Improved Mean Areal Rainfall Estimation by Integration of Radar Data
by
Franz Konecny
Institute of Mathematics and Applied Statistics, University of Agricultural Sciences, Vienna, AUSTRIA

The problem under consideration is rainfall estimation as input to a real time streamflow model. Most important is to estimate mean areal rainfall for a catchment domain . Rain gage data are typically considered to provide accurate point measurements, but usually the networks are too sparse to offer information on the spatial variability of rain events. Better descriptions on the spatial structure of rainfall can be obtained by indirect information from radar estimates of rainfall. These are based on reflectivity measurements Z which are transformed into rainfall R through a calibration process. Due to the uncertainty associated with the Z-R relationship and miscalibration of electronic components the radar estimates are burdened with quite significant errors. Thus it hoped that combining the data from rain gages and radar sensors will result in fairly accurate areal estimates.

Let D denote a domain covered the wheather radar and including the raingages. The radar provides spatially averaged rainfall on a grid of squares ("bins") of 2×2 km. The rain gage data are referred to as the ``hard data'' , whereas the radar estimates are the ``soft data'', specified by intervals (inequalities). The problem is to obtain an estimate [^Y](B) of Y(B) , defined as

Y(B) =  1 |B| ó õ B  Y(x)  dx

where B is some subdomain of B, |B| its area and Y(x) the rainfall at location x. This problem can be solved by multivariate methods of geostatistics [Wackernagel, 1995]).

Although it is possible to directly cokrige the hard data and the spatially averaged soft data, (see Krajewski [1987]), it is computationally more efficient to adopt the following two-step approach of Seo et al. [1990]: (1) block krige the point gage measurements, thus rendering the the spatial scale of the gage measurements compatible with that of the soft data; (2) cokrige the spatially averaged gage rainfall and the radar rainfall as if gage rainfall and radar rainfall were sampled at the center of the bin.

For inequalities as data Langlais [1990] proposed to replace them by exact values. The procedure is to simulate exact data satisfying the given inequalities, proceed to cokriging from the actual and generated data, and finally average the results over several simulations of the inequality data. We will keep Langlais' approach but use an implementation based on the Gibbs sampler for the empirical computation of conditional expectations, given the data.

An alternative methology is to estimate the conditional distribution of the rainfall rate Z(x₀), given the inequality data, by indicator kriging [Journel, 1983]. Thus a discrete approximation of the conditional expectation E[Z(x₀)|data] can be obtained. These estimates substitutes the inequality data in the cokriging predictor of Z(B). This method is compared to the Gibbs sampler approach in application to one simulated and one real data set to identify similarities and differences between both methods.

References

Journel, A.G.[1983]: Nonparametric estimation of spatial distributions. Math. Geology 15(3), 445-468.

Krajewski, W.F. [1987]: Cokriging Radar-Rainfall and Rain Gage Data. Journ. Geophys. Res. 92, 9571-9580.

Langlais, V. [1990]: Estimation sous contraites d'inégalités. Doctoral thesis, E.N.S. des Mines de Paris.

Seo, D.-J., W.F. Krajewski and D.S. Bowles [1990] Stochastic Interpolation of Rainfall Data From Rain Gages and Radar Using Cokriging I. Water Resour. Res. 26(3), 469-477.

Wackernagel, H. [1995]: Multivariate Geostatistics. Springer, New York.

Date received: October 13, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafr-48.