34;True" field Kriged gauge field
3.5 Conditional merging
In Sinclair and Pegram (2004a)a comparison waspresented betweentheBayesian merging technique of Todini (200 I) and the conditional merging technique de- scribed in this section. LaterMazzetti (2004) used a small-scale numerical exper- iment toshow that the Bayesian merging technique outperformed several others (Brandes, 1975;Krajewski, 1987;Koistinen and Puhakka, 1981).The experiment hasbeendiscussed insections3.4.1,3.4.2and3.4.3. Sinclair and Pegram (2004a) showed that conditional mergingperform ed competitivelywith Bayesian merging ina comparisonexperiment(the results ofthat study are repeated here insection 3.6. 1). Further refinement of the conditional mergingtechnique and case studies using observeddata from South Africa and Spain are presented in sections 3.6.2 and 3.6.3 .
Weather radar produces an observation of the unknown rainfall field that is subject to several well-known source of error (e.g Wilson and Brandes, 1979;
Chumchean etal., 2003) but whichretain sthe general covariance structure of the true precipitation field. The spatial inform ation from the radar can be used to condition the spatially limited information obtained byinterp olatingbetween rain gauges and produce anestimate oftherainfall field that contains the appropriate spatialstructure while being constrained to match the rain gauge data (where it is available). The conditional merging technique of Ehret (2002) makes use of ordinary Kriging (e.gJournel and Huijbregts, 1978;Cressie, 1991) to extrac t the unbiased minimum variance rainfall estimate, given the information content of the ob erved data. Figure 3.32 gives an overview of the techn ique (for the one dimensional case)which is adapted fromEhret's (2002) work.
The spatialstructure of the merged rainfall field is obtained from the radar while therainfall valuesare "stitched down" to the gaugeobserva tionsofthe true rainfall field . The approac h taken here is similar to the technique of conditional simulation by Krigingdiscussed in Chiles and Delfiner (1999, pp452). However, the key difference in this case is that the radar rainfall estimate is not simply a simulation, unrelated (except by it's statistical properties) to the rainfall field to be estimated, but is in fact an independent observa tion of the true unknown
(a)
(c)
(b)
(f)
(c) (d)
(g)
Fig.3.32: One dimensional overview of the Conditional Merging algorithm. (a) The rainfall field is observed at discrete points by rain gauges. (b) The rainfall field is also observed by radaron a regular, volume-integrated grid. (c) Kriging of the rain gauge observationsis used to obtain the bestlinear unbiased e timate of rainfall on the radar grid. (d)The radarpixel values at the rain gauge location s are interpolated onto the radar grid using Kriging. (e) At each grid point, the deviation cR(S )between the observed and interpolated radar value is computed.
(j)The field of deviation s obtained from (e) is applied to the interpolated rainfall field obtained from Kriging the rain gauge ob ervations. (g) A rainfall field that followsthe mean field of the rain gauge interpolation ,while preserving the mean field deviationsand the spatialstructure of the radar field isobtained.
rainfall field. This important link meansthatthe radar data providesan estimate of the Kriging error(a conceptwhich isdistinctfrom the Krigingvariance) and in particular thespatialstructure of thiserror.
Therainfall field isobserved at discrete pointsby rain gauge and is also ob- served by radar on a regul ar, volume-integratedgrid(Figure 3.32a,b).Kriging of the rain gauge observations is used to obtain the bestlinear unbiased estim ate of rainfall on the radar grid (Figure3.32c). This can bedescrib ed by
(3. 16)
where Z(s) is the true (unknown) rain fall field at location s and GK(s) is the Kriged (unbiased field ) estimateofZ(s)from the raingaugedata.Thetermcc (s) in equation 3.16 cannot beestimated(except at the gaugeswhere it isexactly zero - if we assume for simplicity that the gauges have no error in estimating Z(s)
93
since Z(s) is unknown. The radar pixel values at the rain gauge locat ions are interpo latedonto the radar grid using Kriging (Figure3.32d), this operation may be describ ed as
(3.17) where R(s)isthe measured radar rainfall estimate and RJ« (s)is the Kriged (un- biased field) estimateof R(s)using the radarvalues at rain gauge locations. At eachgrid point,cn(s) the deviation between the observed and interp olated radar value(Figure3.32e) is computed using equation3.17. On the basis that R(s)is a measurement ofZ(s),the fieldof deviationscn(s)is applied toGK(s)the inter- polated rain fall field obta ined from Kriging the rain gauge observatio ns (Figure 3.32t)to produce M(s),the merged estimate ofZ(s)
M(s)
=
GK(S)+
cn(s) (3. 18)Equation3. 18 resultsinarain fall field that follows themeanfield of therain gauge interp olation, while preser vingthe mean field deviations and the spatialstruc ture of theradar field (Figure3.32g).
Theexpected valueofthe error between themerged estimateand thetrue field is zero if the fields are Gaussian, since the Kriged estimates are unbiased in this case, i.e.
E [Z(.)- M(s)]= E [cc(s) - cn(s)] (3.19) and E [cc (s)] = 0, E[cn( )] = O. The variance of the merged estimationerrors can be examined by consideri ng
Va1"[Z(s) - 1\1(s)]= var [cc(s) - cn(s)]
= (7; C(8)
+
(7;n (8) - 2co v [cc;{ ),c/l(S)]= {3- 2(7Ec (S)(7En (8)P
(3.20) (3.2 1) (3.22)
where{3= (72EC( )8
+
(72en( )8Thevarianceof theerror estimate given by equation3.20 can be decomposed as show n inequations 3.21and3.22. The variance ofthe error is (trivially) zero
cR(S).Ifca(s)andcR(S)are strongly(pos itively) correlated asone would expect since both gauges and radar are measurements of Z(s),then the variance of the error will be significantly less than (3 as suggested in equation 3.22. IfO"cc (s) =
O"cn(s) = 0" and p
>
0.5, then var [Z(s) - M(s)] = 20"2[1- p] showing that thevariance of the error between the merged and real rainfall field will be less than that ofeither field (Kriged gauge or radar estimate) alone. If p = 0 (the errors are uncorrelated ),then the error isthe sum of theerrorsof the individual fieldsas expected.