retrieval of moisture from simulated gps slant-path water...

Retrieval of Moisture from Simulated GPS Slant-Path Water Vapor ObservationsUsing 3DVAR with Anisotropic Recursive Filters

HAIXIA LIU AND MING XUE

Center for Analysis and Prediction of Storms, and School of Meteorology, University of Oklahoma, Norman, Oklahoma

R. JAMES PURSER

Science Applications International Corporation, Beltsville, Maryland

DAVID F. PARRISH

Environmental Modeling Center, National Centers for Environmental Prediction, Camp Springs, Maryland

(Manuscript received 7 November 2005, in final form 9 July 2006)

ABSTRACT

Anisotropic recursive filters are implemented within a three-dimensional variational data assimilation(3DVAR) framework to efficiently model the effect of flow-dependent background error covariance. Thebackground error covariance is based on an estimated error field and on the idea of Riishøjgaard. In theanisotropic case, the background error pattern can be stretched or flattened in directions oblique to thealignment of the grid coordinates and is constructed by applying, at each point, six recursive filters along sixdirections corresponding, in general, to a special configuration of oblique lines of the grid. The recursivefilters are much more efficient than corresponding explicit filters used in an earlier study and are thereforemore suitable for real-time numerical weather prediction. A set of analysis experiments are conducted at amesoscale resolution to examine the effectiveness of the 3DVAR system in analyzing simulated globalpositioning system (GPS) slant-path water vapor observations from ground-based GPS receivers and ob-servations from collocated surface stations. It is shown that the analyses produced with recursive filters areat least as good as those with corresponding explicit filters. In some cases, the recursive filters actuallyperform better. The impact of flow-dependent background errors modeled using the anisotropic recursivefilters is also examined. The use of anisotropic filters improves the analysis, especially in terms of finescalestructures. The analysis system is found to be effective in the presence of typical observational errors. Thesensitivity of isotropic and anisotropic recursive-filter analyses to the decorrelation scales is also examinedsystematically.

1. Introduction

The poor knowledge of finescale spatial and tempo-ral distributions of water vapor is partly responsible forthe slow improvement in quantitative precipitationforecasts (QPF). The International H2O Project(IHOP_2002, 13 May through 25 June 2002; Weck-werth et al. 2004) was conducted over the central GreatPlains of the United States to investigate, as one of itsgoals, the four-dimensional distribution of atmospheric

water vapor and its impact on QPF. DuringIHOP_2002, a network of sensors collected numerouswater vapor data. This network included ground-basedGPS receivers, which measure the delay along the slantpath from GPS satellites to the ground-based receivers.The total integrated water along the slant paths can bederived from the slant-path wet delay (the part of thedelay caused by the water vapor) (Rocken et al. 1993;Ware et al. 1997; Braun et al. 2001).

Because of the integrated nature of the GPS obser-vations, a variational method is the natural choice foranalyzing such data. Most of the earlier work utilizingvariational methods examined the impact of the verti-cally integrated water vapor, or precipitable water(PW), or the zenith total delay (ZTD) data (Kuo et al.

Corresponding author address: Dr. Ming Xue, School of Me-teorology, University of Oklahoma, 100 E. Boyd, Norman, OK73019.E-mail: [email protected]

1506 M O N T H L Y W E A T H E R R E V I E W VOLUME 135

DOI: 10.1175/MWR3345.1

© 2007 American Meteorological Society

MWR3345

1993, 1996; Guo et al. 2000; De Pondeca and Zou2001a,b; Falvey and Beavan 2002; Cucurull et al. 2004).Despite the fact that they all found a positive impact ofassimilating the GPS data on short-range (within 6 h)precipitation forecasts for their specific cases, there is aloss of information content when using derived ZTD orPW data as compared with the original slant-path data,due to the spatial and temporal averaging involved intheir derivation. It should, therefore, be beneficial toexploit the slant-path total or wet delay or slant-pathwater vapor (SWV) data directly. The multiple slantpaths form, in a sense, a “net,” which provides muchbetter three-dimensional (3D) coverage than the verti-cal-only zenith paths. Further discussions on the GPSdata in various forms can be found in Liu and Xue(2006, hereafter abbreviated as LX06). So far, there hasbeen only a very limited number of studies that directlyanalyze slant-path data (MacDonald et al. 2002; Ha etal. 2003; Liu and Xue 2006), all utilizing simulated datafrom hypothetical GPS networks. These works demon-strated the effectiveness of variational procedures inrecovering 3D moisture structures from slant-path wa-ter vapor observations.

Many efforts have been made to develop more accu-rate GPS observation operators to reproduce the slantmeasurements; most variational analysis systems, how-ever, still assume isotropic and static background errorstatistics. For slant-path GPS data, MacDonald et al.(2002) did not use a model background term. Ha et al.(2003) assumed the background error covariance is di-agonal in their four-dimensional variational data as-similation (4DVAR) system. It is well known that flow-dependent background error covariance [sometimescalled the “error of the day” for large-scale applications(see, e.g., Pu et al. 1997)] plays an important role in dataassimilation. Such an error covariance is usually spa-tially inhomogeneous and anisotropic. The backgrounderror covariance builds the relationship between thegrid points and determines how the observation inno-vations are spread in the analysis domain. Conse-quently, the reliable estimation of the degrees and ori-entations of anisotropy in the moisture background er-rors is important for moisture analysis and QPFforecasts because of the high spatial variability in mois-ture.

In our recent study (LX06), a 3D variational dataassimilation (3DVAR) analysis system is developedthat models the flow-dependent background error fieldusing an explicit spatial filter. Better moisture analysisfrom simulated GPS slant-path water vapor data is ob-tained when using an anisotropic spatial filter to modelthe flow-dependent background error covariance. Theexplicit anisotropic filter, when applied over even a

moderate number of grid points in three dimensions, is,however, too expensive in terms of both computationaland memory storage costs, because of the need to cal-culate and store filter coefficients and apply the filterexplicitly at every grid point, in all three directions. Amuch more computationally efficient algorithm is theimplicit recursive filter, whose flow-dependent aniso-tropic version, though much more complex than theisotropic version, has seen significant development re-cently (Purser et al. 2003a,b). The significantly in-creased efficiency using recursive filters makes theirimplementation in operational data assimilation sys-tems practical.

In this paper, the recursive filter with an anisotropicoption is implemented in our 3DVAR system. Re-trieval experiments of GPS slant-path water vapor dataare performed and the results are compared with thoseof explicit filters reported in LX06. We also examinethe impact of flow-dependent background errors real-ized through anisotropic recursive filters on the qualityof 3DVAR analysis. Further, the sensitivity of isotropicand anisotropic recursive-filter analyses to the spatialdecorrelation scales is also examined systematically.We point out here that even though the experimentsperformed in this paper are similar to those presentedin LX06, the implementation of anisotropic recursivefilters for the modeling of truly flow-dependent back-ground error structures represents significant progress.The use of experiments with very similar configurationsas those in LX06 facilitates the comparison with theresults of explicit filters.

The rest of this paper is organized as follows. Section2 describes our 3DVAR system and a scheme for de-termining flow-dependent background error covari-ances. It also describes the procedure for determiningthe recursive filter coefficients, especially for the aniso-tropic case, given the modeled background error cova-riances. Section 3 introduces our hypothetical ground-based GPS observation network and the simulation ofGPS slant-path data. After illustrating in section 4 theeffect of recursive filters on the analyses with singleobservation tests, we apply in section 5 our 3DVARsystem to the 3D moisture retrieval. In section 6, wecheck the effect of filter length scales on the analysisquality and the sensitivity of the analysis to observa-tional errors. Conclusions and suggested future workare given in section 7.

2. 3DVAR system with recursive filters

In this section we present the 3DVAR analysis sys-tem developed and used in this study. It is based on the3DVAR system developed by LX06. It uses a precon-

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ditioner that involves the square root of the back-ground error covariance matrix B, instead of B itself asin LX06. As a result, the control variable of the currentincremental 3DVAR system is different from that usedin LX06. Further, it uses a recursive filter instead of anexplicit spatial filter to model B. The notation of Ide etal. (1997) is used in the equations below.

a. 3DVAR cost function

A variational method is to find an analysis that mini-mizes a predefined cost function. In this study, the costfunction is defined as

J�x� � Jb�x� � Jswv�x� � Jsfc�x� � Jc�x�, �1�

where

Jb�x� �12

�x � xb�TB�1�x � xb�, �2a�

Jswv�x� �12

�Hswv�x� � SWV�TRswv�1 �Hswv�x� � SWV�,

�2b�

Jsfc�x� �12

�Hsfc�x� � q� sfc�TRsfc

�1�Hsfc�x� � q� sfc�,

�2c�

and

Jc�x� �12 �x � |x |

2 �T

Wc�x � |x |2 �. �2d�

In the above equations, superscript T denotes matrixtranspose. The cost function J given in Eq. (1) is com-posed of four terms. The first is the background term,defined in Eq. (2a), and the second and third, Eqs. (2b)and (2c), are, respectively, the observational terms forGPS SWV and regular surface moisture observations.The physical requirement that water vapor is nonneg-ative is expressed in the last term, Eq. (2d), as a weakconstraint. The operational National Centers for Envi-ronmental Prediction (NCEP) spectral statistical inter-polation analysis (Parrish and Derber 1992) also uses aweak constraint, but on relative humidity being positiveand not supersaturated (J. C. Derber 2006, personalcommunication). Being a weak constraint, it does notstrictly enforce the positivity of the analyzed water va-por field, however. The definitions of the variables inthe equations are given as follows:

• The variable x is the analysis or control variable vec-tor, which in our case is the 3D specific humidityfield.

• The variable xb is the analysis background or firstguess of specific humidity in this case.

• The variable q� sfcis the surface observation vector of

specific humidity.• The variable B is the background error covariance

matrix. This study focuses on the effect of B in iso-tropic and anisotropic forms on the analysis. Assum-ing the background error variances are homoge-neous, that is, equal to a constant in the analysis do-main, then B is the product of this constant andbackground error correlation matrix.

• SWV is a vector containing SWV observations.• The variable Hswv is the forward observation opera-

tor that calculates integrated SWV observations be-tween GPS satellites to ground-based receivers fromwater vapor values at grid points.

• The variable Hsfc is the forward observation operatorfor surface observations of specific humidity.

• The variables Rswv and Rsfc are, respectively, the ob-servation error covariances for SWV and surfacemoisture observations. The observation errors are as-sumed to be uncorrelated; therefore the error cova-riances are diagonal. Usually, the diagonal elementsof Rswv (Rsfc) are the SWV (surface moisture) obser-vation error variances, which are assumed here to beindependent of stations, so Rswv (Rsfc) can be simpli-fied as a constant error variance multiplying an iden-tity matrix, that is, Rswv � 2

swvI and Rsfc � 2sfcI.

• Wc is the weighting coefficient (a scalar) for nonneg-ative weak constraint.

Given that Rswv � 2swvI and Rsfc � 2

sfcI, the R�1swv and

R�1sfc in Eqs. (2b) and (2c) can be replaced by �2

svw and�2

sfc , respectively, or by Wswv � �2swv and Wsfc � �2

sfc ,which we call the weighting coefficients for the SWVand surface observation terms. Similarly, the back-ground term in Eq. (2a) is proportional to a weightingcoefficient Wb, which is equal to the inverse of thebackground error variance, that is, Wb � �2

b (see morediscussion later).

In this study, except for one experiment, the simu-lated data are assumed error free; that is, no error isadded to the simulated observations. This assumption,as in LX06, allows us to determine how well, under anidealized error-free condition, a 3DVAR procedure asdeveloped here can retrieve the 3D structure of themoisture field from the slant-path water vapor data,which are integrated quantities along the slant pathsrather than local observations. The answer to this ques-tion was not obvious before our first study (LX06). Thiserror-free assumption is kept in this study for most ex-periments to facilitate direct comparisons of the resultswith those of LX06.


Theoretically, the weighting coefficients of the obser-vation terms should be infinitely large when the obser-vations are error free. While this naively suggests theunimportance of the background and its constraintterm in the cost function because of the relative mag-nitudes of the terms, without the background term ouranalysis problem remains underdetermined. To retainthe effect of the background term, in particular, thestructure information contained in the background er-ror covariance, we choose relative weights of 1, 100,500, and 50, respectively, for the four terms in the costfunction. The much higher weights given to the obser-vation terms reflect the much higher accuracy of theobservations as compared with the background, and thedifferences in the weights are sufficiently large to en-sure a close fit of the analysis to the observations. Herethe surface observations are given more weight becausethey tend to be more accurate. A different set ofweights will be used for one experiment (UB_err; seeTable 1) in which observations contain errors. Finally,we point out that it is the spatial structure of the back-ground error correlation that determines the spatialspread of the observation information (see, e.g., Kalnay2002); the small weight of the background term doesnot diminish this role.

Usually, the dimensionality of matrix B is too large to

directly compute or store in its entirety. To avoid theinversion of B and to improve the conditioning of theminimization problem, a new control variable v is de-fined, which satisfies

�x � Dv, �3�

where x � x � xb is the increment of x, and D isdefined as DDT � B.

The cost function J, can then be rewritten as

J�v� �12

vTv �12

�Hswv�xb � Dv� � SWV�TRswv�1 �Hswv�xb � Dv� � SWV�

�12

�Hsfc�xb � Dv� � q�sfc�TRsfc

�1�Hsfc�xb � Dv� � q�sfc�

�12 �xb � Dv � |xb � Dv |

2 �T

Wc�xb � Dv � |xb � Dv |2 �. �4�

This new cost function contains no inverse of B and theconditioning of the cost function is improved by thisvariable transformation. The iterative minimization al-gorithm used in this study is the limited-memory Broy-den–Fletcher–Goldfarb–Shanno method (Liu and No-cedal 1989). The implementation of this (as well as mostother) minimization method requires that the gradientof the cost function J(v) be evaluated with respect tothe control variable v. The gradient �vJ is given bydifferentiating J in Eq. (4) with respect to v as follows:

�vJ � v � DTHswvTRswv

�1 �HswvDv � dswv�

� DTHsfcTRsfc

�1�HsfcDv � dsfc� � �vJc , �5�

where Hswv and Hsfc are the linearized version of theobservation operators Hswv and Hsfc, respectively; dswv

and dsfc are the innovation vectors of SWV and surfacemoisture observations, which are given by dswv �SWV � Hswv(xb) and dsfc � q� sfc

�Hsfc(xb), respectively,and �vJc is given by �vJc � WcD

T(xb � Dv), which isonly active when the analyzed moisture, typically of theprevious iteration, is negative. In all experiments pre-sented in this paper, the recursive filter models matrixD and is applied to control variable v. No explicit back-ground error covariance matrix is involved in the cal-culation of the cost function or its gradient.

As discussed earlier, the background error covari-ance controls the extent to which values at the gridpoints away from an observation are influenced by theobservation innovation. The effect of this covariancecan be modeled using a spatial filter (Huang 2000) asfollows:

TABLE 1. List of retrieval experiments. In this table, SWV de-notes GPS slant-path water vapor data and “sfc” denotes thesurface moisture observation data. CC is the overall correlationcoefficient and RMSE is the root-mean-square error between thederived moisture increment and the true moisture increment(truth minus background).

ExptObservational

errorsAnisotropic

filterRMSE

(g kg�1) CCCC inLX06

ISO No No 0.35 0.84 0.83*ANISO No Yes 0.28 0.91 0.93UB No Yes 0.34 0.86 0.83UB_err 5% sfc error Yes 0.42 0.80 0.79

and 7%SWV error

* Lr � 4, in units of grid point, is optimal for the ISO experimentin this study, while Lr � 3 is optimal in LX06.

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bij � �b2 exp��

12 � rij

Lr�2�� b

2Cij , �6�

where rij is the 3D distance between two grid points iand j measured in terms of the grid index coordinate,and Lr is the geometric decorrelation length scale interms of the grid intervals and is in practice often linkedto the observation density. Here 2

b is the backgrounderror variance, which we assume to be constant, and Cij

is the background error correlation coefficient betweenthe ith and jth points. This definition in Eq. (6) providesan isotropic background error covariance because thespatial correlation only depends on the distance, notdirection. If Lr is constant, then the covariance is alsospatially homogeneous. With the covariance defined byEq. (6) or later by Eq. (7) and constant 2

b, the back-ground error covariance matrix can be written as B �2

bC, where C is the correlation coefficient matrix madeup of Cij.

An alternative to Eq. (6) is the following anisotropiccovariance:

bij � �b2 exp��

12 � rij

Lr�2� exp��

12 �fi � fj

Lf�2�

� �b2Cij , �7�

where f is a field whose pattern represents that of thebackground error, which we call the error field. HereCij is the corresponding correlation coefficient. Equa-tion (7) formulates a covariance function that is positivedefinite according to the definition 2.2 in Gaspari andCohn (1999) and Schur product theorem (cf. Horn andJohnson 1985, p. 458). In this study, f is chosen to beeither the true error field of the background or a certainestimate of the true error, and Lf is the decorrelationscale in the error field space in units of grams per kilo-gram for our moisture retrieval experiments; it controlsthe degree of the anisotropy. The spatial covariancedefined in Eq. (7) is larger between two nearby pointswith similar background errors. As a result, the spatialcovariance pattern is stretched along the direction of fcontours and squeezed across the contours.

Equation (7) is based on the idea of Riishøjgaard(1998) with an important difference. In his work, it issuggested that the analysis background field be used asthe f, under the assumption that the error field has asimilar pattern as the background field. This may betrue for certain quasi-conservative quantities that areadvected by the flow but not necessarily true for allfields. In our case, f is defined as the error field. Toestimate the f field, one possibility is to first perform anisotropic analysis, then use the difference between this“trial” analysis and the background as an estimate of

the error field. This idea will be evaluated in our analy-sis experiments.

b. Recursive filter

As noted earlier in the introduction, the explicit fil-ters used in LX06 are computationally very expensive.The univariate 3DVAR analyses in LX06, using theexplicit anisotropic filters, where Lr � 4 grid intervalsand Lf � 2 g kg�1 together with the cutoff radii of 10grid intervals in the horizontal and 6 coordinate layersin the vertical, typically require about 160 min of CPUtime to perform 100 minimization iterations while ittakes the recursive filter only about 4 min to do thesame using a single 1.1 GHz IBM Power 4 (Regattap690) processor. The recursive filters are much moreefficient because the filters are typically applied onlyonce or a few times (in the case of multiple filter passes)in each filtering direction while the explicit filter is ap-plied once at every grid point (see also Purser et al.2003a for related discussions). For this reason, recursivefilters, with an anisotropic option, are being imple-mented in a number of operational 3DVAR systems,including those at NCEP, for both global and regionalanalyses (Wu et al. 2002).

In LX06, the background error covariance B, definedby Eqs. (6) or (7), is directly evaluated and applied tothe control variable in the form of an explicit filter. Inthis paper, the effect of B is realized through a recursivefilter representation of the square root of B, or matrixD in Eq. (3), and the recursive filter is applied to controlvariable v. Purser et al. (2003a,b) describe the construc-tion and application of numerical recursive filters forthe task of synthesizing the background error covari-ances for variational analysis. Those two papers, re-spectively, deal with spatially homogeneous and isotro-pic and spatially inhomogeneous and anisotropic cova-riances with recursive filters. Some important details onthe filters are given below.

1) ONE-DIMENSIONAL RECURSIVE FILTER

The 3D filters can be created by the applications ofmultiple one-dimensional filters. The general nth-orderone-dimensional recursive filter has two steps (Haydenand Purser 1995). First is the advancing step:

qi � �pi � �j�1

n

�jqi�j , �8�

where p is the input and q is the output of the advancingstep and n is the order of the recursive filter. Here i isthe gridpoint index and must be treated in increasingorder so that the quantities on the right-hand side of the


equation are already known. The advancing step is fol-lowed by the backing step:

si � �qi � �j�1

n

�j si�j , �9�

where s is the final output from the recursive filter andhere index i must be treated in decreasing order. Pa-rameters j are the filter coefficients and have the fol-lowing relationship with �:

� � 1 � �j�1

n

�j , �10�

in order that the homogeneous forms of these filtersconserve the quantity being filtered.

We note here that when the above recursive filter isapplied to a limited-area domain, as in our case, specialboundary treatment is needed, which is discussed byPurser et al. (2003a) in their appendixes. The recursivefilter operations can easily be performed in the advanc-ing step according to Eq. (8) and the backing step ac-cording to Eq. (9) once the filter coefficients ( and �)are determined. In this study, we use a fourth-orderrecursive filter that effectively models the explicitGaussian filter with a single pass (Purser et al. 2003a).In our application, the input quantities pi in Eq. (8)correspond to the elements of control variable v in Eq.(4).

2) “ASPECT TENSOR” AND “HEXAD” ALGORITHM

For 3D problems in the isotropic case, one simplyapplies three one-dimensional filters, one in each coor-dinate direction. In the anisotropic case, however, thebackground error pattern can be stretched or flattenedin directions oblique to the alignment of the grid coor-dinates. The covariance operator in the anisotropic caseis constructed by applying six recursive filters along sixdirections corresponding, in general, to a special con-figuration or a hexad of oblique lines of the grid. Thishexad can be determined from an aspect tensor, whichis defined to be the normalized-centered second mo-ment of the response of the filter representing the de-sired covariance locally [see Eq. (12)]. It thereforeserves as a convenient way of characterizing the localspatial structure of the covariance. This term goes be-yond the concept of the “aspect ratio,” which only de-scribes the ratio of vertical and horizontal scales in thesimplest horizontally isotropic case, because the tenso-rial character expresses not just the absolute scale butthe dominant shape components.

A so-called hexad algorithm is used to determinefrom the aspect tensor the six filtering directions and

the corresponding hexad “weights,” which are thesquares of the six unidirectional scalar filtering scales.Each weight is, in effect, a one-dimensional aspect ten-sor, in grid space units, associated with the correspond-ing generalized grid line belonging to the hexad con-figuration. The algorithm is described briefly in Purseret al. (2003b) and in much greater technical detail inPurser (2005). While it is inappropriate to reconstructthe detailed technical description of the hexad algo-rithm here, it is important to note that the generichexad of lines is not any arbitrary set of six generalizedgrid lines; the rules for the construction of a legal hexadrequire that the smallest discrete steps, or “generators,”of the first three of a hexad’s directions, expressed ingrid units as integer three vectors, form a parallelepi-ped of unit volume. Furthermore, the remaining threeline generators consist of the three distinct possible dif-ferences amongst the first three generators. By theserules, the six generators of a hexad, together with theirnegatives, collectively define the 12 vertices of a dis-torted “cuboctahedron” embedded within the lattice oflocal grid displacements. As is shown rigorously inPurser (2005), these constitutive rules defining a legalhexad, together with the assumption that the centeredsecond moments of a sequence of filters combine addi-tively, and the further reasonable stipulation that thehexad weights on which the aspect tensor is projectedare each nonnegative, generally ensures a uniquechoice of a hexad of filtering directions and weights,apart from the vacuous ambiguity of directions in spe-cial transitional cases, where one or more hexadweights vanish. (A negative weight would imply a nega-tive amount of diffusion for a Gaussian smoother,which is physically meaningless, but a vanishing hexadweight simply implies the acceptable limiting case of nosmoothing in that direction.)

To find the aspect tensor at each grid point for theflow-dependent background error covariance definedin Eq. (7), it is necessary to rewrite Eq. (7) in matrixand differential form. If the finite difference fi � fj inEq. (7) is approximated by infinitesimal differential dfand distance rij is replaced by the differential dy, thenwe obtain

B � �b2 exp��

12 �dyTdy

Lr2 �

�df �2

Lf2 ��. �11�

This anisotropic background error correlation shouldshrink along the direction of a strong gradient of theerror field compared with the corresponding isotropicone. For example, if the gradient of the error field islarge in a given direction, then the difference in thebackground errors is large, that is, df is large, so that the

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background error correlation is small between points inthat direction, according to Eq. (11). Alternatively, thebackground error covariance can be written in a gen-eral Gaussian form that involves the aspect tensor A, asfollows:

B � �b2 exp��

12

dyTA�1dy�. �12�

Equating Eqs. (11) and (12) then gives

dyTA�1dy �dyTdy

Lr2 �

�df �2

Lf2 . �13�

Differentiating the above equation twice with respectto y and making the assumption that second derivativesof f can be neglected gives the inverse of A as follows:

A�1 �I

Lr2 �

��f �T��f �

Lf2 , �14�

where �f is the gradient vector of f field. For 3D prob-lems, I is a 3 � 3 identity matrix in our case. Note thatin the cases where different geometric length scales areused for the horizontal and vertical directions, term I/L2

r

in Eq. (14) should be a more general diagonal matrix.The aspect tensor A is a 3 � 3 symmetric and positivedefinite matrix and should have six independent com-ponents in general. It is locally defined at each gridpoint but is assumed to vary smoothly in space. Its lin-ear projection onto the appropriate hexad’s weights istherefore also a smooth function over the region ofphysical space to which that particular hexad configu-ration maps. When Lf goes to infinity, the anisotropicfilter reduces to the special case of the isotropic onesince, in this situation, the second term on the right-hand side of Eq. (14) vanishes. The resulting diagonalmatrix A must therefore map to a degenerate hexad inwhich three of the six weights vanish while the filteringthat corresponds to the three nonvanishing weights actsonly along the three coordinate directions. In this spe-cial situation, the effective filtering scale does notchange with grid points when the geometric decorrela-tion scale, Lr, is constant.

c. Estimation of the background error field foranisotropic analysis

The variable f in Eq. (7) represents the backgrounderror field. For simulated observations, we know thetruth so the true background error is also known. Wecan use the true background error as f to obtain theaspect tensor A for flow-dependent background errorcovariance according to Eq. (14). Then valid filteringdirections and associated weights can be determined by

the aspect tensor at each grid point. This is what will bedone in the experiment ANISO, which will be de-scribed in section 5b. But for realistic applications, thetrue background error is not known beforehand. Insuch a case, an estimate of the error is needed beforewe can perform any anisotropic analysis. FollowingLX06, we first perform an analysis using isotropic back-ground error (as in an experiment called ISO). Thedifference between this analysis from the background isthen used as an estimate of the background error field,that is, as f in Eq. (7). An anisotropic analysis is thenperformed and this experiment is referred to as UB,implying “updated background error covariance B.”Such a two-step iterative approach is feasible in prac-tice and is in a sense a double-loop strategy that issimilar in procedure to the double-loop approach com-monly employed by operational systems of variationalanalysis (e.g., Courtier et al. 1994)

3. Generation of simulated observations

Simulated data will be used to test our analysis sys-tem in this study. With simulated data, both the atmo-spheric truth and the error properties of the observa-tions are known. The Advanced Regional PredictionSystem (ARPS; Xue et al. 2000, 2001, 2003) in a gen-eralized terrain-following coordinate is used to simulatethe 3D water vapor field of a dryline at 2000 UTC 19June 2002. The dryline is characterized by a strongmoisture gradient. The generation of simulated obser-vations is the same as in LX06. The truth simulation isbased on the configurations of the Center for Analysisand Prediction of Storms real-time forecast forIHOP_2002 (Xue et al. 2002). The analysis domain cov-ers an area of 1620 km � 1440 km over the centralGreat Plains of the United States. The ARPS 9-kmsimulation subsampled to the 36-km analysis grid isconsidered the truth of the atmospheric water vaporfield. Shown in Fig. 1 is the truth at the fifth terrain-following model level, or about 500 m above theground.

The following formula is used to simulate the GPSslant-path water vapor observations:

SWVij � �ith receiver

jth satellite

q� ds, �15�

where SWVij is the integrated water vapor along theslant path between the ith ground-based receiver andthe jth GPS satellite, and q� is the specific humidityalong the path, interpolated from the gridpoint valuesof the truth. As in LX06, the GPS observation network


is composed of nine irregularly distributed satellitesand 132 ground-based receivers, which are evenly dis-tributed in the analysis domain. There is one receiverstation every four grid points, giving a receiver networkresolution of 144 km in each horizontal direction.

Therefore, for our analysis grid, there are 1188 SWVobservations and 132 surface observations. Figure 2 inLX06 gives a schematic illustration of the GPS obser-vation network.

4. Single observation tests

To assess the performance of the variational methoddescribed above using recursive filters and, more im-portantly, to understand the behaviors of the isotropicand anisotropic recursive filters, two single observationexperiments are performed using two-dimensional iso-tropic and anisotropic recursive filters, respectively. Insuch experiments, the SWV observation term is ex-cluded from the cost function. Only one moisture ob-servation at the surface marked by a filled black dot inFig. 2, with a value of 14.72 g kg�1, is analyzed. Thebackground is set to a constant value of zero and thegeometric decorrelation filter scale Lr is specified asfour grid intervals. Because the filters are two-dimensional, there is no coupling between the verticallevels; therefore the analyses are essentially two-dimensional. We will therefore examine the surfacefield only, which is impacted by the single surface ob-servation.

Figure 2 shows the surface analysis increments fromthese two experiments. Because the background is zero,the analysis increments are the analysis themselves andrepresent the corresponding structures of the back-

FIG. 2. Surface analysis increments from single observation experiments (a) for the isotropic example whose analysis increment is ofcircular shape and (b) for anisotropic example coupled to a reference field (dashed lines). The increment in the latter case is stretchedalong the contours of the reference field showing the strong anisotropy of the analysis. The contour interval for the increment is 2 g kg�1

and the first contour shown is at 2 g kg�1. The filled black dots mark the locations of the single observations. A circle of radius Lr �4 grid intervals and centered at the observation station is overlaid in thick black line in both (a) and (b).

FIG. 1. The q� at the fifth terrain-following grid level, or about500 m above the ground, from ARPS “truth” simulation for anIHOP_2002 case valid at 2000 UTC 19 Jun 2002. The contourinterval is 1 g kg�1. The analysis domain is 1620 km � 1440 km inthe Great Plains. The dryline is represented by the strong gradientof q�. The thick line A–B is at y � 270 km.

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ground error covariances. Figure 2a shows, as expected,the circular shape of the isotropic analysis increment.For the anisotropic case, Lr is also set to be equal tofour grid intervals while Lf is specified as 2 g kg�1. The“true” moisture field is used as the reference field (er-ror field f ) for this analysis (the background is zero inthis case). Strong anisotropy in the analysis increment isclearly revealed by Fig. 2b; the increment is stretchedalong the directions of the contours of the referencefield. This is because the correlation decreases rapidlythrough the region where the reference field has astrong gradient, for example, west of the single obser-vation; in contrast, it decreases much more slowlynortheast of the observation where the gradient issmall. The analyzed values at the observation stationfor both cases are about 14.69 g kg�1, which is veryclose to the observed value 14.72 g kg�1. This result isconsistent with the given ratio, that is, 1:500 of theweighting coefficients of the background and surfaceobservation terms in the cost function; the analysis ismuch closer to the observation than to the background.In the current ideal case where the background errorcovariance is modeled after the known truth, using ananisotropic recursive filter, the analysis should match thetruth very well when a sufficient number of observationsare present. This will be examined in the next section.

5. Three-dimensional moisture analysis

The earlier single observation experiments demon-strate correct behaviors of our analysis system usingrecursive filters. Subsequently, we move to the analysisof full 3D water vapor field from simulated SWV andsurface moisture observations discussed in section 3.Three experiments that use different background errorcovariances are discussed in this section. In all threecases, the observational data are error free, and theanalysis background was created by smoothing thetruth 50 times using a two-dimensional horizontal nine-point filter. Our main purpose in this study is to exam-ine the ability of our 3DVAR system to analyze meso-and small-scale moisture structures from the SWV andsurface observations with an initial guess that does notcontain such structures. The background produced bysmoothing truth is adequate for this purpose. An ex-ample of horizontal and vertical cross sections of thissmoothed field can be found in Fig. 4 of LX06.

In the first experiment, called ISO (Table 1), the cor-relation between any two points is only a function oftheir distance, so the covariance is isotropic; it is mod-eled using the isotropic recursive filter. The covarianceis also assumed to be spatially homogeneous. For thelatter two experiments, the correlation is the function ofnot only the distance but also the error structure; the

covariance is therefore state dependent (or flow depen-dent) and is modeled using an anisotropic recursive fil-ter. For the first of the two, called ANISO (Table 1),the error field is based on the truth field; that is, the f inEq. (7) is equal to the truth minus analysis background.Through this experiment we can determine how wellour analysis system can recover the 3D humidity fieldfrom the SWV and surface moisture observations underan ideal condition. For the second of the two anisotro-pic experiments (called UB; Table 1), the iterative pro-cedure described in section 2c is employed, in which anestimate of the error field, as the difference betweenthe first isotropic analysis and the background, is usedfor the subsequent anisotropic analysis.

a. Analysis with an isotropic background errorcovariance model

The one-dimensional fourth-order isotropic recursivefilter is applied along each coordinate direction usingone pass only (Purser et al. 2003a). The horizontal andvertical filtering scales are chosen to be four grid inter-vals, which are found to be nearly optimal for this case(see later discussion in section 6a). The choices of thegeometric part of the filtering scales are, in practice,often linked to the observational network density; wewant the observation innovations to spread far enoughso that they at least cover the gaps between groundreceiver stations (in the current case the station spacingis four grid intervals). The sensitivity of the analysis tothe value of the decorrelation length scale is examinedin section 6a.

Figure 3 shows the cost function and the norm of thecost-function gradient as functions of the number ofiterations during the minimization procedure for ex-periment ISO. Significant reductions (by at least twoorders of magnitude) occur in both the cost functionand the norm of the gradient during the first 100 itera-tions. In all cases, we run the minimization algorithmfor 100 iterations, which should be sufficient for thedesired accuracy.

Shown in Fig. 4 is the east–west vertical cross sectionof the analyzed 3D water vapor field (dashed lines)versus the truth of the moisture field (solid lines) at y �270 km (along line A–B in Fig. 1). It can be seen thatthe analysis follows the truth reasonably well. The mois-ture contours take an essentially vertical orientation atthe location of the dryline (about x � 360 km) belowthe 1.5-km level. This boundary separates the dry air com-ing from the high plateau to the west and the moist airoriginating from the Gulf of Mexico to the east. Mois-ture ridges and troughs are found at the right locationsand the moisture extremum centers match the truthwell too.


This analysis obtained using an isotropic recursivefilter is actually much better than that obtained using anexplicit isotropic Gaussian filter (cf. Fig. 7 in LX06). Allparameter configurations are the same as experimentSNF in LX06 except for the geometric decorrelationscale Lr that was set to three grid intervals, which wasfound to be optimal for that case. In the case of explicitfiltering, the correlation had to be cut off at a certaindistance to keep the computational cost manageable.This action actually destroys the exact positive definite-ness of the background error covariance1 although the

effect is usually small. In the case of the recursive filter,no cutoff radius is necessary so that the positive definiteproperty can be preserved. In addition, the recursivefilter does have the important advantage of being muchmore computationally efficient than the explicit filter.

The analysis increment at the fifth model level is pre-sented in Fig. 5a, together with the corresponding truthincrement (truth minus background) in Fig. 5b. Theyshow roughly similar patterns and extremum locations.The dryline can be recognized easily in the analysis.The overall correlation coefficient (CC) and the root-mean-square error (RMSE) between the analysis incre-ment and the truth increment are presented in Table 1.The CC is 0.84 and the RMSE is 0.35 g kg�1, bothindicating good analysis. For reference, the RMSE forthe background is 2.70 g kg�1.

b. Analysis with an anisotropic background errorcovariance model

Experiment ANISO is performed to examine howthe anisotropic background error covariance affects theanalysis. The chosen decorrelation scales are four gridintervals for Lr and 2 g kg�1 for Lf. This set of scaleswill be shown to be nearly optimal in section 6a. Theanalysis increment at the fifth grid level shown in Fig.6a matches the truth increment (Fig. 5b) very well. Thedryline pattern is analyzed with pronounced similarityto the truth. Their extremum locations coincide almostexactly. The analysis through the same vertical cross

1 Gaspari and Cohn (1999) have proposed filter functions forlocalizing covariance while preserving the positive definiteness.These functions can be applied to the explicit filter although theydo not solve the problem we have with an anisotropic explicitfilter.

FIG. 3. The variation of the cost function J and the norm of thegradient �J with the number of iterations during the minimizationprocedure for experiment ISO.

FIG. 4. East–west vertical cross section of the retrieved specifichumidity field (dashed lines) from experiment ISO at y � 270 kmvs the truth (solid lines). The contour interval is 2 g kg�1.

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section as in Fig. 4 is shown in Fig. 6b. Obviously, thereis a significant improvement compared with the analysisof ISO. The dryline structure is analyzed very well andthe analyzed moisture contours almost coincide withthe truth. The finescale structures are also capturedwell. For example, the q� � 14.0 g kg�1 contour has asharp downward drop at x � 620 km in the truth, a

finescale feature that is also captured by the analysis.Further, an upward moisture bulge at x � 300 km, as-sociated with upward motion at the dryline, is recov-ered very well by this anisotropic analysis but is missedin the isotropic analysis. The overall CC and RMSE forexperiment ANISO are, respectively, 0.91 and 0.28g kg�1 (Table 1), obviously better than those of ISO.

FIG. 6. (a) Analysis increment of q� at the fifth grid level from experiment ANISO with contour interval 1 g kg�1, where dashed linesare for negative values and solid lines for positive values. (b) East–west vertical cross section of q� at y � 270 km with interval 2 g kg�1.Solid lines are for truth and dashed lines for experiment ANISO.

FIG. 5. (a) Analysis increment of q� at the fifth grid level from the experiment ISO. (b) Truth minus background at the same level.The contour interval is 1 g kg�1. Dashed lines represent negative values and solid lines positive values.


c. Anisotropic analysis based on estimatedbackground error field

Experiment ANISO presented in the previous sub-section has shown clearly that the use of an anisotropicspatial filter improves the results of analysis when thebackground error covariance is modeled based on thestructure of true error field. This suggests that flow-dependent background error covariance can play a sig-nificant role in 3DVAR system. However, in reality, thetrue error field is not known beforehand. One promisingsolution is to use an ensemble of forecasts to estimatethe flow-dependent error covariances (e.g., Buehner2005), but the cost of running the ensemble is high. Here,we present results from experiment UB, which has beenintroduced earlier in this section and also discussed insection 2c and uses a two-step iterative procedure.

The analysis of UB is shown in Fig. 7. The analysisincrement at the fifth grid level (Fig. 7a) is morestretched along the dryline direction in comparisonwith the analysis from ISO (Fig. 5a), and the patternalso agrees much better with that of the true increment(Fig. 5b). The agreement of the analysis with the truthis also excellent in the vertical cross section (Fig. 7b).For example, the 4 and 6 g kg�1 q� contours match thetruth contours down to the finescale details while thefinescale details of the ISO analysis are not as good(Fig. 4). These finescale structures may play an impor-tant role for accurate convective initiation along adryline and the subsequent precipitation forecast. TheCC and RMSE for this experiment are 0.86 and 0.34 gkg�1, respectively, compared with the 0.84 and 0.35 g

kg�1 of ISO. The improvement, though not as dramaticas ANISO, is nevertheless evident. This improvement isin fact more evident in the analyzed fields shown in Fig.7 than these scores reveal.

To provide a quantitative measure of the differencebetween the flow-dependent B derived from the “up-dated” error field (i.e., the updated B used in experi-ment UB) and that derived from the true error field(i.e., the “truth based” B used in experiment ANISO),we calculate the L2-norm of the difference (matrix)between the two matrices. It is found that this differ-ence is a factor of 2.5 smaller than that between thetruth-based B and the isotropic B (i.e., the B used inexperiment ISO), indicating that the updated B is muchcloser to the truth-based B than the isotropic one.

The above procedure for estimating B is an iterativeprocedure. One naturally would ask if additional itera-tions would further improve the estimation of B and thesubsequent analysis. To seek the answer, additional it-erations were performed. It was found that the qualityof analysis after the third iteration remains essentiallythe same; therefore, only one update step or two totaliterations are employed here. This result may be casedependent, however. It should be tested when applyingit to other cases.

6. Sensitivity experiments

a. Sensitivity to decorrelation scales

The quality of an analysis is closely related to thedecorrelation scales used in Eqs. (5) and (6); these

FIG. 7. Same as in Fig. 6 but for experiment UB.

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scales control the spatial extent over which an observa-tion increment is spread. Fixed values of Lr (four gridintervals) and Lf (2 g kg�1) were used in the earlier 3Dmoisture analysis experiments. We examine in this sec-tion how the analysis quality varies with the decorrela-tion scales, as measured by the CC and RMSE of theanalysis increment with the true increment. In thesesensitivity experiments, the weighting coefficientsspecified for the terms in the cost function remain thesame.

In the isotropic case, only Lr is a free parameter.Given that the truth of moisture is known, we can ex-plore the parameter space of Lr in order to find a valuethat yields the best moisture analysis. In Fig. 8, thedotted line shows the RMSEs of analysis as a functionof Lr (in units of analysis grid intervals). We see thatthe optimal decorrelation length scale is equal to fourgrid intervals for this isotropic case. This appears rea-sonable because the 4 grid interval is actually the dis-tance between our uniformly spaced ground receivers.

In the anisotropic case, there are two free param-eters, Lr and Lf, so we explore the parameter space ofthe correlation model to find the set of Lr and Lf thatyields the best analysis. For the experiments that useupdated B, the RMSEs (solid lines in Fig. 8) as a func-tion of Lr are presented for four different values of Lf ,that is, 1, 2, 3, and 4 g kg�1. It can be seen that for theLf � 1 g kg�1 case, the RMSEs are the largest among allcases when Lr � 4 grid intervals. This suggests that thisvalue of Lf is too small when Lr is also small, so that thecombined spatial correlation fails to fill the gaps be-tween observation stations. For other more appropriatevalues of Lf , the optimal value of Lr is four grid inter-vals. Overall, the set of optimal decorrelation scales isLr � 4 grid intervals and Lf � 3 g kg�1 for the updatedB case.

The dashed line in Fig. 8 shows the RMSEs as afunction of Lr for experiments that use the B based onthe true error field. ANISO is one of these experiments;Lf is equal to 2 g kg�1 for these experiments. It is ob-vious that this set of experiments always yields the bestanalysis for a given value of Lr, compared with theother sets of experiments. It is also clear from Fig. 8that when Lr � 4 the isotropic analysis is always worsethan the corresponding anisotropic analyses with thesame Lr, except for the anisotropic case with updated Band Lf � 1 g kg�1. For Lr � 3, the analyses are ofsimilar quality except for the Lf � 1 g kg�1 case. Aspointed out earlier, this value of Lf � 1 g kg�1 is toosmall. The RMSE increases considerably with Lr whenit is larger than four grid intervals for the isotropic case,whereas the worsening is much less significant for theanisotropic cases, indicating a much smaller sensitivity

of the analysis to the choice of Lr in the anisotropiccases; the use of an improperly large Lr that helps to fillthe observation gaps does not hurt as much as the iso-tropic case. This is because in the anisotropic case thefield-dependent covariance imposes an additional con-straint on the spatial spread of the observation incre-ment and helps limit the potential damage of too largean Lr. As long as the choice of Lf is appropriate, theanalysis tends to be good, according to Fig. 8.

b. Sensitivity to typical observation errors

All experiments presented so far use simulated SWVand surface moisture data that include no observationerrors. In practice, the observations would not be errorfree, so it is important to test the sensitivity of theanalysis to the observation errors, in part to test therobustness of our analysis system. We perform experi-ment UB_err, which is the same as experiment UBexcept for the errors added to the observations. Thestandard deviations added to the simulated surface andSWV observations are, respectively, 5% and 7% of theerror-free values and the added errors are normallydistributed with zero means. The SWV errors are con-sistent with the estimate of Braun et al. (2001) for realdata. In this case, because of the presence of observa-tion errors, the weighting coefficients for SWV and sur-face observation terms are specified to be 15 and 29, re-

FIG. 8. The overall RMSE (g kg�1) between retrieved 3D analy-sis increments and the true increment (truth minus background),as a function of the geometric decorrelation scale Lr (in units ofanalysis grid intervals). The dotted line is for the experimentsusing the isotropic recursive filters (with ISO as one of them) andthe dashed line is for the experiments using the truth-based aniso-tropic background error covariance (with ANISO as an example)with Lf � 2 g kg�1. The solid lines are for the experiments withthe anisotropic error covariances based on the updated errorfields (e.g., UB) with Lf � 1, 2, 3, and 4 g kg�1, respectively.


spectively, relative to the 1 of the background term.These weights are specified to be proportional to theinverse of estimated background error variance and thevariances of the errors added to simulated observations(see discussion in section 3a). The analysis from thisexperiment, though not as good as the correspondingerror-free case, still matches the truth reasonably well(Fig. 9). In other words, our 3DVAR analysis proce-dure is able to retrieve the 3D structure of moisturefield from the slant-path water vapor observations(which are integrated values), even though the obser-vations are contaminated by errors of typical magni-tudes. This is also supported by the error statistics. Theoverall CC of the increments decreases from the 0.84 ofthe error-free UB case to 0.80 and the RMSE increasesfrom 0.35 to 0.42 g kg�1 (Table 1).

7. Conclusions

Compared with Liu and Xue (2006), which uses anexplicit spatial filter, a computationally much more ef-ficient algorithm based on the recursive filter is imple-mented and evaluated in this study, for the purpose ofanalyzing 3D water vapor fields from simulated GPSSWV observations. Surface moisture observations col-lected at the ground receiver sites are analyzed at thesame time. This study represents the first time that afully anisotropic recursive filter is used for modelingflow-dependent background errors in the context ofanalyzing GPS observations. The main conclusions arelisted as follows:

1) The analysis, produced by our 3DVAR method withan isotropic recursive filter, captures the main struc-ture of the dryline examined.

2) An anisotropic recursive filter, which is adaptive tothe structure of the background error both insmoothing directions and spatial correlation scales,produces significantly better analyses of the specifichumidity field associated with a dryline than the iso-tropic one, especially in the finescale moisture struc-tures.

3) In the absence of a good knowledge of the back-ground error, a two-step iterative procedure for itsestimation is proposed, in which an isotropic analy-sis is performed first. The difference between thistentative analysis and the background is used to de-fine the error field needed by the flow-dependentbackground error covariance model. This estimatederror field is then used in the second anisotropicanalysis step. Improved analysis, compared with theisotropic one, is obtained through this procedure.

4) Experiments on the sensitivities of the analysis tothe error decorrelation scales are performed. It isfound that the isotropic analysis is more sensitive tothe geometric decorrelation scale Lr and in such acase the range of Lr, with which a good analysis isobtained, is narrow. When the flow-dependent com-ponent of the background error is introduced, theanalysis becomes much less sensitive to Lr. For thecurrent dryline test case, the optimal geometric lengthscale Lr is found to be roughly equal to the meandistance between the GPS ground receiver stations.

FIG. 9. Same as in Fig. 6 but for experiment UB_err.

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5) The analysis procedure is found to be feasible andeffective for this dryline case. The analysis is foundto be not very sensitive to the presence of observa-tional errors of typical magnitudes in the SWV dataand in the surface moisture observations. But moresensitivity tests are necessary to show the statisticalsignificance and general applicability of this finding.

6) Compared with the explicit filters examined in Liuand Xue (2006), the biggest advantage for using re-cursive filters is the computational efficiency. Thecomputational cost of a recursive filter is about 40times smaller than that of the corresponding explicitfilter even when moderate cutoff radii are used. Thequality of analyses using recursive filters are in gen-eral comparable to the analyses obtained with ex-plicit filters. In the isotropic case, the analysis ob-tained using the recursive filter is actually betterthan that of the explicit filter.

Here, we point out that because the formulation ofthe background term is the same in 3DVAR and4DVAR, our method can be applied to 4DVAR aswell. The application of recursive filters to 4DVAR isexactly analogous to, though somewhat more compli-cated than, the application to 3DVAR. Purser (2005)briefly describes the construction of fully 4D anisotro-pic covariance operators via the sequences of line filtersthat generalize the sequences obtained through thehexad method.

In the near future, we will use the retrieved moisturefield to initialize a mesoscale model, such as the ARPS,and to examine the impact of assimilating SWV data onshort-range precipitation forecasts. The assimilationand the examination of the impact of real SWV datacollected during the IHOP_2002 field experiment arealso planned. The results of this current study can bestatistically more significant if the procedure is testedagainst more cases. Further, the effectiveness of thetwo-step procedure for estimating the background er-ror, then using it in the analysis, is worth further inves-tigation as it is applied to more cases.

Acknowledgments. Drs. Keith Brewster, Jidong Gao,and William Martin are thanked for their helpful dis-cussions. The authors thank Drs. Manuel Pondeca andWan-Shu Wu for valuable comments on the initialmanuscript. We also thank Dr. Ross Hoffman and ananonymous reviewer for their thoughtful commentsand suggestions. This work was mainly supported byNational Science Foundation Grants ATM-0129892and ATM-0530814. Xue was also supported by NSFGrants ATM-0331756, ATM-0331594, EEC-0313747, aDOT-FAA Grant via DOC-NOAA NA17RJ1227, and

by “Outstanding Overseas Scholars” awards from Chi-nese Natural Science Foundation (No. 40028504) andfrom Chinese Academy of Sciences (No. 2004-2-7). Su-percomputers at OSCER, University of Oklahoma,were used for most of the experiments.

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