HYDRAULIC CONDUCTIVITY ESTIMATION IN PARTIALLY SATURATED SOILS USING THE ADJOINT METHOD

Transcription

1 HYDRAULIC CONDUCTIVITY ESTIMATION IN PARTIALLY SATURATED SOILS USING THE ADJOINT METHOD J. SANTOS, Y. EFENDIEV, AND L. GUARRACINO Abstract An iterative algorithm based on the adjoint method for the estimation of the saturated hydraulic conductivity k in a partially saturated soil Q is proposed. Groundwater flow in Q is assumed to be described by Richards equation. The optimization problem minimizes the L 2 -error between the pressure head values pk, x, t calculated as the solution of a direct problem and the measured values of the pressure head at discrete points inside the domain Q. The exact gradient of the cost functional is obtained by solving an appropriate adjoint problem, which is derived from the equations of the Gâteaux derivatives of the pressure head with respect to the parameter k. A finite element procedure is used to obtain approximate solutions of the direct and adjoint problems and the Gâteaux derivatives. A discrete form of expression of the gradient of the cost functional at the continuous level is used inside a nonlinear conjugate gradient iteration to solve the optimization problem. A numerical example showing the implementation of the algorithm to estimate the saturated hydraulic conductivity kx during an hypothetical infiltration experiment in a heterogeneous soil is also presented. KEYWORDS: Inverse Problems, Adjoint Methods, Finite Elements. 1. Introduction. In recent years, understanding and quantifying the global hydrologic cycle has become a priority research. Soil moisture, in particular, has gained a lot of attention as it constitutes a key variable in the global hydrologic cycle. Soil moisture is most often described as the moisture in the top several meters of soil that can interact with the atmosphere through evapotransportation, infiltration, and runoff. Soil moisture conditions are important in determining the amount of infiltration, runoff, and groundwater recharge. In addition, land-atmosphere processes critically depend on the state of soil moisture, as soil moisture partitions the energy of fluxes available at the land surfaces into latent and sensible heat fluxes. Accurate assessment of the spatial and temporal variation of soil moisture are advantageous for numerous applications and for answering diverse research questions. Measurement of soil moisture is important for the study and understanding of all surface biogeophysical processes. This includes agriculture, environment, ecology, water resources, climate dynamics, soil strength and soil erosion. For example, in climate dynamics, long-term changes in soil moisture stores have been identified as an indicator of climate change, and soil moisture information can calibrate and validate global climate models. Given the critical role that soil moisture plays in most land-surface processes, it is desirable that soil moisture be monitored with the same accuracy and frequency as other important environmental variables. Department of Mathematics, Purdue University, West Lafayette, IN and Facultad de Ciencias Astronomicas y Geofisicas, Universidad Nacional de La Plata, Paseo del Bosque, S/N, 19 La Plata, Argentina, Department of Mathematics, Texas A & M University, College Station, TX , Facultad de Ciencias Astronomicas y Geofisicas, Universidad Nacional de La Plata, Paseo del Bosque, S/N, 19 La Plata, Argentina 1

2 Numerical modeling of soil moisture requires an accurate knowledge of the hydraulic conductivity and water content functions. These characteristic functions are usually described by empirical mathematical models with different number of fitting parameters, such as Brooks-Corey [1] or van Genuchten [2] models. Model parameters are often difficult or even impossible to measure directly because of instrumentation, scale or conceptual constraints. Thus, inverse modeling of laboratory or field data has become an attractive alternative to direct measurements [3, 4, 5, 6]. In recent years, various optimization methods such quasi-newton [7], Simplex [4], Levenberg- Marquardt [6, 8] and Ant Colony [9] have been used for parameter estimation of characteristic curves. In particular, the estimation of the saturated hydraulic conductivity is rather critical because the groundwater flow is highly sensitive to this parameter [1]. Hydraulic conductivity values are relatively easy to obtain from laboratory methods but these values are often non-representative of in-situ conditions [11]. Traditionally, in-situ moisture measurement techniques provide point measurements. Though these measurements do not account for the spatial variability of the typical soil moisture profiles. Because in-situ soil moisture measurements are generally expensive and often problematic, no large-scale soil moisture networks exist to measure soil moisture at the ideal high frequency, multiple depths, and fine spatial resolution that is needed for complete understanding of soil moisture dynamics. The objective of this paper is to present a nonlinear optimization algorithm to determine the saturated hydraulic conductivity field the parameter to be estimated, based on local measurements. Groundwater flow is assumed to be described by Richards equation [12] in conjunction with the van Genuchten model. The optimization problem minimizes the L 2 -error between the pressure head values px, k, t calculated at the measurements points and the measured values of the pressure head at these discrete points. The gradient of the cost functional in our nonlinear optimization problem is defined at the continuous level using the adjoint of the Gâteaux derivative of the solution with respect to the parameter. Both the Gâteaux derivative and the adjoint are defined at the continuous level as solutions of partial differential equations with appropriate initial and boundary conditions and then discretized using finite element procedures. This approach, known as differentiate-then-discretize, provides an expression for the gradient which is independent of the particular discretization algorithm used to solve the differential problems. This method has been used for example, in [13, 14, 15, 16, 17, 18] to solve parameter estimation problems in geophysics and other applications. For an account of several aspects of parameter estimation such as regularization, identifiability, etc, we refer to [19]. In particular, the proposed adjoint procedure allows for a more accurate calculation of the gradient of the cost functional than the standard discretize-then-differentiate approach consisting in discretizing the differential equations first and then applying optimization techniques to a discrete version as described for example in [2]. The organization of the paper is as follows. Section 2 states Richards equation in terms of the pressure head pk as a function of saturated hydraulic conductivity kx that will be the parameter to be estimated. The analysis of the inverse problem and results on the continuity and differentiability of pk with respect to the parameter kx are presented in Section 3. Section 4 is devoted to the formulation of the adjoint problem and the derivation of an expression of the gradient of the cost functional that will be used in the discrete minimization procedure. In Section 5, a Newton iteration to solve the continuous minimization problem is formulated and analyzed. 2

3 The formulation of the discrete parameter estimation algorithm employing a nonlinear conjugate gradient algorithm is done in Section 6. Section 7 presents numerical experiments showing the application of the proposed algorithm to estimate the saturated hydraulic conductivity in a vertical soil profile. Finally, the Appendix contains the proof of the continuity of the parameter to output mapping k pk. 2. The direct model and the parameter estimation problem. We consider the problem of estimating the saturated hydraulic conductivity kx in a multidimensional bounded variably saturated soil Q with boundary Q. Let Γ be the part of Q associated with the top surface of the soil, i.e., the part of Q, where the rain and evapotranspiration data will be specified and we set Γ = Q \ Γ. It will be assumed that water flow within Q is governed by Richards equation [12] stated in the form D t θpk divkgpkd x pk + x 3 =, x Q, t I =, T, 2.1 with boundary conditions and initial conditions kgpkd x pk + x 3 ν = q, x Γ, t I, 2.2 kgpkd x pk + x 3 ν =, x Γ, t I, pkt = = p x, x Q. 2.3 In the equations above ν denotes the unit outward normal to Q, gp is the relative hydraulic conductivity and the x 3 -axis is considered to be positive upward. In the rest of the paper, it will be assumed that there exists positive constants c 1, c 2, c 3, c 4 such that < c 1 < kgpk < c 2, 2.4 < c 3 < D p θpk < c To solve the differential problem , the functions θp and gp need to be specified. One of the commonly used pairs θp, gp is given by the van Genucthen model [2]: θ s θ r θp = [1 + α vg p n ] m + θ r, for p < 2.6 θ s for p, {1 α vg p n 1 [1 + α vg p n ] m } 2 for p < gp = [1 + α vg p n ] m/2 1 for p, 2.7 where θ r and θ s are the residual and saturated water contents, respectively; n and α vg are shape parameters; and m is given by the relation m = 1 1/n. Next, we formulate our parameter estimation problem. Assume that the pressure head values px ri, t 1 i Nr are recorded at the points x ri inside Q for all t I. 3

4 Then our objective is to use the observation vector p obs t = px ri, t 1 i Nr to infer the actual values of the saturated hydraulic conductivity kx. We will consider the set of admissible parameters to be P = {k : k is measurable, k kx k } endowed with the L 2 Q-topology, where K and K are positive constants. We consider the cost functional J k defined as follows. For each point x ri let B i be a small ball of radius ρ small enough such that B i B j =, i j. Then let us define pk, x ri, t = 1 pk, x, tdx, 2.8 B i B i pk, t = pk, x ri, t 1 i Nr R Nr. Then let J k be defined by J k = 1 2 pk pobs L 2 I,R Nr. 2.9 Our estimation problem solved using a least squares criterion will be minimize J k over P Analysis of the minimization problem. Let us consider the parameter-tooutput mapping that associates to each element k the corresponding solution pk, x, t of Consider 2.1 for two different hydraulic conductivities k 1 and k 2 : D t θpk 1 divk 1 gpk 1 D x pk 1 + x 3 =, D t θpk 2 divk 2 gpk 2 D x pk 2 + x 3 =, 3.1 and subtract them: D t θpk 1 θpk 2 divk 1 gpk 1 D x pk 1 + x divk 2 gpk 2 D x pk 2 + x 3 =. For simplicity of notations, we denote p 1 = pk 1 and p 2 = pk 2, and for any function fpk, fp 1 = fpk 1, fp 2 = fpk 2. After adding and subtracting the term divk 2 gp 2 D x p 1 + x 3 in 3.2, defining and denoting equation 3.2 can be rewritten as ζk 1, k 2 = k 1 [gp 1 gp 2 ] + gp 2 [k 1 k 2 ], 3.3 d p k 1, k 2 = p 1 p 2, D t [θp 1 θp 2 ] divk 2 gp 2 D x d p k 1, k = divζk 1, k 2 D x p 1 + x 3, x Q, t I, 4

5 or in the equivalent form D p θp 2 D t d p k 1, k 2 + [D p θp 1 D p θp 2 ] D t p divk 2 gp 2 D x d p k 1, k 2 = divζk 1, k 2 D x p 1 + x 3, x Q, t I, with the following boundary and initial conditions k 2 gp 2 D x d p k 1, k 2 ν = ζk 1, k 2 D x p 1 + x 3 ν, x Γ, t I, 3.6 k 2 gp 2 D x d p k 1, k 2 ν =, x Γ, t I, 3.7 d p k 1, k 2 t = =, x Q. 3.8 For simplicity of the notations in further analysis, we use the notation d f k 1, k 2 = fpk 1 fpk The following theorem shows the continuity of the parameter-to-output mapping when the set of admissible parameters P is endowed with the L 2 Q-topology. Theorem 3.1. Assume that there exists a positive constant c 5 such that D n t pk L I,L Q c 5, n =, 1, 2, 3 and D x D n t p L I,L Q c 5, n =, 1, 2. Also assume that D p n θp, n = 1, 2, 3 and D p n gp, n =, 1, 2 are bounded and Lipschitz continuous functions of p. Then the solution d p k 1, k 2 of 3.5 with the boundary and initial conditions exists and is unique and satisfies the estimates D n t d p k 1, k 2 L I,L 2 Q + D x D n t d p k 1, k 2 L2 I,L 2 Q c k 1 k 2 L 2 Q, n =, 1, The proof of this theorem is presented in Appendix A. Note that the assumption imposed on p requires higher regularity of k. Consequently, the admissible set P is a subset of L 2 Q. The higher regularity conditions are required for the proof of Theorem 3.4. As a consequence of Theorem 3.1, we obtain the following result. Lemma 3.1. Under the assumptions of Theorem 3.1, the mapping P, L 2 Q L I, H n Q L 2 I, H n+1 Q 3.11 k pk is continuous for n =, 1, 2, where H = L 2. The validity of following theorem concerning the existence of solutions of our least squares problem 2.1 is an immediate consequence of Lemma 3.1. Theorem 3.2. Let A P be a compact set on L 2 Q. Then the problem min J k A has a solution. To solve the parameter estimation problem later we will define an iterative procedure requiring the calculation of the Gâteaux derivative D k pδk of the pressure head pk, x, t with respect to the parameter k. 5

6 It will be assumed that the parameter kx is known in a small neighborhood B of Γ. Thus, the space of perturbations δk of the parameter k will be chosen to be the space δp = {δk L 2 Q such that δkx = for x B }, 3.12 endowed with the L 2 Q-topology. We will show in Theorem 3.4 below that pk + δk = pk + D k pδk + φk, k + δk, 3.13 where D k p is the linear operator defined from δp into L I, L 2 Q L 2 I, H 1 Q and φk, k + δk L I,L 2 Q L 2 I,H 1 Q 3.14 as tends to zero. To demonstrate the validity of , for any δk P, we first define Φ = Φk = D k pδk, as the solution of the equation D t D p θφ divkgpkd x Φ divkd p gφd x p + x 3 = divgpkδkd x pk + x 3, x Q, t I, 3.15 with the boundary condition and the initial condition kgpkd x Φ ν =, x Q, t I, 3.16 Φ, t = =, x Q Since δk δp, a weak formulation for is as follows: find Φ H 1 Q such that D t D p θφ, v + kgpkd x Φ, D x v + kd p gφd x pk + x 3, D x v3.18 = gpkδkd x pk + x 3, D x v, v H 1 Q. Theorem 3.3. Under the assumptions of Theorem 3.1, for any δk δp there exists a unique solution Φ = D k pδk of and satisfies the following estimate D n t Φ L I,L 2 Q + D x D n t Φ L2 I,L 2 Q c δk L2 Q, n =, 1, 2, 3.19 so that in particular D k pδk L 2 I, L 2 Q and D k p : δp L 2 I, L 2 Q is a continuous linear operator. The proof of this theorem is similar to the proof of Theorem 3.1. In particular, following the proof of Theorem 3.1, one can show that 3.19 holds for δk L Q δp. Further, applying the Hahn-Banach theorem, this result can be extended to δk δp. 6

7 To show the validity of , we will demonstrate that φ = φk + δk, k = Φ d pk + δk, k converges to zero as in L I, L 2 Q L 2 I, H 1 Q. Taking k 1 = k + δk, k 2 = k in 3.4, with δk δp, dividing by and subtracting the resulting equation from 3.15, we obtain that φ satisfies the following differential equation. D t D p θφ d θk + δk, k divkgpkd x φ 3.2 = div kd p gφ + gpkδkd x pk + x 3 div [k + δk d gk + δk, k + gpkδk]d x pk + δk + x 3, x Q, t I. Also, subtracting 3.6 for the same choice of k 1 and k 2 from 3.16, we get the following boundary conditions for φ: kgpkd x φ ν =, x Q, t I Moreover, we have the following initial condition cf. 3.8 φ, t = =, x Q Theorem 3.4. Assume the conditions of Theorem 3.1 are satisfied. Then, the solution φ of satisfies as, provided δk L Q δp. Proof. For further calculations, we denote φ L I,L 2 Q + D x φ L2 I,L 2 Q 3.23 [k + δk d gk + δk, k R 1 = [kd p gφ + gpkδk]d x pk + x gpkδk]d x pk + δk + x It can be shown that the equation 3.2 has the following variational form D t D p θφ d θk + δk, k, ψ + kgpkd x φ, D x ψ 3.25 = R 1, D x ψ, ψ H 1 Q. The fact that there are no boundary terms from the integration by parts of the last term can be verified directly. In the proof we will use the following identity which holds for any smooth function Ψp C 2 R, Ψpk + δk = Ψpk + D p Ψpkpk + δk pk+ D 2 p Ψp pk + δk pk 2,

8 where p = pk + δk for some between and. written as This identity can be also d Ψ k + δk, k = D p Ψpkpk + δk pk D 2 p Ψp pk + δk pk 2. Applying 3.27 to d θ in 3.25, we have D p 2 θp D t [D p θφ], ψ D d pk + δk, k 2 t, ψ +kgpkd x φ, D x ψ = R 1, D x ψ Taking ψ = φ in 3.28 we have D p 2 θp D t D p θφ, φ D d pk + δk, k 2 t, φ The equation above can be written as +kgpkd x φ, D x φ = R 1, D x φ. 1 2 D td p θφ, φ + 1 D 2 φ2 p 2 θp, D t D p θ D d pk + δk, k 2 t, φ From 3.29 we have +kgpkd x φ, D x φ = R 1, D x φ D t D p θ 1/2 φ 2 L 2 Q + D xφ 2 L 2 Q 3.3 c φ 2 L 2 Q + R 1 2 L 2 Q d pk + δk, k 4 L 4 Q D td p k + δk, k 4 L 4 Q. To estimate R 1 2 L 2 Q on the right-hand side of 3.3, we write R 1 as R 1 = gpkδkd x d p k + δk, k 3.31 dg k + δk, k +k D p gd k pδk D x pk + k + x 3 +δk d g k + δk, kd x pk + k + x 3 + k D p gφd x d p k + δk, k. Using 3.27 d g k + δk, k = D p gpkd p k + δk, k +D p 2 gp d pk + δk, k 2, 3.32 where p = pk + δk for some between and, we have the following estimate 8

9 for the second term on the right hand side of 3.31 D p gd k pδk d gk + δk, k L 2 Q c D p g[d k pδk d pk + δk, k = c D p gφ L 2 Q + D p 2 gp c φ L 2 Q + c 1 d pk + δk, k 2 L 4 Q, ] L 2 Q + D p 2 gp d pk + δk, k 2 L 2 Q d pk + δk, k 2 L 2 Q since D x pk, D p g and D p 2 g are bounded functions. Next, the using that gp and δk are in L Q, the L 2 -norm of first term on the right hand side of 3.31 can be bounded by c D x d p k + δk, k L 2 Q. Also, since gp is a Lipschitz continuous function and D x p L Q is bounded, the L 2 -norm of the third term on the right hand side of 3.31 can be bounded by c d p k + δk, k 2 L 4 Q + d pk + δk, k L2 Q. To bound the last term on the right hand side of 3.31, first note that from Theorem 3.3 and standard elliptic regularity estimates, it follows that Φ L I, H 2 Q. Because of the continuous embedding of H 2 into L, we obtain Φ L I, L Q. Thus, the last term on the right hand side of 3.31 is bounded by c D x d p k + δk, k L2 Q. Combining all these estimates for R 1 we have R 1 L 2 Q c d p k + δk, k H 1 Q + φ L 2 Q d pk + δk, k 2 L 4 Q Thus, the estimate 3.3 becomes D t D p θ 1/2 φ 2 L 2 Q + D xφ 2 L 2 Q 3.34 c φ 2 L 2 Q + d pk + δk, k 2 H 1 Q d pk + δk, k 4 L 4 Q D td p k + δk, k 4 L 4 Q. Finally, using the assumption D p θ > c 3 and applying Gronwall s inequality we get max φ 2 t L 2 Q + D xφ 2 L 2 I,Q c d p k + δk, k 2 H 1 Q d pk + δk, k 4 L 4 Q D td p k + δk, k 4 L 4 Q dτ

10 Using Theorem 3.1 we can bound the first term on the right hand side of 3.35 by c 2. Next, we show that the second term on the right hand side of 3.35 goes to zero as. For this purpose, we only need to show that d p k + δk, k/ L 4 Qdτ c 6, 3.36 where c 6 is independent of. From Theorem 3.1, we have d p k + δk, k/ L 2 I, H 1 Q and D t d p k+δk, k/ L 2 I, H 1 Q. Consequently, d p k+δk, k/ CI, H 1 Q after possibly being redefined on a set of measure zero. Because of the continuous embedding H 1 Q L 4 Q for n = 2, 3, we obtain d p k + δk, k/ L 4 I, L 4 Q, and 1 2 d pk + δk, k 4 L 4 Q dτ c2, so that 3.36 holds. Similarly, one can show that the third term on the right hand side of 3.35 approaches to zero as. It follows from Theorem 3.1 that D t d p k + δk, k/ L 2 I, H 1 Q and D t D t d p k + δk, k/ L 2 I, H 1 Q. Consequently, D t d p k + δk, k/ is a bounded function of the time variable and D t d p k + δk, k/ CI, H 1 Q after possibly being redefined on a set of measure zero. Thus, D t d p k+ δk, k/ L 4 I, L 4 Q, and we have Hence 1 2 D td p k + δk, k 4 L 4 Q dτ c2. max φ 2 t L 2 Q + D xφ 2 L 2 I,Q c which shows the validity of This completes the proof. The following theorem about the continuity of Φk = D k pδk with respect to k will be needed for the convergence analysis of the iterative estimation algorithms which are defined later. The technique of the proof of this theorem is the same as that in Theorem 3.4 and it will be omitted. Theorem 3.5. Under the assumptions of Theorem 3.1, for any δk δp Φk = D k pδk is Lipschitz continuous with respect to k in the following sense: Φk 1 Φk 2 L I,L 2 Q + D x Φk 1 Φk 2 L2 I,L 2 Q c k 1 k 2 L2 Q Analysis of the adjoint problem. For the algorithm description we need the adjoint of the differential problem for the Gâteaux derivative D k pδk. The assumption that δk P implies that to solve the adjoint problem , we need to find the solution W k of the adjoint differential equation with boundary conditions D p θd t W k divkgpkd x W k 4.1 +kd p gd x pk + x 3 D x W k = f, x Q, t I, kgpkd x W k ν = x Q, t I, 4.2 1

11 and final condition W k, T =, x Q. 4.3 A weak form for is as follows: find W k H 1 Q such that D p θd t W k, v + kgpkd x W k, D x v kd p gd x pk + x 3 D x W k, v = f, v, v H 1 Q. The continuity of W k with respect to the parameter k can be obtained with an argument similar to that given in Theorem 3.1. Lemma 4.1. Under the hypothesis of Theorem 3.1, the solution W k of is Lipschitz-continuous in the following sense: D n t d W k 1, k 2 L I,L 2 Q + D x D n t d W k 1, k 2 L 2 I,L 2 Q c k 1 k 2 L2 Q, n =, 1, In the next two lemmas, we derive an expression for the gradient of our cost functional J k, which is needed for the definition of the discrete iterative estimation algorithm. Lemma 4.2. The value of the adjoint map can be computed by the relation D k p : L2 I, L 2 Q δp Dk pf = gpkd x pk + x 3 D x W kdt, f L 2 I, L 2 Q, 4.6 where W k is the solution of Proof. Take v = Φ = D k pδk with δk P as a test function in 4.4 and integrate the obtained equation from to T : [ D p θd t W k, Φdt + kgd x W k, D x Φ 4.7 ] +kd p gd x p + x 3 D x W k, Φ dt = f, Φdt. Using integration by parts in time in the first term on the left-hand side of 4.7 and the final and initial conditions for W k and Φ the above equation can be written as [ ] W k, D t D p θφdt + kgd x W k, D x Φ + kd p gd x p + x 3 D x W k, Φ dt = f, D k pδkdt = δk, Dk pf. 4.8 Next, choosing W k as a test function in the weak form 3.18 for Φ and integrating the resulting equation from to T we obtain D t D p θφ, W kdt + + kd p gφd x p + x 3, D x W kdt = kgd x Φ, D x W kdt gδkd x p + x 3, D x W kdt.

12 Writing the third term on the left hand side of 4.9 as from we obtain δk, D kpf= = δk, From here, 4.6 follows. Φ, kd p gd x p + x 3 D x W kdt gδkd x pk + x 3, D x W kdt gpkd x pk + x 3 D x W kdt, δk P. Lemma 4.3. The functional J k has a gradient with respect to the parameter k, denoted J k k = D kj k : δp R which can be computed from the identity where fx,t is the residual-related function given by fx, t = N r i=1 J kk = D kpf, B i χ B i x pk, x ri, t p obs x ri, t In 4.11, χ Bi x denotes the characteristic function of the ball B i. Proof. Let δk δp. First note that J k + δk = J k + D k J kδk + Rk + δk, k 4.12 = J k + D k J k, δk L2 Q + Rk + δk, k, with Since Rk + δk, k δk L 2 Q as δk L 2 Q using 3.13 we see that J k = 1 2 pk pobs, pk p obs L 2 I,R Nr, J k + δk = 1 2 pk + D k pδk + φk + δk, k p obs, 4.14 pk + D p kδk + φk + δk, k p obs L 2 I,R Nr = J k + pk p obs, D k pδk dt + 1 R Nr 2 D k pδk 2 L 2 I,R Nr + φk + δk, k, pk + Dk pδk + φk + δk, k p obs L2I,RNr J k + pk p obs, D k pδk dt + Sk + δk, k, R Nr 12

13 where Sk + δk, k = 1 2 D k pδk 2 L 2 I,R Nr + φk + δk, k, pk + Dk pδk + φk + δk, k p obs. L 2 I,R Nr Next, note that using Theorem 3.3, N D r k pδk 2 L 2 I,R Nr = 1 2 D k pδkx, tdx B i dt 4.15 B i i=1 c D k pδk 2 L 2 I,L 2 Q c δk 2 L 2 Q. Next, a similar argument employing Theorem 3.4 shows that φk + δk, k, pk + Dk pδk + φk + δk, k p obs L2I,RNr 4.16 c δk 2 L 2 Q. Thus using in 4.14 we conclude that Sk + δk, k δk L 2 Q as δk L2 Q Then from and we see that Next, note that D k J k, δk L 2 Q = = = pk p obs, D k pδk N r i=1 Q pk p obs, D k pδk dt, δk δp R Nr dt 4.19 R Nr pk, xri, t p obs x ri, t dt 1 D k pδkx, tχ Bi xdx B i B i D k pδkx, tfx, tdxdt = f, D k pδk L 2 I,L 2 Q = D kpf, δk L 2 Q, δk δp. Now the conclusion follows from Now we have the following Corollary. Corollary 4.4. Under the assumptions of Theorem 3.1, the continuous linear functional J k k is given by the equation J kkx = gpkxd x pk, x + x 3 D x W k, xdt, x Q, 4.2 where W k is the solution of with f defined by Also, J k k is Lipschitz continuous with respect to the parameter k in the following sense: 13

14 J k k 1 J k k 2 L 2 Q + D x J k k 1 J k k 2 L 2 Q 4.21 c k 1 k 2 L 2 Q. Proof. Equation 4.2 follows from Lemmas 4.2 and 4.3. Further using the fact that gp is bounded and continuously differentiable as a function of p, and that pk and W k are bounded and Lipschitz continuous thanks to Theorem 3.1 and Lemma 4.1, we conclude the validity of This completes the proof. 5. A Newton iteration at the continuous level. Let LδP, δp denote the continuous linear functionals from δp into itself. Following [16], for k P we define Mk LδP, δp by the rule Mkγ = D k pd kpγ, γ δp. 5.1 Next, for k P such that Mk has an inverse, define Then we define the Newton iteration by Λk = [Mk] 1 D kp pk p obs. 5.2 k j + Λk j, k k j + Λk j k, k j+1 = k, k j + Λk j < k, k, k j + Λk j > k. 5.3 Next, we analyze the convergence of the Newton iteration 5.3. First recall that according to Lemma 3.1 and Theorem 3.3 the mapping k pk is continuous and D k p : δp L 2 I, L 2 Q is a bounded linear operator. Also, from Theorem 3.5 we know that for δk δp, D k pδk is a Lipschitz continuous function of k. Then, the validity of the following theorem can be established with the argument given in Theorem 4.1 of [16]. Theorem 5.1. Assume the conditions of Theorem 3.1 are satisfied. Also assume that there exists a k c P such that D k Jk c = and that Mk c is invertible. Then k c is a point of attraction of the iteration 5.3. Any discrete implementation of the Newton iteration 5.3 is computationally expensive, involving the solution of as many forward problems as parameters chosen to represent the function kx. See for example [21], [22], [23] and [24]. Thus in order to solve our parameter estimation problem in the next sections, we will define a fast conjugate gradient-type iteration using a discrete version of the expression for the gradient J k k obtained in Lemma A discrete parameter estimation algorithm. In order to define a discrete estimation algorithm, as a first step we need to obtain approximations to the solution pk, x, t of , to the Gâteaux derivative D k pδkk, x, t defined in and to the solution W k, x, t of the adjoint problem This is done using finite element procedures as indicated below. Let M h be a finite element subspace of H 1 Q associated with a quasi-uniform partition T h of Q into elements Q j of diameter bounded by h. Let L be a positive integer and t = T/L. Also, set 14

15 u h,n = u h n t, d t u h,n = uh,n+1 u h,n t, u h,n+ 1 u h,n+1 + u h,n 2 =, 2 and for any function up h of p h set u h,n = up h n t. The discrete-time Galerkin procedure to obtain approximations to the solution of is defined using a backward Euler algorithm combined with a modified Picard iteration in time of iteration index i as indicated in what follows. Let pk h,n+1, M h be an initial guess for the Picard iteration to obtain pk h,n+1 pk h,n+1,, which denotes the value of pk h,n+1,i+1 M h after convergence with a prescribed tolerance in the iteration has been achieved. Then, we find pk h,n+1,i+1 M h such that [Dp θ] h,n+1,i pk h,n+1,i+1, ϕ + kg h,n+1,i D x pk h,n+1,i+1, D x ϕ 6.1 t = q,n+1, ϕ θ h,n+1,i [D p θ] h,n+1,i pk h,n+1,i θ h,n Γ t kg h,n+1,i D x x 3, D x ϕ, ϕ M h, n = 1, 2,, L 1,, ϕ with p h,1 M h chosen to be an approximation to the initial condition p x. Next, the approximation to the solution Φ = D k pδk of is defined as follows. Find D k pδk h,n+1 M h such that [D p θ] h,n+ 1 2 dt D k pδk h,n, ϕ + kg h,n+ 1 2 Dx D k pδk h,n+1, D x ϕ k[d p g] h,n+ 1 2 Dx p h,n x3 D x D k pδk h,n+1, ϕ = g h,n+ 1 2 δkdx p h,n x3, D x ϕ, ϕ M h, n = 1, 2,, L 1, D k pδk h,1 =. Finally the approximation W k h,n to the solution W k of the adjoint problem is defined as W k h,n = V k h,l n, n = 1,, L, 6.3 where V k h,n is the solution of the problem: Find V k h,n+1 M h such that [D p θ] h,n+ 1 2 dt V k h,n, ϕ + kg h,n+ 1 2 Dx V k h,n+1, D x ϕ + k[d p g] h,n+ 1 2 Dx pk h,n x3 D x V k h,n+1, ϕ 6.4 = f n, ϕ, v M h, n = 1, 2,, L 1, V k h,1 =. The following theorem can be demonstrated using a discrete analogue of the arguments given in the proofs of Theorem 3.1 and Lemma 4.1 and applying the discrete Gronwall s lemma. 15

16 Theorem 6.1. Under the hypothesis of Theorem 3.1, the solutions pk h,n and W k h,n of 6.1 and 6.4 are continuous with respect to the parameter k in the following sense: max 1 n L ph,n k 1 p h,n k 2 L 2 Q + D x p h,n k 1 p h,n k 2 L 2 Q t n c k 1 k 2 L2 Q, 6.5 max W k 1 h,n W k 2 h,n L 1 n L 2 Q + D x W k1 h,n W k 2 h,n L 2 Q t n c k 1 k 2 L 2 Q The conjugate gradient algorithm. In order to define our conjugate gradient algorithm we need to define a discrete version of the continuous functional J k and its gradient J k k. For this purpose we will choose a finite set P M Q containing the M-points where the values of the parameter k will be iteratively updated. Since we are employing Galerkin finite element procedures, one simple choice is to use the same finite element basis associated with M h to represent our parameter k, so that P M is chosen to coincide with the set of nodal points x m 1 m M associated with the representation of kx in a basis of M h. Another possible choice is to select as P M the set of all centers of the elements Q j T h or a subset of such centers. Next, we define our discrete functional by J h k = 1 2 L N r p h,n k, x ri p obs,n x ri 2 t. 6.7 n=1 i=1 Identifying k with its values kx m 1 m M, J h k can be regarded as a functional from R M into R. Also, it follows from 6.5 and 6.7 that J h k is a continuous linear functional with respect to the parameter k. Next, since D k p h,n : δp L 2 Q let us define D k p h = D k p h,n, 1 n L so that and let D k p h, denote its adjoint: D k p h : δp [L 2 Q] L D k p h, : [L 2 Q] L δp. Now the following relation is a discretized form of 4.2: for any f [L 2 Q] L, D k p h, f, δk = g h,n+ 1 2 Dx p h,n x3 D x W h,n+1 k t, δk, L 2 Q n L 2 Q δk δp,

17 where W h,n k is the solution of 6.4 with right-hand side f = f n 1 n L. Next, writing pk + δk h,n = pk h,n + D k pδk h,n + φk, k + δk h,n, 6.9 J h k + δk = J h k + D k J h kδk + R h k + δk, k, the argument in Lemma 4.3 can be repeated to see that Next, let Dk J h k, δk L 2 Q = L f n x = N r i=1 n=1 D k pδk h,n, pk h,n p obs,n R Nr t, 6.1 δk δp. 1 pk, xri h,n p obs,n x ri χ Bi x, 1 n L, 6.11 B i and note that the right-hand side of 6.1 is L n=1 = = = D k pδk h,n, pk h,n p obs,n R Nr t 6.12 L N r n=1 i=1 L n=1 L n=1 Q 1 D k pδk h,n χ Bi xdx pk, xri h,n p obs,n x ri t B i B i D k pδk h,n xf n xdx t Dk pδk h,n, f n L 2 Q t = D k p h, f, δk, δk δp. L 2 Q Thus, 6.8, 6.1 and 6.12 show that the gradient J h k k = D k J h k can be computed using the discrete analogue of 4.2: J h k kx = L g h,n+ 1 2 xdx pk h,n+ 1 2 x + x3 D x W k h,n+1 x t, n=1 x Q, 6.13 where W k h,n is the solution of 6.4 with the right-hand side defined by Now Theorem 6.1 and 6.13 imply the validity of the following Lemma. Lemma 6.2. Under the assumptions of Theorem 3.1, the continuous linear functional J h k k : δp R is Lipschitz continuous with respect to the parameter k in the sense: Jk h k 1 Jk h k 2 L2 Q + D x Jk h k 1 Jk h k 2 L2 Q 6.14 c k 1 k 2 L 2 Q. Next the Polak-Ribière conjugate gradient iteration is defined as follows: 17

18 1 Give an initial guess k x, compute p h,n k by solving Compute d x m = Jk h k x m, m = 1,, M using Set j= 4 Compute step length α j α j = L n=1 Dkj pd j h,n L n=1 Dkj pd j h,n,, pk j h,n p obs,n Dkj pd j h,n where means the evaluation of the corresponding function at x ri, i = 1,, N r and D kj pd j h,n is the solution of 6.2 for the choice δk = dj. 5 Update k j x at the points x m by the rule R Nr R Nr k j+1 x m = k j x m + α j d j x m, m = 1,, M and use those values to obtain the updated k j+1 P. 6 Compute p h,n k j+1 by solving Compute error, if convergence is achieved, stop. 8 Compute J h k k j+1 x m, m = 1,, M using Compute β j+1 using the Polak-Ribière formula [25] 1 Compute J h βj+1 P R = k k j+1, Jk h k j+1 Jk h k j R M J h k k j, Jk h k j R M 11 d j+1 x m = J h k k j+1 x m + β P R j+1d j x m, m = 1, M J h k k j x m J h k k j+1 x m, d j x m d j+1 x m, m = 1,, M, j = j + 1, go to Note that each iteration j of the conjugate gradient algorithm described above implies the solution of one forward nonlinear problem for pk j h,n and two linear problems to compute D kj pd j h,n and W k j h,n. Also, according to the results stated above in this section, J h k is continuously differentiable and its gradient Jk h k is Lipschitz continuous. Thus, the convergence of the the Polak-Ribière conjugate gradient iteration is insured if a line search is performed to determine the step length α j. See for example Corollary 4.4 in [26] and Theorem 3.5 in [27] for results on the convergence of this procedure. Since performing a line search implies the evaluation of the functional J h k and consequently the solution of the forward problem 6.1, in our numerical experiments instead we used the value of the step length given in item 4 above, corresponding to a quadratic approximation of J k. This step length performed quite well in our numerical simulations. In the next section we present the numerical experiments showing the implementation of the Polak-Ribière conjugate gradient iteration to estimate the saturated hydraulic conductivity kx using an hypothetical infiltration experiment in an heterogeneous soil. 18

19 7. Numerical experiments. The proposed algorithm is implemented to estimate the saturated hydraulic conductivity kx in a vertical heterogeneous soil profile during an infiltration experiment using synthetically generated observations. The observed data p obs are the pressure head values versus time at different depths obtained as the solution of the forward problem. For the numerical test we consider a 25 cm soil profile Q consisting of four layers with the following values of saturated hydraulic conductivity kx = cm/s cm x < 45 cm cm/s 45 cm x < 125 cm cm/s 125 cm x < 25 cm cm/s 25 cm x 25 cm. 7.1 The other hydraulic parameters of van Genuchten model are assumed to be constant over the whole profile with θ s =.368, θ r =.14, n = 2. and α vg =.335 cm 1. In the infiltration experiment, water is uniformly applied on the soil surface x = 25 cm at a rate of cm/s for a period of 1 days. The initial pressure head values are assumed to be constant and equal to -4 cm, corresponding to a relatively dry water content condition in the soil profile. The numerical test is stopped when the infiltration front reaches the bottom boundary where no-flux boundary condition is prescribed. The time step used in the numerical solution of Richard s equation 6.1, the Gâteaux derivative 6.2 and the adjoint problem 6.4 is t = 864 s with a uniform partition T h of Q into elements Q j of size h = 3.33 cm. The pressure head values are assumed to be recorded at discrete times t n at 16 points x ri spaced 15 cm from each other. The set P M, where the hydraulic conductivity values are updated consists of all centers of the elements Q j T h. Figure 1 shows simulated pressure head observations at the recording points x = 27.5 cm, 87.5 cm, cm and cm. The initial guess for kx in the inverse procedure is taken to be constant and equal to cm/s. Figure 2 shows the initial guess and the profile updates of kx after 1 and 5 iterations, where some oscillations in the estimated hydraulic conductivity profiles can be observed. To eliminate these oscillations and stabilize the parameter estimation procedure, a simple post-processing algorithm of the predicted hydraulic conductivity profile is implemented using weighted averages of nearest neighbors. The detailed numerical nature of weighted averaging is not important. We have observed similar numerical results when small weights are assigned to the nearest neighbors of the point where the hydraulic conductivity is estimated. Figure 3 shows the updated hydraulic conductivity profiles of kx after 1, 3 and 5 iterations. Numerical oscillations almost disappear after 1 iterations and the estimated profile is quite accurate except near the global boundaries where convergence is slow. Note that in this numerical example, the hydraulic conductivity values are not assumed to be known near the top surface, as it was assumed in the derivation of our parameter estimation procedure. The algorithm first quickly reaches the true hydraulic conductivity values in the interior of the domain and then slowly adjusts the true hydraulic conductivity profile near the surface and bottom boundaries. Several choices of the observation times t n, ranging from continuous observations to only 1 observation per day during the 1 days of the simulation time, give almost identical estimates of the hydraulic conductivity profile kx. The behavior of the cost functional J k against the number of iterations is shown in Figure 4. This almost monotone decreasing behavior is attained by combining the 19

20 pressure head cm x = cm x = cm x = 87.5 cm x = 27.5 cm t days Fig Simulated pressure head observations at x = 27.5, 87.5, and cm depths..5 5 iterations k cm/s.5 1 iterations.5 initial guess x cm Fig Initial, estimated dashed and true continuous saturated hydraulic conductivity. Polak-Ribière conjugate gradient iteration with a restart procedure as described in [28]. From this example, we can conclude that the proposed algorithm yields a very good estimate of saturated hydraulic conductivity in a stratified medium and becomes a promising method for in situ estimation of this parameter. 2

21 .5 5 iterations k cm/s.5 3 iterations.5 1 iterations x cm Fig Estimated dashed and true continuous saturated hydraulic conductivity. 1e+9 1e+8 Jk 1e+7 1e iterations Fig Cost functional. Appendix A. The proof of Theorem 3.1. Proof. A weak form of equation 3.5 can be stated as follows: D p θp 2 D t d p k 1, k 2, v + [D p θp 1 D p θp 2 ] D t p 1, v + k 2 gp 2 D x d p k 1, k 2, D x v = ζk 1, k 2 D x p 1 + x 3, D x v, v H 1 Q. A.1 Take v = d p k 1, k 2 in A.1 to obtain the equation 21

22 1 2 D t D p θp 2 d p k 1, k 2, d p k 1, k 2 + k 2 gp 2 D x d p k 1, k 2, D x d p k 1, k 2 = 1 D p 2 2 θp 2D t p 2 d p k 1, k 2, d p k 1, k 2 [D p θp 1 D p θp 2 ] D t p 1, d p k 1, k 2 ζk 1, k 2 D x p 1 + x 3, D x d p k 1, k 2 = T 1 + T 2 + T 3. A.2 Now we bound T 1, T 2 and T 3. First, using that D p 2 θp and D t pk are bounded and that D p θp is Lipschitz continuous, T 1 + T 2 c d p k 1, k 2 2 L 2 Q. A.3 Next, using that gp is bounded and Lipschitz continuous and D x pk is bounded T 3 ɛ D x d p k 1, k 2 2 L 2 Q d + c p k 1, k 2 2 L 2 Q + k 1 k 2 2 L 2 Q, A.4 where ɛ > is an arbitrary small constant. Using the estimates in A.3,A.4 in A.2 and that 2.4 we get the inequality 1 2 D t D p θp 2 d p k 1, k 2, d p k 1, k 2 + c 1 ɛ D x d p k 1, k 2 2 L 2 Q c d p k 1, k 2 2 L 2 Q + k 1 k 2 2 L 2 Q. A.5 Since d p k 1, k 2 t = =, choosing ɛ sufficiently small, we integrate A.5 with respect to time and use 2.5 to get the estimate t d p k 1, k 2 2 L 2 Q t + D x d p k 1, k 2 2 L 2 Q τdτ t c d p k 1, k 2 2 L 2 Q τ + k 1 k 2 2 L 2 Q dτ. A.6 Using Gronwall s lemma in A.6 we obtain 3.1 for n =. Next we proceed to obtain the estimate 3.1 for n = 1, i.e., for Zk 1, k 2 = D t d p k 1, k 2. Recall that for any function fpk, we denote d f k 1, k 2 = fp 1 fp 2. Then taking time derivatives in 3.5,3.6, 3.7 and 3.8, we obtain that Zk 1, k 2 satisfies the equations D 2 p θp 2 D t p 2 Zk 1, k 2 + D p θp 2 D t Zk 1, k 2 A.7 +D t d Dpθk 1, k 2 D t p 1 + d Dpθk 1, k 2 D 2 t p 1 div k 2 D p gp 2 D t p 2 D x d p k 1, k 2 div k 2 gp 2 D x Zk 1, k 2 = div D t ζk 1, k 2 D x p 1 + x 3 + div ζk 1, k 2 D x D t p 1, x Q, t I, 22

23 with the boundary conditions k 2 gp 2 D x Zk 1, k 2 ν k 2 D p gp 2 D t p 2 D x d p k 1, k 2 ν = D t ζk 1, k 2 D x p 1 + x 3 ν + ζk 1, k 2 D x D t p 1 ν, x Γ, t I, k 2 gp 2 D x Zk 1, k 2 ν =, x Γ, t I, and initial condition A.8 Zk 1, k 2, t = =, x Q. A.9 A weak form for A.7-A.8 is as follows: D p 2 θp 2 D t p 2 Zk 1, k 2, v + D p θp 2 D t Zk 1, k 2, v + k 2 gp 2 D x Zk 1, k 2, D x v = D t d Dpθk 1, k 2 D t p 1, v d Dpθk 1, k 2 D 2 t p 1, v A.1 k 2 D p gp 2 D t p 2 D x d p k 1, k 2, D x v D t ζk 1, k 2 D x p 1 + x 3, D x v ζk 1, k 2 D x D t p 1, D x v, v H 1 Q. Taking v = Zk 1, k 2 in A.1, we obtain the equation 1 2 D t D p θp 2 Zk 1, k 2, Zk 1, k 2 + k 2 gp 2 D x Zk 1, k 2, D x Zk 1, k 2 A.11 = 1 D p 2 2 θp 2D t p 2 Zk 1, k 2, Zk 1, k 2 D t d Dpθk 1, k 2 D t p 1, Zk 1, k 2 d Dpθk 1, k 2 D 2 t p 1, Zk 1, k 2 k 2 D p gp 2 D t p 2 D x d p k 1, k 2, D x Zk 1, k 2 ζk 1, k 2 D x D t p 1, D x Zk 1, k 2 D t ζk 1, k 2 D x p 1 + x 3, D x Zk 1, k 2 = T 4 + T 5 + T 6 + T 7 + T 8 + T 9. Next we bound T i, i = 4, 5, 6, 7, 8, 9. First, we note that and D t d Dpθk 1, k 2 = D 2 p θp 1Zk 1, k 2 + d D 2 p θ k 1, k 2 D t p 2, A.12 D t ζk 1, k 2 = k 1 [ Dp gp 1 Zk 1, k 2 + d Dpgk 1, k 2 D t p 2 ] +D p gp 2 D t p 2 [k 1 k 2 ]. A.13 Then, using A.12, and the facts that that D p 2 θp, D t pk, and D 2 t pk are bounded and that D p θp, D p 2 θp are Lipschitz continuous functions, we obtain that T 4 + T 5 + T 6 c Zk 1, k 2 2 L 2 Q + d pk 1, k 2 2 L 2 Q. A.14 Next, using the facts that that gp is bounded and Lipschitz continuous and that D t p, D p gp are bounded, we get T 7 + T 8 ɛ D x Zk 1, k 2 2 L 2 Q Zk + c 1, k 2 2 L 2 Q + d p k 1, k 2 2 L 2 Q + k 1 k 2 2 L 2 Q. A.15 23

24 Finally, using A.13, that D p gp is bounded and Lipschitz continuous and that D x p, D t p are bounded, T 9 ɛ D x Zk 1, k 2 2 L 2 Q d + c p k 1, k 2 2 L 2 Q + Zk 1, k 2 2 L 2 Q A.16 + k 1 k 2 2 L 2 Q. Using the bounds for T i, i = 4,, 9 and 2.4 in A.11, for ɛ appropriately chosen we get the inequality 1 2 D t D p θp 2 Zk 1, k 2, Zk 1, k 2 + D x Zk 1, k 2 2 L 2 Q c Zk 1, k 2 2 L 2 Q + d pk 1, k 2 2 L 2 Q + k 1 k 2 2 L 2 Q. A.17 Integrate A.17 with respect to time, using 2.5 and applying Gronwall s lemma in the resulting equation, we obtain 3.1 for n = 1. Finally, we proceed to derive the estimate 3.1 for n = 2. First, we take the time derivative of A.7-A.9 and set and obtain Uk 1, k 2 = D t Zk 1, k 2 D 3 p θp 2D t p 2 2 Zk 1, k 2 + D 2 p θp 2D 2 t p 2 Zk 1, k 2 A.18 +2D 2 p θp 2D t p 2 Uk 1, k 2 +D p θp 2 D t Uk 1, k 2 + D 2 t d Dpθk 1, k 2 D t p 1 +2D t d Dpθk 1, k 2 D 2 t p 1 + d Dpθ k 1, k 2 D 3 t p 1 div k 2 D p 2 gp 2D t p 2 2 D x d p k 1, k 2 2div k 2 D p gp 2 D t p 2 D x Zk 1, k 2 div k 2 gp 2 D x Uk 1, k 2 = div D 2 t ζk 1, k 2 D x p 1 + x 3 + 2div D t ζk 1, k 2 D x D t p 1 +div ζk 1, k 2 D x D 2 t p 1, x Q, t I, with the boundary condition k 2 gp 2 D x Uk 1, k 2 ν 2k 2 D p gp 2 D t p 2 D x Zk 1, k 2 ν k 2 D p 2 gp 2D t p 2 2 D x d p k 1, k 2 ν = D 2 t ζk 1, k 2 D x p 1 + x 3 ν + 2D t ζk 1, k 2 D x D t p 1 ν +ζk 1, k 2 D x D 2 t p 1 ν, x Γ, t I, k 2 gp 2 D x Uk 1, k 2 ν =, x Γ, t I, A.19 and the initial condition Uk 1, k 2, t = =, x Q. A.2 24

25 A weak form for A.18-A.19 is as follows: D p 3 θp 2 D t p 2 2 Zk 1, k 2, v + D p 2 θp 2 D 2 t p 2 Zk 1, k 2, v A D p 2 θp 2D t p 2 Uk 1, k 2, v + D p θp 2 D t Uk 1, k 2, v + D 2 t d Dpθk 1, k 2 D t p 1, v +2 D t d Dpθk 1, k 2 D 2 t p 1, v + d Dpθ k 1, k 2 D 3 t p 1, v + k 2 D p 2 gp 2D t p 2 2 D x d p k 1, k 2, D x v + 2 k 2 D p gp 2 D t p 2 D x Zk 1, k 2, D x v + k 2 gp 2 D x Uk 1, k 2, D x v = D 2 t ζk 1, k 2 D x p 1 + x 3, D x v 2 D t ζk 1, k 2 D x D t p 1, D x v ζk 1, k 2 D x D 2 t p 1, D x v, x Q, t I. Now take v = Uk 1, k 2 in A.21 to obtain the equation 1 2 D t D p θp 2 Uk 1, k 2, Uk 1, k 2 + k 2 gp 2 D x Uk 1, k 2, D x Uk 1, k 2 A.22 = 3 D p 2 2 θp 2D t p 2 Uk 1, k 2, Uk 1, k 2 D p 3 θp 2D t p 2 2 Zk 1, k 2, Uk 1, k 2 D p 2 θp 2D 2 t p 2 Zk 1, k 2, Uk 1, k 2 D 2 t d Dpθk 1, k 2 D t p 1, Uk 1, k 2 2 D t d Dpθk 1, k 2 D 2 t p 1, Uk 1, k 2 d Dpθ k 1, k 2 D 3 t p 1, Uk 1, k 2 k 2 D p 2 gp 2D t p 2 2 D x d p k 1, k 2, D x Uk 1, k 2 = 2 k 2 D p gp 2 D t p 2 D x Zk 1, k 2, D x Uk 1, k 2 D 2 t ζk 1, k 2 D x p 1 + x 3, D x Uk 1, k 2 2 D t ζk 1, k 2 D x D t p 1, D x Uk 1, k 2 ζk 1, k 2 D x D 2 t p 1, D x Uk 1, k 2 i=2 i=1 T i. Next, we bound each T i on the right-hand side of A.22. First, using the facts that D p 2 θp, D p 3 θp, D t pk and D 2 t pk are bounded, we obtain T 1 + T 11 + T 12 c Zk 1, k 2 2 L 2 Q + Uk 1, k 2 2 L 2 Q. A.24 Next, from A.12 we get D 2 t p 2 t d Dpθk 1, k 2 = D p 3 θp 1D t p 1 Zk 1, k 2 + d 2 D p θ k 1, k 2 D 2 [ ] + D p 3 θp 1 Zk 1, k 2 + d 3 D p θ k 1, k 2 D t p 2 D t p 2 +D 2 p θp 1 Hk 1, k 2. Thus, using that D p θp, D p 2 θp, D p 3 θp are bounded and Lipschitz continu- 25 A.23

26 ous functions and D t pk, D 2 t pk, D 3 t pk are bounded, we get T 13 + T 14 + T 15 c d p k 1, k 2 2 L 2 Q + Zk 1, k 2 2 L 2 Q + Uk 1, k 2 2 L 2 Q. A.25 Moreover, since D p gp, D p 2 gp, D t p are bounded, we have T 16 + T 17 ɛ D x Uk 1, k 2 2 L 2 Q A.26 +c d p k 1, k 2 2 L 2 Q + Zk 1, k 2 2 L 2 Q. Next, using A.13, we obtain that [ D 2 t ζk 1, k 2 = k 1 D 2 [ + D 2 ] p gp 1 D t p 1 Zk 1, k 2 + D p gp 1 Uk 1, k 2 ] p gp 1Zk 1, k 2 + d 2 D p g k 1, k 2 D t p 2 +d Dpgk 1, k 2 D 2 t p 2 + A.27 [ ] D p 2 gp 2 D t p D p gp 2 D 2 t p 2 [k 1 k 2 ]. Thus, using A.13, A.27 and the facts that gp, D p gp, D p 2 gp are bounded and Lipschitz continuous functions and D t pk, D 2 t pk, D x pk are bounded, we get T 18 + T 19 + T 2 ɛ D x Uk 1, k 2 2 L 2 Q d + c p k 1, k 2 2 L 2 Q A.28 + Zk 1, k 2 2 L 2 Q + Uk 1, k 2 2 L 2 Q + k 1 k 2 2 L 2 Q. Collecting the bounds for T i, i = 1, 2 and choosing ɛ appropriately, from A.22 we obtain 1 2 D t D p θp 2 Uk 1, k 2, Uk 1, k 2 + D x Uk 1, k 2 2 L 2 Q c Uk 1, k 2 2 L 2 Q + Zk 1, k 2 2 L 2 Q + D xzk 1, k 2 2 L 2 Q + d p k 1, k 2 2 L 2 Q + D xd p k 1, k 2 2 L 2 Q + k 1 k 2 2 L 2 Q. A.29 Integrating A.29 with respect to time, using 2.5 and 2.4, and applying Gronwall s lemma in the resulting equation and using the estimates 3.1 already derived for n =, 1 we obtain D 2 t d p k 1, k 2 L I,L 2 Q + D x D 2 t d p k 1, k 2 L 2 I,L 2 Q c d p k 1, k 2 L 2 I,L 2 Q + D x d p k 1, k 2 L 2 I,L 2 Q + D t d p k 1, k 2 L2 I,L 2 Q + D x D t d p k 1, k 2 L2 I,L 2 Q + k 1 k 2 L 2 I,L 2 Q c k 1 k 2 L2 I,L 2 Q. A.3 This completes the proof. REFERENCES 26

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