Statistics for Spatial Data.

Highway Capacity Manual改訂の動向--テイラ-教授の講演より

Low flow hydrology: a review

Geostatistical approaches for incorporating elevation into the spatial interpolation of rainfall

This paper reviews the role of uncertainty in the identification of mathematical models of water quality and in the application of these models to problems of prediction. More specifically, four problem areas are examined in detail: uncertainty about model structure, uncertainty in the estimated model parameter values, the propagation of prediction errors, and the design of experiments in order to reduce the critical uncertainties associated with a model. The review is rather lengthy, and it has therefore been prepared in effect as two papers. There is a shorter, largely nontechnical version, which gives a quick impression of the current and future issues in the analysis of uncertainty in water quality modeling. Enclosed by this shorter discussion is the main body of the review dealing in turn with (1) identifiability and experimental design, (2) the generation of preliminary model hypotheses under conditions of sparse, grossly uncertain field data, (3) the selection and evaluation of model structure, (4) parameter estimation (model calibration), (5) checks and balances on the identified model, i.e., model “verification” and model discrimination, and (6) prediction error propagation. Much time is spent in discussing the algorithms of system identification, in particular, the methods of recursive estimation, and in relating these algorithms and the subject of identification to the problems of prediction uncertainty and first-order error analysis. There are two obvious omissions from the review. It is not concerned primarily with either the development and solution of stochastic differential equations or the issue of decision making under uncertainty, although clearly some reference must be made to these topics. In brief, the review concludes (not surprisingly) that much work has been done on the analysis of uncertainty in the development of mathematical models of water quality, and much remains to be done. A lack of model identifiability has been an outstanding difficulty in the interpretation and explanation of past observed system behavior, and there is ample evidence to show that the “larger,” more “comprehensive” models are easily capable of generating highly uncertain predictions of future behavior. For the future of the subject, it is speculated that there is the possibility of progress in the development of novel algorithms for model structure identification, a need for new questions to be posed in the problem of prediction, and a distinct challenge to the conventional views of this review in the new forms of knowledge representation and manipulation now emerging from the field of artificial intelligence.

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Water quality modeling: A review of the analysis of uncertainty

Cokriging is used to merge rain gage measurements and radar rainfall data. The cokriging estimators included are ordinary, universal, and disjunctive. To evaluate the estimators, two simulation experiments are performed. The first experiment assumes that high-quality radar rainfall fields are ground truth rainfall fields. From each ground truth rainfall field, multiple combinations of rain gage measurement field and radar rainfall field are artificially generated with varying gage network density and error characteristics of radar rainfall. The second experiment uses a stochastic space-time rainfall model to generate assumed ground truth rainfall fields of various characteristics. Due to the sparsity of rain gage measurements, the second-order statistics required for cokriging can only be estimated with large uncertainty. The adverse effects of this uncertainty, and the point sampling error of rain gage measurements are explicitly assessed by cokriging the ground truth rainfall data and the radar rainfall data with near perfectly known second-order statistics.

Stochastic interpolation of rainfall data from rain gages and radar using cokriging: 1. Design of experiments

Various estimation procedures using ordinary, universal, and disjunctive cokriging are evaluated in merging rain gage measurements and radar rainfall data. The estimation procedures and the simulation experiments were described in part 1 (Seo et al., this issue) of this two-part work. In this part, the experiments are described in detail. An objective comparison scheme, devised to compare a large number of estimators, is also described. The results are presented focusing upon (1) the potential of radar-gage estimation using cokriging over radar-only estimation and gage-only estimation under widely varying conditions of gage network density and the error characteristics of radar rainfall, (2) the potential for using universal or disjunctive cokriging over ordinary cokriging, (3) how the uncertain second-order statistics affect the estimators, due to lack of rain gage measurements, and (4) how the statistical characteristics of ground truth rainfall affect the estimators.

Stochastic interpolation of rainfall data from rain gages and radar using Cokriging: 2. Results

Three-dimensional non-cohesive earthen dam breach model. Part 1: Theory and methodology

The feasibility of linear and nonlinear geostatistical estimation techniques for optimal merging of rainfall data from raingage and radar observations is investigated in this study by use of controlled numerical experiments. Synthetic radar and raingage data are generated with their hypothetical error structures that explicitly account for sampling characteristics of the two sensors. Numerically simulated rainfall fields considered to be ground-truth fields on 4×4 km grids are used in the generation of radar and raingage observations. Ground-truth rainfall fields consist of generated rainfall fields with various climatic characteristics that preserve the space-time covariance function of rainfall events in extratropical cyclonic storms. Optimal mean areal precipitation estimates are obtained based on the minimum variance, unbiased property of kriging techniques under the second order homogeneity assumption of rainfall fields. The evaluation of estimated rainfall fields is done based on the refinement of spatial predictability over what would be provided from each sensor individually. Attention is mainly given to removal of measurement error and bias that are synthetically introduced to radar measurements. The influence of raingage network density on estimated rainfall fields is also examined.

Spatial rainfall estimation by linear and non-linear co-kriging of radar-rainfall and raingage data

A nonparametric resampling technique for generating daily weather variables at a site is presented. The method samples the original data with replacement while smoothing the empirical conditional distribution function. The technique can be thought of as a smoothed conditional Bootstrap and is equivalent to simulation from a kernel density estimate of the multivariate conditional probability density function. This improves on the classical Bootstrap technique by generating values that have not occurred exactly in the original sample and by alleviating the reproduction of fine spurious details in the data. Precipitation is generated from the nonparametric wet/dry spell model as described in Lall et al. [1995]. A vector of other variables (solar radiation, maximum temperature, minimum temperature, average dew point temperature, and average wind speed) is then simulated by conditioning on the vector of these variables on the preceding day and the precipitation amount on the day of interest. An application of the resampling scheme with 30 years of daily weather data at Salt Lake City, Utah, USA, is provided.

David S. Bowles

Papers

Stochastic interpolation of rainfall data from rain gages and radar using cokriging: 1. Design of experiments

Stochastic interpolation of rainfall data from rain gages and radar using Cokriging: 2. Results

Three-dimensional non-cohesive earthen dam breach model. Part 1: Theory and methodology

Spatial rainfall estimation by linear and non-linear co-kriging of radar-rainfall and raingage data

Multivariate Nonparametric Resampling Scheme for Generation of Daily Weather Variables