P. T. M. Vermeulen

Model-Reduced Variational Data Assimilation

Abstract This paper describes a new approach to variational data assimilation that with a comparable computational efficiency does not require implementation of the adjoint of the tangent linear approximation of the original model. In classical variational data assimilation, the adjoint implementation is used to efficiently compute the gradient of the criterion to be minimized. Our approach is based on model reduction. Using an ensemble of forward model simulations, the leading EOFs are determined to define a subspace. The reduced model is created by projecting the original model onto this subspace. Once this reduced model is available, its adjoint can be implemented very easily and can be used to approximate the gradient of the criterion. The minimization process can now be solved completely in reduced space with negligible computational costs. If necessary, the procedure can be repeated a few times by generating new ensembles closer to the most recent estimate of the parameters. The reduced-model-based ...

Inverse modeling of groundwater flow using model reduction

Numerical groundwater flow models often have a very high number of model cells (greater than a million). Such models are computationally very demanding, which is disadvantageous for inverse modeling. This paper describes a low?dimensional formulation for groundwater flow that reduces the computational burden necessary for inverse modeling. The formulation is a projection of the original groundwater flow equation on a set of orthogonal patterns (i.e., a Galerkin projection). The patterns (empirical orthogonal functions) are computed by a decomposition of the covariance matrix over an ensemble of model solutions. Those solutions represent the behavior of the model as a result of model impulses and the influence of a chosen set of parameter values. For an interchangeable set of parameter values the patterns yield a low?dimensional model, as the number of patterns is often small. An advantage of this model is that the adjoint is easily available and most accurate for inverse modeling. For several synthetical cases the low?dimensional model was able to find the global minimum efficiently, and the result was comparable to that of the original model. For several cases our model even converged where the original model failed. Our results demonstrate that the proposed procedure results in a 60% time reduction to solve the groundwater flow inverse problem. Greater efficiencies can be expected in practice for large?scale models with a large number of grid cells that are used to compute transient simulations.

Model inversion of transient nonlinear groundwater flow models using model reduction

Despite increasing computational resources many high?dimensional applications are impractical for model inversions. In this paper, two methods are presented that are promising for high?dimensional model inversion. The methods draw on proper orthogonal decomposition (POD) and yield reduced models that describe a truncated behavior of the original model. We utilize POD differently for the two methods, which differ in efficiency and implementation. The first method (RIGM) applies a POD to an existing partial differential equation; the second method (RISM) applies it to an autoregressive formulation of a discrete model. Both methods were applied to several synthetic cases and a real?world case with synthetic measurements and by comparing them to classic inverse methodologies (i.e., the method of finite differences and the adjoint method). The two POD methods appeared to be computationally robust and more efficient for a wide range of prior estimates. Moreover, the implementation of an adjoint for the RISM method is easy.

Limitations to upscaling of groundwater flow models dominated by surface water interaction

Different upscaling methods for groundwater flow models are investigated. A suite of different upscaling methods is applied to several synthetic cases with structured and unstructured porous media. Although each of the methods applies best to one of the synthetic cases, no performance differences are formed if the methods were applied to a real three-dimensional case. Furthermore, we focus on boundary conditions, such as Dirichlet, Neumann, and Cauchy conditions, that characterize the interaction of groundwater with, for example, surface water and recharge. It follows that the inaccuracy of the flux exchange between boundary conditions on a fine scale and the hydraulic head on a coarse scale causes additional errors that are far more significant than the errors due to an incorrect upscaling of the heterogeneity itself. Whenever those errors were reduced, the upscaled model was improved by 70%. It thus follows that in practice, whenever we focus on predicting groundwater heads, it is more important to correctly upscale the boundary conditions than hydraulic transmissivity.

Reduction of Large-Scale Groundwater Flow Models Via the Galerkin Projection

Abstract In this paper we describe a reduced model structure that describes the hydraulic head h for ground water flow models as a linear combination of a set of spatial patterns P with time-varying coefficients r. We discuss a data-driven technique to extract patterns P (EOFs) that span a subspace of model results that captures most of the relevant information of the original model. We make use of the patterns to obtain a reduced dynamic model for the time-varying coeffecients via a Galerkin Projection. This technique substitutes h within the PDE for groundwater now by the reduced model structure PTr. We acquire a dynamic reduced model for dr/dt by multiplying the outcome with PT. The vector dimension of r is often small compared to the original dimension of h, and a model which operates within a lower dimension requires less computational time. The method has heen evaluated for a realistic case, whereby we achieved a maximal reduction in computational time of ≈ 80. The reduced model has a promising prospect as its efficiency increases whenever the number of grid cells increases and the parameterization of the original model grows in complexity.