Assessing the Performance of the Ensemble Kalman Filter for Land Surface Data Assimilation

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1 2128 M O N T H L Y W E A T H E R R E V I E W VOLUME 134 Assessng the Performance of the Ensemble Kalman Flter for Land Surface Data Assmlaton YUHUA ZHOU, DENNIS MCLAUGHLIN, AND DARA ENTEKHABI Ralph Parsons Laboratory, Department of Cvl and Envronmental Engneerng, Massachusetts Insttute of Technology, Cambrdge, Massachusetts (Manuscrpt receved 4 February 2005, n fnal form 8 July 2005) ABSTRACT The ensemble Kalman flter provdes an easy-to-use, flexble, and effcent opton for data assmlaton problems. One of ts attractve features n land surface applcatons s ts ablty to provde dstrbutonal nformaton about varables, such as sol mosture, that can be hghly skewed or even bmodal. The ensemble Kalman flter reles on normalty approxmatons that mprove ts effcency but can also compromse the accuracy of ts dstrbutonal estmates. The effects of these approxmatons can be evaluated by comparng the condtonal margnal dstrbutons and moments estmated by the ensemble Kalman flter wth those obtaned from a sequental mportance resamplng (SIR) partcle flter, whch gves exact solutons for large ensemble szes. Comparsons for two land surface examples ndcate that the ensemble Kalman flter s generally able to reproduce nonnormal sol mosture behavor, ncludng the skewness that occurs when the sol s ether very wet or very dry. Its condtonal mean estmates are very close to those generated by the SIR flter. Its hgher-order condtonal moments are somewhat less accurate than the means. Overall, the ensemble Kalman flter appears to provde a good approxmaton for nonlnear, nonnormal land surface problems, despte ts dependence on normalty assumptons. 1. Introducton Ensemble-based data assmlaton methods are becomng popular n many of the earth scences, largely because they are easy to use, flexble, and make relatvely few restrctve assumptons (see the revew by Evensen 2003). In partcular, the ensemble Kalman flter has recently been suggested as a practcal opton for real-tme estmaton of land atmosphere fluxes from remote sensng data (Rechle et al. 2002; Marguls et al. 2002; Crow and Wood 2003; Rechle and Koster 2003; Entekhab et al. 2004). Ensemble estmaton procedures have the advantage of provdng dstrbutonal nformaton about uncertan varables, ncludng approxmate margnal dstrbutons, quantles, and hgherorder moments. Ths nformaton s partcularly useful n land surface applcatons, where varables such as sol mosture can be hghly skewed toward the wet or dry ends and can even be bmodal, dependng on the tme Correspondng author address: Denns McLaughln, Bldg , 15 Vassar Street, Cambrdge, MA E-mal: dennsm@mt.edu and space scale consdered (Rodrguez-Iturbe et al. 1991). In such cases, means and covarances alone may not adequately characterze varablty. In sequental flterng the dstrbutonal propertes of an uncertan state x t, gven a set of measurements y 0:t taken through tme t, are conveyed by the condtonal probablty densty p(x t y 0:t ). The random replcates generated by ensemble methods may be used to compute fnte sample approxmatons to ths densty and ts moments. When new measurements become avalable some verson of Bayes s theorem s typcally used to update the replcates (and the correspondng dstrbutonal approxmatons). The accuracy of ths update depends on the assumptons made when applyng Bayes s theorem as well as the number of replcates. The ensemble Kalman flter s partcularly effcent because t reles on normalty assumptons that greatly smplfy the update process. But ths smplfcaton can also lmt the flter s ablty to provde accurate dstrbutonal nformaton. Here we evaluate the accuracy of the ensemble Kalman flter by comparng ts dstrbutonal estmates to those of a less effcent ensemble method that reles on an exact Bayesan update. Ths s done for two examples that provde useful nsght about the en Amercan Meteorologcal Socety

2 AUGUST 2006 Z H O U E T A L semble Kalman flter s performance n land surface applcatons. 2. Ensemble flterng Many of the nonlnear flterng problems encountered n the earth scences rely on dscrete tme state and measurement equatons of the followng form: x t f x t 1, u t and 1 y t g x t v t, where x t s the system state vector wth an uncertan ntal condton x 0, u t s a vector of uncertan model nputs (not necessarly addtve), y t s the measurement vector, and v t s a vector of addtve random measurement errors. In a land surface problem x t could be a vector of sol mosture values n dfferent pxels and layers, u t a vector of precptaton rates, and y t a vector of mcrowave radometer measurements ndrectly related to sol mosture. The uncertan varables x 0, u t, and v t are assumed to have known pror probablty dstrbutons and the measurement error vectors at dfferent tmes are assumed to be ndependent. The functons f() and g() represent spatally and temporally dscretzed models of the system dynamcs and measurement process. The objectve of flterng s to characterze the current state x t from y 0:t, the set of all measurements obtaned at dscrete tmes n the nterval [1, t]. The deal probablstc characterzaton s the condtonal probablty densty p(x t y 0:t ), whch conveys everythng known about x t gven y 0:t. Snce ths multvarate densty s dffcult to compute or nterpret for large problems we typcally focus on partcular propertes of p(x t y 0:t ), such as ts moments and unvarate margnal denstes of p(x t y 0:t ). In flterng applcatons t s convenent to dstngush two sequental estmaton operatons: 1) propagaton of the state from one measurement tme to the next (forecastng) and 2) updatng of the propagated state wth the new measurement (analyss). If the complete densty p(x t y 0:t ) s desred, forecastng s carred out by dervng p(x t y 0:t 1 ) from p(x t 1 y 0:t 1 ) (e.g., usng the Fokker Planck equaton) and analyss s carred out by dervng p(x t y 0:t ) from p(x t y 0:t 1 ) (e.g., usng Bayes s theorem; Jazwnsky 1970). The requred calculatons are generally feasble only for very small problems. Ensemble methods are able to provde a practcal alternatve to exact Bayesan solutons because they rely on dscrete approxmatons of the denstes p(x t y 0:t 1 ) and p(x t y 0:t ). The approxmatons can be expressed as 2 N p x t y 1:t 1 1 N p x t y 1:t 1 w t t 1 w t t x t x t t 1 and 3 x t x t t. These approxmatons replace each contnuous densty by a sum of N Drac delta denstes located at the randomly generated state vectors, or replcates, x t t 1 or x t t, for 1,...,N. The Drac delta terms (and the correspondng replcates) for each approxmaton are assgned dscrete probabltes (or weghts) w t t 1 or w t t, respectvely. If the weghts n each expanson sum to unty, the ntegrals of (3) and (4) yeld stepwse approxmatons of the contnuous cumulatve dstrbuton functons for p(x t y 0:t 1 ) and p(x t y 0:t ), respectvely. The random replcates and correspondng weghts can be generated n a varety of ways. The sequental mportance resamplng (SIR) partcle flter and ensemble Kalman flter dscussed n ths paper are two partcular alternatves. The SIR and ensemble Kalman flters share the same forecastng step. To examne the mechancs of ths step, suppose that replcate at t 1 has the value x t 1 t 1 wth weght w t 1 t 1 1/N. The nonlnear state Eq. (1) can be used to compute the value of ths replcate at tme t from the value at t 1, gvng x t t 1 4 f x t 1 t 1, u t and 5 w t t 1 1 N. Note that ths operaton requres generaton of a random nput replcate u t, whch s a random sample drawn from the specfed pror nput probablty densty p(u t ). Equatons (5) and (6) yeld the followng approxmaton for the forecast probablty densty: N p x t y 1:t 1 1 N 1 N 1 N 1 x t x t t 1 6 x t f x t 1 t 1, u t. 7 Ths forecastng step s just a Monte Carlo-based procedure for dervng p(x t y 0:t 1 ) from p(u t ) and p(x t y 0:t 1 ). The dfferng analyss steps for the SIR and ensemble Kalman flters are dscussed n the followng paragraphs. a. The SIR partcle flter Partcle flters are a class of sequental Bayesan ensemble algorthms that can be derved from a dscrete verson of Bayes s theorem. Arulampalam et al. (2002)

3 2130 M O N T H L Y W E A T H E R R E V I E W VOLUME 134 provde a useful tutoral that shows how several dfferent partcle flterng algorthms may be developed from the perspectve of sequental mportance samplng (SIS). We use the SIR flter here because t s easy to mplement and converges to the exact Bayesan soluton as the number of replcates approaches nfnty. It s also well suted for the land surface applcaton, where uncertan tme-dependent nputs are generally more mportant than ntal condton errors. In other applcatons, other types of partcle flters may gve better performance for a gven number of partcles. The SIR algorthm adopts the approxmaton of (5) and (6) durng the forecastng step of flterng. The analyss step s based on the followng form of Bayes s theorem: p x t y 1:t p y t x t p x t y 1:t 1 cp y p y t y 1:t 1 t x t p x t y 1:t 1, 8 where c s a normalzng constant that ensures that p(x t y 0:t ) ntegrates to one. If we substtute the Drac expansons for p(x t y 0:t 1 ) and p(x t y 0:t ) nto (8) we can relate the analyss densty replcate values and weghts of the unknown analyss densty (left-hand sde) to those of the known forecast densty (rght-hand sde). In the SIR flter the analyss replcate values are ntally kept the same as the forecast values and only the analyss weghts are changed. Ths gves x t t x t t 1 w t t cp y t x t t and w t t 1 c N p y t x t t 1, cp y t x t t 1 w t t where p(y t x t t 1) s the lkelhood functon for the propagated replcate x t t 1 The lkelhood functon can be readly computed f the measurement error s addtve (as assumed here) snce p y t x t t 1 p yt y t x t t 1 p t y t g x t t 1, 11 where p t s the known (e.g., normal) probablty densty of the measurement error v t. The lkelhood functon can be vewed as a measure of the closeness of the replcate x t t 1 to the measurement y t. We could substtute (9) and (10) drectly nto (4) to obtan an approxmaton of the analyss probablty densty but the result may be unsatsfactory unless the number of replcates s very large. Ths s because (10) gves replcates closer to the measurements much more weght than those that are farther away. Ths can result n the collapse of the ensemble to a very small number of replcates wth hgh weghts, gvng a very coarse dscrete representaton of the analyss probablty densty. To prevent ths, the SIR flter resamples the ensemble wth replacement N tmes. The probablty that replcate s selected on sample k s equal to ts weght: p replcate selected on sample k w t t. 12 By constructon, ths resamplng operaton creates a new analyss ensemble of N equally lkely replcates wth the followng values x k t t and weghts w k t t (for k 1,...,N): and x k t t replcate value selected on sample k 13 w k t t 1 N. 14 The new analyss ensemble s a subset of the old analyss ensemble. Old replcates wth hgh weght are more lkely to be repeated n the new ensemble and old replcates wth low weght are more lkely to be omtted. Once the resamplng operaton s completed (13) and (14) can be substtuted nto (4) to gve N p x t y 1:t 1 N k 1 x t x k t t. 15 The new equally weghted resampled replcates can then be propagated from t to t 1, followng the procedure gven n (5) and (6) (wth t replaced by t 1 and k by ). Although many of the resampled analyss replcates at t have the same value, these values dverge n the subsequent propagaton to t 1 because of the nfluence of the random nput nose u t 1. Ths keeps the ensemble from collapsng and s why the SIR approach works best for problems wth random nputs. The SIR flter s ensemble statstcs (margnal denstes, moments, etc.) can be shown to converge to ther exact counterparts as the number of replcates approaches nfnty. The verson of the SIR flter descrbed here assumes that the measurement errors are addtve and ndependent over tme but does not restrct the form of the probablty denstes for x t, u t,or v t or the form of the functons f() and g(). The prmary dsadvantage of the SIR flter s the large number of replcates requred to accurately represent the multvarate condtonal probablty denstes of x t. When the number of measurements exceeds a few hundred the SIR flter s not practcal for land surface problems. However, t provdes a very useful performance benchmark for small problems snce t yelds optmal cond-

4 AUGUST 2006 Z H O U E T A L tonal denstes (as well as condtonal means and other moments) f the ensemble s suffcently large. b. The ensemble Kalman flter (EnKF) Lke the SIR flter, the ensemble Kalman flter uses the Drac expansons of (3) and (4) to approxmate the condtonal probablty denstes of x t and t adopts the approxmaton of (5) and (6) durng the forecastng step. However, the ensemble Kalman flter makes more assumptons at the analyss step. The Kalman flter analyss step can be derved from varous perspectves. Here we take a Bayesan or dstrbuton-orented perspectve because we are nterested n the flter s ablty to estmate propertes of the condtonal densty p(x t y 0:t ). It s generally very dffcult to derve an exact closed form expresson for p(x t y 0:t ) from Bayes s theorem, especally for problems wth nonlnear dynamcs and measurement operators. However, t s possble to obtan an exact soluton when the forecast states and measurements are jontly normal. Ths typcally occurs only when the state and measurement equatons are lnear and all sources of uncertanty are normally dstrbuted. In ths specal case the analyss densty gven by (8) s normal and completely defned by the followng mean and covarance, whch are the update expressons of the classcal Kalman flter: x t t x t t 1 K t y t g x t t 1 and 16 C xx,t t C xx,t t 1 K t C yx,t t 1, 17 where the overbars ndcate expected values, the Cs are covarances, and K t C xy,t t 1 C yy,t t 1 C,t In practce, adoptng a jont normalty assumpton s equvalent to assumng that the forecast and measurement denstes are adequately characterzed by ther means and covarances (.e., hgher-order moments are gnored n the analyss step). It s possble to use the Kalman update expressons even when the jont normalty assumpton does not apply. In ths case the condtonal statstcs produced by the Kalman flter may not match the true values but they may be close enough to be useful. In an ensemble verson of the Kalman flter we need to generate an ensemble of analyss replcates at t, for propagaton from t to t 1. The sample mean and covarance of ths ensemble should converge to the mean and covarance of (16) and (17) n the lmt as the number of replcates approaches nfnty. There are many ways to generate analyss replcates that satsfy ths requrement. In nonlnear applcatons t s best to use an ensemble generaton method that preserves at least some of the nonnormal characterstcs of the forecast ensemble when normalty assumptons do not apply. One way to accomplsh ths s to generate an analyss ensemble drectly from the forecast ensemble, usng the followng algorthm (Evensen 1994, 2003): x t t w t t x t t 1 w t t 1 1 N, K s,t y t v t g x t t 1 and where v t s a sample drawn from the measurement error probablty densty p(v t ) and K s,t s a sample estmate of the Kalman gan K t, T T K s,t X t t 1 Y t t 1 Y t t 1 Y t t 1 C,t The columns of the sample matrces X t t 1 and Y t t 1 are constructed from the mean-removed replcates of x t t 1,g(x t t 1 ), and v t. The ensemble Kalman flter algorthm of (19) through (21) produces analyss replcates that converge to the exact Bayesan soluton for normal states and measurements. When there are devatons from normalty the flter s suboptmal but the replcates are able to nhert nonnormal propertes from the forecast. There are a number of other versons of the ensemble Kalman flter that use dfferent approaches for generatng nonnormal ensembles that conform to (12) and (13) (Tppett et al. 2003). Although each of these has advantages n certan stuatons the basc concepts are smlar to Evensen s (1994) classcal ensemble Kalman flter. Moreover, we have found that the performance dfferences for land surface applcatons are not substantal. For that reason we focus on the classcal verson of the flter descrbed above. 3. A smple nonlnear land surface data assmlaton example Sol mosture s one of the key states controllng the parttonng of water and energy fluxes at the land atmosphere boundary. It s lkely to be skewed to the wet end (after precptaton) or the dry end (after a prolonged drydown perod). Here we use a smple scalar example motvated by sol mosture behavor to llustrate the two nonlnear flters descrbed n the prevous secton. Suppose a scalar sol mosture value x at a partcular measurement tme has the followng forecast probablty densty:

5 2132 M O N T H L Y W E A T H E R R E V I E W VOLUME 134 FIG. 1. Estmates of scalar sol mosture state statstcs for a skewed pror probablty densty where y s the actual observaton. (a) Pror probablty densty functon (pdf): p(x) 27.7 exp( x/0.1), 0.1 x 0.5; (b) posteror pdf for y 0.15, R 0.05, estmated wth SIR and ensemble Kalman flters; (c) Posteror mean vs y ; and (d) posteror covarance vs y. The theoretcal Bayesan soluton s also plotted. p x 27.7 exp 0.1 x 0.1 x Ths truncated exponental densty s shown n Fg. 1a. The assocated mean x s 0.19 and the varance s C xx (0.08) 2. We suppose that a sngle measurement y x w s taken, where w s a zero mean normally dstrbuted addtve error ndependent of x wth standard devaton C (0.05) 2. For ths problem the analyss probablty densty p(x y) may be derved n closed form from Bayes s theorem, p x y cp y x p x. 23 Ths exact analyss densty s plotted n Fg. 1b for a measurement value y , together wth the results obtaned from an SIR flter and an ensemble Kalman flter, each usng replcates (ths large sample sze essentally elmnates samplng error problems). The SIR flter closely approxmates the skewed exact analyss densty. The ensemble Kalman flter analyss densty s much more normal n shape, reflectng the nfluence of the normally dstrbuted measurement perturbatons v added n the update step. As the measurement error becomes larger the Kalman gan eventually becomes very small, the forecast replcates domnate, and the Kalman analyss densty becomes more skewed. The exact, SIR flter and ensemble Kalman flter analyss means are plotted versus the measurement value y 0 n Fg. 1c. The Kalman flter analyss mean devates only slghtly from the exact and SIR flter means. Fgure 1d shows that the analyss standard devatons for the two flters behave qute dfferently. The SIR flter standard devaton depends on the measurement value whle the ensemble Kalman flter standard devaton does not. So the Kalman flter underestmates uncertanty for mdrange measurements and overestmates uncertanty for low or hgh measurement values. Although ths scalar example s very smple t suggests that dfferences between the SIR and ensemble Kalman flters for land surface problems may be more apparent n the hgher-order moments than n the analyss means. We nvestgate ths hypothess further n the next secton. 4. Formulaton of an Observng System Smulaton Experment (OSSE) In ths secton we descrbe a land surface smulaton experment that enables us to compare the performance of the suboptmal ensemble Kalman flter to an optmal SIR flter for a realstc land surface applcaton. The problem s to characterze sol mosture and evapotranspraton from remotely sensed passve mcrowave (radometer) measurements. Land surface dynamcs are descrbed by the Communty Land Model (CLM, verson 2.0; Bonan 1996; Bonan et al. 2002). Radometer measurements are descrbed by a nonlnear radatve transfer model (Njoku et al. 2002). Input uncertantes and measurement errors are descrbed by

6 AUGUST 2006 Z H O U E T A L statstcal models that are ntended to realstcally represent natural varablty. These models determne how the replcates of the ensemble flters are generated. a. The land surface and radatve transfer models The CLM s a nonlnear spatally dstrbuted model that descrbes energy, momentum, water, and CO 2 exchange between the land and the atmosphere. Dynamc nputs to the model nclude precptaton, wnd speed, ar temperature, pressure, humdty, and solar radaton. Tme-nvarant nputs nclude sol and vegetaton classfcatons. The model s dscretzed nto square pxels that are each dvded nto several sol layers. Mosture and heat can only move vertcally wthn ndvdual pxels. Further detals are dscussed n Bonan (1996) and Bonan et al. (2002). Although mosture does not flow between pxels the states n dfferent pxels are correlated by vrtue of ther dependence on spatally correlated nputs such as precptaton and vegetaton. The study regon for our computatonal experment reflects condtons at the southern Great Plans (SGP97) ste n eastern Oklahoma. Ths km km regon s shown n Fg. 2. It s dscretzed over a6 6 grd of (approxmately) 3.12 km 3.12 km estmaton pxels wth eght sol layers n each pxel. The study regon s small enough to be feasble for a SIR flter assmlaton whle large enough to reveal the mpacts of horzontal correlaton. The land use s assumed to be cropland wth a loam sol and the sol layers have thcknesses (from top to bottom) of 2, 3, 5, 8, 12, 20, 57, and 88 cm, respectvely. The CLM model states nclude sol mosture and sol temperature n the center of each sol layer as well as surface sol temperature, canopy temperature, and canopy-ntercepted water, for a total of 684 states n our 36-pxel grd. The CLM derves evapotranspraton from these states. The study perod corresponds to a 28-day feld campagn conducted from 0000 UTC 19 June 1997 through 1500 UTC 16 July 1997 (Marguls et al. 2002). Input data are generated and the CLM s run for a 1-h tme step. Synthetc radobrghtness measurements can be related to sol mosture through sol reflectvty, as descrbed by the Fresnel equaton. For our experment ths process s descrbed by the followng expresson for brghtness temperature (Njoku et al. 2002): T b T s 1 r H e T c 1 1 e 1 r H e, 24 FIG. 2. Multple scales used n the land surface OSSE. Precptaton data are avalable n a sngle 100 km 100 km (GPCP) pxel, synthetc radometer measurements are generated n a sngle km km pxel, and estmates are computed n 36 pxels, each 3.12 km 3.12 km, nested nsde the radometer pxel. where T s and T c are the surface and canopy temperature (K), respectvely, and r H s the horzontal polarzaton sol reflectvty. For L-band (1.4 GHz) mcrowave, the vegetaton can be consdered predomnantly absorbng wth a small sngle scatterng albedo, and the vegetaton opacty along the slant path s gven by (Jackson and Schmugge 1991) bw cos, 25 where w s vegetaton water content (kg m 2 ), b s a vegetaton-specfc parameter, and s the ncdence angle. The vegetaton water content s derved from normalzed dfference vegetaton ndex data (Jackson et al. 1999). Rough surface reflectvty s derved from the procedure descrbed by (Choudhury et al. 1979) r H r H exp h cos 2, 26 where r H s the smooth surface reflectvty and h s a vegetaton-specfc parameter. In our experment, w, h, and b have the values 0.3 kg m 2, 0.1, and 0.04, respectvely. The vew angle s set to zero and the scatterng albedo s b. Uncertan model nputs and measurement errors The prmary sources of uncertanty n land surface applcatons are tme-nvarant sol propertes and tmedependent meteorologcal nputs, ncludng precptaton and ntal condtons. In the ensemble approach, random replcates for each of the uncertan nputs are provded to the CLM, whch generates random replcates of the land surface states. Correspondng rado-

7 2134 M O N T H L Y W E A T H E R R E V I E W VOLUME 134 TABLE 1. Summary of uncertan nputs and measurement errors for the land surface smulaton experment. Varable Specfed nomnal value Uncertantes n replcates Sol fractons (sand slt clay) Loam over entre study regon Unformly dstrbuted ponts n loam secton of sol trangle Vegetaton Cropland wth LAI 1.6 and SAI 0.4 (June); LAI 1.3 and SAI 0.8 (July) Spatally uncorrelated multplcatve unform nose U(0.85, 1.15) for LAI Humdty, solar radaton, wnd speed Oklahoma Mesonet tme seres at El Reno, assumed to apply over entre study regon Spatally and temporally uncorrelated multplcatve unform nose: relatve humdty: U(0.9, 1.1); solar radaton: U(0.9, 1.1); wnd speed: U(0.7, 1.3) Ar temperature Oklahoma Mesonet tme seres at El Reno, assumed to apply over entre study regon Spatally and temporally uncorrelated addtve unform nose U( 4, 4 K) Precptaton GPCP 1 daly data for SGP97 regon Nomnal GPCP values downscaled n tme from daly to hourly values wth random pulses model. Temporally downscaled replcates downscaled n space from 100- to 3.1-km pxels wth multplcatve cascade model Intal sol mosture (at t 10 days) Specfed sol mosture profles Spatally uncorrelated addtve Gaussan nose N(0.0, 0.3) Intal sol temperature (at t 10 days) Specfed temperature profles Spatally uncorrelated addtve Gaussan nose N(0.0, 4 K) Radobrghtness measurement Smulated true value at 18.3 km 18.3 km scale Temporally uncorrelated addtve Gaussan nose N(0.0, 3 K) brghtness values at the estmaton pxel scale are generated by the radatve transfer model of (24). The tme-dependent random nputs can cause the ensemble to spread durng the propagaton step whle assmlaton of radobrghtness measurements can cause the ensemble to narrow at the analyss step. These effects are moderated by the physcs of the problem, whch constrans the states to le n lmted ranges (e.g., the volumetrc sol mosture must le between 0.0 and the porosty, whch s less than 1.0). The uncertan nputs are generated by transformng nomnal nput values to obtan sets of physcally realstc replcates. Ths s done n varous ways, dependng on the varable. Table 1 lsts the uncertan nputs and measurement errors consdered n our smulaton experment. Note that dfferent methods are used to ntroduce randomness for dfferent nputs. The sol, vegetaton, and precptaton nputs deserve some elaboraton. The nomnal sol s assumed to be loam throughout the study regon. Loam corresponds to a certan secton of the classcal slt sand clay sol trangle. The sol propertes assocated wth dfferent replcates and dfferent pxels are obtaned by selectng random ponts n the loam secton and then readng off the correspondng slt, sand, and clay fractons that are used by CLM. Dfferent ndependent random samples are taken n dfferent pxels and sol layers so sol property fluctuatons are not correlated over space. The vegetaton type s assumed to be cropland through the study regon. The CLM characterzes landuse types n terms of the leaf area ndex (LAI) and the stem area ndex (SAI). It uses these ndces to compute varous model vegetaton parameters that control net radaton, energy parttonng, and ntercepted water capacty. In our experment the nomnal LAI for cropland s 1.6 for June and 1.3 for July. The nomnal SAI s 0.4 for June and 0.8 for July. Indvdual LAI replcates are generated by multplyng the nomnal value by a unformly dstrbuted random varable n the range (0.85, 1.15). SAI s treated as a determnstc nput. Precptaton dsplays sgnfcant correlaton n tme and space and has a patchy pattern that cannot be reproduced wth smple multplcatve or addtve perturbatons to spatally unform nomnal values. A more realstc opton s to generate small-scale replcates by downscalng (or dsaggregatng) larger-scale measurements of real precptaton over both tme and space (S. A. Marguls 2004, personal communcaton). Downscalng reles on statstcal models of small-scale varablty. Our smulaton experment uses nomnal precptaton data from the Global Precptaton Clmatology Project (GPCP). These daly data are avalable n the SGP97 regon at a spatal resoluton of 1 (100 km 100 km). The GPCP tme seres for SGP97 durng the 28-day tme perod of nterest n our experment s shown n Fg. 3d. We need to downscale ths GPCP data from daly to hourly values n tme and from 100 km 100 km to 3.1 km 3.1 km values n space, as ndcated n Fg. 2. Our temporal downscalng procedure s based on a

8 AUGUST 2006 Z H O U E T A L FIG. 3. Spatal and temporal ranfall dsaggregaton model. (a) RPM for temporal dsaggregaton of daly ranfall, (b) multplcatve cascade model for ranfall spatal dsaggregaton, and (c) one realzaton from the random cascade model over a grd. probablstc rectangular pulses model (RPM; Marguls and Entekhab 2001) that s constraned to reproduce observed daly totals (see Fg. 3a). The RPM treats ranfall events as random rectangular pulses wth an exponentally dstrbuted constant ntensty r, and duraton t r and a unformly dstrbuted arrval tme between 0 and 24 t r. Dfferent RPM replcates have dfferent hourly ranfall values. A gven replcate may have no ranfall n any partcular hour but ts 24 hourly values must add up to the observed GPCP daly value. In our experment the RPM mean ntensty s 3.2 mm h 1 for June and 2.3 mm h 1 for July, and the mean tme between storms s 5.0 h for June and 8.0 h for July. These were estmated from clmatologcal data (Hawk and Eagleson 1992). The temporal downscalng procedure provdes 1-h precptaton replcates at the 100 km 100 km GPCP measurement scale. These coarse-resoluton replcates can be downscaled to the 3.1 km 3.1 km estmaton pxel scale f we suppose that ranfall follows a multplcatve cascade model that relates ntenstes at dfferent scales (Gupta and Waymre 1993; Gorenburg et al. 2001). Ths model can be portrayed as a sx-level tree composed of groups of pxels (nodes) coverng regons of dfferent areas (see Fg. 3b). The top (root) node defnes the coarsest scale (one GPCP pxel) whle the bottom nodes defne the fnest scale (the 1024 estmaton pxels contaned n the GPCP pxel). The ranfall value at a gven node s obtaned by multplyng the value at the next coarsest node (the parent) by a random lognormally dstrbuted coeffcent W(s), W s exp w s w 2 s 2, 27 where w(s) N[0, 2 w(s)], w (s) 2 0.3(s 1), and the scale ndex s ncreases from 1 at the root node of the cascade to 6 at the fnest scale. A typcal realzaton from ths random multplcatve cascade s shown n Fg. 3c. The cascade model generates ranfall that has a patchy pattern and s correlated over space. In our smulaton experment the cascade model generates spatally downscaled ranfall on a grd wth a fnest scale resoluton of about 3.1 km, enforcng the same spatal pattern for each replcate n each hour of a gven rany day but allowng the ranfall ntensty to change every hour. Ranfall ntenstes at the fnest scale are normalzed for each replcate to ensure that the total ranfall at ths scale s equal the total ranfall at the GPCP scale. A 6 6 porton of ths grd provdes the 3.1 km 3.1 km ranfall data needed by the CLM model. The CLM model s started at t 10 days wth random ntal condtons generated by perturbng unform sol mosture and temperature profles. Each replcate s run forward wth the model for 10 days to t 0 to allow mosture n ndvdual pxels to redstrbute n accordance wth local sol propertes. The resultng sol mosture and temperature replcates ntalze the SIR and Kalman flter ensemble smulatons. c. Smulaton experment specfcatons For our synthetc experment truth s defned by the state from a sngle CLM run obtaned for a partcular set of sol, vegetaton, meteorologcal, and ntal condton replcates, as descrbed above. The CLM states and assocated sol propertes for ths truth replcate are then used n (24) to generate a synthetc

9 2136 M O N T H L Y W E A T H E R R E V I E W VOLUME 134 FIG. 4. Ensembles from SIR soluton at pxel 9 and the assocated GPCP ranfall data: (a) ensemble of the frst-layer sol mosture ; (b) ensemble of evapotranspraton and 1 std dev of the ensemble; (c) ensemble of surface temperature; and (d) GPCP 1DD daly ranfall data tme seres. The astersks on tme axs of (d) represent the measurement tmes. brghtness temperature measurement at 1500 UTC every day durng the 28-day smulaton perod. Ths measurement s defned at a coarser scale than the model states, reflectng the lower resoluton of antcpated satellte mcrowave radometer measurements. In partcular, we assume that the mcrowave measurement covers an 18.3 km 18.3 km area (6 6 pxels) and s the average of the 36-pxel-scale brghtness values computed from (24). At each measurement tme a zero-mean normally dstrbuted random perturbaton s added to the averaged brghtness temperature to account for the effect of measurement nose. Ths set of nosy measurements s provded to the two ensemble flters. 5. Results of the smulaton experment It s useful to start our comparson of the SIR and ensemble Kalman flters by examnng surface (top layer) sol mosture, surface sol temperature, and evapotranspraton replcates produced by the SIR flter at a typcal pxel (pxel 9). These tme seres are shown n Fg. 4, together wth the applcable 1 daly (1DD) GCPC precptaton record for the study perod. The red lne s the true replcate, the thck blue lne s the mean of the SIR flter ensemble, and the thn cyan lnes are the ndvdual SIR flter replcates. Before t 100 uncertanty n sol mosture, ndcated by the ensemble spread, s manly due to uncertantes n ntal condtons, sol propertes, and LAI. After ranfall events occur, the uncertantes n surface layer sol mosture prmarly reflect uncertantes n precptaton. Note that the ensemble spread s narrower durng the dry perods, when the absence of ranfall makes t easy to nfer that sol mosture values are low, even wthout the added nformaton provded by radobrghtness measurements. By contrast, the ground temperature ensemble spread s wder durng dry perods and narrower durng wet perods. The spread of the evapotranspraton ensemble depends strongly on tme of day, peakng just after noon. Ths s more apparent n the evapotranspraton ensemble standard devaton plot ncluded just below the replcate plot. Replcates from the ensemble Kalman flter have a very smlar structure.

10 AUGUST 2006 Z H O U E T A L It s mportant that our comparson of the SIR and ensemble Kalman flters s based on enough replcates to ensure that samplng error s not a sgnfcant factor. Fgure 5 shows the spatal root-mean-squared error (rmse) for the top-layer sol mosture, computed over all analyss tmes, where error s defned as the dfference between the analyss ensemble mean and the true value. Also plotted are error bars that show plus or mnus one standard devaton of the rmse, computed over q flter runs started wth dfferent flter ensemble random seeds. The truth and measurements are kept the same for these runs. Clearly, the SIR flter needs more replcates to converge, although t eventually gves nearly the same rmse as the Kalman flter. Ths s not surprsng, consderng that the converged SIR flter needs to resolve hgher-order dstrbutonal propertes that are gnored by the Kalman flter. Fgure 6 shows margnal forecast (left) and analyss (rght) probablty denstes for pxel 9 surface sol mosture for some typcal analyss tmes. Open-loop (uncondtonal) denstes are also shown for comparson. At the frst analyss tme, just pror to the frst measurement (t 15 h), both flters and the open loop share the same forecast densty wth a skewness of 0.2 and kurtoss of 3.5 (snce there have not yet been any measurements). The dfference n the SIR and Kalman flter analyss denstes at ths tme s mnmal. The benefcal effect of the measurement s best revealed by a comparson of the open-loop and analyss denstes. The denstes plotted at tmes t 231 and 279 show condtons durng two rany perods. The skewness to the left n both of the forecast denstes reflects the effects of the precedng drydown. The measurements at both tmes move the densty notceably toward the wet end, producng sgnfcant dfferences between the open-loop and fltered denstes. Here agan, dfferences between the SIR and Kalman flter analyss denstes are mnor. Also, at t 351, after a drydown perod all of the forecast denstes reveal bmodal behavor. Ths bmodalty s lkely because of the propertes of the multplcatve cascade ranfall model, whch tends to produce replcates wth wet or dry patches. The analyss denstes for the SIR and ensemble flters at t 351 are notceably dfferent. After a long perod of drydown, at t 471, the forecast and analyss denstes are all skewed to the dry end and the radobrghtness measurement does not provde much addtonal nformaton about the surface sol mosture. The margnal denstes shown n Fg. 6 llustrate the advantages of takng a dstrbutonal perspectve n data assmlaton. Ensemble means and even means plus FIG. 5. Averaged spatal rmse of surface layer sol mosture at measurement tmes vs replcate numbers. Error bars show 1 std dev of the rmse, computed over q flter runs started wth dfferent flter ensemble random seeds. For ensemble sze n 10, 80, 800, 3200, and , and the run tmes q 40, 10, 8, 6, and 4, respectvely. varances do not always tell the whole story. Physcal condtons such as prolonged wettng or dryng can lead to skewed denstes where the means are much dfferent than the most probable values (modes). Although SIR flters provde accurate nformaton on margnal dstrbutons they are not practcal for large problems. Fortunately, the ensemble Kalman flter seems able to convey much of ths dstrbutonal nformaton, despte ts smplfyng normalty assumptons. Ths s a drect result of the ensemble Kalman flter s ablty to update each replcate rather than just the ensemble mean. Indvdual replcate updatng s able to preserve some skewness and multmodalty, even when the analyss step s suboptmal. To assess global performance, rather than performance at a sngle pxel, we examne n Fg. 7 the tme seres of the dfferences between the ensemble mean and the true replcates for surface sol mosture, evapotranspraton, and ground temperature, all averaged over the entre doman. The errors are shown for the SIR flter, ensemble Kalman flter, and open-loop estmates. The abrupt change n sol mosture error due to assmlaton of brghtness temperature can be observed at analyss tmes for both flters but the mpact of measurements s less clear for the ground temperature. Ths reflects the fact that brghtness temperature s more senstve to sol mosture than to ground temperature. Although evapotranspraton s a dagnostc varable rather than an updated state we can see that the SIR and ensemble Kalman estmates of evapotranspraton beneft from radobrghtness measurements. Both of these generally have lower errors than the open-loop

11 2138 M O N T H L Y W E A T H E R R E V I E W VOLUME 134 FIG. 6. (left) Margnal forecast, (rght) analyss probablty denstes for pxel 9 surface sol mosture for some typcal analyss tmes, and (bottom) daly ranfall seres. Open-loop (uncondtonal) denstes are also shown for comparson. estmate. A closer look at the plots suggest that the study perod can be roughly dvded nto two stages, before and after at t 250. Durng the frst stage, sol mosture s relatvely hgh, so the evapotranspraton s controlled by avalable energy rather than sol mosture. Hence, the open-loop estmates of evapotranspraton and ground temperature are nearly as good as the flter estmates. Durng the second stage, there s a long drydown perod, evapotranspraton s mosture lmted, and open-loop errors are larger than flter errors. Assmlaton of brghtness temperature s clearly more benefcal durng ths stage. Surface brghtness temperatures can be used to estmate subsurface sol mosture profles that are dffcult to observe at large scales. Fgure 8 shows the ensemble mean of the ntegrated sol water depth above 50 cm deep over the entre doman. The ntegrated sol water depth could be vewed as a rough measure of the water avalable to a plant wth a root depth of 50 cm. Here agan, the ensemble Kalman flter gves results that are nearly as good as the SIR flter. Fgure 9 provdes some ndcaton of the dstrbutonal dfferences between the two flters by comparng tme seres of the hgher-order moments (standard devaton, skewness, and kurtoss) of the surface sol mosture for pxel 9. Dfferences between the hgher-order moments produced by the two flters are greater than dfferences between the means. Both flters are able to capture the sgnfcant reducton n varance and ncrease n skewness experenced durng the extended drydown perod after t 300. Heavy ranfall events seem to reduce dfferences between the two flters. It should be noted that the random pulse and multplcatve cascade models tend to generate very nonnormal surface sol mosture densty functons, as shown by the skewness and kurtoss n Fgs. 9b and 9c. The ensemble

12 AUGUST 2006 Z H O U E T A L FIG. 7. Tme seres of the dfferences between the ensemble mean and the true replcates for (a) surface sol mosture, (b) evapotranspraton, (c) ground temperature, all averaged over the entre doman, and (d) GPCP ranfall tme seres. Kalman flter captures much of ths nonnormal behavor, at least for our applcaton. The results of our land surface data assmlaton experment are summarzed n Table 2, whch lsts rmse values obtaned from SIR flter, ensemble Kalman flter, and open-loop means for the four varables of most nterest. It s obvous that the SIR and ensemble Kalman flter errors are comparable n all cases. Taken FIG. 8. The ensemble mean of the ntegrated sol water depth above 50-cm depth over the entre doman.

13 2140 M O N T H L Y W E A T H E R R E V I E W VOLUME 134 FIG. 9. Tme seres of (a) standard devaton, (b) skewness, (c) kurtoss of surface layer sol mosture at pxel 9, and (d) GPCP ranfall tme seres. The thck straght lne n (a) and (b) shows the trend of standard devaton and skewness durng drydown perod. together, our results strongly support the use of the Kalman approxmaton n land surface applcatons of ensemble data assmlaton. 6. Dscusson and conclusons Ths paper consders the performance of the ensemble Kalman flter n a partcular context: land surface data assmlaton. Land surface problems have several dstnctve characterstcs. In partcular, the state equaton s nonlnear and dsspatve and the states are confned to relatvely small ranges, wth probablty dstrbutons that change over tme and are often nonnormal. Precptaton nputs are ntermttent and hghly varable over space and tme, and other nputs, such as sol propertes, are uncertan and dffcult to observe over large regons. The measurement equaton s also nonlnear. Ensemble Kalman flters have been appled to land surface data assmlaton wth reasonable success, despte ther dependence on assumptons that may not apply. Our objectve here has been to better understand the reasons for ths success and to obtan a more complete pcture of the strengths and weaknesses of the ensemble Kalman flterng approach. The smple example descrbed n secton 3 shows that the SIR flter s condtonal moment and margnal densty estmates are very close to ther exact counterparts f the replcate sze s large enough. The ensemble Kal- TABLE 2. Rmse (over tme) of spatally averaged top-layer sol mosture, evapotranspraton, ground temperature, and water depth above 50 cm wth respect to the truth. SIR EnKF Open loop Top-layer sol mosture Evapotranspraton (W m 2 ) Ground temperature (K) Water depth above 50 cm (mm)

14 AUGUST 2006 Z H O U E T A L man flter s condtonal mean estmate s also qute close to the exact value but ts margnal densty and varance are notceably dfferent. Ths example suggests that a converged SIR flter provdes a good bass for evaluatng the ensemble Kalman flter when an exact soluton s not avalable. Sectons 4 and 5 descrbe a more realstc land surface estmaton example that reles on state and measurement models used n operatonal settngs. In ths OSSE we generate hypothetcal true states and measurements so that flter estmaton errors can be evaluated exactly. The example problem s kept small so that the SIR flter s computatonally feasble. The number of replcates needed for ths flter to converge becomes very large when the state and measurement dmensons ncrease much beyond the values used n our example. Ths s why the ensemble Kalman flter, whch converges for much smaller ensemble szes, s preferable to the SIR flter n practcal applcatons. The results of our land surface example reveal that the ensemble Kalman flter performs nearly as well as the SIR flter for most condtons smulated. The surface sol mosture forecast denstes obtaned from the Kalman flter can be qute skewed and even multmodal and are generally smlar to those obtaned from the SIR flter. The unvarate denstes of Fg. 6 make t clear that the normalty assumptons that must be met n order for the ensemble Kalman flter to yeld optmal pont estmates do not prevent t from generatng nonnormal ensembles. Ths s further emphaszed n Fg. 9, whch shows that the skewness and kurtoss of the ensemble Kalman flter sol mosture ensembles can dffer sgnfcantly from those assocated wth normally dstrbuted varables. The ensemble Kalman flter s especally good at reproducng the correct sol mosture condtonal mean. Ths appears to be a consstent result at all tmes and pxels n our experment and t s observed both at the surface (Fg. 7) and ntegrated over the sol column (Fg. 8). Smlar performance s obtaned for evapotranspraton, whch benefts most from radobrghtness measurements when t s lmted by sol mosture. It s worth notng that the structure and tmng of precptaton exert a domnant nfluence on the land surface system. Ths nfluence tends to reduce dfferences between alternatve data assmlaton algorthms that make smlar assumptons about ranfall. The RPM and multplcatve cascade dsaggregaton models used here tend to create very nonnormal sol mosture durng rany perods. In these perods sol mosture s skewed to the hgh end. As the surface mosture decreases through nfltraton and evaporaton, the skewness and kurtoss tend to decrease, makng the ensemble flter s normalty assumptons more approprate. However, the skewness and kurtoss tend to ncrease agan when sol dres and sol mosture s lmted at the low end. Sol propertes also have a strong nfluence on the behavor of the land surface system and the performance of alternatve flters. Open-loop (uncondtonal) predctons of sol mosture are usually better for rapdly nfltratng sandy sols than for less permeable loam or clay sols. Also, sol mosture updates have less mpact on evapotranspraton for sandy sols. In such stuatons dfferences between optmal and suboptmal flterng algorthms are less lkely to be dramatc. Even takng these dstnctve problem features nto account, our overall concluson s that the ensemble Kalman flter provdes surprsngly good performance n the land surface applcaton. Ths apples both to the flter s ablty to characterze nonnormal dstrbutonal propertes and ts ablty to provde accurate condtonal means. We beleve these results support prevous studes that ndcate the ensemble Kalman flter s a good estmaton opton for land surface applcatons. It would be useful to see the results of computatonal experments smlar to ours n other applcaton areas. 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