Application of inverse modeling methods to thermal and

Application of inverse modeling methods to thermal anddiffusion experiments at Mont Terri Rock laboratory

A. Cartalade a,*, P. Montarnal a, M. Filippi a, C. Mugler b, M. Lamoureux a,J.-M. Martinez c, F. Clement d, Y. Wileveau e, D. Coelho f, E. Tevissen g

a CEA/Saclay, DEN-DM2S-SFME-MTMS, 91191 Gif-sur-Yvette cedex, Franceb CEA/Saclay, DSM-LSCE, 91191 Gif-sur-Yvette cedex, France

c CEA/Saclay, DEN-DM2S-SFME-LETR, 91191 Gif-sur-Yvette cedex, Franced INRIA/Rocquencourt, BP105, 78153 Le Chesnay cedex, France

e ANDRA, Meuse/Haute-Marne Laboratory, BP 3, 55290 Bure, Francef ANDRA, DS/TR, Parc de la croix blanche, 92298 Chatenay-Malabry, Franceg CEA/Saclay, DEN-DPC-SECR-L3MR, 91191 Gif-sur-Yvette cedex, France

Received 19 April 2005; received in revised form 19 January 2006; accepted 19 August 2006

Available online 27 October 2006

Abstract

The Opalinus clay parameter identification at Mont Terri Underground Rock Laboratory (URL) is performed with inverse modeling.This article focuses on a comparison between the global and local inverse approaches, in a deterministic framework, applied to the ther-mal (HE-C) and diffusion (DI) experiments respectively. Each experiment presents a similar diffusion process described by the same equa-tion. This simple forward model makes easier the comparison of each inverse approach and allows to easily apply different methods suchas the neural networks and the adjoint state method. The synthesis yields to valuable information about their merits and flaws, and high-lights the importance of the parametrization. The singular value decomposition of the Jacobian matrix is presented to identify the bestparametrization. In each experiment, the numerically found parameter values are in good agreement with the experimental methods. InDI experiment a spatial variability description of the medium is a hypothesis to explain the rapid decrease of the tracer in the injectionchamber.Ó 2006 Published by Elsevier Ltd.

Keywords: Mont Terri Underground Rock Laboratory; HE-C and DI experiments; Inverse modeling; Sensitivity; Optimization; Global and local

approaches; Neuronal network; Adjoint state; Singular value decomposition; CAST3M; NeMo; Kalif

1. Introduction

Various countries are considering consolidated clay for-mations as suitable host rocks for the deep disposal ofradioactive waste. The main purpose of experiments per-formed at the Mont Terri Underground Research Labora-tory (Mont Terri Project – URL), in Switzerland, is todevelop experimental tools and modeling methods to char-acterize the properties of the clay formations. The Mont

Terri Laboratory is located in a tunnel and the porousmedium is composed of Opalinus Clay.

Among all the experiments carried out, HE-C is dedi-cated to the characterization of the thermal behaviour ofthe rock, and DI is a diffusion tracer test. The HE-C exper-iment has consisted in measuring the time evolution of therock temperature submitted to a heating source during 250days in order to determine the thermal conductivity param-eters of the Opalinus clay. The main components of theexperiment (Fig. 2) were a heating source with a heaterpower regulation unit and several temperature sensors withthe required data acquisition and control components. Theheating source was first put in a long steel tube and then set

1474-7065/$ - see front matter Ó 2006 Published by Elsevier Ltd.

doi:10.1016/j.pce.2006.08.043

* Corresponding author. Fax: +33 169 085 242.E-mail address: [email protected] (A. Cartalade).

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Physics and Chemistry of the Earth 32 (2007) 491–506

up in a deep vertical borehole located close to the gallerywall. Temperature sensors were set up in the surroundingrock mass in two vertical boreholes. These two boreholeswere drilled parallel to the main borehole containing theheating source. The design of the experiment and the firstresults are described in Wileveau (2002).

In DI experiment, tritiated water (HTO) and stableiodine (127Iÿ) were injected in a borehole between twopackers (3). The tracer concentration was weakly measuredand several readjustments of concentrations in the testinterval were necessary. Concentration has been monitoredin the injection system for 324 days. An over-coring of this

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Fig. 1. Set of profiles for the 3D model DI (Montarnal et al., 2002).

492 A. Cartalade et al. / Physics and Chemistry of the Earth 32 (2007) 491–506

diffusion experiment has been performed at the end of theexperiment to recover the whole rock volume around theexperimental area. The aim was to sample it, to measureseveral concentration profiles of HTO and I in differentdirections. These profiles were used as the basis for model-ing an in-situ diffusion tensor for HTO and I. The fieldwork and the monitored data are presented in Fierz(1999). A description of the sampling method and proce-dure carried out in the over-core and the concentrationprofile extractions can be found in Mori et al. (2000). A

synthesis on the modeling of DI experiment is presentedin Tevissen and Soler (2003) (see Fig. 3).

Phenomena such as temperature and concentrationpropagation in the porous medium are described by a par-tial differential equation which links parameters to statevariables. Initial and boundary conditions being given,we consider the forward problem as solved when theparameters are known and we wish to compute the mea-surable quantity such as the temperature or the concen-tration. However, in several problems, and particularly in

Fig. 3. Design of the DI experiment (Tevissen and Soler, 2003).

Fig. 2. Design of the HE-C experiment (Wileveau, 2002).

A. Cartalade et al. / Physics and Chemistry of the Earth 32 (2007) 491–506 493

geosciences, it is easier to measure the state variables thanto measure parameters. So in HE-C and DI experiments,the temperature and concentration values are known andwe search to estimate the best parameter values to carryout reliable simulations. This is what we call an inverseproblem.

Inverse methods are applied in several scientific area anda lot of works have been done and many books publishedon this subject. Let us cite Sun (1994) and de Marsily et al.(1999) for the groundwater flow and contaminant trans-port problems, and Ozisik et al. (2000) for the inverseheat transfer problems. A lot of approaches are welldocumented for solving the inverse problem. They can beclassified in two main frameworks: probabilistic or deter-ministic. In the probabilistic one, all the variables are seenas random functions. The solution of the inverse problem isto determine an a posteriori probability density function.All the variables of the problem (mean value, correla-tion, . . .) can be deduced from this function. A descriptionof probabilistic inverse approach can be found in Tarant-ola (1987) and OFTA (1999). A comparison on sevengeostatiscally based inverse methods is presented in Zim-merman et al. (1998). Another framework we use in thispaper is the maximum likelihood estimator (Carrera andNeuman, 1986). The approach is inspired from the Bayes-ian approach but the interpretation is quite differentbecause there is no underlying stochastic model. The solu-tion of the inverse problem is a deterministic function, themost probable in relation to the data.

In spite of these two different frameworks to solve theinverse problem, the solution is usually found using anidentical approach. A criterion is defined to measure thedifference between computed and measured values andthe aim is to find the set of parameters associated to thelowest criterion value. The methods used to find these «bestparameters» can be local or global. In the local approach astarting point is given and a sequence of new parameters isgenerated from this point until the criterion is minimized.The stationary point found by the local method is a localminimum depending on initial values. To deal with theproblem of local minima, we can use a global approachbased on a great number of simulations using several val-ues from the parameter space. This work focuses on a com-parison between global and local approaches to find thelowest criterion value in the deterministic framework. Eachof them has been applied on HE-C and DI experimentsrespectively. After a presentation of the forward models(Section 2), the main steps of each method are described(Section 3). A critical discussion on advantages and incon-venients is exposed in Section 4. The discussion will focusnext on the parametrization and the sensitivity analysisusing the Singular Value Decomposition tool.

HE-C and DI experiments are interesting to performsuch a comparison because the physical situations aredescribed by the same partial differential equation, wellknown and easy to solve. This simplicity for solving theforward problem will allow us to focus on each step of

the global and local approaches. In spite of this simplicity,the equations exhibit a non-linearity between the state vari-ables and the parameters and complicates the computationof the gradients. Modelings are performed on data mea-sured with two different experimental protocols. As a con-sequence we will have to deal with different forwardmodelings and different inverse approaches. The descrip-tion of each step will make the designation «inverse model-ing» more comprehensible.

This work is a synthesis of several CEA and INRIAreports about the inverse modeling of these experiments.More details about the models and the numerical tools willbe found in Filippi (2003) for the neural network buildingand the presentation of the global approach. The three fol-lowing reports are dedicated to DI inverse modeling. A 3Dlocal approach is described in Montarnal et al. (2002). Acomplete description of the adjoint state method appliedin a one-dimensional case can be found in Cartaladeet al. (2003) and the Singular Value Decomposition methodis presented in Clement et al. (2004).

2. Forward model

This section presents the common characteristics of thetwo physical phenomena and the equation to be solved foreach experiment. The same notations will be used in bothmodelings. Physical meanings of each term and hypothesisapplied in HE-C and DI models will be described in nextsubsections. For more details about the numerical tools(number of meshes, convergence, physical values for eachmodels and so on) the reader can refer to Mugler (2003)for the HE-C forward modeling using the Finite Elementmethod, Montarnal and Lamoureux (2000) for a DI MixedHybrid Finite Element modeling and Cartalade et al.(2003) for the DI Finite Difference modeling.

2.1. General equation

DI and HE-C experiments were designed so that themain process is the diffusion of temperature or concentra-tion in the porous medium. This process is described by apartial differential equation similar for the two phenomena.Two main properties of the medium, the anisotropy andthe presence of a bedding plane, require a three-dimen-sional modeling. An energy (resp. mass) balance is carriedout in a thermal problem (resp. diffusion). The equation tobe solved is a parabolic-type equation, linear with respectto the state variable F(x, t). It has the following form:

WðxÞotF ðx; tÞ ¼ ÿ ~r �~qðx; tÞ þ Sðx; tÞ: ð1Þ

Integrated on a volume, Eq. (1) expresses the decrease intime of the total conservative quantity, W(x)F(x, t), whichis equal to the outgoing flux through the surface surround-ing the volume. In (1), F(x, t) is the state variable dependingupon position x = (x,y,z)T and time t [s]. The symbolot means the partial derivative with respect to the time,


S(x, t) is a source term indicating a disappearance of theconservative quantity if it is negative, and an addition ifit is positive. In HE-C and DI, this source term has two dif-ferent forms and will be described more precisely below,W(x) is the first parameter of the equation. It is a functionof space. The flux vector,~q, is given by the Fourier’s law inthe thermal problem and by the Fick’s law in the diffusionproblem. The mathematical form of these laws is identical.The flux is given by the product of a medium property ��vðxÞand the gradient of the state variable (2). The negative signmeans that the flux direction is the opposite of the gradientdirection

~qðx; tÞ ¼ ÿ��vðxÞ ~rF ðx; tÞ: ð2ÞIn expression (2), ��vðxÞ is a second order symmetric tensorto take into account the anisotropy of the medium. It ispossible to establish this tensor form by using the transfor-mation rules of the tensor components in a rotation of thecartesian coordinate system. Merging the Oy axis of thetwo coordinate systems this tensor takes the followingform:

��vðxÞ ¼vl cos

2 hþ vt sin2h 0 ðvl ÿ vtÞ sin h cos h

0 vl 0

ðvl ÿ vtÞ sin h cos h 0 vl sin2hþ vt cos

2 h

0

B@

1

CA;

ð3Þwhere h is the angle between the axis Ox and Ox 0 (Fig. 4),vl(x) and vt(x) the longitudinal and transverse properties,respectively. vl(x) and vt(x) are the other parameters andalso depend on the x-position.

To ensure the uniqueness of the solution, Eq. (1) mustbe supplied with initial and boundary conditions. In thetwo problems considered in this paper, the initial conditionis supposed to be constant on the whole domain X

F ðx; 0Þ ¼ F 0: ð4ÞBoundary conditions are the following:

boundary condition depending on the experiment on C1;

U ¼ 0 on C2;

�

ð5Þthe second boundary condition in relation (5), applied tothe external surface of the system C2, is null for the two

problems. The surface C1 and the boundary condition ap-plied on it are different in the two models. They will be bothdescribed in the next subsections.

Let us finish this section by indicating an importantproperty of diffusion problem. In isotropic and homoge-neous media, it is possible to introduce a new quantityffiffiffiffiffijt

p½m� where j is the diffusivity [m2 sÿ1] and t the time

[s]. It is possible to show that the solution of the diffusionequation depends only on this quantity. It means that aninitial perturbation propagates on a distance of the orderl ¼

ffiffiffiffiffijt

pafter a time t. The linear ratio between the mean

diffusion distance and the time square root is an essentialcharacteristic of a diffusion problem. Such a property canbe represented by an adimensional number called theFourier number and defined by the ratio between js

and L2 where s is a characteristic time and L a character-istic length. The Fourier number characterizes the propa-gation of the diffusion (heat or concentration) in themedium.

The general equation, similar for the two experiments,being introduced, let us emphasize now the main differ-ences between HE-C and DI models.

2.2. Thermal model applied to the HE-C experiment

In the HE-C modeling, F(x, t) is the temperature [K],W(x) is the product of the mass per volume unit q [kg mÿ3]and specific heat Cp [J kgÿ1 Kÿ1] of the porous mediumcomprising water plus solid, ��vðxÞ is the thermal conductiv-ity tensor [W Kÿ1 sÿ1], and S(x, t) is the power per volumeunit given by the source.

The HE-C modeling supposes an homogeneous med-ium, so the parameters W and ��v are not space-dependent.According to this modeling, S(x, t) is given by the productof a dimensionless coefficient describing a loss of power aand a time-dependent power per volume unit Q(x, t). Theequation has the following form:

WotF ðx; tÞ ¼ ~r � ð��v~rF ðx; tÞÞ þ aQðx; tÞ: ð6Þ

The initial temperature is supposed to be the same in thewhole domain and is equal to the mean value of the tem-perature measured by the various probes at time t = 0:F0 = 285.9 K (12.756 °C). The first boundary condition isapplied on the surface C1 separating the porous mediumand the tunnel gallery. Some measurements have shownthe temperature gallery influence on the porous mediumtemperature, this is why the following boundary conditionis used:

Uðx; tÞ ¼ hðF ðx; tÞ ÿ F gðtÞÞ on C1; ð7Þ

with h the transfer coefficient [W mÿ2 Kÿ1] and Fg(t) thetime-dependent temperature in the gallery.

Eqs. (6) and (7) are solved numerically with Cast3M, aCEA code. A finite element method is used to discretizethe spatial term and a Crank-Nicholson approximationfor the temporal term.

z

x

z′

x′

y′

y

θ

Fig. 4. Rotation of the cartesian coordinate system.


2.3. Diffusion models applied to the DI experiment

In the diffusion problem, F(x, t) is the concentration[mol mÿ3], W(x) is the accessible porosity (dimensionless),��vðxÞ is the effective diffusion tensor [m2 sÿ1] and S(x, t)expresses the tracer disappearance in the medium due tothe radioactive decay of the tracer symbolized by the coef-ficient k [sÿ1]: S(x, t) = ÿkW(x)F(x, t).

The boundary condition on C1 is given by a mass bal-ance performed in the injection chamber taking intoaccount the radioactive decay:

F ðx; tÞ ¼ F ðx; t0Þÿ1

V

Z t

t0

Z

C1

~qðx; sÞ �~ndCdsÿ k

Z t

t0

F ðx; sÞds;

ð8Þwhere V is the total volume of the mixture water plus tracerin the whole system comprising the injection chamber, thepipes and the tanks. The concentration F(x, t) at the time tis obtained from the initial concentration F(x, t0) minus thequantity disappeared in the porous medium (second termin the right hand side) and the quantity desappeared dueto the radioactive decay (last term). Two different modelshave been applied for DI inverse modeling.

2.3.1. 3D modeling

The first one was performed in a three-dimensionalhomogeneous case. The equation to be solved is then sim-ilar to Eq. (6) with another expression for the source term:

WotF ðx; tÞ ¼ ~r � ð��v ~rF ðx; tÞÞ ÿ kWF ðx; tÞ: ð9ÞComputations were performed with Cast3M using theMixed Hybrid Finite Element method (Dabbene, 1995).At the initial time the concentration is supposed to be zeroin the whole domain (F(x, 0) = 0). The boundary condition(8) is applied. A simulation result using a 40° bedding planeis presented in Fig. 5. We notice the bedding plane effect onthe concentration field at the top and the bottom of theinjection interval.

2.3.2. 1D modeling

Although the Mont Terri geometry requires a three-dimensional forward modeling because of the beddingplane presence, the second model applied was only onedimensional. This is because the inverse approach usedfor this problem requires to develop a new system whichis much more easily and rapidly deduced and implementedon a one-dimensional model. The 1D approximation ishowever sufficient to introduce the parametrization con-cept and to show the improvement on the results broughtby the zonation. The model is now one dimensional buttakes into account the spatial variability of parametersW(r) and v(r).

In a one-dimensional cylindrical problem the diffusionequation can be stated "r 2 [r1, r2]:

WðrÞotF ðr; tÞ ¼1

rorðrvðrÞorF ðr; tÞÞ ÿ kWðrÞF ðr; tÞ; ð10Þ

with the null initial condition (F(r, 0) = 0). The boundarycondition on C1 expressed in one dimension is given bythe following relation:

F ðr1; tÞ ¼ F ðr1; t0Þþrvðr1ÞZ t

t0

orF ðr1;sÞdsÿk

Z t

t0

F ðr1;sÞds;

ð11Þ

where C1 is the surface between the injection chamber inthe borehole and the porous medium. In the Eq. (11), ris the ratio between the injection chamber surface and thevolume V. Eqs. (10) and (11) are solved with a finite differ-ence scheme for the spatial term and an implicit scheme forthe temporal term.

Let us mention that a semi-analytical solution in a one-dimensional homogeneous case, taking into account thisboundary condition, can be found in Novakowski andven der Kamp (1996). We did not use this analytical solu-tion in this work in order to take into account the variabil-ity of parameters with the position.

3. Inverse models

Before beginning the description of the inverseapproaches, let us summarize the parameters to be identi-fied in each model. For HE-C, three scalar values have tobe estimated: the longitudinal and transverse thermal con-ductivities vl and vt and the a coefficient. In the three-dimensional model for DI, three scalar parameters havealso to be estimated: the effective porosity W and the longi-

H

>3.69E09–9

< 6.29E05–5

4.87E07

3.43E06

6.38E06

9.33E06

1.23E05

1.52E05

1.82E05

2.11E05

2.41E05

2.70E05

3.00E05

3.29E05

3.58E05

3.88E05

4.17E05

4.47E05

4.76E05

5.06E05

5.35E05

5.65E05

5.94E05

6.24E05

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

–5

Fig. 5. Concentration field at the final time of the DI experiment.


tudinal and transverse effective diffusion vl and vt. For theDI one dimensional model, two functions of the positionhave to be identified: W(r) and v(r).

3.1. Parametrization

One of the practical and difficult aspects in inverse prob-lems is related to the description of the spatial variability ofthe parameters W(r) and v(r). The number of parameters toestimate depends upon the dimension of the problem andthe parametrization chosen to approximate the functionsW(r) and v(r). The amount of data is often not sufficientto identify one parameter per mesh and the method usedto decrease the number of parameters is called the param-etrization of the problem. Several methods have beenreviewed in Sun (1994) and an enlightening discussion onadvantages and inconveniences on different parametriza-tions can be found in McLaughlin and Townley (1996).The parametrization presents two main difficulties: theparameter structure and its corresponding values insideit. In most of inverse modelings the parameter structureis supposed to be known and only the parameter valuesare identified. However the parameter structure is impor-tant because a wrong structure leads to a wrong parameterestimation (Sun and Yeh, 1985).

In this work, the parametrization used is the classicalzonation method. The parameter structure is the numberand the position of each zone. For example inside a systemcomprising N mesh points, the parameters WN ¼ðw1;w2; . . . ;wN ÞT; �vlN ¼ ðvl1; vl2; . . . ; vlNÞ

Tand �vtN ¼ ðvt1; vt2;

. . . ; vtN ÞT will be replaced by new vectors WM ¼ ðw1;

w2; . . . ;wMÞT, �vlM ¼ ðvl1; vl2; . . . ; vlMÞT and �vtM ¼ ðvt1;

vt2; . . . ; vtMÞ

Twith M very small compared to N. M is the

number of zones, and each zone contains several meshpoints.

In practice, the choice of M, called the dimension ofparametrization, and the positions of each zone are difficultand depend upon the physical problem and the availabledata. In this work the parameter structure is supposed tobe known, and only the parameter values inside them areidentified. A solution to get the best parametrization usingthe measured data is given by the Singular Value Decom-position (SVD) method and will be presented in Section 4.

The importance of the zonation, number and position ofeach zone, will not be highlighted on the three-dimensionalHE-C and DI models because they are considered homoge-neous. So the vectors W, �vl and �vt are reduced to a scalarvalue w and vl and vt applied on the whole domain. The dif-ficulties of parametrization will be more easily pointed outwith the one dimensional model.

In all the rest of this paper the parameter to identify willbe noticed by a vector p. For HE-C modeling the compo-nents are the following: p = (vl,vt,a)

T. For the three dimen-sional model of DI p expresses the vector (w,vl,vt)

T and forthe one dimensional model p ¼ ðWM ; �vMÞT. Although theDI forward model is one dimensional compared to the

three dimensional models, the dimension of p could begreater if MP 2.

3.2. Global approach applied on the HE-C experiment

3.2.1. Objective function

Inverse modeling requires to define a criterion called anobjective function (or performance function) measuring thedifference between experimental and computed data. InHE-C and DI experiments, the objective functions are bothbased on the least squares but are slightly different due tothe difference between the available data and the hypothe-sis made in each modeling. In the thermal problem, theobjective function has the following form:

j ¼ 1Pns

j¼1lj

Xns

i¼1

liðF ðxiÞ ÿ bF ðxiÞÞ2; ð12Þ

where ns is the number of sensors, bF ðxiÞ is the ith measuredtemperature, bF ðxiÞ is the ith computed temperature, xi, isthe position. li is a positive weight, different for each exper-imental value, describing the degree of confidence on thismeasure. We emphasize the fact that the objective functionis not time-dependent. A steady state inverse problem hasbeen solved in this study. Once the objective function is de-fined, the problem to solve is the following: find the threecomponents of p = (vl,vt,a)

T to obtain the lowest valueof J.

The global strategy to find p is simple and almost intu-itive. Lower and upper bounds of each component are fixedand a great number of pk (k = 1, . . . ,N) are chosen insidethis parameter space. The choice of a parameter pk+1 isindependent from the previous one pk. A statistical lawcan be used to generate all the pk in order to have an uni-form repartition inside the parameter space. The objectivefunction is then computed for each parameter pk and whenall simulations are achieved, the set of parameters givingthe lowest cost function value is the solution of theproblem.

3.2.2. Neural network

The large computation time required by the globalapproach leads us to build a Cast3M approximation usinga neural network, especially in this application because theforward model is solved by a numerical method (3D finiteelement). For each parameter pk, the neural network com-putes a temperature according to the following equation:

F s ¼ fs0 þXNh

i¼1

fsifXN in

j¼1

wsijpj þ ws

i0

!

; ð13Þ

with Fs the computed temperature at the considered sensors. The superscript s indicates that one neural network isbuilt for each temperature sensor. Ten sensors were avail-able in the experiment. fsi , w

sij and ws

i0 are some weights,Nh is the number of hidden neurones, pj indicates the inputparameter (pj = vl,vt,a), and Nin their number. f ðxÞ ¼ 1

1þeÿx

is the logistic activation function.


The neural networks were built with NeMo (CEA soft-ware, Dreyfus et al. (2002)) from a sample training of 1100Cast3M simulations. These simulations give some indica-tions to choose the number of hidden neurones and foreach of them to fit the weights fi, wij and wi0. A simple neu-ral network includes one hidden layer made of three hiddenneurones (6). The most simple neural network with no hid-den layer has been tested but gave no satisfactory results.(Fig. 6)

The very fast neural networks allowed to approximatethe Cast3M code and to compute the objective functionfor 500,000 different values for the parameterp = (vl,vt,a)

T. 100 values of vl in the range [1,4], 100 valuesof vt between [0.5,3] and 50 values of a in the range [0.7,1].This number of 500,000 instantaneous computations per-formed with the neural network has to be compared withthe 1100 Cast3M preliminary direct simulations necessaryto the training of the neural network.

3.2.3. Results for HE-C experiment

The values of the thermal conductivities were estimatedseparately using the probes located on each side of theheating source. The triplet of optimized values p is quitedifferent on each side of the heating source. On one side,we obtain vl = 1.84 ± 6%, vt = 055 ± 9% and a = 077 ±3%, although we obtain on the other side vl = 1.90 ± 7%,vt = 1.07 ± 11% and a = 0.77 ± 3%.

In order to check the validity of this method, some for-ward numerical Cast3M simulations were performed withvarious values of parameters p. Fig. 7 displays the temper-ature evolution versus time measured with the five probeson each side of the heating source, and calculated withthree value triplets on each side, equal to p

C21 ¼

ð1:727; 0:525; 0:718ÞT, pC22 ¼ ð1:848; 0:550; 0:773ÞT and

pC23 ¼ ð1:970; 0:601; 0:816ÞT on one side (C2 line) and equal

to pC31 ¼ ð1:788; 1:005; 0:718ÞT, pC3

2 ¼ ð1:939; 1:081; 0:773ÞTand p

C33 ¼ ð2:030; 1:131; 0:816ÞT on the other side (C3 line).

Experimental and numerical results are in good agreement.The additional direct simulations prove that the neural net-

p2=χtp

1=χ

lp

3=αp

0=1

4

1 2

Fs

3

Fig. 6. Neural network with one hidden layer.

Fig. 7. Set of temperatures for the 3D model HE-C (Filippi, 2003).


work correctly simulates Cast3M computations and thatparameters p obtained from the global approach yield tothe correct measured temperatures.

The functional values in the parameter space are pre-sented in Fig. 8. The intervals of parameter values givingthe lowest objective function is represented by the yellowzone.

3.3. Local approach applied on the DI experiment

3.3.1. Objective function

The inverse approach used in DI experiment is based onthe same deterministic theory but two main differences existbetween DI and HE-C and modify the inverse modeling.The first one concerns the experimental protocol and thedata sampling. Two sets of concentration data are availablein DI experiment. The concentrations bF ðxi; tf Þ measured atthe final time tf of the experiment and varying with the posi-tion xi belong to the first set of data. The concentrationsbF ðx1; tjÞ measured in the injection chamber x1 and varyingwith the time tj belong to the second set of data. The relation(14) gives the objective function for the DI problem:

J ¼XI

i¼1

l2i ðF ðxi; tf Þÿ bF ðxi;tf ÞÞ2þ

XJ

j¼1

b2j ðF ðx1; tjÞÿ bF ðx1; tjÞÞ2:

ð14ÞTwo terms are embedded in the objective function, eachone describing one set of experimental data. In the firstterm of the right hand side, the experimental concentra-tions are described by a function depending on the posi-tion, and in the second term the concentrations aredescribed by a function depending on the time. I is thenumber of space data, J the number of temporal data, l2

i

and b2j are positive weights.

3.3.2. Optimization

The minimization of criterion (14) is performed with anoptimization algorithm called an optimizer. A lot of books

were published on this subject: Fletcher (1987), Bonnanset al. (1997), and Nocedal and Wright (1999). Many opti-mization methods exist in the literature and it is out ofrange of this paper to describe all of them. We onlymention the gradient algorithms, more robust and lesstime-consuming than the others. Let us mention that theoptimization methods are found in many problems wherea stationary point of an objective function is searchedfor. These methods are sometimes extended to the resolu-tion of non-linear systems. For example, another applica-tion is given by the coupling between the transportequation and the chemistry (Bouillard et al., 2005).

The principle of these algorithms is simple. A searchsequence (p1, . . . ,pk, . . . ,pN) is generated from a startingpoint p0, such as for the iterate k + 1, J(pk+1) < J(pk), pkis the parameter vector at iterate k. The search sequencebetween the iterate k and k + 1 is not independent contraryto the global approach. The relation is given by pk+1 =pk + nkdk where nk is a positive scalar indicating the lengthof the step and dk is the kth descent direction.

A first class of gradient methods are the conjugate gra-dient algorithms. The descent direction is given by dk =ÿrk + bkdkÿ1 where rk is the residual and the scalar bk isto be determined by requirements that dk and dk+1 mustbe conjugate with respect to the discretization matrix ofthe problem. Several forms of bk depending upon $Jk =$J(pk) exist and distinguish the methods. The Fletcher–Reeves and the Polak–Ribiere (PR) methods are the mostpopular conjugate gradient algorithms. In practice, thePR method with automatic restarts out performs theother methods (Bonnans et al., 1997; Nocedal and Wright,1999).

In a second class of gradient methods, dk is given by therelation dk ¼ ÿBÿ1

k rJ k with Bk a symmetrical matrix and$Jk the gradient of the objective function. In the steepestdescent method, Bk is simply the identity matrix, whereasin the Newton method, Bk is exactly the Hessian matrix$2Jk. In the quasi-Newton methods, Bk is an approxima-

tion of the Hessian matrix. The BFGS formula is one ofthe most popular Hessian approximation. The Byrd et al.algorithm (1995) was applied in DI inverse modeling. Thealgorithm uses a BFGS method and a limited memory toapproximate directly the inverse Hessian matrix. Lowerand upper bounds can also be specified to impose a positiveconstraints on parameters. Optimization with bound con-straints is more difficult.

The Levenberg–Marquardt method is a very efficientalgorithm but it requires the computation of the sensitivitymatrix (or Jacobian matrix). It is possible to compute it forthe one-dimensional model but it is difficult for two or threespace dimensions. In Section 4 we will compute the Jaco-bian matrix but in few points, whereas the Levenberg–Mar-quardt optimization method needs to compute it for eachiteration. A good review of all these algorithms (conjugategradients, BFGS, Newton methods, Levenberg–Marquardt. . .) with or without bound constraints, and a critical discus-sion on the advantages and inconvenients about the main

0.7 0.72 0.74 0.76 0.78 0.8 0.82

1.5

1.6

1.7

1.8

1.9

2

2.1

λl (

W/m

.K)

Optima set (λl, λt, α) | fcout(∆Τ) < fcout(∆Tl)

0.7 0.72 0.74 0.76 0.78 0.8 0.82α (-)

0.4

0.5

0.6

0.7

0.8

λt (

W/m

.K)

C2+B1 : ∆TC2

= 0,15˚C, ∆TB1

= 1,5˚C

C2+B1 : ∆TC2

= 0,10˚C, ∆TB1

= 1˚C

C2 : ∆TC2

= 0,25˚C

λ l,λt : thermal conductivity α : thermal power rate ∆T = Texp. - Tcal.

CAST3M proof

Approximate solution zone

Fig. 8. Functional values in the parameter space (Filippi, 2003).


classes of many methods (trust-region and line search meth-ods) can be found in Nocedal and Wright (1999).

3.3.3. Results for the 3D inverse model

A first application of the local approach with a BFGSalgorithm has been performed on the 3D modeling usingCast3M. The gradients of the cost function were computedby the finite difference method and the optimization soft-ware used was Kalif (Martinez et al., 2002). We refer Tevis-sen et al. (2004) for a complete comparison between thein situ and laboratory diffusion studies from the MontTerri. The laboratory measurements on the centimetriccore samples are described. We present in this paper onlythe results for the HTO tracer in order to focus the discus-sion on the parametrization in the following of this paper.

The simulations performed have shown that the entireset of HTO data is consistent with due account to only adiffusive process in the rock (Palut et al., 2003). The 3Dmodel explains the disappearance of the tracer from theinjection system over the time (Fig. 9), and its distributionin the rock versus the distance from the borehole (Fig. 10).

The best estimate values for HTO are vl = 18 ·

10ÿ11 m2 sÿ1, vt = 1.2 · 10ÿ11 m2 sÿ1 and W = 12%. Theappendix presents the comparisons between all the experi-mental profiles and the computed concentrations.

On the monitoring curve, only the first data are not wellfitted. Those differences between the experimental data andthe model, could be explained by the effect of a disturbedzone around the injection borehole where the porosityand diffusion coefficients are higher. As the tracers diffusefurther into the rock, the diffusion slows down. The tracerdisappearance in the injection borehole is more affected bythe disturbed zone than profiles in the rock, because thetracer is mostly located near the borehole as illustratedby the shape of the profiles. This hypothesis will be testedin Section 3.3.5.

3.3.4. Adjoint state method

Another way to compute the gradients of the cost func-tion, required by the BFGS algorithms, is to apply theadjoint state method. It is based on the optimal control the-ory (Lions, 1968) and was used in parameter identificationproblems by Chavent (1971, 1975). The method computesaccurate gradients with respect to the parameters to iden-tify. A complete description of the adjoint state methodis presented in Carrera and Neuman (1986) and Sun(1994). The application is performed on a classical diffusionequation and a convection–dispersion equations.

In DI problem, the adjoint state is slightly different dueto the boundary condition, which depends on the diffusionto identify and to the time. Another difference for theadjoint deduction comes from the objective function for-mulation. Let us remember that the cost function in thisproblem presents two terms, the first one for the monitor-ing data and the second one for the profile data. Theadjoint system, deduced from the forward problem, hasto be solved according to the following equations:

ÿWðrÞot� ðr; tÞ ¼1

rorðrvðrÞor� ðr; tÞÞ ÿ kWðrÞ� ðr; tÞ; ð15Þ

� ðr; tf Þ ¼XI

i¼1

2l2i

WrðF ðri; tf Þ ÿ bF ðri; tf ÞÞ; ð16Þ

1aot� þ vor� jr1 ¼ ÿ

PJ

j¼1

2b2jr1ðF ðr1; tjÞ ÿ bF ðr1; tjÞÞ;

vor� jr2 ¼ 0:

8><

>:ð17Þ

This system is adjoint-qualified because the operators in itsdefinition are adjoint of state equation operators: ÿot asadjoint of ot and

1rorðrvðrÞorÞ is the same because it is self

adjoint. c(r, t) is called the adjoint state of the concentra-tion F(r, t). A final time (and not initial time) is given inEq. (16), consequently Eq. (15) must be solved backwardsin time. Because data are measured at the end of the exper-iment (profile data) and in the injection chamber (monitor-ing data), differences between computed and observedconcentrations are present in final condition and in the firstboundary condition respectively.Fig. 10. Experimental and computed profile.

75

80

85

90

95

100

0 50 100 150 200 250

Conce

ntr

atio

n i

n i

nje

ctio

n c

ham

ber

(B

q/c

m3)

Time (day)

Experimental data and simulations - monitoring

SimulationsExperimental data

Fig. 9. Experimental and computed monitoring (Montarnal et al., 2002).


When F(r, t) and (r, t) are computed, the gradients of theobjective function can be deduced from the followingrelations

dJ

dv¼ ÿ

Z tf

0

Z r2

r1

orF ðr; tÞor� ðr; tÞrdrdt; ð18Þ

dJ

dW¼ ÿ

Z tf

0

Z r2

r1

� ðr; tÞðotF ðr; tÞ þ kF ðr; tÞÞrdrdt: ð19Þ

Two possibilities are available to deduce the discretized ad-joint system. The first one is to discretize the relations (15)–(17) and (18), (19). Another way is to find the adjoint sys-tem with the variational method from the primary problemalready discretized. This latter method is better as indicatedin Chavent (1979) and has been applied in this work.

The success of the gradient method depends strongly onthe possibility to accurately compute the gradients. Theadjoint state method is based upon an analytical derivationof the gradients and allows to compute them efficiently.This is the main advantage of adjoint state method com-pared to the finite difference method. Indeed, in a compar-ison between the gradients computed by an adjoint stateand a finite difference method several significant values ofthe steps required by the finite difference, must be adjustedto obtain the same values of the gradients. In this work,this comparison has been performed to verify the develop-ment of the relations (15)–(19).

3.3.5. Results for the 1D inverse model

The one-dimensional inverse model using the adjointstate method has been applied on the DI experimentaldata. Several inverse simulations have been performedusing one, two and three zones respectively. The lowest val-ues of the cost function have been obtained with threezones with two parameters per zone: one diffusion andone porosity coefficient. Six parameters were optimized.The inverse simulations performed with three zones gaveseveral admissible solutions. Figs. 11 and 12 present oneof them chosen from some physical considerations. Asalready mentioned in Section 3.3.3, the diffusion and

porosity parameters could be higher near the injectionborehole.

The forward simulation results performed with theseparameter values are presented in Figs. 13 and 14.The improvements brought by the spatial variability of

1e-11

1e-10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

D [

m2/s

]

r [m]

Diffusion values

Fig. 11. Diffusion parameters estimated (Cartalade et al., 2003).

0

5

10

15

20

25

30

35

40

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

ω [

%]

r [m]

Porosity values

Fig. 12. Porosity parameters estimated.

65000

70000

75000

80000

85000

90000

95000

100000

105000

0 50 100 150 200 250 300 350

Conce

ntr

atio

ns

Time [day]

Concentrations in the injection chamber

Experimental data

Model

Fig. 13. Experimental and computed monitoring.

10000

20000

30000

40000

50000

60000

70000

0 2 4 6 8 10 12 14

Conce

ntr

atio

ns

r [cm]

Concentration profile (t final)

Experimental dataModel

Fig. 14. Experimental and computed profile.


parameters v(r) and W(r) on the monitoring are presentedin Fig. 13.

In spite of the one-dimensional modeling, the parametervalues found seem to be coherent with the experimentaldata and the 3D inverse modeling. The range of the diffu-sion is between 10ÿ11 and 1010 m2 sÿ1 for the diffusion. Theporosity presents a mean value of 5% high value over thefirst two centimeters. The first two centimeters present ahigh value of the porosity (greater than 35%) which couldbe explained by the disturbed zone.

4. Discussion

The way to explore the parameter space } in order tofind the solution of the minimization problem is the maindifference between the global and local approaches.

In the global strategy } is scanned with a large numberof parameters. The larger is the set of {pk/k = 1, . . . ,N}, thebetter is the chance to find the solution. However whenthe simulations are performed with a numerical code, theapproach increases the computation time because one sim-ulation must be carried out for each pk to obtain the corre-sponding value of the cost function.

On the contrary, the local strategy searches to reduce thecomputation time by scanning only a region of }, aroundan initial point p0 given by the modeler. The choice ofpk+1 at iterate k + 1, is not independent from the previousone pk. The local terminology is now clear. All the sequencepk (k = 1, . . . ,N) is dependent from the starting point p0and its choice is important for the final solution. Indeed,if the cost function presents more than one stationarypoint, an initial point p0 could lead to a local minimum.The flaw of this approach is the local character of the solu-tion. To ensure the finding of the global solution, severaloptimizations should be performed with different valuesof p0.

Let us emphasize that the problem is ill-posed: it maynot have an unique solution, or the solution may dependnon continuously on the data. Hence it leads to numericaloscillations when the number of parameters is too high,and Tichonov regularization by adding a penalizing termcures the problem.

Another issue about the efficiency of the optimizationalgorithms is the computation of the derivatives of the costfunction, or of the forward model. There are several waysto compute derivatives: approximated or analytical, director reverse mode, automatic or manual implementation.The approximate computation of derivatives by finite dif-ferences is very simple as it does not need any further codedevelopment, but it provides the direct mode. Hence, it isadvisable to restrict its use to the validation of analyticalderivatives computations. The complexity of the direct(or forward) mode is proportional to the number of inputs,e.g. to the number of parameters, and the complexity of thereverse (or backward, or adjoint) mode is proportional tothe number of outputs, eg to the number of measures.So, when dealing with gradients, the reverse mode is obvi-

ously the most efficient way since there is only one output:the value of the cost function. But when computing a Jaco-bian (or sensitivity matrix), the choice of the cheapest modedepends on which is higher between the number of linesand the number of columns. Automatic differentiationopens either through code transformation as Tapenade(Hascoet and Pascual, 2004), or Adifor (Bischof et al.,1992) or through operator overloading as AdolC (Waltherand Griewank, 2004). These high-level tools are very prom-ising and their use is highly recommended for the directmode, but there are still some limitations: choice of theprogramming language, access to the reverse mode, sourcecode requirements. Manual differentiation of the discreteequations in the reverse mode is not difficult in principlewhen using variational numerical schemes, but its imple-mentation, and its validation, can be a long and tedioustask for large codes. Furthermore, it is even possible to dis-cretize the continuous adjoint equations, but then one hasto be careful, eg by refining the mesh when getting closer tothe solution. We have chosen to compute the derivatives inthe reverse mode, and to implement them manually. Evenwhen computing the full Jacobian matrix for the singularvalue decomposition in the next section.

No supplementary development is required in the globalapproach. The direct simulation code is sufficient to obtaina good estimation of the parameters. However the interpre-tation is difficult especially when the dimension of theparameter space is greater than 3. It is then not possibleto interpret graphically the results and it can be extremelyhard to distinguish several small values of the cost functionand to choose between a set of parameters and another.Moreover the exploration of the parameter space with alarge number of p does not guarantee that the found solu-tion is the global solution of the problem. But it highlightsrapidly the family of probable solutions and can be testedafterwards with an optimization algorithm. A genetic algo-rithm with multicriteria optimization techniques, could beused to avoid these difficulties and interpret the results(Gaudier and Dumas, 2005).

Because the neural network is efficient, it allows to scana large region of the parameter space and gives rapidlysome preliminary results. However its implementation isnot fast because four steps have to be performed: (1) train-ing simulations, (2) neural network building, (3) verify thesuitability of the neural network with the simulation code,(4) finding the minimum of the objective function, and (5)validation of solutions with a simulation code. The firststep is the longest because it requires a lot of direct simula-tions. Each neural network is next quite easy to build. Letus emphasize that the relation (13) is a combination of non-linear function depending on the parameters. It is a supple-mentary approximation of a code which is already anumerical approximation of physical equations. The initialchoice of the parametrization and the stationarity/transientproblem is important because the built neural network willbe dependent from these hypothesis. A modification of thezonation requires to perform again the training simulations


and to build new neural networks. Nevertheless, let usremember that the neural network is an interesting toolto obtain some results quickly. The solution of the minimi-zation problem should be next validated with the simula-tion code.

4.1. Singular value decomposition

The parametrization, introduced in Section 3.1, raisesseveral difficulties. As already mentioned, two notions arehidden in the parametrization concept: the structure ofparameters and the values inside it. In this paragraph, theposition and the number of zones (in DI inverse modeling),were supposed to be known. The presentation of methodsand the discussion were focused on the identification ofparameter values. However, in practice, the parameterstructure is not always known. So, to overcome this diffi-culty, two possibilities exist. The first one tries to estimateat the same time the structure of the zonation and theparameter values inside it. Such a method has been appliedby Ben Ameur et al. (2002) in a two-dimensional case usingthe refinement and coarsening indicators for an adaptiveparametrization.

A second possibility to find a good zonation, or even tocompare several parametrizations, is based on a sensitivityanalysis of the concentration measures with respect to theparameters. The study of singular values and vectors, givenby the decomposition (SVD) of the Jacobian matrix, is apowerful tool to define the identifiable degrees of freedomand is very efficient to quantify the sensitivity of the system.It gives indications to choose the best parametrization, totest a new acquisition system, and to evaluate the non lin-earity of the model. However, conclusions are valid onlylocally around the considered point.

In DI experiment, the components of the Jacobianmatrix are the first partial derivatives of the concentrationwith respect to the porosity and diffusion parameters. TheJacobian matrix is computed row by row with the adjointstate method. The difference between the adjoint equationgiving the gradient of the cost function is the right handside of the equation. The singular value decomposition is

a generalization of the diagonalization concept applied torectangular matrices. The singular values of the Jacobianmatrix A with I rows and J columns are the positive squareroots of the eigenvalues of the symmetric matrix ATA.More precisely, the SVD of A is

A ¼ USVT; ð20Þ

with V, the singular vector matrix in the parameter space,U, the singular vector matrix in the measure space, andS, the diagonal singular value matrix in decreasing order.Singular values are nonnegative and U and V are unitarymatrices.

Many simulations have been performed to test the sen-sitivity of different parameter values and zonations. Param-etrization with 20 and 3 zones have been studied. In the lastcase, the sensitivity of zone positions have been studiedtoo. We focus the discussion in this paper on the interpre-tation of singular value decomposition. We refer Clementet al. (2004) for a presentation of other results.

We consider a parametrization with three zones. Fig. 15displays the porosity and diffusion values associated witheach zone. 175 values of monitoring values and 22 profilevalues (Fig. 16) are next computed with these 6 parametersusing the 1D simulation code. The derivatives of the 197(=175 + 22) measures with respect to the 6 parametersare then computed by the adjoint state method. A matrixA with 197 rows and 6 columns is obtained. The singularvalue decomposition (20) gives the unitary square matricesU and V. The order of U and V is 197 and 6, respectively.The dimension of the matrices A and S are identical. Butonly the main diagonal of the matrix S is potentially nonnull. It is composed of positive terms classified in decreas-ing order.

The interpretation of the SVD is the following: fori = 1,2, . . . , 6, a modification of the parameter in the vidirection will affect the measures in the ui direction, propor-tionally to the singular value si. Fig. 17 displayed on thelogarithmic scale presents the six relative singular values.The ratio between the values are 30 meaning that the com-ponent of parameter on m1 will be found with the samenoise level than the measures. The component on m6 will

0

1e-10

2e-10

3e-10

4e-10

5e-10

0 10 20 30 40 50

Dif

fusi

on (

m2/s

)

r (cm)

reference

0

5

10

15

20

25

0 10 20 30 40 50

Poro

sity

(%

)

r (cm)

reference

Fig. 15. Diffusion (left) and porosity (right) values (Clement et al., 2004).


be found with a noise level 30 times higher. So the more thesingular value curve is near the constant curve 1, the morethe inversion will be better. In Fig. 17 the first four are in aratio 6.

Fig. 18 displays the singular vectors in the parameterspace VT. The influence of the parameters is the following:

porosity in the zone 1, porosity in the zone 3, diffusion inthe zone 1, the porosity in the zone 2, difference of the dif-fusion in the zones 2 and 3; mean of the diffusion in thezones 2 and 3. Fig. 19 displays the first six singular vectorsin the measure space. The classification in the decreaseorder of the sensitivity of measures is the following: differ-ence between the mean in the zone 1 of the profile at initialtime and the injection peaks in the monitoring; mean valuein the zone 3 of the profile at final time; mean between themean in the zone 1 of the profile at the final time and theinjection peaks in the monitoring; mean in the zone 2 ofthe profile at the final time; difference between a mean atthe beginning of the zone 2 of the profile at the final timeand one value at the end of this zone; mean between thebeginning and the end of the zone 2 of the profile at thefinal time.

Let us mention that the singular value decompositionresults depend greatly upon the weights l and m in the costfunction (14). They also depend upon normalization valuesof porosity x0 and D0. So, before beginning the sensitivityanalysis, several simulations have been performed tochoose the best weight values l and m and the best normal-ization values of diffusion and porosity. The simulation

0

20000

40000

60000

80000

100000

50 100 150 200 250 300 350

Conce

ntr

atio

n (

Bq/l

)

Time (days)

reference

5000

10000

15000

20000

0 10 20 30 40 50

r (cm)

reference

Fig. 16. Concentration monitoring (left) and profile (right) simulated.

Rel

ativ

e si

ngula

r val

ue

0.0001

0.001

0.01

0.1

1

1 2 3 4 5 6Singular value index

Fig. 17. Singular values.

-1

-0.5

0

0.5

1

1 2 3

Parameter index (porosity)

123456

-1

-0.5

0

0.5

1

4 5 6

Parameter index (diffusion)

123456

Fig. 18. Singular vectors in the parameter space.


results shown in this paper are obtained with l = 1,m = 10ÿ1, x0 = 10ÿ1 and D0 = 10ÿ10 m2/s. The choice ofthese weight and normalization values is important and iscomputation time consuming.

5. Conclusion

The existence of various methods to identify numericallythe parameters in a partial derivative equation complicatesthe manner to proceed to solve the inverse problem. Theapproach to use, global or local, and the optimizationmethods to apply (algorithms with or without gradientsand methods of computation associated) depends on thephysical problem, the framework and the numerical toolsfor modeling. The objective function is written differentlyaccording to the experimental design and the availabledata. Of course the overview is far to be exhaustive andmany methods have not been investigated. Let us mentionthe genetic algorithms and all the methods based on theBayesian theory.

Global and local approaches have been applied to HE-Cand DI experiments, respectively. Rock properties weredetermined using complementary laboratory measure-ments on rock samples. Parameter values found with theinverse modeling are in good agreement with the experi-mental methods. This contribution allows us to synthesizedifferent approaches for inverse modeling and yields valu-able information about their advantages and weaknesses.In the global approach, interpretation in the parameterspace becomes difficult when the number of parameters isgreater than three. The global approach requires a greatnumber of computations (500,000). In order to drasticallyreduce the CPU time of computations, a neural network,using NeMo (software CEA), is built to approximate theforward model. The neural network training requires1100 forward simulations but a new parametrization offorward model requires to build a new neural network.In the local approach, the set of parameters is a local solu-

tion and is not necessarily the global minimum. To acceler-ate convergence it is better to have gradients of theobjective function. It is time consuming when computedby finite differences, particularly when the number ofparameters is important. Adjoint state method gives moreaccurate gradients, but the adjoint system building requiresa long programming effort.

Appendix. Comparison between the data and models

for the DI experiment

See (Fig. 1).

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indicators for adaptive parametrization: application to the estimation

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