Automatic Prediction of High-Resolution Daily Rainfall Fields for Multiple Extents: The Potential of Operational Radar

Transcription

1 1204 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 8 Automatic Predictio of High-Resolutio Daily Raifall Fields for Multiple Extets: The Potetial of Operatioal Radar J. M. SCHUURMANS, M. F. P. BIERKENS, AND E. J. PEBESMA Departmet of Physical Geography, Utrecht Uiversity, Utrecht, Netherlads R. UIJLENHOET Hydrology ad Quatitative Water Maagemet Group, Wageige Uiversity, Wageige, Netherlads (Mauscript received 6 Jue 2006, i fial form 3 Jauary 2007) ABSTRACT This study ivestigates the added value of operatioal radar with respect to rai gauges i obtaiig high-resolutio daily raifall fields as required i distributed hydrological modelig. To this ed data from the Netherlads operatioal atioal rai gauge etwork (330 gauges atiowide) is combied with a experimetal etwork (30 gauges withi 225 km 2 ). Based o 74 selected raifall evets (March October 2004) the spatial variability of daily raifall is ivestigated at three spatial extets: small (225 km 2 ), medium ( km 2 ), ad large ( km 2 ). From this aalysis it is show that semivariograms show o clear depedece o seaso. Predictios of poit raifall are performed for all three extets usig three differet geostatistical methods: (i) ordiary krigig (OK; rai gauge data oly), (ii) krigig with exteral drift (KED), ad (iii) ordiary collocated cokrigig (OCCK), with the latter two usig both rai gauge data ad rage-corrected daily radar composites a stadard operatioal radar product from the Royal Netherlads Meteorological Istitute (KNMI). The focus here is o automatic predictio. For the small extet, rai gauge data aloe perform better tha radar, while for larger extets with lower gauge desities, radar performs overall better tha rai gauge data aloe (OK). Methods usig both radar ad rai gauge data (KED ad OCCK) prove to be more accurate tha usig either rai gauge data aloe (OK) or radar, i particular, for larger extets. The added value of radar is positively related to the correlatio betwee radar ad rai gauge data. Usig a pooled semivariogram is almost as good as usig evet-based semivariograms, which is coveiet if the predictio is to be automated. A iterestig result is that the pooled semivariograms perform better i terms of estimatig the predictio error (krigig variace) especially for the small ad medium extet, where the umber of data poits to estimate semivariograms is small ad evet-based semivariograms are rather ustable. 1. Itroductio Raifall is the mai iput variable for hydrological models. Hydrologists use spatially distributed hydrological models to gai isight i the spatial variability of soil moisture cotet, groudwater level as well as the discharge of catchmets. As the spatial iformatio o surface elevatio, lad use, ad soil properties icreases, hydrologists icrease the spatial resolutio of these models. However, up to ow the spatial resolutio of the raifall iput iformatio lags behid. To Correspodig author address: J. M. Schuurmas, Departmet of Physical Geography, Faculty of Geoscieces, Utrecht Uiversity, P.O. Box 80115, 3508 TC Utrecht, Netherlads. h.schuurmas@geo.uu.l properly model soil moisture cotet ad groudwater level at high resolutio, hydrologists require raifall iformatio to be at high resolutio as well. The raifall iformatio that is readily available for hydrologists i the Netherlads comes from both rai gauges ad meteorological radar. All the raifall data are collected ad distributed by the Royal Netherlads Meteorological Istitute (KNMI). There are two rai gauge etworks, of which the etwork with the highest desity has approximately 1 gauge (100 km 2 ) 1 with daily raifall measured. The operatioal radar product is a processed composite field of daily raifall with a resolutio of 2.5 km 2.5 km from two C-bad Doppler radars. If operatioally available raifall data (i.e., raifall fields from radar ad rai gauge data) could be used to DOI: /2007JHM America Meteorological Society

2 DECEMBER 2007 S C H U U R M A N S E T A L FIG. 1. Locatios of weather radars ad rai gauges i The Netherlads: two C-bad Doppler radars, voluteer etwork with 330 rai gauges (temporal resolutio of 1 day), automatic etwork with 35 tippig-bucket rai gauges (temporal resolutio of 10 mi), ad experimetal etwork with 30 tippig-bucket rai gauges (equipped with evet loggers). The three spatial extets studied are also show. predict high-resolutio raifall automatically, hydrologists would probably be more willig to use these i their modelig. I this study we therefore focus o usig operatioal radar products that ca be readily obtaied olie from the KNMI. Moreover, we cocetrate o predictio procedures that ca be automated (i.e., they have to be reliable ad robust such that without additioal itervetio daily predictios are guarateed). Of course, at the same time the resultig predictios have to be sufficietly accurate to be of ay use. Cosequetly, the mai objective of this study is to provide for automatic predictio of raifall at a high (withi a radar pixel) spatial resolutio usig operatioal daily raifall products. Our most importat research questio is the followig: what is the added value of operatioal radar with respect to rai gauges i terms of raifall predictio? To predict at a high spatial resolutio we eed iformatio about the spatial variability of daily raifall at a small extet. Therefore we have set up a experimetal high-desity rai gauge etwork of 30 rai gauges withi a area of 225 km 2 i the middle of the Netherlads. Although we kow that there is a space time correlatio i raifall we restrict ourselves to daily raifall for two reasos. First, withi the Netherlads we have a relatively slow hydrological respose to stratiform-domiated raifall, which meas that a temporal resolutio of 1 day is already very iformative. Secod, the operatioal radar products we use i this study are available at a daily time step. We selected 74 raifall evets ad studied for each evet the spatial variability of raifall at three differet extets: the small (225 km 2 ), medium ( km 2 ), ad large ( km 2 ) extet (i.e., the meso- ad meso- scale; Orlaski 1975). We used three geostatistical predictio methods, oe usig rai gauge data oly ad the other two combiig rai gauge ad radar data, ad compared the results. The orgaizatio of this paper is as follows. I sectio 2 we describe the data ad the evet selectio procedure. I sectio 3 the methods are described, startig with variogram modelig, followed by the three geostatistical predictio methods used. The results are give i sectio 4, startig with the variography (i.e., spatial variability) of raifall, followed by a case study to show the predictio methods used ad fially the cross-validatio results are show. Sectio 5 deals with the ucertaities of this study. I sectio 6 we summarize the mai coclusios. 2. Data ad data processig a. Rai gauge etwork We used two differet kids of rai gauge etworks; oe permaet etwork, which is operated by the KNMI, ad oe experimetal etwork. Figure 1 shows the locatio of all the rai gauges of these two etworks. The largest etwork cosists of 330 statios ad has a desity of approximately 1 statio (100 km 2 ) 1. This etwork is maitaied by voluteers who report the raifall depth daily at 0800 UTC. These data are available olie at the KNMI. The experimetal high-desity etwork cosists of 30 tippig-bucket rai gauges withi a area of 225 km 2 i the cetral part of the Netherlads. The choice for this particular area was based o the fact that several other hydrometeorological experimets are ogoig withi this area as well [i.e., the Cabauw Experimetal Site for Atmospheric Research (CESAR)], which may be mutually beeficial. Differet from the Hydrological Radar Experimet (HYREX), which took place i the Uited Kigdom betwee May 1993 ad April 1997 (Moore et al. 2000), the purpose of our etwork was ot to give the best estimate of mea raifall over a radar pixel but i additio to asses the small-extet variability of raifall (i.e., to estimate the semivariograms of the

3 1206 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 8 rai fields), i particular, for short lag distaces. Therefore, at five locatios two rai gauges were placed at very close distace (1 5 m) from each other, a setup that is also recommeded by Krajewski et al. (2003). The rest of the gauges were set up i such a way that we had may differet itergauge distaces. Krajewski et al. (2003) state that kowledge of raifall structure at spatial extets betwee a few meters ad a few kilometers is still poor. The operatioally available rai gauge etwork i the Netherlads oly gives iformatio for distaces larger tha approximately 10 km, which implies that predictio of high-resolutio raifall fields (smaller tha 10 km) is very ucertai. Our highdesity experimetal etwork ca be compared with the high-desity etworks of HYREX, which had 49 tippig-bucket rai gauges withi 132 km 2 (Moore et al. 2000), ad the etwork of Iowa Istitute of Hydraulic Research, which has 15 rai gauges with separatio distaces ragig from 10 to 1000 m (Krajewski et al. 1998). Our etwork is therefore uique i the Netherlads ad provides valuable iformatio o the spatial structure of raifall at short distaces. For the experimetal etwork we used ARG100 tippig buckets (developed by the Cetre for Ecology ad Hydrology i the Uited Kigdom), which are desiged to reduce their sesitivity to wid speed ad directio, through their aerodyamic desig. We placed the rai gauges i ope area, free from obstacles. All gauges were equipped with evet loggers, that record the mometary tippig evet, storig the time ad date of each evet with a time resolutio of 0.5 s. The omial raifall accumulatio per tippig was 0.2 mm, but a laboratory-derived itesity-depedet correctio was made. Approximately every moth the loggers were read out ad the rai gauges maitaied. b. Radar data The KNMI operates two C-bad Doppler radars, oe at De Bilt ad oe at De Helder, Netherlads (Fig. 1), which both record 288 pseudo CAPPI (800 m) reflectivity fields each day (i.e., every 5 mi) after removal of groud clutter (Wessels ad Beekhuis 1997). The resolutio of these fields is 2.5 km 2.5 km. The measured radar reflectivity factor Z (mm 6 m 3 ) of each resolutio uit is coverted to surface raifall itesity R (mm h 1 ) usig the Marshall Palmer Z R relatioship, which has bee foud to be most suitable for stratiform-domiated raifall evets (Batta 1973): Z 200R 1.6. For both radars, the surface raifall itesities are accumulated from 0800 to 0800 UTC the followig day 1 for each pixel. It is kow that there is a distacerelated uderestimatio of surface raifall by weather radars due to spatial expasio of the radar beam ad due to atteuatio of the radar sigal. Also overestimatio due to the bright bad (vertical profile of reflectivity) may occur. Therefore, data from the rai gauges of the voluteer etwork, from the same period, are used to perform a rage correctio for each radar separately every day (Hollema 2004). This is a ragedepedet bias correctio ad is doe as follows (Hollema 2003). Collocated radar (R) ad gauge (G) observatios form the variable RG, which is oly calculated if both the radar ad rai gauge measured more tha 1-mm raifall: RG 10 log R G. The available RG values are plotted as a fuctio of distace from the radar (r) ad a parabola is fitted through this data: RG r a br cr 2 with a, b, ad c beig the fittig parameters After the rage correctio, a composite field is costructed by averagig the pixel values of the two radars up to a radius of 200 km away from each radar. Withi a radius of 15 km from oe radar, the iformatio of the other radar is used. This composite radar field is a operatioal product of KNMI ad is used i this study. c. Evet selectio The rai gauges of the experimetal etwork were istalled at the begiig of Betwee March ad October 2004 we selected the 74 evets with mea daily raifall depth of all rai gauges of the experimetal etwork exceedig 1 mm. Raifall was accumulated from 0800 to 0800 UTC (over a 24-h period) to match the measurig period of the voluteer etwork ad the weather radar. Krajewski et al. (2003) ad Steier et al. (1999) have already metioed the problems that ca occur with rai gauges ad we experieced these same problems, resultig i the fact that it was seldom that all 30 rai gauges of the experimetal etwork yielded reliable data simultaeously. We performed a quality check to filter outliers. For each evet we made boxad-whisker plots, showig the distributio of the data. Data from rai gauges further tha 1.5 times the iterquartile rage from the earest quartile were marked as suspicious poits. Oly if it was clear from the fieldwork that these rai gauges showed problems (e.g., clogged up, problems with logger, etc.) they 2 3

4 DECEMBER 2007 S C H U U R M A N S E T A L FIG. 2. Gaussia quatile quatile plots of ormalized ozero raifall data of the 74 evets for three differet extets (small, medium, ad large extet) ad for three trasformatio scearios: o trasformatio (o), log trasformatio (log), ad square root trasformatio (sqrt). were removed from the dataset, otherwise the outlier was attributed to spatial variability of the true raifall field. d. Data trasformatio For krigig a (multivariate) Gaussia distributio of the data is preferred. I that case the krigig predictor is the same as the coditioal mea ad the krigig variace is the same as the variace of the coditioal distributio, which makes it possible to calculate exceedece probabilities from krigig predictios (Goovaerts 1997). To approximate a Gaussia distributio, we tried both a log-trasformatio ad a square root trasformatio o the data. Oly measuremets with ozero raifall were take ito accout ad were ormalized for each evet. Figure 2 shows these trasformatios for the three differet extets. As the square root trasformatio gave the best results i approximatig a Gaussia distributio at all three extets, we chose this trasformatio for further study.

5 1208 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 8 3. Methods a. Variogram estimatio The semivariogram, from ow o simply called the variogram, describes i terms of variaces how spatial variability chages as a fuctio of distace ad directio (Isaaks ad Srivastava 1989). The variogram is eeded for krigig. To get isight ito the multiextet spatial variability, we distiguished three differet spatial extets (Fig. 1); small (225 km 2 ), medium ( km 2 ), ad large extet ( km 2 ). The extet is defied i this study as the area over which predictios are made, followig Bierkes et al. (2000). I case of o failure, we had data from 30 rai gauges at the small extet, 103 rai gauges at the medium extet (icludig the 30 from the small-extet etwork), ad 330 rai gauges at the large extet (icludig the 103 from the medium-extet etwork). 1) INDIVIDUAL VARIOGRAMS For each of the 74 evets we calculated the experimetal variogram of the square root trasformed rai gauge data. The experimetal variogram is calculated as half the average squared differece betwee the paired data values, so i case of square root trasformed rai gauge measuremets G(x i ), ˆ h 1 N h 2N h G x i G x i h 2, i 1 i which N(h) is the umber of poit pairs ad h the separatio distace. The maximum separatio distace cosidered is take as 1 3 of the extet (i.e., 8, 50, ad 150 km for the small, medium, ad large extet, respectively). The reaso for that is that poit pairs with a larger separatio distace are too highly correlated (Jourel ad Huijbregts 1978). Up to the maximum separatio distace we chose 15 distace itervals ito which data poit pairs were grouped for semivariace estimates. To assure that the krigig equatios have a uique ad stable solutio (i.e., to force the variogram to be positive defiite) we have to fit a suitable variogram model (Isaaks ad Srivastava 1989) to the experimetal variogram. Amog several models, we chose the widely used spherical variogram model, defied as h C 0 1 k h C 3h 2a 4 h3 2a 3 0 h a C 0 C h a, 5 i which the Kroecker delta fuctio k (h) is1for h 0 ad 0 for h 0. The parameter C 0 is the ugget variace, C is the partial sill, ad a is the rage of the spherical variogram model. The parameters of the spherical variogram models were fitted automatically with oliear regressio, usig weights N(h)/h 2 with N(h) as the umber of poit pairs ad h as the distace. This criterio is partly suggested by theory, ad partially by practice (Pebesma 2004). Whe it was ot possible to fit a uique variogram model, the rage was forced to be beyod the extet. All geostatistical operatios were carried out usig the package gstat (Pebesma 2004) withi R (R Developmet Core Team 2004). 2) POOLED VARIOGRAMS Without the experimetal rai gauge etwork, hydrologists should extract iformatio about the smallscale raifall variability from the KNMI etwork. As metioed i sectio 2 these etworks do ot give isight ito the spatial variability of raifall at distaces smaller tha approximately 10 km. To be able to make high-resolutio predictios of raifall i spite of that, we computed a sigle pooled variogram for each extet based o all the 74 selected evets, icludig the iformatio of the experimetal rai gauge etwork. For each evet we performed krigig predictios usig both the idividual variogram model for that evet as well as the pooled variogram model ad compared the results. This way we ca draw coclusios about whether it is permitted to use oe stadard variogram model, which would be helpful for automated predictio. b. Predictio methods This sectio itroduces briefly three geostatistical predictio methods that were used for this study. The first method uses oly rai gauge iformatio while the secod ad third method use both rai gauges ad radar iformatio but i a differet maer. For more detailed iformatio readers are referred to Isaaks ad Srivastava (1989), Goovaerts (1997), ad Cressie (1993). 1) ORDINARY KRIGING Geostatistical predictio is based o the cocept of a radom fuctio, whereby the ukow values are regarded as a set of spatially depedet radom variables. Krigig is a geeralized least squares regressio techique that allows oe to accout for the spatial depedece betwee observatios, as revealed by the variogram, i spatial predictio. Krigig is associated

6 DECEMBER 2007 S C H U U R M A N S E T A L with the acroym BLUP, meaig best liear ubiased predictor (Cressie 1993). It is liear as the estimated values are weighted liear combiatios of the available data. It is ubiased because the expectatio of the error is 0 ad it is best as it miimizes the variace of the predictio errors. The ordiary krigig predictio of square root raifall (Rˆ OK) at the usampled locatio x 0 is a liear combiatio of the eighborig square root trasformed rai gauge observatios G(x i ): Rˆ OK x 0 i 1 i G x i. The weights i must be such that the predictor Rˆ OK is 1) ubiased (i.e., givig o systematic uder or overestimatio) ad 2) optimal (i.e., with a miimal mea squared error). These weights are obtaied by solvig j 1 j x i x j x i x 0 j 1 j 1, i 1,..., with beig the Lagrage parameter accoutig for the ubiasedess costrait o the weights. The oly iformatio eeded to solve the krigig system [Eq. (7)] are the semivariogram values, which ca be calculated from the fitted variogram model [Eq. (5)]. 2) KRIGING WITH EXTERNAL DRIFT Beyod usig iformatio from the rai gauges G(x i ) oly, krigig ca use secodary iformatio to improve the krigig predictio (Goovaerts 1997). I case of raifall, a iformative secodary data source is the square root raifall estimated by the weather radar (R). We used two krigig algorithms that icorporate exhaustively sampled secodary data: 1) krigig with exteral drift (KED) ad 2) ordiary collocated cokrigig (OCCK). KED is comparable to uiversal krigig (krigig with a tred): we assume that we kow the shape of the tred ad krigig is the performed o the residuals while the tred parameters are implicitly estimated. I uiversal krigig the tred is ofte a fuctio of iteral variables (coordiates) while i KED the tred surface is based o a tred through the secodary data (exteral variables). Mathematically they are idetical. Although the tred ca be a liear regressio through several exteral variables, we oly cosidered radar R(x) as secodary data: m x a 0 a 1 R x The tred coefficiets (a 0 ad a 1 ) are implicitly estimated through the krigig system withi each search eighborhood usig geeralized least squares. I our case the search eighborhood was ot fixed but depeded o the available data poits. At maximum 40 eighborig poits are take ito accout. The krigig with exteral drift predictio of square root raifall (Rˆ KED) at the usampled locatio x 0 is Rˆ KED x 0 i 1 i G x i. The weights of krigig with exteral drift are obtaied by solvig the system: j 1 j res x i x j 0 1 R x i res x i x 0 j 1 j 1 i 1,..., j 1 j R x j R x 0, 9 10 with 0 ad 1 beig the Lagrage parameters accoutig for the ubiasedess costraits o the weights. Krigig with exteral drift is performed usig res as the variogram of the residuals from the tred. 3) ORDINARY COLLOCATED COKRIGING Aother algorithm for takig ito accout secodary iformatio is OCCK. Differet from KED i which the secodary data provides iformatio o the tred oly, i OCCK the secodary data, which is ow cosidered as a radom variable as well, iflueces the krigig predictio directly. I additio, OCCK accouts for the global liear correlatio betwee primary ad secodary variables, whereas with KED the secodary iformatio teds to strogly ifluece the predictio, especially whe the estimated slope or itercept of the local tred model is large. A more or less similar ad more well-kow krigig method usig secodary iformatio is ordiary cokrigig (OCK). Several studies have used OCK to merge rai gauge ad radar raifall: Creuti et al. (1988), Fiorucci et al. (2001), Krajewski (1987), ad Seo et al. (1990a,b). Goovaerts (2000) icorporated elevatio as secodary iformatio ad tested several krigig methods that accout for secodary data. OCCK is preferred to OCK for the followig four reasos (Goovaerts 1997): 1) OCCK avoids istability caused by

7 1210 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 8 highly redudat secodary data, 2)OCCK is faster tha OCK as it calls for a smaller cokrigig system, 3) OCCK does ot call for a secodary covariace fuctio at distaces larger tha 0, ad 4) OCCK does ot require modelig of the cross-covariace fuctio by usig the Markov-type approximatio (i.e., depedece of the secodary variable o the primary is limited to the collocated primary datum). The ordiary collocated cokrigig predictio of square root raifall (Rˆ OCCK) at the usampled locatio x 0 is a liear combiatio of the eighborig square root trasformed rai gauge observatios G ad oe collocated square root trasformed radar observatio R: Rˆ OCCK x 0 i 1 i G x i R x 0, 11 with the costrait that the weights ( i 1 i ) sum to 1. I case the expected value of the primary ad secodary data are ot equal, Eq. (11) must be adjusted, so the secodary data is bias corrected. I our case we assumed the expected values of the rai gauges ad radar to be the same because the operatioal radar are already bias corrected (sectio 2b). The ordiary collocated cokrigig weights are obtaied by solvig the system: j 1 j GG x i x j GR x i x 0 GG x i x 0 i 1,..., j GR x 0 x j RR 0 GR 0 j 1 j 1, j 1 12 with beig the Lagrage parameter accoutig for the ubiasedess costraits o the weights, GG is the direct variogram of the rai gauge data, GR is the cross variogram of rai gauge ad radar data, ad RR is the direct variogram of the radar data. The three variograms are modeled as a liear combiatio of the same basic model, the fitted spherical variogram model of the ormalized square root trasformed rai gauge data [Eq. (5)]. The direct variogram of the rai gauges ( GG ) was calculated by multiplyig the stadardized variogram with the variace of the square root trasformed rai gauge measuremets. The direct variogram of the radar data ( RR ) was calculated by multiplyig the stadardized variogram by the variace of the square root trasformed FIG. 3. Box-ad-whisker plots of the stadard deviatio of ozero raifall at the three spatial extets for the 74 evets. The black dot deotes the media, solid boxes rage from the lower to the upper quartile, ad dashed whiskers show the data rage. Data that are further tha 1.5 times the iterquartile rage from the earest quartile are show as ope bullets. radar data. The cross variogram ( GR ) was calculated by multiplyig the direct variogram of the rai gauges ( GG ) with the correlatio betwee the collocated square root trasformed rai gauge ad radar data, assumig the Markov-type approximatio (Goovaerts 1997). c. Back trasformatio ad zero raifall Raifall ca be cosidered as a biary process, it either rais or it does ot. As show i Fig. 2 the measuremets of ozero daily raifall closely follow a Gaussia distributio after a square root trasformatio. Krigig performs best whe data are (multivariate) Gaussia distributed ad we therefore applied krigig to the ozero, square root trasformed (ostadardized) raifall measuremets. This meas that the ordiary krigig predictio ad collocated cokrigig predictio are best i predictig the square root raifall give that it rais. To obtai back-trasformed raifall values we caot simply take the square of the krigig predictio based o the square root trasformed raifall data. The reaso for this is that if a Gaussia distributio is squared, it becomes positively skewed, resultig i a mea larger tha the media. Simply squarig the krigig predictio of square root trasformed raifall data would uderestimate the co-

8 DECEMBER 2007 S C H U U R M A N S E T A L FIG. 4. Small-extet ormalized variograms of square root trasformed daily raifall with fitted spherical model for each of the 74 evets i 2004 (header i moth moth day day, e.g., 0306 meas 6 March). For 17 ad 21 Aug 2004, the automatically fitted variogram model is outside the plotted semivariace rage. ditioal mea of raifall, especially i case of large krigig variaces. To back trasform predictio values we therefore calculated the percetiles of the coditioal distributio of square root trasformed raifall, assumig this distributio to be Gaussia with mea equal to the krigig predictio ad variace equal to the krigig variace. After that we back-trasformed (squared) these percetiles, whose rak ad percetile value do ot chage with trasformatio. From this ew distributio fuctio we calculated the mea ad the variace. By ot cosiderig the umber of zeroes i the dataset valuable iformatio would be lost. Therefore we forced the predicted raifall values to cotai the same percetage of zero s as i the dataset. The predicted raifall amouts were arraged i icreasig order ad a threshold was calculated that correspoded with the percetage of zeroes i the raifall dataset. All

9 1212 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 8 FIG. 5. Same as i Fig. 4, but for the medium extet. the predictio locatios with raifall smaller tha this threshold were set to zero. 4. Results a. Variograms To compare the spatial structure of each evet i oe plot we calculated the variograms of the ormalized (i.e., variace equals 1) square root trasformed raifall data. The variace i raifall however, differs betwee both the evets ad extets. Figure 3 shows i a box-ad-whisker plot for each extet the stadard deviatio of ozero (oormalized ad otrasformed) raifall for all the 74 evets. This figure shows a positive tred i the variability of ozero raifall from a small to larger extet. Figure 4, 5, ad 6 show for each evet the ormalized experimetal variograms of the square root trasformed daily raifall, as well as the automatically fitted spherical variogram model of respectively the small, medium, ad large extet. I Fig. 4 the fitted variogram model for 17 ad 21 August (0817 ad 0821, respec-

10 DECEMBER 2007 S C H U U R M A N S E T A L FIG. 6. Same as i Fig. 4, but for the large extet. tively) is outside the plotted semivariace rage. Durig those days there was a high semivariace betwee poit pairs with the smallest separatio distace, causig our automated procedure to fit a variogram model with a large ugget. Due to the larger data availability at larger extets, the experimetal variograms become less ambiguous ad the variogram models fit better. Figure 7 shows the fitted rages of the spherical variogram models for each evet for the small, medium, ad large extet. This figure shows a large variability i fitted rages across evets, but a seasoal effect is ot clear. Figure 7 also shows the fitted rage of the spherical variogram model for the small ad medium extet to be ofte 25 ad 150 km, respectively, which are the values that were eforced whe the oliear variogram fittig procedure was ot successful. Figure 8 shows the experimetal pooled variograms as well as the fitted spherical variogram models of the small, medium, ad large extet. For the medium ad large extet we fitted a ested spherical variogram model (i.e., a sum of two spherical variogram models; Deutsch ad Jourel 1998). The parameters

11 1214 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 8 FIG. 7. Automatically fitted rages (km) of spherical variogram models for small, medium, ad large extet as a fuctio of the evet date. of the fitted spherical variogram models are give i Table 1. The pooled variogram model was calculated usig ormalized square root trasformed daily raifall data. For the krigig predictio, the ugget ad sill of the variogram model should be multiplied by the variace of the square root trasformed daily raifall data. b. Raifall predictio 1) CASE STUDY Figure 9 shows the accumulated daily composite rage-corrected radar field for the period March October 2004 at the small extet. Durig this period, 22 radar images were either missig or icomplete. From this figure is ca be see that eve for a accumulatio period of 7 moths, there still is a differece of 10% raifall withi the small extet. This corroborates our argumet that for the Netherlads a temporal resolutio of 1 day is importat eve at a small extet. To demostrate the krigig methods described i the precedig paragraphs we selected two of the 74 raifall evets: 4 April ad 1 May Figure 10 shows the composite rage-corrected radar fields for the selected dates. These two evets were chose because they represet two differet raifall types with differet correlatio betwee rai gauge data ad radar. The evet of 4 April 2004 is a example of a stratiform evet because it rais over a large area ad extremely high raifall areas caot be detected. The evet of 1 May 2004 is a example of a covective evet because the raifall area is smaller ad high raifall values are preset. Figure 11 shows the predictio of raifall depth at the small extet for 4 April 2004 accordig to the rage

12 DECEMBER 2007 S CHUURMANS ET AL FIG. 8. Pooled variogram models of the small, medium, ad large extet, calculated from the stadardized ozero square root trasformed daily raifall data. For the medium ad large extet a ested spherical variogram model is fitted. corrected radar, the ordiary krigig predictio, krigig with exteral drift predictio, ad the ordiary collocated cokrigig predictio, respectively. The krigig predictios are made at a high-resolutio poit grid with distaces of 100 m. For the medium ad large extet we oly used the 40 earest data poits istead of the complete dataset for krigig. For this evet the correlatio betwee the raifall measured by the rai gauges ad the collocated radar at the small extet was Because of this low correlatio, the tred surface of the radar has a very small effect o the KED predictio. For the ordiary collocated cokrigig predictio, however, the radar field ca be clearly see. This is due to the fact that for each predictio locatio withi a radar pixel this same collocated radar value is take ito accout [Eq. (11)]. Withi the radar pixel itself, the raifall depths are iterpolated. To overcome the problem of sudde trasitios i raifall depth from oe radar pixel to the other, the radar field ca be presmoothed before executig the KED or OCCK. We TABLE 1. Parameters of fitted pooled spherical variogram models. Here C 1 ad a 1 are the partial sill ad rage, respectively, of the first variogram model. I case of a ested variogram model C 2 ad a 2 are the partial sill ad rage, respectively, of the secod variogram model [see Eq. (5)]. Parameter Small extet Medium extet Large extet C C C a 1 (km) a 2 (km)

13 1216 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 8 FIG. 9. Total raifall durig March October 2004 at a small extet accordig to daily rage-corrected radar. retrieved good results by smoothig the radar with iverse distace iterpolatio usig the four closest grid cell ceter poits of the radar field. Figure 12 shows the raifall fields usig the same krigig methods for the 1 May 2004 evet. For this evet the correlatio betwee the raifall measured by the rai gauges ad the collocated radar at the small extet was I Figs. 11 ad 12 we show the predicted raifall depths usig OK, KED, ad OCCK for the small extet. We assumed that for all evets it raied throughout the whole small extet so we did ot have to cope with the problem of zero raifall. To illustrate how our method deals with zero raifall, we show the raifall depths of the differet methods for 1 May 2004 at the large extet i Fig. 13. Durig this evet 114 rai gauge statios out of the 211 reported zero raifall. We forced the same percetage (54%) of the predictio locatios to have zero raifall. From this figure it ca be see that, usig ordiary krigig (OK), raifall is predicted i the orther part of the Netherlads where the radar did ot detect raifall, whereas i the souther part ordiary krigig does ot predict raifall where the radar does detect raifall. KED ad OCCK combie the two sources of iformatio, but i a slightly differet maer. I this case, the predictios of both methods follow the spatial raifall structure as show by the radar. The raifall measured by the rai gauges i the orther part of the Netherlads was so low that it was set to zero. 2) CROSS VALIDATION To compare the accuracy of the three differet krigig forms (OK, OCCK, ad KED) as well as the use of either the idividual variogram models or pooled variogram models, we used cross validatio. The idea of cross validatio is to remove oe data poit at a time from the dataset ad repredict this value from the remaiig data. For each extet ad evet we performed cross validatio, usig the three differet krigig forms. For ordiary krigig as well as ordiary collocated cokrigig we used both the idividual variogram model of that evet (e.g., Fig. 4) as well as the pooled variogram model (Fig. 8) of that extet. Fially, for each evet we calculated the root-mea-squared error (rmse) resultig from the cross-validatio calculatios. FIG. 10. Composite rage-corrected radar fields of (left) 4 Apr ad (right) 1 May 2004.

14 DECEMBER 2007 S C H U U R M A N S E T A L FIG. 11. Predictio of raifall depth at small extet for 4 Apr 2004 accordig to the rage-corrected radar (radar), OK pred, KED pred, ad OCCK pred. Gree triagles show the rai gauge locatios. Radar rai gauge correlatio is I our case study of 74 evets, we had 7 icomplete radar fields, due to techical malfuctio or maiteace of oe of the radars. These evets were ot take ito accout i the cross validatio. The distributios of the rmse resultig from the cross validatio usig the differet krigig methods are plotted i box-adwhisker plots (Fig. 14). We also calculated the rmse of the differece betwee the raifall measured by the rai gauges ad the collocated radar pixel ad these results (radar) are show as well i Fig. 14. Figure 14 shows that takig ito accout radar as secodary variable, usig either KED or OCCK, leads to better results tha oly takig ito accout data from rai gauges for the medium ad large extet. Also, for the medium ad large extet, radar performs better tha rai gauge data aloe (OK). For the small extet, however, the rai gauge data aloe perform better ad the added value of takig ito accout radar is ot so clear. This is ot surprisig as we have a dese etwork of rai gauges at the small extet. The use of a pooled variogram model for all evets istead of idividual variogram models for each separate evet makes little differece. For the small extet the pooled variogram model performs less tha the idividual variogram models whereas for the medium ad large extet it is the other way aroud. Figure 15 shows the ratio betwee the rmse as calculated with OK ad KED (ratio.rmse.ok.ked, show as ) as well as the ratio betwee the rmse as calculated with OK ad OCCK (ratio.rmse.ok.cck, show as o) as a fuctio of the correlatio betwee rai gauges ad radar for the small, medium, ad large extet. If the ratio is higher tha 1, it meas that KED or OCCK performs better tha OK. Agai we see that the larger the extet, the higher the added value of the radar. Figure 15 also shows the positive effect of the correla-

15 1218 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 8 FIG. 12. Same as i Fig. 11, but for 1 May Radar rai gauge correlatio is tio betwee radar ad rai gauge ad the ratio of rmse, especially at the medium ad large extet. We also looked at the z score of the cross-validatio exercise, which is the residual divided by krigig stadard error. The z score should have zero mea ad uit variace. If the mea z score deviates from zero, we have a biased predictio, ad if the variace is higher tha 1, we uderestimate the krigig variace. For each evet we calculated the mea z score usig the three differet krigig methods. The results are show i a box-ad-whisker plot (Fig. 16). It ca be see that for almost all evets ad krigig methods, the mea z score is close to zero, except for OCCK. This is probably due to our assumptio that the expected value of the rai gauge data ad radar data are the same. Figure 17 shows i a box-ad-whisker plot the variace of the z score that we calculated for each evet usig the three krigig methods. It is most strikig that the pooled variogram used for both OK ad OCCK leads to lower z-score variaces ad thus better estimatio of the predictio error variace. 5. Discussio I this sectio we deal with some remaiig ucertaities ad possible improvemets of the methods preseted i this paper. This paper deals with daily raifall. For hydrological applicatios it would be iterestig to also be able to geerate high spatial resolutio raifall fields with a higher time resolutio (e.g., 3 h). I that case the mai problem we would have to cope with is the reductio of the amout of rai gauge measuremets, as the largest operatioal rai gauge etwork i The Netherlads cosists of voluteers who oly report the daily raifall depth. Cosequetly the preset rage correctio of the weather radar caot be performed o a higher time resolutio, as this method uses the voluteer et-

16 DECEMBER 2007 S C H U U R M A N S E T A L FIG. 13. Same as i Fig. 11, but for the Netherlads o 1 May work as well. With a low-desity rai gauge etwork our method to correct for zero raifall is also ot suitable because a idividual rai gauge reportig zero raifall would have too much ifluece. Besides the fact of the decrease i rai gauge statios we would also have to deal with the followig problems: (i) the correlatio betwee raifall measured by the rai gauges ad radar is kow to decrease at small time scales, especially whe (e.g., i covective raifall) there is a large space time variability of raifall; (ii) reestimatio of the variogram models as it is kow that for raifall averaged over larger spatial scales ad itegrated over

17 1220 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 8 FIG. 14. Box-ad-whisker plots of the rmse of the residuals from the compariso betwee operatioal radar ad rai gauges (radar) as well as the rmse of the residuals from the cross validatio usig three krigig methods: OK, OCCK, ad KED. For both OK ad OCCK we used the idividual variograms (idiv) as well as the pooled variograms (pooled). loger periods the correlatio distace is typically larger; ad (iii) recosideratio of the square root trasformatio of raifall data to make its distributio closer to Gaussia. Possibly for shorter periods a logarithmic trasformatio would be more suitable. For shorter time steps, the spatial cotiuity of radar measuremets becomes a major advatage compared to rai gauge etworks. This however does ot preclude the thorough radar data processig [e.g., correctio for vertical profile of reflectivity (VPR) ad atteuatio] required to improve as far as possible the radar data quality. Besides all the advatages of usig radar data, it is importat to recogize the iheret limitatios of radar data quality, especially as a fuctio of rage. The average rage limit to keep i mid for hydrological use of weather radar is o the order of 80 km. Results from HYREX show that distributed hydrological models are sesitive to rai gauge locatio ad hece to the spatial variability of raifall over the catchmet, especially durig covective raifall (Bell ad Moore 2000). I the Netherlads, covective raifall maily occurs durig summer as a result of local ascet of warm air ad is characterized by heavy raifall with a small spatial extet ad a short duratio. Durig witertime, stratiform raifall evets domiate caused by frotal systems. They have a larger spatial extet tha covective raifall, as well as a loger lifetime. For this reaso, our iitial purpose was to divide the evets ito

18 DECEMBER 2007 S C H U U R M A N S E T A L FIG. 15. Ratio of the rmse as calculated with ordiary krigig ad krigig with exteral drift (rmse.ok/ rmse.ked, show as ) as well as the ratio of the rmse as calculated with ordiary krigig ad ordiary collocated cokrigig (rmse.ok/rmse.cck, show as ) as fuctio of the correlatio betwee rai gauges ad radar for the small, medium, ad large extet. Dashed lie represets the value 1. two raifall types (covective ad stratiform) ad to pool the variograms for each raifall type, istead of usig oly oe pooled variogram model for each extet. To idetify covective areas we applied the algorithm proposed by Steier et al. (1995) to 5-mi CAPPI radar fields usig the same criteria as stated i their paper. I case the majority of the grid cells withi the extet were idetified at least 10 times as covective durig that day, we labeled the evet as covective, otherwise it was classified as stratiform. However, whe we pooled the variograms per raifall type we did ot fid sufficiet differeces betwee the form of the pooled variograms to justify the separated modelig. Further research is required to ivestigate whether a separate modelig of stratiform ad covective evets is required ad if so, how to better distiguish betwee stratiform ad covective evets. Although we are aware of the directioal variability i raifall fields we did ot cosider aisotropy i our variogram model fits, mostly because we had too little data poits withi our small extet. The 30 data poits we had at maximum for each evet at the small extet are the absolute miimum to fit a omidirectioal variogram, but provide isufficiet iformatio to estimate directioal variograms. This was also the case for the medium extet. A additioal reaso ot to use directioal variograms is that estimatig them ad fittig suitable models, is difficult to recocile i a au-

19 1222 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 8 FIG. 16. Box-ad-whisker plots of the mea z score from the cross validatio usig three krigig methods: OK, OCCK, ad KED. Dashed lie represets the value zero. For both OK ad OCCK we used the idividual variograms (idiv) as well as the pooled variograms (pooled). tomatic fashio. Moreover, usig a krigig method that takes ito accout the secodary data of the radar, will take ito accout the existig aisotropy preset i the radar image. Krigig predictios were made usig either the idividual variogram models or the pooled variogram model. We did ot take ito accout the quality of the variogram model fit. This ca be doe by usig for istace Markov Chai Mote Carlo techiques (Diggle et al. 1998). Probably, the effect o krigig predictio would ot be large if we have eough data but it becomes relevat for sparse rai gauge etworks. We applied OCCK assumig that the Markov-type approximatio (Goovaerts 1997) holds, i order to make the automatic predictio procedure faster. It should be possible to automatically calculate ad fit the cross-covariace fuctio ad implemet this i the OCCK procedure. This is, however, beyod the scope of this paper. Krigig with exteral drift as applied i this paper, assumes the secodary data to be free from errors. A possible improvemet could be to use exteral drift krigig with ucertai covariates (Va de Kassteele ad Stei 2006). 6. Coclusios We show that krigig with exteral drift ad ordiary collocated cokrigig successfully take ito accout ra-

20 DECEMBER 2007 S C H U U R M A N S E T A L FIG. 17. Same as i Fig. 16, but for the variace of the z score. Dashed lie represets the value 1. dar as a secodary iformatio source ad are more accurate tha ordiary krigig (rai gauge iformatio oly), especially for larger extets with lower desities of rai gauges. The added value of radar is positively related to the correlatio betwee the raifall measured by the rai gauges ad the collocated radar pixel. The use of a pooled variogram model istead of a idividual variogram model for each evet does ot lead to a loss of accuracy i raifall predictio, so these pooled variogram models ca be used whe there is lack of data or whe a automatic predictio procedure is implemeted without variogram estimatio. We also show that the pooled variogram is preferred over evet-based variograms i terms of correct assessmet of predictio ucertaity (z score variace of 1) for the small- ad medium-extet cases, where the umber of data is small ad evet-based variograms are rather ucertai. Aother coclusio is that KED ad OK are more robust with respect to mea z scores (o average zero) tha OCCK. This may be due to a bias i the radar data. Ackowledgmets. The authors would like to thak the Royal Netherlads Meteorological Istitute (KNMI), i particular Iwa Hollema ad To Doker, for their help ad for providig us with their data. J. M. Schuurmas was fiacially supported by TNO ad R. Uijlehoet was fiacially supported by the Netherlads Orgaizatio of Scietific Research (NWO) through Grat We also ackowledge the reviewers for their valuable commets.

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