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Inverse Problem for Transient Unsaturated Flow Identifiability and Non Uniqueness K. S. Hari Prasad & M. S. Ghidaoui Department of Civil and Structural Engineering The Hong Kong University of Science and Technology Email : hari@wcmailust.hk & GHIDAOUI@usthk.ust.hk Abstract In the present study, the identifiability and non uniqueness of soil hydraulic properties in unsaturated flow systems are discussed. Necessary and sufficient conditions for identifiability are derived. A parameter estimation algorithm is developed by combining the Adjoint State Theory with Conjugate Gradient Method and Levenberg-Marquardt method. These two methods are compared through few numerical examples. It is shown that Levenberg-Marquardt algorithm is superior to Conjugate Gradient Method. However, Levenberg-Marquardt algorithm also fails to converge to the global minimum when the initial guess parameters are far from the true values. Introduction Moisture flow through the unsaturated zone is usually analysed by solving Richards Equation [4]. Solution of Richards equation requires knowledge of soil hydraulic conductivity and water content versus water pressure functions. In recent years, many researchers have investigated the feasibility of determining these functions simultaneously from transient flow by numerical inversion of the governing initial value problem. In such approaches, the unknown parameters are estimated by minimizing deviations between the observed and predicted flow attributes such as water contents, pressure heads or infiltration rates. The inverse problem is often ill-posed which results in non unique and unstable solutions. Application of parameter estimation in unsaturated

34 Computer Methods in Water Resources XII flow are few, consisting analysis of inflow and outflow experiments on laboratory soil cores and some field studies. Kool et al.[3] reviewed the parameter estimation methods in the unsaturated zone. The objective of the present study is to (i) develop theoretical analysis for identifiability of parameters used in different soil hydraulic relationships (ii) to develop a parameter estimation algorithm using Adjoint State Method and Conjugate Gradient Method and Levenberg - Marquardt methods (hi) performing numerical studies to test the robustness of these methods. Governing Equation The pressure head based formulation of Richards equation for one dimensional, single phase, non hysteretic transient vertical flow of water through a rigid and homogeneous unsaturated soil can be written as (Celiaetal.[l]) where ij./ is the pressure head, K is the hydraulic conductivity, C is the soil moisture capacity, 9 is the volumetric moisture content, z is the vertical co-ordinate taken positive upwards, t is the time co-ordinate and L is the length of the soil. In order to solve eqn(l), constitutive relationships between vj/ and nonlinear terms K and C have to be specified. The empirical relationships proposed by Gardner[2] and Van Genuchten[7] are considered in the present study. Gardner's Relationship : Van Genuchten's Relationship : (2) (3) for H/<0 ^ = for > > 0

Computer Methods in Water Resources XII 35 (5) In eqns (2)-(5), (3, a and n are unsaturated soil parameters with m = - l/«, IQat is the saturated hydraulic conductivity and S^ is the effective saturation defined as where 6^ is the saturated moisture content and 0, is the residual moisture content. Identifiability Identifiability refers to the forward problem. The parameters are said to be unidentifiable if more than one set of model parameters produce identical outputs. If two sets (K^CJ and (K^A) produce the same responses i then The parameters are not identifiable, if 5K * 0 and 5C * 0. A necessary, but not sufficient condition for identifiability is 5K = 0 and 5C = 0. Case : dj/dt = 0 and (dy/dz + ) = 0 In this case, i = -z, which corresponds to a no flow problem. In such a case (8K,5C) need not necessarily be equal to (0,0) making K and C unidentifiable. Since, there is no flow in and out of the system, no information can be obtained from such a data for identifying K and C. Case 2 : dj/dt = 0 and (dy/dz + ) * 0 In this case 8K = 0 and 6C need not be zero indicating that the steady state data is not capable of identifying the parameter C. However, the parameter K can be identified from such type of data.

36 Computer Methods in Water Resources XII Case 3 : dv //dt # 0 and (di/dz + ) = 0 In this case v / = -z, which usually does not arise in transient unsaturated flow. Hence this type of data is also not useful for identifying the parameters. Case 4 : dy/dt # 0 and (df/dz + ) * 0 In this case, the data is the transient flow data. In such a case, for identifiability, the necessary condition is 5K = 0 and 5C = 0. Thus, the transient data can be used to identify the parameters K and C. However, is 5K - 0 and 5C = 0 a sufficient condition for identifiability? To illustrate this consider the following two sub cases. Case 4a Let us assume that K and C depend on i and two parameters PI and P2- Then 5K = 8P, + 6P? = 0 5P, ' 8P; ' dc dc 8C = ^ W^k 8P, + <^TTk 5P,=0 / (8) / Generally, (9K/9Pi,9K/9P2) and (9C/9Pi,6C/9P2) are two linearly independent vectors. For eqn(8) to be true, we should have 5Pi = 0 and 8?2 = 0, i.e., PI and?2 are identifiable. In the case of Gardner's model K and C depend upon two parameters K^t and p. Let the ratios of dc/dk^ and 9K/(3Ksat & 9C/3(3 and 3K/3(3 be denoted by aj and a2 respectively. If (5C/5Ksat,3C/ap) and (SK/dK^SK/dp) are linearly independent, then a2 = Aai, where A is a constant. It is clear from eqn(2) that aj is zero. In figs.(l) and (2), the ratios a^ and &2 are plotted against pressure head for different values of K^ and P respectively. It is clear from these figures that at pressures very near to saturation, these two curves tend to become parallel to each other, indicating the linear dependence. Hence, if the data consists of pressure heads near to saturation, K^t and P may not be estimated accurately. Case 4b Let us assume that K and C depend upon j and three parameters PI,?2 and Pg. In such a case 0P, a?2 6P; 6K - 8P, + 5?2 + SPj = 0 (9) 5P ' a* ^ ap ^

Computer Methods in Water Resources XII 37 This system of equations will either have no solution or will have many possible solutions. Hence, in this case, all the three parameters are not identifiable simultaneously. However, fixing one of these parameters enables one to identify the other two parameters. Van Genuchten's relationship has three parameters K^t, a and n for describing the soil hydraulic properties. These three parameters can't be identified simultaneously from transient flow data. Inverse Problem The commonly used objective function G (Russo et al.[5])can be written as : LT 00 ^" (io) where L is the length of the soil profile, T is the total time of simulation, Qm and qc are the measured and computed infiltration rates. The parameter estimation involves the estimation of the parameters K^t, a and n for Van Genuchten relationship. The sensitivity coefficients at each iteration of the parameter estimation are obtained using Adjoint State Theory. Following Sun and Yeh[6], the adjoint equation for eqns () and (0) can be obtained as +AK(^LO,t.. = 0, t = T,0<z<L; X = 0, t < T, z = 0..^ T () The adjoint problem consists of solving for A, after solving the direct problem for j. Conjugate Gradient and Levenberg - Marquardt methods are used for the minimization. To test the robustness of these methods, several numerical tests are performed. A hypothetical data consisting of infiltration rates is generated by solving eqn(l), with the following set of parameters; K^t = 5cm/hr, a = 0.02 cm"% % = 2.3, 6^ = 0.38, 8, = 0.08. The length L of the soil profile is considered as m and the initial moisture content(8j throughout the soil profile is taken as 0.2. The moisture content at the bottom is held at 0.2 while a moisture content equal to 9^ is applied at the top. Infiltration data are obtained by solving the direct problem (eqn(l)) for a period of about 8 minutes. This data

38 Computer Methods in Water Resources XII are used as measured infiltration rates in all the test runs. 8g and 9,. are considered as constants during the parameter estimation. With only one parameter (K^t, oc or n) as an unknown, both Conjugate Gradient method and Levenberg - Marquardt algorithm converge to the true solution. A comparison of Conjugate Gradient method and Levenberg-Marquardt algorithm is presented in Table.. Table. Comparison of Conjugate Gradient and Levenberg - Marquardt Methods Iteration 2 3 4 5 6 7 2 3 4 5 2 3 4 6 Conjugate Gradient Method Levenberg - Marquardt Method Ksatrinn = 9cm/hr, K^truei = 5cm/hr 7.07 6.94 5.87 5.36 5.29 5.02 5.08 5.003 5.02 5 5.005 5 tt(ini) ~ 0.07 Cm"ttftrue) ~ 0.02 Cm" 0.042 0.022 0.033 0.0202 0.025 0.02 0.0207 0.02 "(inn = 3.5 "(true) = 2.3 3.3 3.08 2.84 2.59 2.3 2.23 2.28 2.29 2.3 It is clear from Table. that the Levenberg - Marquardt algorithm converges to the true solution in fewer iterations as compared to Conjugate Gradient Method. For the case, when two parameters are unknown, Conjugate Gradient method fails to converge to global minimum where as Levenberg - Marquardt algorithm converges to the true values. However, Levenberg - Marquardt algorithm also fails to converge, when the initial parameters are far away from the true parameters. Fig.3 shows the parametric space through which the global

Computer Methods in Water Resources XII 39 minimum can be obtained for the case in which K^t and a are considered as unknown parameters. It is clear from Fig. 3 that the non uniqueness is quite pronounced in the estimation of parameters in unsaturated flow systems. Conclusions It has been shown that, with the unsteady flow data, the soil hydraulic model with two parameters can be identified uniquely, where as a model with three parameters are unidentfiable. Adjoint equations are derived for the Richards equation and are coupled with Conjugate Gradient and Levenberg - Marquardt algorithm for the estimation of soil parameters. Levenberg - Marquardt algorithm is found to be superior to Conjugate Gradient Method in estimating the parameters. References. Celia, M.A., Bouloutas, E.T. and Zarba, R.L., A General Mass Conservative Numerical Solution for the Unsaturated Flow Equation, z, 26(7), pp. 483-496, 990. 2. Gardner, W. R., Some Steady State Solutions of Unsaturated Moisture Flow Equations with Application to Evaporation from a Watertable, Soil Science, 85, pp. 228-232, 958. 3. Kool, J.B., Parker, P.C. and Van Genuchten, M.T., Parameter Estimation for Unsaturated Flow and Transport Models - A Review, Journal of Hydrology, 9, pp. 255-293. 4. Richards, L.A., Capillary Conduction of Liquids Through Porous Medium, Physics,, pp. 38-333, 93. 5. Russo, D., Bresler, E., Shani,U. and Parker, J.C., Analysis of Infiltration Events in Relation to Determining Soil Hydraulic Properties by Inverse Problem Methodology, Water Resources z, 27(6),pp. 36-373, 99. Sun, N.Z. and Yeh, W.W.G., Coupled Inverse Problems in Groundwater Modeling,. Sensitivity Analysis and Parameter Identification, Water Resources Research, 26(0), pp. 2507-2525, 990.

320 Computer Methods in Water Resources XII 7. Van Genuchten, M. T., A Closed Form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils, 6W &v'gmce 6bcz'e(y of 'ca JowrW, 44, pp. 892-898, 980..08 -.06 - tt- 04 - H 02- @ ^ in 000 - ct re -02- nt -.04 - -.06 - -.08 -. m -... i I a, (=0) (3 0.0cm"'' P 0.02 cm"^ 0.03 cm'' 0.04 cm'' p = 0.05 cm'' A " > ' / W' / - /'?^ : // / / V / 200-50 -00-50 0 Pressure Head (cm) Fig. : Variation of a, and a% with pressure head - K^ = 3cm/hr -00-50 Pressure Head (cm) Fig. 2 : Variation of a, and a% with pressure head - p = 0.03 cm"'' Global minimum 000 0.02.03.04.05.06 Fig. 3 : Parametric Space of K^ and a - Global and Local Convergence