Abstract.We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a χ 2 distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value decomposition of the system matrix is found to be very efficient for parameter estimation, requiring on average about 10 Newton steps. Additionally, the theory and algorithm apply for Generalized Tikhonov regularization using the generalized singular value decomposition. The performance of the Newton algorithm is contrasted with standard techniques, including the L-curve, generalized cross validation and unbiased predictive risk estimation. This χ 2 -curve Newton method of parameter estimation is seen to be robust and cost effective in comparison to other methods, when white or colored noise information on the measured data is incorporated. Tikhonov regularization, least squares, regularization parameterAMS classification scheme numbers: 15A09, 15A29, 65F22, 62F15, 62G08 Submitted to: Inverse Problems, 24 October 2008This is an author-produced, peer-reviewed version of this article. The final, definitive version of this document can be found online at Inverse Problems, published by Institute of Physics. Copyright restrictions may apply.

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Linear least squares problems with box constraints are commonly solved to find model parameters within bounds based on physical considerations. Common algorithms include Bounded Variable Least Squares (BVLS) and the Matlab function lsqlin. Here, the goal is to find solutions to ill-posed inverse problems that lie within box constraints. To do this, we formulate the box constraints as quadratic constraints, and solve the corresponding unconstrained regularized least squares problem. Using box constraints as quadratic constraints is an efficient approach because the optimization problem has a closed form solution. The effectiveness of the proposed algorithm is investigated through solving three benchmark problems and one from a hydrological application. Results are compared with solutions found by lsqlin, and the quadratically constrained formulation is solved using the L-curve, maximum a posteriori estimation (MAP), and the χ 2 regularization method. The χ 2 regularization method with quadratic constraints is the most effective method for solving least squares problems with box constraints.

A field-scale infiltration experiment was conducted in coarse conglomeratic soil with high gravel fraction. Unsaturated flow properties were estimated from modeling of infiltration using the van Genuchten–Mualem model and a Metropolis–Hasting optimization scheme. Results provide optimal unsaturated flow parameters for a soil type that is underrepresented for vadose zone flow. Conglomeratic alluvial sediments (sand–gravel–cobbles) are common in fluvial, periglacial, and tectonically active regions but have received little attention with respect to unsaturated flow, specifically moisture–tension–conductivity relationships, due to difficulty in making measurements in the field or laboratory and lack of agricultural value. We used a field-scale infiltration experiment, a one-dimensional layered forward model, and parameter estimation modeling to examine in situ flow behavior between residual and partial saturation in a four-layer system under steady infiltration (0.84 cm h−1) for 19 h. Prior information from ground-penetrating radar, grain-size distributions from core samples, and long-term tension (ψ) and moisture (θ) monitoring were used to define geologic structure, simulate test behavior, and provide initial parameter estimates. Vertically distributed measurements of ψ(t) and θ(t) from the experiment were matched using four parameters (θs, α, n, and Ks) of the van Genuchten–Mualem (VGM) relationships for each material layer and a Metropolis–Hastings (MH) search with multiple, independent-chain runs to 106 samples each. Scale reduction factors indicated convergence of independent chains for 11 of 16 parameters. Final distributions of individual parameters varied from normal to nearly uniform distributions, and some parameter pairs showed high cross-correlation (R2 \u3e 0.9). Results showed that (i) VGM relationships can be applied to these coarse, conglomeratic soils to characterize unsaturated flow behavior across the natural range of partial saturation, (ii) even under high sustained infiltration rates, these coarse conglomeratic soils remain well drained, despite relatively low porosity and significant cobble fraction, and (iii) high uncertainty and nonconvergence of MH chains does not lead to significant misfit of the observed data. These findings imply that a significant cobble fraction does not markedly reduce infiltration at low saturation levels that develop under natural recharge conditions