Bias-corrected nonparametric correlograms for geostatistical radar-raingauge combination

Transcription

1 ERAD 00 - THE SIXTH EUROPEA COFERECE O RADAR I METEOROLOGY AD HYDROLOGY Ba-corrected nonparametrc correlogram for geotattcal radar-rangauge combnaton Renhard Schemann, Rebekka Erdn, Marco Wll, Chrtoph Fre Marc Berenguer, Danel Sempere-Torre Federal Offce of Meteorology and Clmatology MeteoSw, Kraehbuehltrae 58, P.O. Box 54, 8044 Zurch, Swtzerland, renhard.chemann@meteow.ch Centro de Recerca Aplcada en Hdrometeorologa, Unvertat Poltècnca de Catalunya, C/Gran Captà, -4, Edfc EXUS 0-06, Barcelona, Span Preentng author Chrtoph Fre. Introducton (Dated: 5 September 0 Geotattcal method have been wdely ued for quanttatve precptaton etmaton (QPE) baed on the combnaton of radar and rangauge obervaton. They are flexble and accurate and allow for radar-rangauge combnaton n real-tme. Even wthn the area of geotattcal method, however, a wde range of choce have to be made when plannng for a partcular applcaton. Thee choce regard, for example, the actual combnaton method (e.g., krgng wth external drft, cokrgng), the krgng neghbourhood (global v. local), the technque ued to etmate the parameter of the geotatcal model (e.g., leat-quare, maxmum-lkelhood etmaton), and the tranformaton of the precptaton varable. In addton to thee ue, there are a number of opton for modelng patal dependence n the precptaton data. Correlogram (varogram) for krgng are cutomarly one-dmenonal, but two- or hgher-dmenonal correlaton map are alo ued and are one way of takng patal anotropy nto account. Furthermore, correlogram can be parametrc or nonparametrc, they can be obtaned from the radar or the rangauge data, and they can be etmated flexbly on a cae-by-cae ba or wth data from a longer perod of tme. Recently, nonparametrc correlogram baed on patally complete radar ranfall feld have been ued n combnng radar and rangauge data []. Here, we compare the etmaton of nonparametrc correlogram wth the etmaton of parametrc emvarogram model conventonally ued n geotattcal applcaton. We dentfy and explan a ba of the nonparametrc correlogram toward too low range, and ugget a correcton for th ba.. Semvarogram etmaton The emvarogram of a patal proce Z defned a (for greater detal ee [], whoe notaton we largely follow): () γ, ) = Var( Z( ) Z( )) ( For a nd order tatonary proce Z, th equvalent to () γ ( ) = E(( Z( ) Z( )) ) ; and (3) γ ( ) = σ ( ρ ( )), where σ = C ( = Var( Z), and ρ ) ( are the varance and the correlaton functon of the proce Z, and E( ) denote an expectaton value. The wdely-ued Matheron-etmator for the emvarance read (we denote etmator wth a hat to dtnguh them from theoretcal quantte): (4) ˆ( γ ) = ( Z( ) Z( )), where ( ) ( ) ( - ) denote the et of all par of obervaton at a gven lag dtance and ( - ) the number of uch par. For complete radar grd of dmenon Fg.. Semvarogram and correlogram etmaton. (a) One-dmenonal ynthetc data ample, (b) emvarogram cloud, (c) emprcal emvarogram and ftted parametrc model, (d) theoretcal and etmated correlogram.

2 ERAD 00 - THE SIXTH EUROPEA COFERECE O RADAR I METEOROLOGY AD HYDROLOGY th number equal to ( -k) ( -l), where k, l, are the component of the lag dtance vector n unt of the grd pacng. The cutomary procedure for etmatng a emvarogram model llutrated by mean of ynthetc data n Fg. a-c. Fg. a how a ngle realzaton of a one-dmenonal Gauan random proce wth varance and an exponental correlaton functon (the practcal range,.e. the lag at whch the correlaton decay to 0.05, equal 0.6 for th proce). The ample emvarogram (or the o-called emvarogram cloud) hown n Fg. b. It how emvarogram ordnate for all par of obervaton. Snce thee value catter ubtantally, t cutomary to mooth the ample emvarogram by calculatng the etmate (4) after poolng the emvarogram ordnate nto a number of lag dtance clae. Th yeld the o-called emprcal emvarogram hown n Fg. c (open crcle). Fnally, a parametrc model ft to the emprcal emvarogram. Here, a curve-fttng technque ha been ued to etmate an exponental emvarogram model (dahed lne n Fg. c). Equaton (3) yeld the correlaton functon correpondng to the ftted emvarogram model (Fg. d, dahed lne). The theoretcal correlaton functon hown by the old black lne n Fg. d. The dfference between the etmated and the theoretcal correlaton due to amplng varablty and a ba of the etmator and wll be dcued later. The ue of a parametrc model ha a number of reaon. Frt, the parametrc model are choen uch that they fulfll the property of potve defntene. Correlaton functon wth th property can be ued n geotattcal nterpolaton (krgng; ee relevant text uch a [] for detal). Addtonally, the parametrzaton further moothe the emprcal emvarogram and allow to etmate the correlaton at unoberved lag dtance. 3. Etmaton of nonparametrc correlogram The nonparametrc etmate of the correlaton functon gven by (5) Z( ( ) ) Z Z Z ˆ( ρ ) =, where ( ) ) 0 0 ) Z = Z( ) and Cˆ ( 0 ) = = ( Z( ) Z ) = are the ample (alo called plug-n) mean and varance, and the number of obervaton (e.g., radar grd pont). Th etmator can be convenently computed n term of the dcrete Fourer tranform (DFT). In fact, the Wener Khnchn theorem affrm that the magntude of the DFT of the tandardzed obervaton the pectral repreentaton of the (auto-)correlaton etmate computed n (5). Thu, (5) can be obtaned rather mply by computng the DFT, multplyng wth the complex conugate and computng the nvere DFT of the product. Th ha two man advantage. Frt, the fat Fourer tranform allow computng (5) much fater than by mean of explct ummaton. Therefore, the complete radar grd can be taken nto account. In contrat, the complete emvarogram etmator (4) cannot be convenently computed for zeable twodmenonal radar grd, and practcally obtaned from thnnedout ubample of the entre feld. Second, the etmated correlaton functon ha, by contructon, a real and potve pectral denty. Accordng to Bochner theorem, t therefore a potve defnte functon (called lct n [3]) and can be drectly appled n geotattcal predcton (krgng). In prncple, no further fttng of a parametrc covarance model or manpulaton of the pectral denty neceary. (Th correct a remark on th ue made n [4], ecton 3.6..). The nonparametrc etmate (5) of the correlaton functon for the ynthetc one-dmenonal data of Fg. a hown n Fg. d (dotted lne). 4. Comparon of etmator Both etmate of the correlaton functon n Fg. d exhbt horter range than the theoretcal correlaton. Of coure, th could be completely due to amplng varablty and we cannot conclude from the etmate for a ngle realzaton (Fg. a) on the behavour of the etmator. Therefore, we extend the experment a follow: For each of three Gauan procee wth unt varance and exponental correlaton functon wth practcal range of 0., 0.6, and.5, we draw 00 realzaton and etmate a parametrc (exponental) emvarogram model and the nonparametrc correlaton for each of the realzaton. Each realzaton ampled n the doman [0,]. The medan etmated Fg.. Behavour of emvarogram-baed and nonparametrc correlogram etmator for Gauan patal procee of dfferent range. Dahed black lne: Medan ftted emvarogram model for a Gauan proce of practcal range 0.. Dotted black lne: Medan nonparametrc correlogram etmate for a Gauan proce of practcal range 0.. Red and blue lne: the ame for procee of larger practcal range (0.6,.5). All dahed and dotted lne how the medan of etmate for 00 realzaton of the Gauan proce. Sold lne: theoretcal correlaton (for all range; the abca caled by the practcal range).

3 ERAD 00 - THE SIXTH EUROPEA COFERECE O RADAR I METEOROLOGY AD HYDROLOGY emvarogram-baed model for the proce wth practcal range 0. hown by the black dahed lne n Fg.. Th lne very cloe to the theoretcal correlaton (old black lne). A a matter of fact, the etmator (4) known to be unbaed. For fnte-ze ample of correlated data, however, t only approxmately unbaed. In the preent example, the potve autocorrelaton caue the varance of the proce (the emvarogram ll) to be underetmated. A a conequence, alo the range of the emvarogram underetmated. Th effect the more pronounced the larger the practcal range compared to the doman ze,.e. keepng the doman ze contant (here equal to ), the ba wll be larger for larger range (red and blue dahed lne n Fg. ). The dotted lne n Fg. how the nonparametrc correlaton etmate from (5) baed on the ame 00 realzaton of the three Gauan procee. For mall lag and a practcal range of 0., the etmate (black dotted lne) tll farly cloe to the theoretcal correlaton. If the practcal range on the order of the doman ze, however, the nonparametrc correlaton trongly baed toward too mall value (red and blue dotted lne). The ba n the nonparametrc correlogram etmate much larger than n the correpondng emvarogram etmate. (ote: At leat for mall lag, the dfferent normalzaton, ( - ) v., n (4) and (5) are only a mnor contrbuton to the dfference between both etmate.) In order to undertand th obervaton, we rewrte equaton (5) a follow: (5) ˆ ρ ( ) = ( ˆ( ρ )) = ( Z( ) Z )( Z( ( ) ) Z ) = + ( Z ( ) Z )( Z( ) Z ) ( ) For lag dtance that are much maller than the doman dmenon, we can approxmate Cˆ ( ( ) ( Z ( ) Z ) and ( - ). Thu, ( ) (5) ˆ( ) ρ ( ( ) ) ( ( ) ) ( ( ) )( ( ) ) Z Z + Z Z Z Z Z Z ) ( ) 0 ( ) and fnally ˆ( γ ) (6) ˆ( ρ ). Equaton (6) how that the computaton of the nonparametrc correlogram approxmately equvalent to the etmaton of a emvarogram, and the ubequent converon of the emvarogram to a correlogram ung the mple plug-n etmate of the varance. From the pont of vew of conventonal geotattc, th a rather far-fetched procedure, whch manly motvated by the convenence of the etmator (5). For potvely correlated data, the etmator Ĉ( underetmate the varance much more than the emvarogram ll, nce the latter largely determned by the emvarance value correpondng to the larget lag dtance and the extrapolaton performed by fttng the parametrc emvarogram model. Th explan the larger ba of (5) compared to (4). 5. Ba correcton Fg. 3. A Fg. but for ba-corrected nonparametrc correlogram. Equaton (6) alo ugget an approxmate ba correcton for the correlaton functon. Gven an alternatve etmate ˆ σ of the varance, aumed to be uperor to the ample varance Ĉ(, the correpondng etmate of the correlaton functon ˆ( γ ) (7) ˆ ρ c( ) ( ˆ( ρ )). ˆ σ ˆ σ The correcton of the correlaton functon equvalent to calng the emvarance functon by a contant factor Ĉ(/ ˆ σ, and therefore preerve potve defntene. For the ynthetc data of our ntroductory example, we have ued the ll of the

4 ERAD 00 - THE SIXTH EUROPEA COFERECE O RADAR I METEOROLOGY AD HYDROLOGY parametrc emvarogram (Fg. c) for ˆ σ n (7) and the corrected correlaton functon obtaned n th way the dah-dotted lne n Fg. d. Repeatng the experment from ecton 4 wth the ba-corrected etmator (7) yeld the reult hown n Fg. 3. Indeed, the correcton work and the ba-corrected nonparametrc correlogram are very cloe to the emvarance-baed correlogram for mall lag dtance. Wth ncreang lag dtance, the approxmaton the ba-correcton baed on deterorate. Th can be clearly een for the example wth larget practcal range (blue dotted lne n Fg. 3). So far, we have analyzed the behavour of the two etmator for ynthetc one-dmenonal data. In the followng, we apply them to meocale hourly radar precptaton feld n Swtzerland. We conduct the followng experment: We collect 0 hourly radar precptaton feld n a doman of km (Fg. 4) between January and March 009. Thee feld are elected uch that the precptaton amount larger than 0.5 mm n at leat a quarter of the doman. For each of the feld, we etmate correlogram n four dfferent way and repreent the medan acro the etmate for all feld n Fg. 5a-d: a) A ubample of 000 grd cell choen randomly from the feld. From the ubample, a parametrc (exponental) onedmenonal emvarogram model ft a llutrated n Fg. a-c and converted to a correlaton functon a llutrated n Fg. d. b) A two-dmenonal emprcal emvarogram determned accordng to (4) and by averagng n two-dmenonal bn of lag dtance clae (th yeld the two-dmenonal analogue of the open crcle n Fg. c). Th emprcal emvarogram converted nto a correlogram ung the emvarogram ll etmated n tep a). c) The unmodfed nonparametrc correlogram etmate (5). d) The nonparametrc correlogram etmate corrected accordng to (7) ung the emvarogram ll etmated n tep a). The etmate a) and b) are tradtonal emvarance-baed etmate; the latter alo repreent the domnant anotropy of the precptaton feld (for th doman largely determned by the orentaton of the man Alpne rdge). The nonparametrc correlogram alo capture th anotropy, but the range of the etmated correlogram conderably maller than that of the emvarogram. Fnally, the corrected nonparametrc correlogram etmate (Fg. 5d) much more mlar to the emvarance baed etmate (Fg. 5b). Of coure, we cannot compare to a theoretcal reference correlaton for the oberved precptaton feld, but the fact that the etmator behave n an analogou way to what wa found for the ynthetc data ugget that mlar mechanm act here. If we correct for the dfferent normalzaton n (4) and (5), the agreement of the ba-corrected correlogram wth the emvarance-baed etmate further mproved (not hown). However, th renormalzaton of the correlaton functon doe not preerve potve defntene and therefore not put forward here. Fg. 6 a catter plot that compare the two varance etmate for all the cae condered. Indeed, the emvarogram ll conderably larger than the plug-n varance for many cae. Th llutrate that the dfference between the two etmator can be much larger for ndvdual cae, than the medan correlogram n Fg. 5 ugget. Furthermore, Fg. 6 how that the dfference between the two etmator trongly dependent on the precptaton tuaton (and mut alo be expected to be ubect to conderable amplng uncertanty). Accordng to the above analy, both etmator hould agree better for tuaton where the correlaton length mall compared to the doman ze. Arguably, th the cae for the pont n the vcnty of the dentty lne n Fg. 6. Fg. 4. Doman (00 00 km ) of radar compote ued n preent analy. The grd pacng km. Fg. 5. Semvarogram and nonparametrc correlogram etmate for hourly radar precptaton feld n Swtzerland. Medan from 0 feld of (a) correlogram from exponental emvarogram ft, (b) from emprcal twodmenonal emvarogram, (c) nonparametrc correlogram, (d) bacorrected nonparametrc correlogram. Lag are n km.

5 ERAD 00 - THE SIXTH EUROPEA COFERECE O RADAR I METEOROLOGY AD HYDROLOGY 6. Summary and dcuon We have hown that the etmaton of nonparametrc correlogram, whle computatonally very convenent, uffer from a hort-range ba that can be much larger than the ba n conventonal parametrc emvarogram etmaton. The dfferent performance of the two etmator due to the fact that the nonparametrc correlogram etmaton mplctly ue the mple plug-n etmator of the ample varance of the patal feld under conderaton. When correlated patal feld are oberved n a fnte doman, th can ubtantally underetmate the varance. In contrat, n the conventonal etmaton of parametrc emvarogram, the varance of the proce determned by the ll of the emvarogram model. For potvely correlated feld th alo underetmate the varance, but much le o than the ample varance. It ha alo been hown that the nonparametrc correlogram can be corrected n a traghtforward way f the proce varance etmated from the emvarogram ll. The relevance of the ba dcued here and the necety for ba correcton wll depend trongly on the data under conderaton and on the context n whch the correlaton map are to be appled. In tuaton where the correlaton length of the data mall compared to the doman ze, and where the focu on tmely calculaton and on the bet etmate of a patally nterpolated feld, t may well be utfed to opt for the uncorrected correlogram etmator. Great care hould be taken, however, when ung uncorrected nonparametrc correlogram for trongly correlated feld oberved n comparatvely mall doman, and when the focu of the applcaton not only on the bet etmate of the nterpolated feld, but alo on the etmaton of the nterpolaton uncertanty (e.g., krgng varance). Analye of the knd preented above can help n utfyng the choce of an approprate etmator. Reference Fg. 6. Scatterplot of emvarogram ll v. ample varance. [] Velaco-Forero C. A., Sempere-Torre D., Caraga E. F., Gómez-Hernández J. J., 009: A non-parametrc automatc blendng methodology to etmate ranfall feld from ran gauge and radar data. Advance n Water Reource, 3, [] Schabenberger O., Gotway C., 005: Stattcal method for patal data analy. Taylor & Franc, 488pp. [3] Yao T., Journel A., 998: Automatc modelng of (cro) covarance table ung fat Fourer tranform. Mathematcal Geology, 30, [4] Velaco-Forero C. A., 009: Optmal etmaton of ranfall feld for hydrologcal purpoe n real tme. PhD The, Centro de Recerca Aplcada en Hdrometeorologa, Unvertat Poltècnca de Catalunya, 8pp.