Modellering van afhanklikheid in die lineêre model : 'n meteorologiese toepassing

Abstract

As deel van die weermodifikasie-eksperiment in Bethlehem, Suid-Afiika, is 'n reenmeternetwerk geinstalleer, en word die neerslagwaardes R; wat by 43 reenmeterstasies waargeneem is, vergelyk met die waargenome radar reflektiwiteit Z;. Alhoewel radar ruimtelike en tydskontinue metings van reflektiwiteit bied wat onmiddellik by een sentrale punt beskikbaar is, is die akkuraatheid van radar om reenval te meet onseker as gevolg van verskeie potensiele foute in die omskakeling van reflektiwiteit na reenval. Dit word aanvaar dat reenmeters akkurate puntwaarnemings van reenval gee en daar bestaan eenstemmigheid dat die kombinasie van die twee metodes beter is as enigeen van die metodes afsonderlik. In hierdie studie ondersoek ek die toepassing van die veralgemeende lineere model as 'n beramingstegniek. Vorige studies gebruik die log-log transformasie, d. w.s. logZ = logA + b(logR) van die Z = ARb verwantskap om die koeffisiente A en b met behulp van kleinste-kwadrate-regressie te bepaal. Die implisiete aanname hiermee is dat die foute ongekorreleerd is. Met die inverse verwantskap R = czd d.w.s. logR = logC + d(logZ) neem ek aan dat die waarnemings nie onafhanklik is nie sodat die regressiekoeffisiente bereken word met behulp van die metode van die veralgemeende lineere model. Om die ruimtelike afhanklikheid van die reenmeterwaarnemings te modelleer, word eksperimentele variogramme uit die data bereken en gepas met teoretiese variogramme wat gebruik word om die variansie-kovariansiematriks te vu!. "Gemiddeld" vaar hierdie metode beter as gewone regressie vir analises wat reenmeters wat verder as 45km vanaf die radarstel is, insluit. Residu-stipping wys dat die afstand van die meter vanaf die radarstel as 'n afsonderlike onafhanklike veranderlike in die regressievergelyking ingesluit behoort te word, d.w.s. die beraming verbeter met logR = 3-0 + a,(logZ) + a2(afstand). Hierdie meervoudige regressiemodel stem ooreen met die teoretiese model van Smith en Krajewski omdat e -- afstand as 'n praktiese manifestasie van die foutproses [e.,, (ij)] beskou kan word. Omdat E(ez) = eE<ZJ e'"a' as Z 'n lognormaalverdeling het, kan die sydigheid wat ontstaan wanneer antilogaritmes geneem word, reggestel word deur die beraamde reenval met e112 "' te vermenigvuldig. Die studie !ewer 'n bydrae met die afleiding van 'n beramingstegniek wat die beraming van neerslag uit radar betekenisvol verbeter.

In a study of a rain-gauge network that was installed for a weather modification experiment in Bethlehem, South Africa, precipitation values R; observed at 43 gauging stations are compared to the observed radar reflectivity Z;. Although radar provides spatial and temporal measurements of reflectivity that are immediately available at one location, the accuracy of radar estimation of rainfall is uncertain due to various potential errors in the conversion from reflectivity to rainfall. Rain-gauges are assumed to give accurate point measurements of rainfall and there is general agreement that the combination of systems is better than either system alone. In this study I explore the application of the general linear model as an estimation technique. Previous studies have used the log-log transform, i.e. logZ = logA + b(logR) of the Z = ARb relation, and applied least-squares regression analysis to determine the coefficients A and b. This implicitly assumes that the disturbances are uncorrelated. Working with the inverse relation R = czd i.e. logR = logC + d(logZ) and assuming that the observations are not independent we compute the regression coefficients using generalised least squares. To model the spatial dependence of the rain-gauge observations we compute experimental variograms from the data and fit them with theoretical variograms which are then used to fill the variance-covariance matrix. "On average" this method performs better than ordinary regression for the analyses that included rain-gauges further than 45km from the radar set. Residual plotting revealed that distance of the rain-gauge from the radar set should be included as a separate independent variable in the regression equation, i.e. logR = ao + a1(logZ) + a1(distance) improved the estimation of rainfall as it performs better than ordinary regression. This multiple regression model agrees with the theoretical model of Smith and Krajewski in the sense that e "'distance is a practical manifestation of the error process [ e,, (ij)]. Showing that E( ez) = el!.(!.) e 112 "' if Z has a lognormal distribution, the bias when taking antilogs can be removed by multiplying estimated rainfall by e1 ' 2a'. The contribution of this study is the derivation of an estimation technique which significantly improves the estimation of rainfall from radar