concentration and its temporal moments which, in most ®eld applications, hardly di€er from the respective ¯ux concentration and its moments [20]. The zeroth moment m0of the ¯ux concentration is the solute mass divided by

the discharge passing the observation point, whereas the ®rst moment m1 is the zeroth moment times the mean

arrival time.

The moment-generating equations are steady-state transport equations and can be solved much faster than the transient transport equation (5). The right-hand side of Eq. (12) is zero for the zeroth moment. Assuming small transverse dispersivities, the distribution of the zeroth moment is almost binary: Inside the plume, the zeroth moment is about its value at the in¯ow boundary; outside the plume it is zero. The transition at the edge of the plume is rather abrupt. Obviously, if we want to apply inverse methods that use gradient-based opti-mization, matching the zeroth-moment will be dicult. By contrast, the ®rst moment is monotonically increas-ing with the travel distance. The gradient in the direction of ¯ow is inversely proportional to the velocity and therefore sensitive to the hydraulic conductivity.

3. Stochastic partial-di€erential equations

operators are applied to the state variables of interest [29]. In Section 4, we will derive sensitivities of heads and temporal moments using the adjoint states. The approach of continuous adjoint states has been applied to coupled ¯ow and transport problems (using concen-tration measurements) by Sun and Yeh [30]. The fol-lowing description follows that of Sun [29], but we introduce a modi®cation that leads to higher stability in the numerical evaluation.

3.1. Groundwater ¯ow

In the following, we will use the log-conductivity Y _ˆlnK_†rather than the conductivityK. Let us assume that Y~ is our estimate of Yand /~the solution of the hydraulic head associated withY~. We are interested in the change of/if we changeY~ by a small perturbation Y0

Y ˆY~‡Y0; …15†

/_ˆ/~_‡/0: …16_†

From the perturbation of the log-conductivity Ywe derive that of the conductivityKby linearization

KˆK~…1‡Y0† …17†

in whichK~ˆexpY~†. Substituting Eqs. (16) and (17) into Eq. (1) and dropping products of perturbations yields

o

Note that in the linear approach of inversing Y~ is the prior mean, which is usually a constant, whereas in the quasi-linear theory of inversingY~ is the best estimate of Ybased on the last iteration. Taking expected values of (18) yields

subject to the boundary conditions:

~

/_ˆ/^ onC1; …20†

ni o_/~

o_xiK~ˆq^ on C2: …21†

Subtracting Eq. (19) from Eq. (18) yields the linearized stochastic partial di€erential equation of the hydraulic heads

The boundary conditions are:

/0_ˆ0 onC1; …23†

Eq. (22) relates the head-perturbations to the log-conductivity perturbations. A weak form of Eq. (22) may be derived by introduction of the trial functionw/

Z

Application of Green's theorem yields

ÿ

Considering the boundary conditions (Eqs. (23) and (24) related to Eq. (22), the boundary integrals vanish if the following boundary conditions are chosen for w/:

w/ ˆ0 on C1; …27†

niK~ o_w

/

o_xi ˆ0 on C2: …28†

If we do not apply Eq. (28) as boundary condition on

C2, the following boundary integral remains on the

left-hand side of Eq. (26):

It should be noted that some boundary condition for

w/ onC2 is needed in all cases. As we will see later, for

coupled problems, Eq. (28) may be replaced with a more expedient choice.

Substituting Eqs. (16) and (17) into Darcy's law yields

in whichqi is the speci®c discharge. The linearized mean

speci®c discharge is given by

~

qiˆ ÿK~ o_/~

o_xi: …31†

Subtracting Eq. (31) from Eq. (30) gives the linearized governing equation for the velocity perturbations

q0_iˆ ÿKY~ 0o/~ o_xiÿK~

o_/0

o_xi: …32†

3.2. Temporal moments

viˆ~vi‡v0i; …33†

mkˆm~k‡m0k; …34†

mkÿ1ˆm~kÿ1‡m0_kÿ1: …35†

The e€ects of the spatial variability of the pore-scale dispersion tensor on the mass ¯ux are small in com-parison to the velocity ¯uctuations. Therefore we follow the common praxis of the stochastic linear theory, and neglect the perturbations of the dispersion tensor [9]. Again neglecting products of perturbations and taking expected values yields

subject to the boundary conditions:

niDij

Subtracting Eq. (36) from Eq. (12) gives the linearized governing equation for thekth moment perturbations

o

In the derivation of Eq. (36), it was considered that, due to the continuity of the volumetric ¯uxes, the expression

…o_m~k=o_xi†v0

A weak form of Eq. (39) may be derived by introduction of a trial functionw_k

Z Application of Green's theorem yields

Z

In their approach of inversing transport data, Sun and Yeh [30] applied Green's theorem also to the ®rst term in Eq. (43). This simpli®es the treatment of total-¯ux boundary conditions. As we will discuss in the

follow-ing, however, the calculation of the adjoint state of the hydraulic heads and the evaluation of the sensitivities is less a€ected by numerical errors if we keep the derivative of m_~k rather than of wk. We introduce the following

boundary condition forwk:

niDij

In the case of advection-dominated transport or if the gradient of thekth moment is small, the moments of the prescribed concentration in the in¯ow m_^k are almost

identical to the moment of the concentrations at the boundary m~k, so that the remaining boundary integral

in Eq. (46) is negligible. Substituting Eq. (32) into Eq. (46) (neglecting the boundary integral) and applying Green's theorem yields the stochastic pde relating the perturbationsY0_,/0_,m0

OnC1 the boundary integral vanishes because/0 equals

zero. As will be discussed in Section 4, we add Eq. (26) to Eq. (47) for the zeroth to kth moment in order to calculate the sensitivities of the kth moment. Then we can eliminate the sum of the remaining boundary inte-grals in Eqs. (29) and (47) by choosing the following boundary condition forw/ onC2:

For the case that C2 is a no-¯ow boundary, the latter

belongs to Cn Cin and the principal directions of the

normal to the boundary. As a consequenceo_m~k=o_xiˆ0

at this boundary and the condition forw/ becomes

ni o_w/

o_xi ˆ0: …49†

4. Calculation of sensitivities

We want to ®nd the sensitivities of the hydraulic head

/as well as the zeroth and ®rst temporal moments of a tracer m0 and m1 at a given location x` on the log-hydraulic conductivityYk at multiple locationsxk

o_/l

We use the continuous adjoint-state method as de-scribed by Sun and Yeh [30] and Sun [29]. These authors derived the equation on the basis of a general perfor-mance function. Since we are interested in the speci®c case of calculating the sensitivity matrix, we can simplify the derivation.

Consider the simulated value of the measured quan-tity (hydraulic head or temporal moment) denoted byJ at locationx`

This is formally equivalent to

J…/;m0;m1;x`† ˆ

The variation J0_{can be derived by}

J0ˆ

0_{dV if sensitivity of /;}

R

However, the dependency ofJ0_onY0

kis not given in (53).

subject to the boundary conditions:

niDij

Eqs. (55)±(57), are known as the adjoint partial di€er-ential equations to Eqs. (12) and (1).w1,w0andw/ are

the adjoint states to m1, m0 and /. Due to the

depen-dencies, Eqs. (55)±(57) must be solved in the order of the equations' numbers. If w1, w0 andw/ meet the adjoint

pde's, Eq. (54) simpli®es to:

J0ˆ

Since Yk is the constant conductivity within the

sub-volumeVk;the sensitivity ofJwith respect toYkis given

o_J From the equations given above, we can derive the sensitivities of hydraulic heads, zeroth and ®rst temporal moments by:

· _{In case of a head measurement, bothw0 and w1 are} zero throughout the domain. The only adjoint pde to be solved is Eq. (57) in which a point-source is con-sidered at the location of the measurement x`. The boundary conditions simplify to a ®xed value of zero at boundaries on C1 (Eq. (60)) and no-¯ux

bound-aries ofw/ onC2 (Eq. (61)).

· _{In case of a zeroth-moment measurement, onlyw}

1 is

zero throughout the domain. Two adjoint pde's are to be solved: Eq. (56) for the adjoint state of the ze-roth momentw₀ with a point-source at the location of the measurement x`, succeeded by Eq. (57) for the adjoint state of the headw_/ in which only the ze-roth-moment related term is considered as source. · _{In case of a ®rst-moment measurement, all three}

ad-joint states must be calculated. First we solve for the adjoint state of the ®rst momentw1by Eq. (55) with a

point-source at the location of the measurement x`. Then we solve for the adjoint state of the zeroth mo-ment by Eq. (56) with the value ofw1 as distributed

source. Finally we solve for the adjoint state of the heads w/ by Eq. (57) with the distributed source

re-lated to both moments.

The improvement of our formulation in comparison to that of Sun and Yeh [30] may be illustrated by a ze-roth-moment measurement. If we had applied Green's theorem to the ®rst term of Eq. (43) as supposed by Sun and Yeh [30], the termo_m~0=o_{xi†w0would be replaced by}

…ÿo_w0=o_{xi†m~0. Since the distribution ofw0 is dominated}

by the point-source atx`, it varies much stronger than

~

m0, and its gradient is more prone to numerical errors.

Therefore, it is computationally advantageous to con-sider the source-term

in the governing equation for w/. This may illustrated

for the case of a homogeneous-¯ux boundary condition of m_~0 at all in¯ow boundaries. In this case, m~0 is

con-stant throughout the domain regardless of the log-conductivity distribution. Consequently, the sensitivity

of m~0 with respect to Yis zero everywhere. Using the

expression derived in the present paper, this behavior is revealed: with o_m~0=o_{xiˆ0 there is no source-term for} w/, and the latter is zero everywhere. Also the

sensitiv-ities o_m0`=o_Yk are zero because o_m~0=o_xiˆ0 and

o_w/=o_xiˆ0. By contrasto_w0=o_xi6ˆ0 so that there is a source-term for w/ if one uses the method of Sun and

Yeh [30]. Only if the numerical scheme is extremely ac-curate, the e€ects ofo_w

0=oxi andow/=oxi will cancel out

in the evaluation of the sensitivities. The same argu-ments apply to the sensitivities of ®rst-moment mea-surements. Sun [29] recommended the application of higher-order FEM for the calculation of the for-ward and adjoint transport problems. This is no more necessary with our approach.

5. Numerical implementation

The numerical implementation described in the fol-lowing is restricted to the two-dimensional case. Ex-tension to a third spatial dimension is straightforward. We use the standard FEM for the discretization of the groundwater ¯ow equation and its adjoint./andw_/are approximated by bilinear functionsNwithin rectangular elements of constant size and the hydraulic conductivity is assumed constant within an element.

The system of equations is assembled from the ele-ment-related matrices M/_e` that are identical for all ele-ments except for the element-wise constant conductivity. The same matrices are used for the evaluation of the sensitivities of head-measurements

at the vertices of the element related to Yk. For the

so-lution of advective±dispersive equations, we use the Streamline-Upwind Petrov±Galerkin (SUPG) method [2] which is identical to the Galerkin Least-Square (GLS) method [13] when using bilinear elements. Starting point is the steady-state advection±dispersion equation

in which the index of the moment has been omitted and possible source-terms are expressed byr. For the adjoint states, the principal structure of the governing equation is identical, but the velocity vector points into the op-posite direction. Let … †; V denote the inner product in

boundary integral, then the SUPG formulation of Eq. (66) is [2]

… ÿvm‡D_rm;rN†V ‡…vmÿDrm;N†oV

‡…vrm;sv rN†V

ˆr;N_{†V ‡}r;sv _rN_†V67_† in which Green's theorem has been applied to both transport terms. At the in¯ow boundaries of the do-main, the advective part of the boundary integral may be multiplied by the vector of nodal in¯ow momentsmin.

At out¯ow boundaries of the domain, the boundary integral leads to extra terms on the main diagonal of the system of equations. The stabilization term acts like streamline-di€usion but is treated consistently. The stabilization factorsis determined by [2,8,13]:

sˆ hel

in which hel is an e€ective grid-spacing, Peel is the

el-ement-related Peclet number, n…Peel† an upwind

coe-cient, _{k k} the Euclidean norm, h and k are the dimensions of the element in thex- andy-directions. It is important to notice that the SUPG stabilization does not only in¯uence the advective transport term but also the source term. This is of particular interest for the adjoint-state equations. Boundary conditions, however, are not a€ected. Thus we need to evaluate the element-matrices for advective±dispersive transportMad_el and for the source termMr_el:

The integration is done by Gaussian quadrature using three points per direction. The element-matrices are as-sembled to the matrices of the entire system.

We restrict the boundary conditions to advective-¯ux conditions. That is, all di€usive ¯uxes are set to zero at the boundaries of the domain. This simpli®cation leads only to minor errors, since we consider the advection-dominated case. Also we do not consider ®xed temporal moments (although the implementation would be straightforward). Then we have to distinguish between in¯ow and out¯ow boundaries. At in¯ow boundaries,

the temporal moments of the ¯ux concentrations must be prescribed. At out¯ow boundaries, the mass-¯ux leaving the domain equals the volumetric ¯ux times the concentration: Calculating the sensitivity matrix requires solving for one steady-state ¯ow equation and two steady-state transport equations for the forward problem, one ¯ow equation for each head measurement, and two transport plus one ¯ow equation for each ®rst-moment measure-ment. If enough memory is available, all adjoint states can be stored so that the solution of the last iteration can be used as initial guess for the solver. If memory is limited, the adjoint states may be kept only until the sensitivity of the speci®c measurement is calculated.

6. Quasi-linear method of inversing

The adjoint-state method for the calculation of sen-sitivities may be applied to any gradient-based inversion scheme. However, it is particularly attractive in geo-statistical models in which the log-conductivity is modeled as a random spatial function discretized at the same level as the heads and the concentration moments. In this framework, we consider that the spatial distri-bution of log-conductivity has the following linear structure:

Y…x_{† ˆ}X p

kˆ1

fk…x†bk‡…x† …77†

in which the ®rst term is the deterministic part with known spatial functions fk…x† and unknown linear

co-ecients b_k. The simplest case is that of an unknown constant mean,pˆ1,f1…x†equals unity throughout the

domain, and b_k is the mean.…x_†is a random function with zero mean and characterized through a generalized covariance function. After discretization, Y is rep-resented by anm1 vectorsin whichmis the number of elements. The mean vector and covariance matrix ofs

are:

E_{‰ Š ˆ}s Xb; …78_† Eh…sÿXb†… ÿs Xb†TiˆQ; …79_† where X is a knownmp and b arep unknown drift coecients. Q is a known mm matrix. Let y be the n1 vector of observations (both heads and temporal moments) consisting of part h…s†predicted bysand an error componentv

The observation error is assumed to be a random, nor-mally distributed function with zero mean and known covariance matrixR. The most common model forRis based on uncorrelated measurement errors so that

Rˆdiagr2

1;r22;. . .;r2n†with the vector of measurement

errors r. In the quasi-linear method of inversing, we linearize Eq. (80) about the last estimate of s denoted by~s:

h…s† ˆh…~s† ‡H s~ ÿ~s …81†

in which H~ is the Jacobian or sensitivity matrix the calculation of which has been explained in the previous sections

~ H_ˆ oh

o_s

sˆ~s

: …82_†

We estimatesby the continuous function^s[18]

^s_ˆXb_‡QHTn 83_†

in whichbis the 1pvector of theb-estimate andnis a 1n vector of weights associated with the measure-ments. We ®ndbandnby solving the function-estimate form of the linearized cokriging system [18]

~

R HX~ ~

HX

T

0

" #

n b

ˆ yÿh…~s† ‡H~~s

0

…84_†

with

~

RˆHQ~ H~T‡R: …85†

Since H~ depends on values chosen for s estimated by Eq. (83), the procedure has to be repeated until ^s is practically equal to ~s. We use a weighted root mean-square criterion for quantifying convergence

e_ˆ



1 n

Xn

iˆ1 ^

siÿ~si

2

~

V…i;i†

v u u u

t …86†

in whichV~ is thea posterioricovariance matrix of esti-mation. Typically, we require e to be smaller than 0.1 or 0.25. V~ can be calculated from the inverse of the cokriging matrix:

Pyy Pyb

PT_yb Pbb

" #

ˆ

~

R HX~ ~

HX

T

0

2

4

3

5

ÿ1

; …87_†

wherePyy isnn,Pyb isnp andPbbispp. Then,

~

VˆQÿQH~TPyyHQ~ ÿXPbbXTÿXPT_ybHQ~

ÿQH~TPybXT: …88_† Typically, the values of the ®rst moment measurements are several orders of magnitude larger than those of the head measurements, leading to poorly conditioned cokriging matrices. This problem can be overcome by

multiplication of the measured ®rst moments and the corresponding measurement errors with a constant leading to values in the same order of magnitude as that of the heads. The same factor must be applied to the sensitivities of the ®rst-moment measurements.

While we refer to the method above asiterative cok-riging, it is identical to the Gauss±Newton method for maximizing the restricted likelihood function of the Y-®eld [16]. Therefore thea posterioricovariance matrix of estimation V~ is unbiased. The method can easily be extended to include the estimation of the structural parameters as described by Kitanidis [16].

7. Application

In order to illustrate the method, we have created an arti®cial two-dimensional test case. All parameters are listed in Table 1. Lx and Ly are the dimensions of the

aquifer. An injection well of speci®c strength Qw is

lo-cated at xw;yw. No ¯ow is allowed across the top and

bottom boundaries, whereas the heads are ®xed at the left-hand and right-hand side boundaries. The ``true'' log-conductivity distribution was generated on a rect-angular grid using the spectral approach of Dykaar and Kitanidis [5]. An anisotropic exponential covariance function for the log-conductivities was chosen. This conductivity ®eld is shown in Fig. 1(a).

Based on the ``true'' log-conductivity distribution, we simulated the hydraulic heads, the zeroth and the ®rst temporal moments of a tracer introduced into the in-jection well. The zeroth moment of the in¯ow concen-tration in the well m^w

0 was set to unity and the

corresponding ®rst momentm^w

1 to zero. A random error

with zero mean and standard deviationr/ was added to the head-values at the grid-points of xˆ5;10;15;

20;25;30;35 m and yˆ4;8;10;12;16 m. These values were taken as head measurements. The true head-dis-tribution and the locations of the head ``measurements'' are shown in Fig. 3(a). The zeroth and ®rst moments were calculated for a tracer test in which the tracer is injected as a pulse into the injection well. A random error with zero mean and standard deviationrm1 of 10% of the actual value was added to the ®rst-moment values at the grid-points of xˆ10;15;20;25;30;35 m and y ˆ8;10;12 m except the well-location itself. These values were taken as ®rst-moment measurements. The true zeroth-moment distribution is shown in Fig. 4(a) and the ®rst-moment distribution in Fig. 5(a).

Fig. 1. Distribution of log-conductivityY ˆlnK†(Kin (m/s)). (a) True distribution. (b)±(d) Best estimates using indicated type of data for inversing.

…†Locations of head measurements;_†locations of ®rst-moment measurements. Table 1

Parameters of the test case

Geometric parameters and discretization

Lxˆ40 m Lyˆ20 m Dxˆ0:2 m Dyˆ0:2 m

Geostatistical parameters of the lnK_†®eld (Kin m/s)

Y_{ˆ ÿ}7 r2

Yˆ1 kxˆ4 m kyˆ2 m

Boundary conditions for ¯ow

D/ˆ0:4 m xwˆ10 m ywˆ10 m Qwˆ1:510ÿ4m2=s

Transport parameters

hˆ0:25 a`ˆ0:05 m atˆ0:01 m Dmˆ10ÿ9m2=s

Measurement error

covariance matrix were identical to the true solution. The measurement errors were assumed uncorrelated with the actual values of the standard deviations used for the creation of the ``measurements''. As initial guess, the log-conductivity was assumed to be _ÿ7 throughout the domain, however, the method con-verged also with di€erent initial values. Using both types of measurements, the method converged to the convergence criterione_ˆ0:25 as de®ned in Eq. (86) in four iterations and to e_ˆ0:1 in six iterations. Using only the head measurements, e_ˆ0:1 was reached within three iterations, whereas this criterion could not be met within twenty iterations when only the ®rst-moment data were used. The results shown for inversing only the ®rst-moment data refer to eˆ0:12 which was reached after seven iterations.

The best estimates of the log-conductivity distribu-tion are shown in Fig. 1(b)±(d) and the corresponding variance of estimation^r2

Y ˆdiag V~

ÿ

in Fig. 2. The best

estimate using only the head data shown in Fig. 1(b) already includes most of the large-scale features. As one anticipates from a best estimate, it is a smoother ®eld than the true log-conductivity ®eld shown in Fig. 1(a). It should be mentioned that the assumed value of 0.5 mm for the standard deviation of head measurements is ex-tremely optimistic for ®eld measurements. Such accu-racy in the measurement of hydraulic heads can be achieved normally only under well-controlled labora-tory conditions. Had we used a larger error criterion, less features would have been detected.

The best estimate based on ®rst-moment data alone shown in Fig. 1(c) does not recover any features outside of the plume since there is hardly any sensi-tivity of concentration data to these conducsensi-tivity val-ues. The sensitivity of ®rst-moment data to a narrow stripe of log-conductivity values makes the inversion procedure less stable and the best estimates less reli-able than that of head data. These ®ndings are in

Fig. 2. Estimation variance of log-conductivity _^r2

Y …m

2_=s2

agreement with the study of Varni and Carrera [35] on groundwater ages.

Combing head and ®rst-moment data in inversing as shown in Fig. 1(d) leads to an improvement of the es-timate based on head data alone (Fig. 1(b)). In addition to the large-scale features that had been already detected by the head inversion, details at a smaller scale are recovered within the plume. The improvement is also evident from the variance of estimation shown in Fig. 2. In comparison to the variance from head-inversion (Fig. 2(a)), the additional information of mean solute arrival times leads to tighter error bounds in the interior of the plume (Fig. 2(c)).

For quantitative analysis, we calculated the di€erence

DY between the estimated log-conductivity and the ac-tual value for all inverse solutions. Table 2 lists the root mean square ofDY, the standard deviation of estimation

^

rY and the weighted error DY=r^Y. Including

®rst-moment data improves the absolute error RMS(DY) by Table 2

Root mean squares (RMS) of errors in the inverse solution using di€erent types of data

Data used RMS

(DY)

RMS (r^Y)

RMS (DY=r^Y)

Only heads 0.982 0.831 1.176 Only ®rst moments 1.128 0.928 1.216 Heads and ®rst moments 0.896 0.782 1.153 Fig. 3. Distribution of hydraulic head / (m). (a) Based on true

distribution of log-conductivity. (b) Based on best estimate using head and ®rst-moment data. …† Locations of head measurements;

…†locations of ®rst-moment measurements.

Fig. 5. Distribution of normalized ®rst momentm1=m^w

0 (s). (a) Based on true distribution of log-conductivity. (b) Based on best estimate using head and ®rst-moment data. …† Locations of head measure-ments;…†locations of ®rst-moment measurements.

Fig. 4. Distribution of normalized zeroth moment m0=m^w

10%. It should be noted, however, that this quantity refers to the log-conductivity values in the entire domain, whereas the ®rst-moment measurements a€ect essentially only the conductivities in the interior of the plume.

Figs. 3(b), 4(b) and 5(b) show the distribution of the hydraulic heads, zeroth and ®rst moment based on the best estimate of the log-conductivity distribution using both types of data. The distributions are smoother than the true values shown in Figs. 3(a), 4(a) and 5(a), but they are very similar, and the ments are reproduced within the prescribed measure-ment error.

8. Discussion and conclusions

Tracer data contain valuable information that should be included in inverse methods whenever they are available. So far, however, this has been done mainly in the context of regression-like techniques [1,23]. Includ-ing tracer information into geostatistically based models has been discussed by various authors [10,15,29,30], but the majority of the studies on geostatistically based in-versing have been restricted to head-measurements. A particular reason may be the high computational e€ort of the direct di€erentiation approach for the evaluation of sensitivities. The direct di€erentiation approach is appropriate for regression-like techniques because they have only a few parameters. This is not the case in geostatistical inverse problems in which the conductivity values in all elements or at all nodes are to be estimated. For such problems, adjoint-state methods are much more ecient. However, the continuous adjoint-state method for coupled problems of Sun and Yeh [30] is sensitive to numerical errors, requiring high-order Finite Element methods [29].

A second obstacle in using tracer data for inversing is that individual concentrations are not very informative [6]. Concentrations at each observation point must be measured at many times and then processed in order to identify the conductivity distribution. Using temporal moments rather than directly concentrations does not only decrease the computational e€ort and the memory requirements, it also leads to more sensitive information [6,10]. The moment-generating equations derived by Harvey and Gorelick [11] and applied in the present study, allow to compute temporal moments by solving steady-state transport equations. Together with the im-proved adjoint-state method presented here, this makes it feasible to perform multiple evaluations of the sensi-tivity matrix in large domains.

With the improved eciency of our method we can apply tracer data to the quasi-linear method of invers-ing. The linear method used so far for inversing

tem-poral moments [10,15] is restricted to the case of a small variance. In our application, the estimate of the ®rst iteration that is identical to the solution of the linear method was considerably di€erent from the ®nal result.

Our modi®cation of the adjoint-state method of Sun and Yeh [30] seems minor but its e€ect on the numerical accuracy is remarkable. In the approach of Sun and Yeh [30], the source-term for the adjoint state of the heads is evaluated by di€erentiating the product of the concen-tration and the gradient of its adjoint state. Since the adjoint state of the concentration (in our case of the temporal moments) is the solution of a point-source problem, its gradient can be calculated accurately only with high-order methods. By contrast, the gradients of the temporal moments are approximated more accu-rately, since the temporal moments are relatively smooth functions. By reversing the order of di€erentiation, we can therefore use linear function spaces for the discret-ization of heads and temporal moments without intro-ducing major numerical errors.

For the stabilization of advection-dominated trans-port, we use the SUPG method [2]. It is well known that the standard Galerkin method or central di€erentiation in Finite Di€erences leads to excessive oscillations in the vicinity of discontinuities. The point-source problem associated with the adjoint states of the temporal mo-ments is exactly the type for which the standard Galer-kin method would fail. It should be mentioned, however, that the SUPG method is not monotonic, that is, spurious oscillations remain. In the application shown, this is the case in the forward-problem near the stagnation point, and in the adjoint-state problem par-allel to the adjoint-state ``plume''. These small oscilla-tions appear also in the sensitivity ®elds. Fortunately, they do not cause stability problems, since the multi-plication of the sensitivity matrix with the covariance matrix of the log-conductivities smoothes small-scale ¯uctuations. This would have been di€erent if the inte-gral scale of the conductivity had been of the size of the grid spacing. Oscillations could be suppressed entirely by the introduction of a nonlinear discontinuity-capturing operator [3,14,24]. However, we decided against these techniques in order to avoid additional linearization.

We did not use zeroth-moment information in the inversing procedure since the sensitivities of the zeroth moment were essentially zero. The almost binary in-formation content of the zeroth moment (within the plume versus outside the plume) may be used in sta-tistical procedures not relying on gradients. Further research will be necessary to combine these type of data with the gradually varying heads and arrival times.

convergence problems. Arrival-time data are very sen-sitive to the log-conductivities along a narrow stripe upstream of the measurement location, whereas hy-draulic heads are in¯uenced from conductivities in all directions. We recommend therefore to combine the more stable head data and the more speci®c arrival-time data in inversing e€orts.

Our method can easily be extended to higher-order temporal moments. For the second central moment, however, the dispersion parameters will have a great impact. The small-scale variability of the log-conduc-tivity ®eld enhances dilution of the plume and there-fore increases the growth of the second central moment [4]. The best estimate of the log-conductivity ®eld is smoother than the actual distribution so that results of the second-central moment will be biased. E€ective dispersion parameters may be estimated jointly with the log-conductivity distribution using hydraulic heads as well as ®rst and second temporal moments. The meaning of these parameters for dilution and reactive mixing may be investigated in follow-up studies.

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