correcting first-order errors in snow water equivalent ...dust.ess.uci.edu/ppr/ppr_dum07.pdf ·...

Correcting first-order errors in snow water equivalent

estimates using a multifrequency, multiscale

radiometric data assimilation scheme

Michael Durand1 and Steven A. Margulis1

Received 22 September 2006; revised 1 February 2007; accepted 27 April 2007; published 13 July 2007.

[1] A season-long, multiscale, multifrequency radiometric data assimilation experimentis performed to test the feasibility of snow water equivalent (SWE) estimation.Synthetic passive microwave (PM) observations at Advanced Microwave ScanningRadiometer-Earth Observing System frequencies and 25 km resolution and synthetic nearinfrared (NIR) narrowband albedo observations corresponding to Moderate ResolutionImaging Spectroradiometer band 5 (1230–1250 mm) and 1 km resolution are assimilatedinto a land surface model snow scheme using the ensemble Kalman filter. First-ordersources of model uncertainty, including error in precipitation quantity, grain sizeevolution, precipitation spatial distribution, and vegetation leaf area index are modeled.SWE remote sensing retrieval schemes would be of limited value for these scenarioswhere snow depth ranged between 1.0 and 2.0 m, grain size varied in space, significantvegetation was present, and the snowpack sometimes contained liquid water. Nevertheless,the true basinwide SWE is recovered, in general, within a root-mean-square error (RMSE)of approximately 2 cm. Synergy is observed between the PM and NIR measurementsbecause of the complementary nature of the multiscale, multifrequency measurements.Results from the assimilation are compared to those from a pure modeling approach andfrom a remote sensing retrieval approach. The effects of model uncertainty, measurementerror, and ensemble size are investigated.

Citation: Durand, M., and S. A. Margulis (2007), Correcting first-order errors in snow water equivalent estimates using a

multifrequency, multiscale radiometric data assimilation scheme, J. Geophys. Res., 112, D13121, doi:10.1029/2006JD008067.

1. Introduction

[2] Snowpack is a critical freshwater reservoir and playsan important role in both the global water and energycycles. While the need to forecast springtime runoff hasmotivated efforts to estimate snow water equivalent (SWE)in mountainous regions for the past fifty years, the task ofdeveloping robust methods for characterizing hydrologicreservoirs and snowpack in particular has been made moreurgent by the prospect of climate change. For instance,Moteet al. [2005] found that western U.S. snowpack has steadilydeclined for the past half century, and linked this withchanging temperature patterns. According to Goodison etal. [1999], better cold regions model parameterizations andbetter snowpack characterization to monitor such trends aretwo crucial areas of cryospheric research. Two quantities ofparticular interest are the SWE and the snow grain size;grain size controls the albedo, thus exhibiting a significantclimatic control through the surface energy balance.[3] In situ measurement techniques, physical and empir-

ical models of snow accumulation and ablation, and remote

sensing instruments have all been employed for character-ization of snow parameters. These various methodologiesexploit different, but complementary, streams of data, eachof which contains different information about the snowpack,is available at different spatial and temporal resolutions, andhas different levels of uncertainty. A robust estimate ofsnowpack characteristics ideally would merge these datastreams, exploiting their respective strengths. Data assimi-lation (DA) is an ideal framework for merging remotesensing observations and modeled estimates because it isnot site-specific and because it allows for simultaneousassimilation of any number of measurements at differentspatial resolutions and wavelengths. Additionally, it pro-vides a means of weighing the relative uncertainty of themeteorological input data, model parameterizations, andremote sensing observations.[4] The goal of this work is to demonstrate the applica-

tion of the Ensemble Kalman Filter (EnKF) [Evensen, 2003]DA framework to merge synthetic remote sensing measure-ments at multiple scales and frequencies with a land surfacemodel (LSM) in order to characterize SWE over the courseof the accumulation season. This study is a follow-on to aone-dimensional synthetic test in which it is shown that theEnKF methodology can be used to successfully recover thetrue SWE under conditions which generally limit existingapproaches [Durand and Margulis, 2006].

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, D13121, doi:10.1029/2006JD008067, 2007ClickHere

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1Department of Civil and Environmental Engineering, University ofCalifornia, Los Angeles, California, USA.

Copyright 2007 by the American Geophysical Union.0148-0227/07/2006JD008067$09.00

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http://dx.doi.org/10.1029/2006JD008067

[5] The second section of this paper describes some of themain sources of uncertainty associated with modeling SWEand grain size, and reviews some of the major problemsassociated with remote sensing inversion approaches. Thethird section draws from this background to lay out therationale of this project, discussing motivation, sciencequestions and hypotheses. The fourth section describes thedata, models, filter and experimental design used to test thehypotheses. In the fifth and sixth sections, the analysisresults are described and discussed, some conclusions aredrawn in response to the hypotheses, and future researchdirections are suggested.

2. Background

2.1. Snow Modeling

[6] Models have been used successfully for many years topredict the evolution of snowpack states and to predict thetiming of melt [e.g., Jordan, 1991; Xue et al., 2003].Nevertheless, models are subject to the uncertainty ofvarious inputs (e.g., precipitation data), model parameter-izations, and model simplifications (i.e., model structuralerror).2.1.1. Modeling SWE[7] According to Mote et al. [2003], the key in modeling

SWE accumulation at a point is to force the model withaccurate precipitation data. During the accumulation seasonwhen minimal snowmelts, SWE uncertainty may be mod-eled on the basis of precipitation uncertainty since sublima-tion processes generally remove less than 15% of thesnowpack [Hood et al., 1999], and precipitation estimatesare often much more uncertain than 15%. Although satellite-based remote sensing approaches show promise [e.g.,Skofronick-Jackson et al., 2004; Bindschadler et al.,2005], in situ precipitation gages still form the primary sourcefor estimating snowfall. Goodison et al. [1981] indicatethat gage efficiency of less than 0.5 is not uncommon formany gages because of wind and other complications,however. Mountainous terrain tends to further reduce pre-cipitation gage accuracy [Groisman and Legates, 1994].[8] Extending point-scale precipitation measurements to a

spatially continuous field (e.g., for use in a distributedhydrology model) introduces additional error into the esti-mates. In order to perform the disaggregation, an empiricalrelationship between snow cover properties and other geo-physical quantities is often used. The spatial distribution ofsnow at the hundreds of meters scale can be modeled as afunction of elevation [e.g., Sloan et al., 2004;Dingman, 1981],but not without introducing uncertainty due to the manyadditional factors that influence precipitation distribution.[9] The forest canopy plays an important role in the mass

and energy balance at the snow surface [Marks et al., 2006].This presents an additional difficulty, because there isuncertainty associated with vegetation data sets as well asuncertainty associated with how vegetation modulates radi-ative fluxes. Both of these sources of uncertainty propagateinto the modeled estimate of both SWE and grain size.2.1.2. Modeling Grain Size[10] The average grain size in a snowpack layer increases

monotonically as vapor diffuses through the snowpackdriven by the temperature gradient between different partsof the pack [Jordan, 1991] and other physical processes

[Flanner and Zender, 2006]. Details of the physics of graingrowth processes are in general not well understood, despiterecent advances [e.g., Legagneux et al., 2004]. Thus anymodel estimate of grain growth will be uncertain because oferror in empirical parameters used in model formulation,and because of uncertainty in the formulation itself (i.e.,structural error). In addition, vegetation uncertainty plays animportant indirect role in modeling grain size through thesurface energy balance.[11] Thus the accuracy of precipitation gage estimates,

models used to disaggregate gage-based precipitation esti-mates, vegetation data, and grain size parameterizations arefirst-order sources of uncertainty with respect to modelingSWE and grain size. Because of these sources of uncertainty,many scientists have attempted to exploit remote sensingmeasurements for SWE and grain size estimation.

2.2. Inversion

2.2.1. Estimation of SWE From PM Measurements[12] Much work has been done to estimate SWE by

directly inverting remote sensing observations. Chang etal. [1987] and Hallikainen and Jolma [1992], for instance,have developed linear relationships relating remote sensingmeasurements to radiative transfer model (RTM) output andto in situ observations, respectively. As discussed byDurand and Margulis [2006], a number of difficulties havebeen encountered in these efforts, including (1) the dynamic,many-to-one nature of the relationship between the PMsignal and SWE (due especially to the unknown snow grainsize), (2) the masking of the PM signal by vegetation, (3) thecoarse spatial scale of the observations compared with thespatial scale at which SWE varies, and (4) saturation ofPM signals for deep and wet snow.[13] The many-to-one problem has been addressed by

conditioning retrieval algorithms on information about grainsize and vegetation, [e.g., Kelly et al., 2003; Tait, 1998;Pulliainen et al., 1999; Foster et al., 2005]. Theseapproaches incorporate a priori knowledge about the snow-pack, but as regression-based techniques, they generallyassume a static relationship between SWE and the PMsignal or apply simplistic corrections to account for grain size.They generally utilize information from only two PMchannels, despite the fact that there are 14 PM channels mea-sured by the Advanced Microwave Scanning Radiometer–Earth Observing System (AMSR-E) instrument. Exploitingmore channels could overcome the many-to-one issues anddeep snow saturation issues, since lower frequency channelshave greater saturation depths [Surdyk, 2002]. The maskingissue by forest has been dealt with by developing separateregression relationships [Foster et al., 2005;Goı̈ta et al., 2003]for different vegetation types. These approaches are notinherently transferable to other locations, however. Effortsto address the coarse spatial resolution issue by, e.g., down-scaling the PM observations to obtain estimates of SWEat a finer spatial scale have not, to our knowledge, beenreported.2.2.2. Estimation of Grain Size From Near-InfraredMeasurements[14] The high reflectance of snow in the visible spectrum

is regularly exploited in order to map snow covered extent[e.g., Dozier, 1989; Hall et al., 1995]; recent research hasfocused on retrieval of subpixel snow covered area [e.g.,

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Painter et al., 2003]. Because snow reflectance is moresensitive to grain size in the near-infrared (NIR) than in thevisible spectrum, estimates of grain size at high resolutionfor local areas have exploited the behavior of albedo as afunction of grain size in the NIR spectrum. Fily et al. [1997]utilize Landsat Thematic Mapper NIR measurements toestimate grain size, but reported significant error betweenin situ and remotely sensed estimates. Nolin and Dozier[2000] utilize the multispectral behavior of snow reflectancein the NIR to estimate grain size from Airborne Visible/Infrared Imaging Spectroradiometer (AVIRIS) measure-ments; their approach is limited to instruments with highspectral resolution. Painter et al. [2003] use a multiple end-member mixing analysis to estimate subpixel snow coveredarea and grain size from AVIRIS measurements. Kay et al.[2003] map snow grain size and other snow properties usingModerate Resolution Imaging Spectroradiometer (MODIS)/Advanced Spaceborne Thermal Emission and ReflectionRadiometer (ASTER) Airborne Simulator measurementsnear 1.8 and 2.2 mm; they reported a low bias in the grainsize retrievals and suggest the use of MODIS measurementsnear 1.1 mm to deal with the low signal-to-noise ratio theyobserved.

2.3. Assimilation

[15] The difficulties associated with remote sensing in-version for snowpack characterization has motivated theapplication of DA schemes. Assimilation methodologiesmerge various data streams such as modeled and remotelysensed estimates of snow properties by weighing the asso-ciated uncertainties. Previous work that reports assimilationof remote sensing measurements into a prior SWE estimateare reviewed here; see Slater and Clark [2006] for anexample of assimilation of in situ SWE observations. Inaddition to a few early studies referred to by Durand andMargulis [2006], several authors report inverting remotesensing measurements to obtain SWE estimates using themethodologies described in section 2.2.1, then assimilatingthe SWE into an LSM. Sun et al. [2004] report such asynthetic test (using the Extended Kalman Filter), and Donget al. [2007] extend this to assimilate satellite-derived SWEestimates. Andreadis and Lettenmaier [2006] successfullyuse the EnKF to estimate SWE by assimilating MODISmeasurements into a macroscale model using a snowdepletion curve. They also assimilate AMSR-E-derivedSWE estimates, but report a degradation of the LSMestimate ostensibly due to the AMSR-E SWE estimateerror. Indeed, such ‘‘snow data assimilation’’ studies inwhich the radiometric quantities are inverted and assimilatedinto an LSM are subject to all the limitations of the retrievalmethods discussed above. Some authors suggest thatradiometric quantities be assimilated directly [Mätzler,2003; Cline, 2006], applying an RTM to relate radiometricmeasurements to snow states within the assimilationscheme.[16] In line with these suggestions, Pulliainen [2006]

reports assimilating PM radiometric measurements, andconsiders the relative uncertainty of each data stream in ameaningful way in a Bayesian scheme. In that study, in situmeasurements are regional a priori SWE and grain sizeestimates, and PM data are measurements on this prior,

updating both SWE and grain size. A one layer snowpackRTM [Pulliainen et al., 1999], and RTMs of the vegetationand atmosphere are employed to relate the radiometricquantity to the states. In our previous study [Durand andMargulis, 2006], synthetically generated radiometric PMand albedo quantities are assimilated into a lumped, 14-statesnow model using the EnKF. A three-layer RTM of thesnowpack and RTMs of the vegetation and atmosphere areapplied. Despite vegetation, deep snow, and wet snow asubstantial bias in the forcing data is overcome. In thecurrent study, the same technique is applied in a distributedcase. The next section describes how the current studyrelates to existing methods, and lists the questions andhypotheses to be addressed in this study.

3. Rationale

3.1. Motivation

[17] Assimilation of all available satellite measurementsdiscussed in section 2.2 is motivated by their complemen-tary nature. The lowest frequency AMSR-E channels ‘‘seethrough’’ the vegetation and snowpack and provide ameasurement on ground temperature. As frequencyincreases, the PM channels become more sensitive to snowdepth and grain size throughout the snowpack. As frequencyincreases further and approaches the 89 GHz PM channels,snow depth sensitivity decreases, approaching saturation,while the sensitivity to the surface grain size increases.As frequency increases past the microwave region into thethermal infrared region, the MODIS instrument provides ameasurement on surface temperature; as frequency increasesfurther into the NIR region, MODIS albedo measurementscorrected for atmospheric and vegetation effects [Klein andStroeve, 2002] are fairly sensitive to the surface grain size. Inthe visible region, MODIS measurements can provide anestimate of snow covered extent. Our contention is that all ofthis information is correlated to some degree with the SWEand melt state of the snowpack.[18] More generally, assimilation frameworks are attrac-

tive because remote sensing, in situ, and meteorologicalmeasurements have complementary qualities. Global PMremote sensing observations (from which precipitation orSWE estimates are obtained) are available daily at coarsespatial resolution, while gage-based precipitation and othermeteorological data are available more continuously atselect locations. In situ measurements of depth are veryaccurate where they are taken; on the other hand, there issignificant uncertainty associated with how snowpack emitsand transmits PM radiation even at a point [e.g., Tedesco etal., 2006]. Nevertheless, PM measurements represent theradiation emitted over an entire spatial area, while interpo-lating local in situ or gage measurements is a difficult task.[19] The complementary nature of the available data

streams motivates us to demonstrate the feasibility of amultiscale, multifrequency radiometric DA approach, inwhich model-based a priori snow state estimates areupdated on the basis of multifrequency, multiscale PMand NIR measurements throughout the accumulation sea-son. Since this analysis has not been attempted, this syn-thetic feasibility test consists of a first-step evaluation of thepotential of this methodology. Specific hypotheses and


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science questions to be answered in this study are discussedin the following section.

3.2. Science Questions and Hypotheses

[20] The overarching question motivating this work is:How accurately can SWE be characterized by mergingremote sensing observations with LSM estimates? Drawingfrom the discussion in section 2.1, we consider the uncer-tainty associated with (1) estimating precipitation frommeasurements, (2) estimating grain size in a model, (3)scaling precipitation estimates to model pixel resolution,and (4) vegetation data and how vegetation modulatesradiative fluxes. Four key science questions and hypothesesare thus stated; the construction of experiments to test thesehypotheses is discussed in section 4.8.[21] 1. How does precipitation accuracy affect the filter

ability to recover SWE? It is hypothesized that merging PMmeasurements with LSM estimates can correct an underes-timation in the precipitation estimate mimicking the resultsof the previous study [Durand and Margulis, 2006], but fora spatially distributed case.[22] 2. How does grain size uncertainty affect the ability

to recover SWE, and how much grain size information iscontained in the NIR measurements? It is hypothesized thatuncertainty in grain size will decrease the ability to estimateSWE, but that the 1 km NIR measurements can be used toaid in SWE estimation by correcting grain size modelestimates.[23] 3. How does the spatial heterogeneity of the SWE

field affect the estimation of SWE? How much SWE

information is lost because of incorrect assumptions aboutthe spatial distribution, combined with the coarse spatialscale of the measurement? It is hypothesized that error inthe spatial distribution of SWE will decrease the accuracy ofthe integrated SWE estimate.[24] 4. How does vegetation uncertainty propagate into

modeled radiative flux estimates and PM and NIR measure-ments? It is hypothesized that vegetation uncertainty willdegrade SWE estimates.

4. Data, Models, and Methods

4.1. Study Area, Elevation Data, and Vegetation Data

[25] The Rabbit Ears Pass mesocell study area (MSA) inColorado of the Cold Land Processes Experiment (CLPX)(http://www.nohrsc.nws.gov/�cline/clpx.html) is chosen forthe study area because of the large number of snow andremote sensing measurements available during the CLPXwhich will be useful for follow-on studies. The Rabbit EarsMSA is shown in the context of the CLPX study area inFigure 1. The 625 km2 domain has the footprint dimensionsof a single PM pixel. The 3-arc second Shuttle RadarTopography Mission digital elevation model (DEM) dataare used, resampled to 1 km spatial resolution: there isapproximately 1300 m of relief in the resampled DEM. The30-m USGS Colorado Land Cover Data Set compiled from1992 Landsat Thematic Mapper images is used, resampledto 1 km spatial resolution. In this resampled data set, around41% of the domain is classified as evergreen forest, and22% is classified as deciduous forest. The 1 km forestdensity data set (vegetation fraction) of Powell et al. [1993]

Figure 1. (a) The Rabbit Ears MSA is shown within a DEM of the CLPX study area in Colorado. A1 km DEM wireframe of the Rabbit Ears MSA is shown overlaid with maps of (b) vegetation type and(c) vegetation fraction.


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and Zhu [1994] is utilized; vegetation fraction ranges from100% to 0% over the domain. The vegetation cover types andvegetation fraction for Rabbit Ears are shown in Figure 1. TheDEM is not treated as uncertain; vegetation uncertainty isdiscussed in section 4.5.

4.2. Spatial Uncertainty Model

[26] In our formulation of this problem, snow states fromeach pixel in the domain are simultaneously updated on thebasis of all of the multifrequency measurements and states.The interpixel or ‘‘horizontal’’ correlations [Reichle andKoster, 2003] must be modeled explicitly. The state andmeasurement interpixel correlations are controlled by meansof the forcing and parameter ensembles into the state model(see section 4.7 for details on how the state and measure-ment correlations are accounted for in the EnKF). In order togenerate the forcing and parameter ensembles, a spatialcovariance matrix relating the strength of the correlationbetween each pixel is specified. In this study, we make theassumption that the covariance function is stationary, andmodel the dependence of the correlation between two pointsseparated by a distance h by first relating the covariance to theexponential form of the geostatistical variogram [Isaaks andSrivastava, 1989]; we thus express the covariance function as

C hð Þ ¼ s2 exp �3ha

� �; ð1Þ

where C(h) is the covariance between points separated bydistance h, s2 is the variance of the variable at a point inspace, and a is the correlation length, or distance beyondwhich the variogram predicts that there is negligiblecorrelation between states. Thus s and a must be specifiedfor each uncertain input.

4.3. Forcing Data

[27] Estimates of the mean and error covariance ofmeteorological forcing variables are needed to drive theLSM and the filter. Estimates of the mean forcing variablesat 1/8� (approximately 8 km) resolution are obtained fromthe North American Land Data Assimilation System(NLDAS) [Mitchell et al., 2004]; NLDAS precipitationestimates utilize the gage-based NCEP precipitation dataset of Higgins et al. [2000, available at http://www.cpc.ncep.noaa.gov/research_papers/ncep_cpc_atlas/7/index.html]. Unfortunately, there is no uncertainty estimatepublished along with the NLDAS forcing data, so this mustbe estimated. As we are focusing on winter snow accumu-lation and grain size uncertainty is handled separately (seesection 4.5), uncertainty associated with the energy inputssuch as shortwave and longwave radiation and temperatureis neglected, and only precipitation forcing uncertainty isincluded. Since this methodology is applied in the moun-tainous Rabbit Ears Pass, we can assume in our uncertaintyanalysis that no radar data is used to complement the gagedata, and we consider two sources of uncertainty associatedwith these data. The first is the intrinsic uncertainty of thegage at each location; according to Goodison et al. [1981],gage undercatch is a function of wind speed. The averagewind speed during precipitation events in this area estimatedby the NLDAS product is around 2.0 m s�1, adjusted for thepossibility that wind speeds near the gages (often at lower

elevations) are slightly lower than the pixel average. FromGoodison et al. [1981], these wind speeds will lead toaverage catch efficiency of around 0.8. Assuming a lognor-mal error distribution for precipitation, and a 95% confi-dence in the wind/undercatch relationship, the theoreticalcoefficient of variation if the model were run at a gagelocation would be approximately 0.25. This value is used inthe following section to estimate the precipitation uncer-tainty for disaggregating precipitation data to 1 km.

4.4. Forcing Disaggregation Models

[28] The second source of uncertainty is due to theinterpolation of the (often spatially sparse) gage data to aregular grid. Spatial interpolation of either precipitation dataor snow depth data in general is difficult [Fassnacht et al.,2003]. For simplicity, it is assumed that there is a linearrelationship between elevation and precipitation [Dingman,1981]. The effective NLDAS precipitation lapse rate overthis area is estimated by regressing precipitation estimatesand elevation. This linear model is then used to disaggregatethe 1/8 � NLDAS precipitation estimates to the 1 km modelresolution. For simplicity, it is assumed that disaggregationuncertainty is equal to gage uncertainty; thus we specify theprecipitation coefficient of variation to be 0.5. This uncer-tainty is itself an unknown; the sensitivity of the results tothis parameter are investigated in section 5.5. This estimatecould be improved by generating a new precipitation dataset from nearby gages and estimating the uncertainty by across-validation method similar to that of Clark and Slater[2006]. The temperature lapse rate (which is not treated asuncertain in this study) used to disaggregate temperaturedata is �6.5 K km�1 as in the study by Mitchell et al.[2004]. Other 1 km model forcing data are assigned thevalue at the nearest NLDAS pixel. The correlation lengthfor the precipitation forcing data is determined by construct-ing a variogram from all of the NLDAS precipitation acrossthe CLPX domain and noting that the correlation length isapproximately 150 km.

4.5. State and Measurement Models

4.5.1. Land Surface Model[29] The simple snow–atmosphere–soil transfer (SAST)

model is used, which is an intermediate complexity, three-layer energy-balance snow scheme [Sun et al., 1999]. TheSAST model is embedded within the simplified simplebiosphere (SSiB) [Xue et al., 1991] model, and run at1 km spatial resolution. The state variables include the snowdepth and ground temperature, as well as the three-layersnow density, snow temperature, liquid water content, andsnow grain size, for a total of 14 state variables per pixel.For details on the temperature-gradient grain size schemeadapted from Jordan [1991] and how the grain size param-eters enter the model, see Durand and Margulis [2006].These parameters are tremendously uncertain, as the modelis a highly simplistic description of actual snowpack physics.The default grain size parameters from Jordan [1991] areused; a variogram from the snow pit grain size measure-ments taken during the CLPX campaign [Cline et al., 2002]is constructed and the parameter coefficient of variation andcorrelation length are chosen to in order reproduce thevariogram shape. Each vegetation cover data type fromthe vegetation data set is mapped to one of 13 SSiB land


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cover types [Xue et al., 1991] in order to assign leaf areaindex (LAI), vegetation reflectivity, etc., for use in the LSM.In order to model vegetation uncertainty, it is assumed that10% of the true LAI value represents two standard devia-tions; the coefficient of variation is chosen accordingly.4.5.2. Microwave Radiative Transfer Models[30] Radiative transfer emission models of the snowpack,

vegetation, and atmosphere must be utilized to predict thesatellite observation. The PM snowpack emission is pre-dicted by the Microwave Emission Model of LayeredSnowpacks (MEMLS) [Wiesmann and Mätzler, 1999;Mätzler and Wiesmann, 1999]. The vegetation is modeledaccording toWegmüller et al. [1995], following the approachgiven by Tigerstedt and Pulliainen [1998]. The vegetationbiomass is the crucial input into this RTM, and is estimatedfollowing Arora and Boer [2005] by the product of LAIand specific leaf area (SLA), where the SLA data are basedon the work of Reich et al. [1998] for each vegetation type.The clear-sky atmosphere is modeled according to Ulaby etal. [1981] as implemented by Tigerstedt and Pulliainen[1998]. For further details on these RTMs, see Durand andMargulis [2006]; however, the models are updated slightlycompared to the previous study. For a pixel completelycovered by vegetation, the downwelling radiative flux (orboundary condition for MEMLS, TBC

b ) at the snow surfaceof each 1 km pixel is given by

TbBC ¼ Tbspacetatmtcan þ Tbatmtcan þ Tbcan; ð2Þ

where Tspaceb is the downwelling cosmic radiation, Tatm

b isatmosphere brightness temperature, Tcan

b is the vegetationbrightness temperature, and tatm and tcan are the atmosphereand vegetation transmissivities. This boundary conditionwould apply over the vegetated portion of each pixel; theboundary condition brightness temperature for a partiallyvegetated pixel as a function of the vegetation fraction (Vc)is given by:

TbBC ¼ 1þ tcan � 1½ �Vcð Þ Tbspacetatm þ Tbatm� �

þ VcTbcan: ð3Þ

The MEMLS is then used to compute the upwellingbrightness temperature at the snowpack (TMEMLS

b ) at each

1 km pixel using the boundary condition T BCb and the snow

properties. For a fully vegetated pixel, the brightnesstemperature at each pixel (Tpixel

b ) is computed by:

Tbpixel ¼ TbMEMLStcantatm þ Tbcantatm þ Tbatm: ð4Þ

This would also hold for the vegetated part of each pixel;the pixel brightness temperature for a partially vegetatedpixel is given by:

Tbpixel ¼ 1þ tcan � 1ð ÞVc½ �TbMEMLSVcTbcan� �

tatm þ Tbatm: ð5Þ

The measurement over the 625 km2 domain is computed bysimply averaging the 625 T pixel

b values; this could beimproved in future studies [e.g., Mätzler, 2000].4.5.3. NIR Reflectance Model[31] The MODIS broadband albedo is less sensitive to

grain size than MODIS band 5 centered at 1240 mm. Kleinand Stroeve [2002] report correcting MODIS band 5 foratmospheric effects and for scattering anisotropy. We as-sume that this measurement could be corrected for atmo-spheric effects, but not adjusted for anisotropy. Then wemodel the bidirectional reflectance distribution function(BRDF) observed by MODIS using the Discrete OrdinatesRadiative Transfer model (DISORT) [Stamnes et al., 1988].In order to drive the DISORT model, single-scatter albedoand phase function for various grain sizes at 1240 mm arederived following the approach of Wiscombe and Warren[1980] utilizing the ice NIR refractive index data of Warren[1984], which is used by Flanner and Zender [2006]. At1240 mm, given the latitude of the Rabbit Ears MSA, thetime of year of our test, the solar zenith angle assuming a1 PM overpass time, and the MODIS observation angle(55�), a static relationship between grain size and the BRDFat the surface is derived and referenced as a lookup table;these values are plotted in Figure 2. The measurement ofsnow reflectance (rsnow) is partially obscured by vegetationcover; the satellite-measured reflectance rmeas is modeled asan average weighted by vegetation fraction Vc:

rmeas ¼ rsnow 1� Vcð Þ þ rsnowt2veg þ rveg� �

Vc; ð6Þ

where rveg and tveg are the reflectivity and transmissivity ofthe vegetation. These values are taken from the SSiB modelinputs for the corresponding vegetation types.

4.6. Synthetic Measurements

[32] The above models are applied in a synthetic DAframework where synthetic measurements generated usingthe measurement models with specified error characteristicsare assimilated instead of satellite observations. The pur-pose of a synthetic test is to verify that the assimilationscheme is feasible before applying it with real data. Night-time AMSR-E PM measurements are generated at 25 kmspatial resolution and daytime MODIS NIR reflectancemeasurements are generated at 1 km. The measurementerror standard deviation associated with AMSR-E is as-sumed to be 2 K. An uncertainty estimate for the broadbandMODIS albedo from Klein and Stroeve [2002] is between8% and 15%; here, it is assumed that the measurement error

Figure 2. Bidirectional reflectance distribution function(BRDF) is plotted versus surface snow grain diameter. TheDISORT approach is used to model the viewing geometryof the MODIS band 5 NIR reflectance measurement.


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standard deviation for a single MODIS band would beslightly lower, around 5%. In the absence of publishederror covariance values, the sensitivity of the results tomeasurement uncertainty is investigated in section 5.5.

4.7. Ensemble Kalman Filter

[33] The EnKF [Evensen, 2003] DA framework is chosensince it is a flexible approach for dealing with the variousnonlinearities and uncertainties in the systems. An ensembleof model inputs is generated on the basis of the coefficientof variation and correlation length estimates of the forcingdata and the model parameters discussed above. The modelis then run for each of the input realizations. During thispropagation step, the LSM is used to forecast the wholeensemble of state estimates forward until the measurementtime. At the update time each state is updated on the basis ofits correlation to all of the measurements predicted by theRTMs. Though linear, this update analysis takes intoaccount the first two moments of the joint distributionbetween the states and measurements as estimated fromthe ensemble.[34] All 14 states at all of the 625 pixels are updated on

the basis of all of the measurements simultaneously; thusthe state vector y has a length of 8750. The measurementvector z has a length which alternates from 625 for daytimeNIR measurements to 12 for nighttime PM measurements.The Kalman update equation for each ensemble member orreplicate, k, is given by

yposteriork ¼ y

priork þ K zþ vk � z

predictedk

� �; ð7Þ

where y prior is the state estimate prior to the incorporationof the new measurement, z is the measurement at the currenttime, v is random error that is added to the measurements toprevent the introduction of correlations among the replicates[Burgers et al., 1998], z predicted is the RTM prediction of themeasurement as a function of the prior, y posterior representsthe state estimate conditioned on the new informationavailable from the current measurement, and K is theKalman gain. The Kalman gain is computed from the errorcovariance of the measurements Cv and two samplestatistics obtained from the ensemble: the covariance ofthe states with the predicted measurements, Cyz, and thecovariance of the predicted measurements Czz:

K ¼ Cyz Czz þ Cvð Þ�1: ð8Þ

As pointed out by Reichle and Koster [2003], the ensemblesize and resulting computational expense can be drasticallyreduced by suppressing interpixel correlations for pixelsseparated by large distances. This is accomplished bymultiplying the covariance matrices in equation (8)elementwise by a compactly supported covariance matrixas described by Houtekamer and Mitchell [2001]. Here wemultiply Cyz and Czz elementwise by the covariance matrixgiven by Gaspari and Cohn [1999] such that pixelsseparated by a greater distance than the correlation lengthwill have zero covariance. Figure 3 illustrates schematicallyhow all of the available prior information, models, andsynthetic measurements described in sections 4.1–4.6 arecombined in equation (7).

4.8. Experiment Design

[35] Before the setup of the experiments is discussed,several simplifying assumptions are noted. First, it could beargued that during the accumulation season (the focus ofthis study) hydrologic models act to essentially integrate themass and energy inputs. Under this assumption, onlytemporally persistent biases in the precipitation or energyinputs will lead to systematic errors in the SWE or the grainsize estimates. Thus all inputs for a given replicate areperturbed in a temporally constant way. Second, subpixelvariability is an important issue in modeling snowpack at1 km but for simplicity we do not model subpixel variabilityexcept through the vegetation fraction of each pixel. Third,terrain geometry influences the brightness temperaturesmeasured from space [Mätzler, 2000], but here we ignoreterrain effects for simplicity and compute an integrated PMmeasurement as the simple spatial mean of pixelwise PMcalculations. Fourth, energy forcing inputs influence SWEthrough snowmelt and influence grain size through vaporflux, but in this study we do not randomize any energyforcing inputs. Fifth, we attempt to restrict the truth to snowcovered cases over the whole domain, though some pixelshave intermittent snow cover during the beginning of thesimulation. Sixth, it is assumed throughout that the corre-lation length and coefficient of variation are known. Sensi-tivity of the results to the range and coefficient of variationwill be the topic of a future study.[36] Analysis is performed from 1 November 2002

through 1 March 2003. This period encompasses enoughwarm temperatures in the NLDAS data set to ensure thatsome liquid water is present in the snowpack, but does notcontinue into the spring when the melt season begins.Precipitation values are scaled to a moderate depth ofapproximately 1.0–2.0 m depth so that the snowpack isdeep enough so that retrieval algorithms would be of littleuse, but not so deep that all of the PM signals saturate. Themodel is run for twenty four days before the first update. Itis assumed that random variables follow lognormal distri-butions. The state variables are represented with an ensem-ble of 80 replicates; sensitivity of the results to the numberof replicates is discussed in section 5.5.[37] The philosophy adopted for experiment design is to

identify first-order sources of uncertainty associated with boththe models and the inversion of PMmeasurements to estimateSWE. Scenarios are designed which test the capacity of thefilter to overcome sources of model uncertainty while subjectto conditions which would limit retrieval algorithm useof PM measurements. In other words, error which wouldbe encountered in a nonsynthetic study is modeled (corre-sponding to the science questions of section 3.2) then over-come by assimilating multiscale measurements. In each offour scenarios, one or two significant inputs are treated asuncertain in the filter and in the truth. We use an unlikelyrealization of the input parameters as the truth in order toensure that the feasibility test is rigorous.[38] For each scenario, a spatial field of an input value

(either a forcing variable or a parameter in the state orradiative transfer model) is generated with spatial statisticalstructure specified by the covariance matrix, which isconstructed as described in section 4.2. A spatial field witha statistically unlikely (usually 90th or 10th percentile)spatial mean is chosen to represent the truth. The ensemble


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members and the truth are chosen from the same probabilitydistribution function (pdf).[39] The four scenarios are shown in Table 1. In scenario 1,

true SWE is created by choosing a precipitation perturba-tion with a spatial mean in the 90th percentile of theensemble distribution; on average, the truth receives 63%more precipitation than the ensemble members. This largedifference represents the sort of error that will likely beencountered in a real situation [e.g., Pan et al., 2003]. Inscenarios 2, 3, and 4, the robustness of the filter to estimateSWE in the presence of multiple sources of error isinvestigated. In scenario 2 both precipitation and snowgrain size are uncertain. The parameters that govern graingrowth and snowfall grain size are treated as randomvariables; the grain size error will degrade the PM mea-surement on SWE. True grain size parameters are chosenfrom the 90th percentile of the pdf of the spatial parametermean resulting in true parameter values with a spatial meanapproximately 30% larger than filter parameters spatialmean. In the third and fourth scenarios, for simplicity andbecause no information is readily available to estimate thecorrelation length, inputs (precipitation lapse rate or LAI) areperturbed uniformly in space (spatially variable precipitationerror is still applied in these scenarios). In scenario 3, bothprecipitation and the precipitation lapse rate used to disag-gregate the NLDAS precipitation data to 1 km are uncertain.

The true lapse rate is selected from the 90th percentile,resulting in a true lapse rate twice as large as the meanensemble lapse rate. This will add additional error in thespatial distribution of the SWE estimate, degrading theoverall SWE estimate. In scenario 4, precipitation and LAIare treated as uncertain; the LAI impacts how the shortwaveand longwave radiative fluxes are attenuated by the canopybefore they reach the snow surface. More importantly forthis study, the LAI also impacts the emission and transmis-sion of radiation used to measure snow properties. It wasfound in initial tests (not shown) that a positive LAI biasresults in an increase in the amount of snow in the filterestimate. Since the filter SWE generally underestimates thetruth in these tests, if the true LAI were chosen to be greaterthan the filter mean, the LAI uncertainty would effectivelycompensate for the precipitation error. Thus we use the 10thpercentile from the LAI pdf to represent the true LAI.

5. Results and Discussion

[40] Results from the four scenarios are presented, fol-lowed by several sensitivity tests. Throughout this section,the ability of the filter to recover true state variables isevaluated. The filter estimate is compared with the controlmodel run where no assimilation is performed (the ‘‘openloop’’ results) and with the SWE estimate from applying a

Figure 3. This schematic illustrates how the prior information, models, and synthetic measurements aremerged using the EnKF and equation (7).


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commonly used retrieval algorithm [Chang et al., 1987] (the‘‘retrieval’’ results) with the synthetic observations.

5.1. Scenario 1: Uncertain Precipitation

[41] In scenario 1, error is introduced only into theprecipitation forcing data, and synthetic PM measurementsat all 12 AMSR-E channels are assimilated; NIR measure-ments are assimilated only in the scenarios with multiplesources of error, as we hypothesize that they will not havesignificant correlation with SWE in most cases. The arealmean snow depth and snow density (for the bottom snowlayer) estimated from the assimilation are shown in Figure 4.The sharp drops in the bottom layer snow density are due toprecipitation events when snow of lower density is incor-porated into the model’s bottom layer. As expected, theresult of the true precipitation selection leads to underesti-mation of snow depth in the open loop case, and the EnKFestimate improves upon the open loop model run when theupdates begin on 25 November.[42] The 1 March true and filter snow depth spatial maps

are shown in Figures 5a and 5b, respectively. The depen-dence of the modeled snow depth on elevation is apparentfrom Figure 5, as the maximum depths are along the ridgethat runs north and south in the center of the domain. Forcomparison, Figures 5c and 5d show the open loop estimateand the snow depth estimated by a retrieval approach[Chang et al., 1987], respectively: From inspection, thefilter estimate is closer to the truth than the open loop orretrieval estimates. It should be noted that the retrievalestimate is spatially uniform, while the filter is able toeffectively downscale the PM measurement based on theprior SWE estimate. The domain-integrated SWE timeseries is computed from the depth and density variablesand compared with the open loop model run and a retrievalestimate in Figure 6. The EnKF-estimated SWE is clearlymuch closer to the truth than the open loop or retrievalestimates; in addition, the EnKF effectively provides anerror bound or an uncertainty estimate (the second moment)of the state variables. The domain-integrated SWE RMSE iscomputed from the ensemble mean time series in Figure 6and shown in Table 2 (referred to as ‘‘run 1’’); the filterRMSE is 82% less than the open loop (referred to as ‘‘run 0’’)RMSE. The SWE RMSE could also be computed from thetime series at each pixel in the domain; the spatial mean andmedian of this ‘‘pixelwise’’ SWE RMSE are shown inTable 2; the filter median RMSE is 74% less than the openloop RMSE and the filter spatial mean RMSE is 55% lessthan the open loop. The domain-integrated RMSE value isless than the spatial mean RMSE values since the domainSWE is more directly corrected by the coarse PM measure-ments; the coarse PM measurements are of limited or no usefor correcting the SWE spatial distribution. The fact thatthe median RMSE improves more than the mean RMSE

gives the impression that a small number of outlier pixelsstrongly influence the mean RMSE results and that mostpixels improve significantly. The mean SWE bias is alsoshown in Table 2; the bias shows an 84% reduction over theopen loop case.

5.2. Scenario 2: Grain Size Parameter Error

[43] In scenario 2, significant error is added to grain sizeparameters and NIR measurements are assimilated alongwith the PM measurements to help to reduce noise intro-duced in the SWE and PM measurement relationship by thegrain size error. Surface grain size, bottom grain size, andSWE statistics are all improved by the updates. The updatesreduce the spatial mean RMSE, median RMSE, and meanbias in the surface grain size estimates by 31%, 58%, 92%,respectively. The fact that themedian RMSE improves nearly

Table 1. Parameters Used to Construct the Four Scenariosa

Scenario Source of Uncertainty Coefficient of Variation Correlation Length, km

Scenario 1 precipitation 0.5 150Scenario 2 precipitation and grain size parameters 0.5/1.0 150/10Scenario 3 precipitation and precipitation lapse rate 0.5/1.0 150/N/AScenario 4 precipitation and LAI 0.5/0.05 150/N/A

aNo correlation length is given for scenarios 3 and 4 because spatially uniform error is applied in those cases.

Figure 4. (a) Spatial mean snow depth and (b) bottom layerdensity are shown from assimilation of AMSR-E observa-tions by EnKF methodology. The truth is the dashed line, theopen loop is the dotted line, the solid line is the EnKFestimate (ensemble mean), and the solid thin lines are themean plus and minus two standard deviations. The firstupdate on 25 November is indicated by a vertical dashed line.


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twice as much as the mean RMSE reinforces the idea thatoutliers strongly influence the mean statistic. The fact that thebias reduces so significantly points to potential synergybetween the PM and NIR measurements; the coarse PMmeasurements are correlated to the spatial mean of the grainsize for each ensemble member, and so they work to correctspatial bias, while the NIR measurements contain informa-tion about the spatial distribution of surface grain size.[44] The grain size results show a strong spatial depen-

dence on the vegetation fraction; results on 1 March for thetrue, filter, and open loop surface grain size are shown inFigures 7a, 7b, and 7c, respectively. It is clear that grain sizevalues are much greater in the true simulation than in the

open loop; this is expected as a result of choosing an outlierto represent the truth. From comparing the true and filterestimates, it is also apparent that the filter is able to recoverthe true grain size much more effectively for pixels wherevegetation fraction values are lower (Figure 7d); this isexpected as a result of the fact that the vegetation masks theNIR measurement (see equation 6). In order to examine thedependence of the results on vegetation fraction, all pixelsare binned according to the vegetation fraction, and thepercentage of pixels showing an improvement in the meanbias is calculated separately for each bin and plotted inFigure 8. From Figure 8, pixels with lower vegetation covervalues are likely to have a strong improvement (100% of

Figure 5. The spatial snow depth on 1 March is shown imposed on an elevation wireframe for the(a) truth, (b) filter ensemble mean estimate, (c) open loop, and (d) a commonly used retrieval algorithm.

Table 2. Run Descriptions Along With Four SWE Statistics

Run DescriptionDomain SWERMSE,a cm

Mean SWERMSE,b cm

Median SWERMSE,c cm

Mean SWEBias,d cm

0 open loop 9.69 9.70 12.86 �9.101 uncertain precipitation, PM assimilated 1.787 4.36 3.36 �1.4782 uncertain precipitation and grain size, PM and NIR assimilation 1.242 4.27 3.18 �0.7403 uncertain precipitation and precipitation lapse rate, PM assimilated 1.784 4.36 3.36 �1.4754 uncertain precipitation and LAI, PM assimilated 2.27 4.44 4.12 �1.9035 uncertain precipitation and LAI, PM and NIR assimilated 1.170 3.57 1.961 �0.8446 uncertain precipitation, PM assimilation (no canopy) 1.448 4.07 3.17 �0.8227 sensitivity: high measurement error, low state error 1.209 3.05 1.843 �0.8228 sensitivity: low measurement error, high state error 1.234 5.55 6.41 �0.703

aDomain SWE RMSE is given by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nt

Pntt¼1

1np

Pnpp¼1

SWEestp;t � SWEtruep;th i !2vuut , where t and p are the time and pixel indices, respectively, nt and np are the total

number of time steps and pixels, respectively, and SWEest and SWEtrue are the SWE estimate and truth, respectively.

bMean SWE RMSE is given by 1np

Pnpp¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nt

Pntt¼1

SWEestp;t � SWEtruep;th i2s !

.

cMedian SWE RMSE is given by Median

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nt

Pntt¼1

SWEestp;t � SWEtruep;th i2s !

.

dMean SWE Bias is given by 1np

Pnpp¼1

1nt

Pntt¼1

hSWEp,t

est � SWEp,ttruei!

.


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pixels with vegetation less than 20% showed an improve-ment), whereas pixels with higher vegetation cover valuesare approximately 60% likely to show an improvement.Thus we can hypothesize that for pixels with bare snow (orthin vegetation) surface grain size can be estimated quiteeffectively.[45] Because of the intercorrelation of the model-estimated

layered grain size at each pixel, some of the NIR updateinformation propagates to grain size estimates throughoutthe pack; updates utilizing both PM and NIR informationreduce the mean RMSE, median RMSE, and mean bias inthe bottom grain size estimates by 40%, 63%, and 3%,respectively. Propagation of measurement information fromobserved to unobserved states in this way is not possible intraditional retrieval algorithms. The first two statistics prob-ably show greater improvement because the open loopbottom grain size mean bias is only 7% of the mean spatialopen loop bottom grain size estimate, which is quite low.[46] The NIR information helps to correct the grain size

estimates and thus leads to improved SWE estimates (see

run 2 in Table 2); despite the grain size noise, the mean biasis approximately half of the previous scenario where nograin size error is added, and all the SWE statistics showimprovement. In addition to improving the mean SWE andsnow depth estimates, the NIR measurements reduce theuncertainty in the snowdepth estimates at each pixel. Figure 9shows spatial maps of the snow depth coefficient ofvariation on 1 March for run 1 (precipitation uncertainand PM measurements assimilated) and for run 2 (precip-itation and grain size uncertain and PM and NIR measure-ments assimilated). It is clear visually that assimilation ofthe NIR measurements greatly reduced the uncertainty inthe snow depth estimate. Indeed, the spatial median coeffi-cient of variation of the 1 March snow depth reduces by34% from run 1 to run 2.[47] Synergy between the two measurements is clear in

Figure 10, where results are shown from pixel 501, locatedin the northwest part of the domain. The filter estimate (blueline) is close to the truth before the updates start on25 November, but the estimate at this pixel degradesbecause of the PM updates and the fact that the snow depth

Figure 6. The snow water equivalent estimate is shown.The truth, open loop, and filter estimates are as in Figure 4;the dash-dot line represents the SWE estimated from thesynthetic observations using a commonly used retrievalalgorithm.

Figure 7. The surface grain size estimate on 1 March is shown for the (a) truth, (b) filter estimate, and(c) open loop. (d) The vegetation fraction is shown for reference.

Figure 8. The fraction of pixels showing improvement inthe surface grain size season-long bias is plotted versus thevegetation fraction.


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is underestimated for most other pixels. Near the beginningof December, all of the snow at pixel 501 in the truesimulation (red line in Figure 10) melts. The NIR measure-ment reflects this information, and the filter assimilatingboth PM and NIR measurements is able to partially correctthe snow depth (dashed blue line in Figure 10) in a way thatis impossible using only the PM measurements (solid blueline in Figure 10). The error on 8 December for the casewith multifrequency assimilation is less than half the errorfor the case with only PM assimilation. This is an excellentexample of the sort of potential synergy that can existbetween the fine-scale NIR and coarse PM measurementsand be exploited using the multiscale DA scheme.

5.3. Scenario 3: Spatial Distribution Error

[48] In scenario 3, the true precipitation lapse rate used todisaggregate the NLDAS precipitation data is chosen to bean outlier around twice as large as the mean lapse rate.Despite this, the SWE error for run 3 in Table 2 does notincrease substantially for the case when PM measurementsare assimilated. The reason for this is that although the lapserate more than doubles, the overall spatial pattern does notchange significantly. In a real situation, the truth couldpotentially have an entirely different spatial pattern thanthe estimate, i.e., one that is not linearly dependent onelevation. In this study, however, we are investigating onlycases in which the ensemble contains the truth. Morecomplex scenarios in which the ensemble does not containthe truth at all will be investigated in a follow-on study.

5.4. Scenario 4: Leaf Area Index Error

[49] In scenario 4, the LAI is treated as a random variableand the true LAI is chosen as a low outlier (10th percentile).From run 4 in Table 2, the SWE error statistics all increasewhen LAI is treated as uncertain as compared to the runwhere LAI is treated as known; the mean SWE biasincreases by 29%. When the NIR and PM measurementsare both assimilated (run 5), all SWE error statistics dropsignificantly; the domain SWE RMSE, spatial mean SWERMSE, median SWE RMSE, and mean SWE bias arereduced by 48%, 20%, 52%, and 56% respectively. It ishypothesized that correlations between the different statesand various physical processes related to LAI at each pixellead to the significant improvements. For example, thebottom layer grain size values in this model are oftencorrelated to the snow depth by virtue of the temperature

gradient [Durand and Margulis, 2006]. Indeed, the bottomlayer grain size mean RMSE decreases by 44% from thecase where only PM measurements are assimilated to thecase where the NIR measurement is also assimilated. It ishypothesized that this more accurate grain size leads tomore effective SWE updates.

5.5. Sensitivity Tests

[50] The filter sensitivity to several crucial inputs, includingthe presence of the canopy, number of replicates, and relativemagnitude of state and measurement error is explored.5.5.1. Vegetation[51] In order to understand the effects of the canopy on

the amount of information contained in the measurements,the canopy is removed entirely and scenario 1 is performedover bare snow; the SWE error statistics are given for run 6in Table 2. The mean SWE bias value is a factor of 1.8larger for the case with vegetation; this is not surprisingsince over half the domain is vegetated with thick ever-greens with LAI values near 8.0. When scenario 2 isrepeated without the canopy, the grain size results improveas well; the mean surface and bottom grain size bias are afactor of 2.03 and 2.99 larger, respectively, for the case withvegetation than the case without vegetation.

Figure 9. Uncertainty (coefficient of variation) maps of snow depth are shown on 1 March for (a) run 1where precipitation is uncertain and PM measurements are assimilated and (b) run 2 where precipitationand grain size are uncertain and PM and NIR are assimilated.

Figure 10. Snow depth throughout the first two weeks ofthe assimilation is shown for two cases: one in which onlyPM measurements are assimilated and one in which PM andNIR measurements are both assimilated. The dashed line isthe truth, the dotted line is the open loop, the sold line is thefilter without NIR measurements, and the dash-dot line isthe filter with NIR measurements.


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5.5.2. Ensemble Size[52] Figure 11 shows the results of the assimilation run of

scenario 1 with different numbers of replicates. None of thefour statistics improve monotonically with ensemble size,but seem to oscillate slightly. Nonetheless, the oscillationclearly becomes less significant as the ensemble sizeincreases; one interpretation of this behavior is that as theensemble size decreases, the results are highly dependentupon the random numbers used to generate the ensemble.The relative stability of the error statistics in Figure 11 forensembles of 64, 80, and 160 members inspires confidencethat the results based on an ensemble of 80 members in thispaper accurately support the conclusions.5.5.3. Measurement Error and State Uncertainty[53] The measurement standard deviations are specified

to be 2 K for the PM brightness temperatures, and 5% forthe albedo measurements; the state uncertainties are set withcoefficients of variation of 0.5 and 1.0 for precipitation andgrain size parameter, but these values are all unknown. Firstwe increase the measurement error by 25% and decrease thestate error by 25% and repeat the assimilation run ofscenario 2, weighing the state variables more heavily; thisresults in a significant improvement in all four SWEstatistics for run 7 versus run 2. Finally, we decrease themeasurement error by 25% and increase the state error by25% and repeat the assimilation run of scenario 2, weighingthe measurements more heavily; this results in a significant

degradation of the mean and median SWE RMSE values forrun 8 versus run 2, although the domain SWE RMSE andbias are fairly stable. The reason for the dramatic changesare probably mostly due to the precipitation coefficient ofvariation value. When that value is decreased, it stronglyreduces the spatial variability of the SWE field, making theSWE easier to estimate with a spatially averaged PMmeasurement (run 7). When the state error is increased,we see the opposite effect (run 8). Thus sub-PM pixel SWEestimation is difficult even with the added benefit of the1 km NIR measurements, and is very sensitive to the stateuncertainty. We postulate that the feasibility of this projectmay be ultimately tied to the precipitation error statistics.

6. Conclusions and Future Work

[54] The simultaneous assimilation of multifrequency,multiscale synthetic measurements into an LSM for recov-ery of model-generated true SWE and other snow statevariables is demonstrated. First-order sources of modeluncertainty are modeled by constructing scenarios in whichthe truth is chosen to be a statistical outlier and in whichmultiple sources of error are present. The true snowpack isfairly deep and contains some liquid water at different timesin the winter, which limits the usefulness of PM measure-ments. Nevertheless, information about the true SWE at1 km is still extracted from the spatially coarse PMbrightness temperatures, and an underestimated SWE esti-mate is corrected. The spatially fine NIR snow reflectanceobservations contain valuable information about the grainsize in particular and about snowpack states in general;assimilation of PM and NIR measurements leads to thecorrection of both precipitation and grain size error. TheNIR measurements significantly reduced the snow depthestimation uncertainty; in one instance the NIR measure-ments prove useful for correcting the SWE estimate whenthe true snowpack melts away and the filter prior estimatestill has snow. These examples of the synergy betweenmultifrequency, multiscale measurements to correct multiplesources of error reveal the potential of DA methodologiesto characterize snowpack.[55] This study supports the conclusion that the estima-

tion of SWE using this methodology is feasible given thevalidity of the assumptions about model input uncertainty.However, as is often the case, better error characterization ofkey inputs is necessary to fully exploit the potential of theDA scheme. A follow-on study will investigate the robust-ness of the filter to the various assumptions about the modelinputs, especially the correlation length, as well as thepotential to use an adaptive filtering scheme to correct thevarious input uncertainties.

[56] Acknowledgments. This work has been sponsored by NASAEarth Science Systems Fellowship NNG05GQ86H, U. C. Water ResourcesCenter/U.S. Department of Interior grant SA6863, and NASA New Inves-tigator Program (NIP) grant NNG04GO74G. We thank Yongkang Xue,Christian Mätzler, Jouni Pulliainen, Noah Molotch, and Mark Flanner forgiving us their codes or data sets, as well as their advice and supportthroughout this project.

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Figure 11. Four error statistics are shown for scenario 2for various ensemble sizes: (a) the mean domain-averagedRMSE, (b) mean spatial season-long RMSE, (c) medianspatial season-long RMSE, and (d) mean bias.


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correcting first-order errors in snow water equivalent ...dust.ess.uci.edu/ppr/ppr_dum07.pdf ·...

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