For the reconstruction and interpolation of precipitation fields, we present the application of a stochastic approach called Random Mixing. Generated fields are based on a data set consisting of rain gauge observations and path‐averaged rain rates estimated using Commercial Microwave Link (CML) derived information. Precipitation fields are received as linear combination of unconditional spatial random fields, where the spatial dependence structure is described by copulas. The weights of the linear combination are optimized such that the observations and the spatial structure of the precipitation observations are reproduced. The innovation of the approach is that this strategy enables the simulation of ensembles of precipitation fields of any size. Each ensemble member is in concordance with the observed path‐averaged CML derived rain rates and additionally reflects the observed rainfall variability along the CML paths. The ensemble spread allows additionally an estimation of the uncertainty of the reconstructed precipitation fields. The method is demonstrated both for a synthetic data set and a real‐world data set in South Germany. While the synthetic example allows an evaluation against a known reference, the second example demonstrates the applicability for real‐world observations. Generated precipitation fields of both examples reproduce the spatial precipitation pattern in good quality. A performance evaluation of Random Mixing compared to Ordinary Kriging demonstrates an improvement of the reconstruction of the observed spatial variability. Random Mixing is concluded to be a beneficial new approach for the provision of precipitation fields and ensembles of them, in particular when different measurement types are combined.
This paper presents a novel methodology for inverse modeling of groundwater flow and transport problems in a Monte Carlo framework, i.e., multiple solutions to the inverse problem are generated. The methodology is based on the concept of random mixing of spatial random fields. The conditional target hydraulic transmissivity field is obtained as a linear combination of unconditional spatial random fields. The corresponding weights of the linear combination are selected such that the spatial variability of the hydraulic transmissivities as well as the actual observed transmissivity values are reproduced. The constraints related to the hydraulic head and contaminant concentration observations are nonlinear. In order to fulfill these constraints, a specific property of the presented approach is used. A connected domain of fields fulfilling all linear constraints is identified. This domain includes an infinite number of realizations, and in this domain, the head and concentration deviations are minimized using standard continuous optimization techniques. The methodology uses spatial copulas to describe the spatial dependence structure. A combination with multiple point statistics allows inversion under specific structural constraints.
Abstract. Knowledge of spatio-temporal rainfall patterns is required as input for distributed hydrologic models used for tasks such as flood runoff estimation and modelling. Normally, these patterns are generated from point observations on the ground using spatial interpolation methods. However, such methods fail in reproducing the true spatio-temporal rainfall pattern, especially in data-scarce regions with poorly gauged catchments, or for highly dynamic, small-scale rainstorms which are not well recorded by existing monitoring networks. Consequently, uncertainties arise in distributed rainfall–runoff modelling if poorly identified spatio-temporal rainfall patterns are used, since the amount of rainfall received by a catchment as well as the dynamics of the runoff generation of flood waves is underestimated. To address this problem we propose an inverse hydrologic modelling approach for stochastic reconstruction of spatio-temporal rainfall patterns. The methodology combines the stochastic random field simulator Random Mixing and a distributed rainfall–runoff model in a Monte Carlo framework. The simulated spatio-temporal rainfall patterns are conditioned on point rainfall data from ground-based monitoring networks and the observed hydrograph at the catchment outlet and aim to explain measured data at best. Since we infer a three-dimensional input variable from an integral catchment response, several candidates for spatio-temporal rainfall patterns are feasible and allow for an analysis of their uncertainty. The methodology is tested on a synthetic rainfall–runoff event on sub-daily time steps and spatial resolution of 1 km2 for a catchment partly covered by rainfall. A set of plausible spatio-temporal rainfall patterns can be obtained by applying this inverse approach. Furthermore, results of a real-world study for a flash flood event in a mountainous arid region are presented. They underline that knowledge about the spatio-temporal rainfall pattern is crucial for flash flood modelling even in small catchments and arid and semiarid environments.
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