spatiotemporal hierarchical bayesian modeling: tropical ...tropical ocean surface winds...
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Spatiotemporal Hierarchical Bayesian Modeling:Tropical Ocean Surface Winds
Christopher K. Wikle, Ralph F. Milliff, Doug Nychka, and L. Mark Berliner
Spatiotemporal processes are ubiquitous in the environmental and physical sciences. This is certainly true of atmospheric and oceanicprocesses, which typically exhibit many different scales of spatial and temporal variability. The complexity of these processes and thelarge number of observation/prediction locations preclude the use of traditional covariance-based spatiotemporal statistical methods.Alternatively, we focus on conditionally speciďż˝ ed (i.e., hierarchical) spatiotemporal models. These methods offer several advantages overtraditional approaches. Primarily, physical and dynamical constraints can be easily incorporated into the conditional formulation, so thatthe series of relatively simple yet physically realistic conditional models leads to a much more complicated spatiotemporal covariancestructure than can be speciďż˝ ed directly. Furthermore, by making use of the sparse structure inherent in the hierarchical approach, as wellas multiresolution (wavelet) bases, the models can be computed with very large datasets. This modeling approach was necessitated bya scientiďż˝ cally meaningful problem in the geosciences. Satellite-derived wind estimates have high spatial resolution but limited globalcoverage. In contrast, wind ďż˝ elds provided by the major weather centers provide complete coverage but have low spatial resolution.The goal is to combine these data in a manner that incorporates the space-time dynamics inherent in the surface wind ďż˝ eld. This is anessential task to enable meteorological research, because no complete high-resolution surface wind datasets exist over the world oceans.High-resolution datasets of this type are crucial for improving our understanding of global airâsea interactions affecting climate andtropical disturbances, and for driving large-scale ocean circulation models.
KEY WORDS: Climate; Combining information; Conjugate gradient algorithm; Dynamical model; Fractal process; Gibbs sampling;Numerical model; Ocean model; Satellite data; Turbulence; Wavelets.
1. INTRODUCTION
Fierce storms in California, ďż˝ oods in Peru, drought inAustralia and Indonesiaâjust a few of the extreme weatherevents attributed to the 1997â1998 El NiĂąo event (e.g., Kerr1998). This El NiĂąo brought unprecedented public attentionto the interaction between the tropics and extratropics, andperhaps more important, the interaction between the oceanand the atmosphere. These interactions have been a focus ofclimate research over the past decade. Changes in weatheraround the world, such as those associated with the recent ElNiĂąo, have been linked to variations in the atmospheric circu-lation that at a fundamental level are affected by exchanges inheat, moisture, and momentum between the atmosphere andocean. This exchange across the air/sea boundary is criticallyrelated to small-scale spatiotemporal features of sea-surfacewinds.
Climatologists and oceanographers use wind information intwo main ways: (1) to improve fundamental knowledge aboutatmospheric phenomena such as El NiĂąo (e.g., Liu, Tang, andWu 1998), tropical cyclones (e.g., Gray 1976), and large-scaletropical oscillations (e.g., Madden and Julian 1994), and (2)to provide input (forcing) for deterministic models of the cou-pled ocean/atmosphere system (e.g., Milliff, Large, Morzell,
Christopher K. Wikle is Assistant Professor, Department of Statistics, Uni-versity of Missouri, Columbia, MO 65211. Ralph F. Milliff is Scientist,Colorado Research Associates, Boulder, CO 80301. Doug Nychka is SeniorScientist, Geophysical Statistics Project, National Center for AtmosphericResearch, Boulder, CO 80307. L. Mark Berliner is Professor, Department ofStatistics, Ohio State University, Columbus, OH 43210. The authors thankT. Nakazawa for discussions regarding the validation of the model and forproviding the GMS cloud imagery used in this presentation, T. Hoar forassistance with the acquisition of datasets, J. Tribbia for helpful discussionspertaining to linear wave theory, and N. Cressie for comments on an earlydraft. The authors also thank the reviewers, the editor, and the associate edi-tor for valuable comments. Support for this research was provided by theNational Center for Atmospheric Research (NCAR) Geophysical StatisticsProject, sponsored by the National Science Foundation (NSF) under grantDMS93-12686, and by the NCAR NSCAT Science Working Team coopera-tive agreement with the National Aeronautics and Space Administrationâs JetPropulsion Laboratory.
Danabasoglu, and Chin 1999 and references therein). In bothcases, one must know something about the behavior of the sur-face wind ďż˝ eld and its horizontal derivatives at small scales.For example, it has been shown through the use of simu-lated datasets that deterministic models of the ocean are sen-sitive to both the temporal (Large, Holland, and Evans 1991)and spatial (Milliff, Large, Holland, and McWilliams 1996)resolution of the surface wind forcing (see also Chen, Liu,and Witter 1999). Indeed, although the deterministic coupledocean/atmosphere models used for prediction of the 1997â1998 El NiĂąo were more accurate than for previous El NiĂąoevents, indications are that many of these models would haveperformed better had uniformly high-resolution tropical windďż˝ elds been available (Kerr 1998).
Unfortunately, there are no spatially and temporally com-plete high-resolution observations of surface winds over thetropical oceans. Thus the major scientiďż˝ c challenge here isthe development of physically realistic high-resolution trop-ical wind ďż˝ elds. Our fundamental scientiďż˝ c contribution isthe development and implementation of a statistical approachto generate high-resolution wind distributions over largeexpanses of the tropical ocean. To that end, we developa hierarchical Bayesian spatiotemporal dynamic model thatcombines wind data from different sources, and backgroundphysics, to produce realizations of high-resolution surfacewind ďż˝ elds. The Bayesian approach is ideal for this applicationbecause (1) it provides a mechanism for combining data fromvery different sources, (2) it provides a natural framework inwhich to include scientiďż˝ c knowledge in the model, and (3) itprovides posterior distributions on quantities of interest thatcan be used for scientiďż˝ c inference.
Our statistical analyses use two strikingly different datasets.The ďż˝ rst dataset involves satellite-derived wind estimates that
Š 2001 American Statistical AssociationJournal of the American Statistical Association
June 2001, Vol. 96, No. 454, Applications and Case Studies
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Wikle et al.: Spaciotemporal Hierarchical Bayesian Modeling 383
have high resolution in space but are limited in areal coverageat any given time. Milliff and Morzel (2001) demonstratedthat the information from a single instrument of this type isnot sufďż˝ cient to resolve all of the meteorological events in thesurface wind ďż˝ eld. The second dataset comprises wind esti-mates, known as analyses, provided by the major weather cen-ters. These provide complete wind ďż˝ elds but have low spatialresolution. Although the large-scale features of the tropicalatmosphere are generally well represented by these analysisďż˝ elds, these ďż˝ elds are unable to resolve many of the small-to medium-scale features in the wind ďż˝ elds needed to under-stand the dynamics of the tropical ocean and atmosphere (e.g.,Milliff et al. 1996). Hence, in isolation, neither of these twodatasets provides the breath of scientiďż˝ c information soughtby climatologists. Our Bayesian model combines these datato yield information about winds at a useful spatial scale, andin a manner that incorporates physical theory about the spa-tiotemporal dynamics inherent in tropical surface winds.
We demonstrate (see Fig. 5) that our posterior wind ďż˝ eldscontain much more ďż˝ nely resolved features than do the cur-rent state-of-the-art weather center wind ďż˝ elds over the trop-ics. Furthermore, based on external veriďż˝ cation with remotelysensed cloud imagery, these higher-resolution features in thewind ďż˝ elds correspond to physically meaningful features ofthe atmosphere. We emphasize that until satellite wind dataare assimilated adequately into numerical weather predictionmodels of similar resolution, a Bayesian procedure of the kindthat we derive here provides the only source of high-resolutiontropical wind ďż˝ eld information sufďż˝ cient for many aspects ofresearch regarding airâsea interactions and their effects on cli-mate. Furthermore, the probability distributions for wind ďż˝ eldsthat we provide will for the ďż˝ rst time allow scientists to con-sider the distributional nature of phenomena that depend onairâsea interaction.
The datasets used here are described in Section 2. The phys-ically based spatiotemporal model that we have developed isdescribed in Section 3. By âphysically based,â we mean thatsubstantial physical modeling and background science wereused in both development of the model and speciďż˝ cations ofpriors on model parameters. The Bayesian implementation andspeciďż˝ c computational issues related to our analysis are dis-cussed in Section 4. The huge datasets used and the largenumber of unknowns modeled necessitated the development ofspecial algorithms. These developments are of general inter-est in large-scale Bayesian analyses. Model veriďż˝ cation andinference based on our wind model are described in Section 5.A brief discussion is presented in Section 6.
2. WIND DATA
Because winds are vector quantities, they can be split intoorthogonal components. We use the standard decomposition inwhich u represents the eastâwest (âx-directionâ) componentand v represents the northâsouth (ây-directionâ) component.Although other decompositions are possible, we selected thisCartesian decomposition for physical reasons; the equator is afundamental line of symmetry in the equatorial dynamics thatgovern weather in the tropics and is a source of anisotropythat discourages the use of any coordinate system other thanCartesian (e.g., Gill 1982, pp. 436â463). We consider surface
wind components over a spatial domain in the western Paciďż˝ cOcean from 107ďż˝ to 170ďż˝ E longitude and 23ďż˝S to 24ďż˝N lat-itude, as shown in Figure 1. This portion of the equatorialPaciďż˝ c contains the âwarm pool regionâ and is critical tothe forcing and maintenance of many weather and climate-scale phenomena (e.g., Philander 1990). We focus on 6-hourincrements during the 2-week period from October 28, 1996,through November 10, 1996. Tropical variability consistentwith these scales include, for example, westerly wind bursts,equatorial Rossby wave propagation, and tropical storms. The2-week time period is sufďż˝ cient to capture up to ďż˝ ve suchevents.
Although some in situ observations of ocean surface windsare made from buoys and ships, they are rather sparsely dis-tributed in space and time relative to land-based observationnetworks. The worldâs major meteorological centers take thesefew observations and insert them into global-scale numeri-cal weather prediction models to produce tropical wind ďż˝ eldanalyses (e.g., Daley 1991). Hence the resulting data are notmeasurements or observations in the traditional sense, butrather are statistics computed as highly complex functions ofobservations.
We consider weather center wind ďż˝ elds from the NationalCenters for Environmental Prediction (NCEP). These data rep-resent surface winds (actually, 10 m above the surface) andhave a reporting period of 6 hours and spatial resolution ofnearly 2ďż˝, or about 200 km in equatorial regions. NCEP u-winds for three consecutive 6-hour periods in early November1996 are shown in Figure 2(a).
Wind data from the NASA scatterometer (NSCAT) instru-ment are also used here. A scatterometer is a satellite-borneinstrument that emits radar pulses at speciďż˝ c frequencies andpolarizations toward the sea surface, where they are backscat-tered by surface capillary waves (e.g., Naderi, Freilich, andLong 1991). The backscattering is detected and related,through a âgeophysical model function,â to wind speed anddirection near the surface (usually 10 m; see, e.g., Stoffelenand Anderson 1997; Wentz and Freilich 1997; Wentz andSmith 1999). That is, as in the case of analysis ďż˝ elds, thesedata are not direct measurements of winds, but rather are func-tions of backscatter detections.
Due to the polar orbit of these satellite platforms, the tem-poral resolution of these data are relatively sparse and, overthe span of several hours, the spatial coverage area is relativelysmall; see Figure 2(b). Each âsnapshotâ in time includes allobservations within a 6-hour window centered on the corre-sponding analysis time. The NSCAT surface (i.e., 10-m) winddata used here were produced by the NSCAT-1 model function(Wentz and Freilich 1997). These data have a 50-km nomi-nal spatial resolution, although the reported winds are actuallyderived by applying the model function to an average of sev-eral backscatter observations within a 50-km by 50-km obser-vational âcell.â
Notation. Let Va4ri3 t5 and Ua4ri3 t5 denote the NCEPanalysis northâsouth and eastâwest, wind components at spa-tial location 8ri 2 i D 11 : : : 1m9 and time 8t 2 t D 11 : : : 1 T 9.The scatterometer (NSCAT) northâsouth (eastâwest) windcomponent is denoted by Vs4Qrj3 t5 (Us4Qrj3 t5) at location8Qrj 2 j D 11 : : : 1 pt9 and time 8t 2 t D 11 : : : 1 T 9. (The number
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384 Journal of the American Statistical Association, June 2001
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Figure 1. Prediction Grid Locations Over the Equatorial Paciâ c Study Region.
of NSCAT observations, pt can be highly variable, see Fig. 2.)We deďż˝ ne the âtrueâ (i.e., noiseless) wind components asv4si3 t5 and u4si3 t5 at spatial locations 8si 2 i D 11 : : : 1 n9 andtimes 8t 2 t D 11 : : : 1 T 9. The cumbersome notation of index-ing spatial locations is needed because we are faced with aâchange of supportâ problem: The NCEP and NSCAT datarepresent different spatial scales, both of which differ from thedesired prediction sites si .
In the present example, we choose a 1-degree regular pre-diction grid (Fig. 1) and consider 54 6-hour time incrementsover the period from 0600 UTC (Coordinated Universal Time)on October 28, 1996, to 1200 UTC on November 10, 1996.We neglect small displacements in the prediction lattice thatare due to the curvature of the earth.
Next, let Vt denote an m C pt vectorization of the northâsouth weather center and scatterometer observations at time t.Similarly, Ut is the combined list of the data correspondingto the eastâwest component. Also, let vt and ut be n vectorsof the âtrueâ northâsouth and eastâwest wind components, atprediction locations at time t.
Finally, we use the following notation to denote matricescomposed of columns of vectors representing intervals of time:let 8V9B
A be the collection of vectors 8Vt 2 t D A1 : : : 1 B9.
3. HIERARCHICAL SPACE-TIME MODELS
A major difďż˝ culty in the application of statistical spatiotem-poral models in geophysical problems has been adequatelydescribing the complicated spatiotemporal covariance struc-tures inherent in these contexts. (For an overview of traditionalspatiotemporal modeling approaches, see Wikle and Cressie1999.) These methods are not suitable to the present prob-lem in that they cannot easily account for propagation ofsynoptic-scale weather disturbances, ďż˝ ll âgapsâ in the observa-tions with realistic variance at all spatial scales, include mul-tiple measurement errors and change of support for differentdata sources, or incorporate huge amounts of data.
3.1 The Hierarchical Approach
Hierarchical models are ideal for extremely complex and/orhigh-dimensional problems. In essence, the strategy is basedon the formulation of three primary statistical models orstages:
ÂĄ Stage 1: Data model: [data|process, Ë1]ÂĄ Stage 2: Process model: 6processâË27ÂĄ Stage 3: Prior on parameters: 6Ë11 Ë27
Here the bracket notation denotes probability distribution(e.g., Gelfand and Smith 1990), and Ë1 and Ë2 generically
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Wikle et al.: Spaciotemporal Hierarchical Bayesian Modeling 385
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Figure 2. NCEP and NSCAT Sampling Locations and u-Wind Component Value (ms 1) Within 6-Hour Time Windows Centered on 1200 UTCon November 6, 1996, 1800 UTC on November 6, 1996, and 0000 UTC on November 7, 1996.
represent parameters introduced in the modeling. The idea isto approach complex problems by breaking them into piecesâin this case, a series of conditional models (e.g., Berliner1996). The stage 2 model for the process (in our case, truewinds) can itself be speciďż˝ ed as a product of physically moti-vated conditional distributions. By treating the spatiotemporalvariability as a series of relatively simple, yet physically based
conditional models, we can obtain spatiotemporal dependencestructures that are much more complicated (and more real-istic physically) than could be speciďż˝ ed directly. Bayesiananalysis relies on the posterior distribution of the processof interest and parameters given data: [process, Ë1, Ë2 âdata]. Recent examples of hierarchical Bayesian spatiotem-poral models include that used by Waller, Carlin, Xia, and
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386 Journal of the American Statistical Association, June 2001
Gelfand (1997) for mapping disease rates. An overview ofhierarchical spatiotemporal dynamic models along with a geo-physical application has been provided by Wikle, Berliner, andCressie (1998).
3.2 Stage 1: Data Model
We expect the wind data to be replete with complicated spa-tiotemporal dependencies. However, conditional on the truewinds, we expect the complexity of this dependence to bedramatically reduced. That is, stage 1 models only measure-ment errors, not that portion of the complex structure presentin the data due to the structure of the true winds. The funda-mental assumptions are that, conditional on the true process8u9T
1 1 8v9T1 , the data are independent with respect to time, and
the set of U observations is independent of the set of V obser-vations. Speciďż˝ cally, we have that
68V9T1 1 8U9T
1â8v9T
1 1 8u9T1 3 Ë17 D
TY
tD1
6Vtâvt3 Ë176Ut
âut3 Ë170 (1)
In particular, we assume normally distributed errors,
Vtâvt1èt Gau4Ktvt1èt5
and Utâut1 èt Gau4Ktut1èt51 (2)
where Gau4ďż˝1A5 refers to a multivariate Gaussian distributionwith mean ďż˝ and covariance matrix A. We assume that thecovariance matrices èt are diagonal with unknown variancesâ 2
B , for NCEP data at sites on the boundary of the NCEP grid,â 2
I for NCEP data at interior sites on the NCEP grid, and â 2
for NSCAT observations; that is, the ďż˝ rst m diagonal elementsof èt are equal to either â 2
I or â 2B , and the remaining pt are
equal to â 2. Further, for each t, Kt is a speciďż˝ ed 4mC pt5ďż˝ n
matrix that maps the prediction grid locations to the observa-tion locations.
Several issues surround our assumptions about the data-acquisition process. First, we assume that conditional ontrue winds, the scatterometer errors and the NCEP analy-sis errors are independent. This is quite plausible, becausethe NCEP did not use scatterometer data in producing windďż˝ elds. Second, evidence in the literature points to the plau-sibility of our assumptions that the scatterometer errors aremutually conditionally independent and have homogeneousvariance, and that the eastâwest and northâsouth componenterrors are independent. For example, Freilich (1997) demon-strated that an independent and normally distributed randomerror model for scatterometer velocity components is con-sistent with observed distributions for wind speed. Freilichand Dunbar (1999) considered comparisons between collo-cated satellite wind estimates and direct measurements fromocean buoys in a validation study. They concluded that theindependent-component error model, with standard deviationsequal to 1.3 m/s (for both components), is appropriate forNSCAT data. Furthermore, over the relatively small geograph-ical region considered here, these references suggest that thehomogeneous variance assumption is reasonable. Haslett andRaftery (1989) showed that application of a square-root trans-formation may enhance both homogeneity and normality in
wind measurements. As suggested by Freilich (1997), such ahomogeneity-of-variance transformation of wind speed is con-sistent with the independent, homogeneous normal randommeasurement error model for the Cartesian wind components.Finally, the assumption that NCEP analysis errors are mutu-ally independent seems to be the least tenable assumption inview of the complex nature of the numerical and statisticalmethods used in the production of such information. The for-mulation of genuine covariances for analyzed ďż˝ elds is a majorresearch area in its own right, and well beyond the scope ofthis article. We believe that the independence assumption isnot critical for our results.
Mapping Matrices. We partition the mapping matrices asKt
D 6K0a1K0
s4t570, where Ka and Ks4t5 are m ďż˝ n and pt
ďż˝ nmatrices. Because the prediction grid is at a ďż˝ ner resolutionthan the NCEP data, Ka acts by assuming that the condi-tional means of the data are smoothed versions of the âtrueâwinds on the lattice. This âchange-of-supportâ approach isfurther justiďż˝ ed because NCEP data have been shown to betoo smooth at large scales (e.g., Milliff et al. 1999; Wikle,Milliff, and Large 1999). Speciďż˝ cally, the Ka matrix consid-ers the nearest nine prediction grid locations within some dis-tance D (D D 165 km) and weights those locations linearlyby wi
D 4D Ćdi5=w Ăź , where di is the distance between the ithprediction grid location and the NCEP datum location and w Ăź
normalizes the weights to sum to 1.Each Ks4t5 is an incidence matrix of 0âs and 1âs that simply
maps the conditional mean of an NSCAT observation to thenearest grid process location. The error induced by this map-ping is related to the chosen prediction grid resolution. Effec-tively, by using the mapping matrix, Kt , we allow the windprocess to âliveâ on a ďż˝ ne-resolution regular grid. The reso-lution of this grid could be so high as to allow the NSCATdata points to each correspond to a unique lattice location.Practically, a balance must be sought between computationalexpense, grid resolution, and resolution of the physics that oneis seeking to describe or model. More-complicated approachesto parameterizing both Ka and Ks4t5 are possible (see Wikleand Berliner 2001). However, for computationally purposes,these mapping matrices must be very sparse (see Sec. 4.1).
3.3 Stage 2: Priors on the Process
Our task is to formulate a joint probability model for thegridded wind process, 8u9T
1 1 8v9T1 . We begin by decomposing
each of the wind processes into three physically meaningfulcomponents. The decomposition and models for the resultingcomponents were developed based on our physical and sta-tistical understanding of the problem. After a review of thatreasoning in the following section, we present the speciďż˝ c sta-tistical models used for each of the three components.
3.3.1 Decomposition of the Wind Process. In the equa-torial region, much of the large-scale variability in wind ďż˝ eldscan be represented by treating the atmosphere as a thin ďż˝ uid;that is, the depth of the atmosphere is much smaller thancharacteristic horizontal length scales (e.g., Gill 1982; Holton1992). However, the thin-ďż˝ uid approximation is incompletein that it excludes small-scale motions that are fundamentallythree-dimensional, and it is based on a zero-mean background
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Wikle et al.: Spaciotemporal Hierarchical Bayesian Modeling 387
ďż˝ ow. The following decompositions for our statistical modeladdress these deďż˝ ciencies while retaining the convenience ofthe thin-ďż˝ uid approximation:
utD Ĺu
C uEt
C Qut (3)
andvt
D ĹvC vE
tC Qvt 0 (4)
Here Ĺu and Ĺv are spatial means for the respective windcomponents, uE
t and vEt are the component contributions from
the thin-ďż˝ uid approximation, and Qut and Qvt represent small-scale motions.
We assume that the components 8Ĺu1Ĺv1uEt 1vE
t 1 Qut1 Qvt9
are mutually independent. The assumption of independencebetween the elements of ut and vt requires physical justiďż˝ ca-tion, which is discussed in Section 3.3.2.
Large-Scale Wind Components. The thin-ďż˝ uid approxima-tion for large-scale tropical dynamics also involves compan-ion approximations. Important among these are the neglect ofnonlinear terms in the momentum equations and the simpli-ďż˝ cation of spherical effects to a linear dependence on lati-tude. These approximations lead to a system referred to in thegeophysical literature as the âlinear shallow-water equationson the equatorial beta planeâ (e.g., Gill 1982; Holton 1992).Looking for solutions in the form of two-dimensional wavesin the Cartesian 4x1 y5 plane leads to an ordinary differentialequation for vE4x1 y3 t5, from which corresponding solutionsfor uE4x1 y3 t5 can be derived. The solutions for vE4x1 y3 t5
can be written as
vE 4x1 y3 t5 DX
p
X
l
vEl1 p4x1 y3 t51 (5)
where the vEl1 p4x1 y3 t5 represent the equatorial normal mode
(ENM) orthogonal basis set (Matsuno 1966). The waves asso-ciated with individual ENMs are identiďż˝ able in observations(e.g., Wheeler and Kiladis 1999), and they form the founda-tion for much of our understanding of tropical dynamics inthe atmosphere and ocean.
In practical applications, the inďż˝ nite series (5) is often trun-cated to a few leading modes, such that
vE4x1 y3 t5PX
pD1
LX
lD0
vEl1 p4x1 y3 t5 (6)
for some choice of P and L; here we use set of P D 2 andL D 3, yielding eight modes for vE . The ENM theory applies tomotions with length scales as long as the circumference of theplanet. The prediction domain size limits the maximum lengthscale in our problem to a small fraction of the circumference.In theory, energy can be distributed across an inďż˝ nity of modesin the series (5). But Wheeler and Kiladis (1999) demonstratedthat most of the energy is distributed in clusters of a relativelyfew modes, suggesting that the truncation used in (6) is nottoo severe.
It can be shown that each mode can be written as
vEl1p4x1 y3 t5 D Vl4y5 cos4kpx Ć âl1pt51 (7)
where Vl4y5 describes the northâsouth structure of the lthmode; kp
D 2ďż˝ p=Dx, where p is the eastâwest wave numberand Dx is the eastâwest domain length; and âl1 p is the dis-persion frequency of the 4l1p5th wave mode solution (i.e., itdescribes the propagation speed and direction of the ENM).Further, the northâsouth structure can be shown to be propor-tional to Hermite polynomials that are exponentially dampedaway from the equator (e.g., Gill 1982),
Vl4y5 D Hl4yĂź 5 exp4Ć05y Ăź 251 (8)
where Hl45 is the lth Hermite polynomial (with l correspond-ing to the number of nodes in the northâsouth direction) andy Ăź is the ânormalizedâ latitudinal distance from the equator.Speciďż˝ cally, y Ăź D â0y=4
pghe=â05
05, where â0 is a constantrelated to the ratio of the earthâs angular velocity to its radius,g is the gravitational acceleration, and he is the âequivalentdepthâ of the thin ďż˝ uid. Of the parameters considered here,Dx (and thus kp), â0, and g are ďż˝ xed and known. The disper-sion frequency (âl1p) and the equivalent depth parameter (he)cannot be precisely determined from the thin-ďż˝ uid approxi-mation theory; however, plausible values can be estimated viadata analysis (e.g., Wheeler and Kiladis 1999). In the caseof the dispersion frequency, we consider a reparameteriza-tion using random components (as discussed later) with pri-ors determined from historical data analysis (see Sec. 3.4.1).For the equivalent depth parameter, it is natural (as Bayesians)to model he as random and use this historical information toconstruct a prior distribution. But in view of the complex wayin which he enters the model through the Hermite polynomi-als and the already complicated scope of our model, a fullyBayesian analysis seems prohibitive. We simply set he
D 25 mwhich is what our prior mean would be based on the discus-sion of Wheeler and Kiladis (1999). Fortunately, the analysisdoes not seem particularly sensitive to the value of he (seeSec. 4.4).
An elementary trigonometric identity allows us to rewrite(7) as
vEl1 p4x1 y3 t5 D cos4âl1pt56Vl4y5 cos4kpx57
C sin4âl1 pt56Vl4y5 sin4kpx570 (9)
In view of the approximations associated with this devel-opment, that real winds are very unlikely to propagate likeperfect sinusoids as suggested in (9). Furthermore, theseexpressions were obtained in continuous space and time; ourstatistical model is for gridded winds de� ned on a limiteddomain. To account for such sources of uncertainty, we embedthe physical modeling into a stochastic model. Speci� cally,we replace the leading cosine and sine terms in (9) with ran-dom coef� cients. That is, for each of our grid points si ²4xi1 yi51 i D 11 : : : 1 n, we let
vEt 4si5 D
PX
pD1
LX
lD0
8al1p314t56Vl4yi5 cos4kpxi57
C al1p324t56Vl4yi5 sin4kpxi5791 (10)
where al1 p3 14t5 and al1p3 24t5 are assumed to be random coef-ďż˝ cients. Allowing these parameters to be random greatly
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388 Journal of the American Statistical Association, June 2001
increases the ďż˝ exibility of our model. In addition, we see thatthe cosine and sine terms that they replace suggest a natural,physically based prior. The model for the aâs is described inthe following section.
Our stochastic version of (6) takes the form
vEt
D ĂŞavt 1 (11)
where vEt is the vector of vE winds for all prediction grid
locations at time t and avt is a vector of pairs of aâs for
each of the J D P ďż˝ 4L C 15 combinations of p and l (recallthat P D 2 and L D 3, so av
t is of length 16). The matrixĂŞ is obtained by evaluating the ENM basis functions at gridpoints. Speciďż˝ cally, for a total of J combinations, ĂŞ is ann ďż˝ 2J matrix with columns â24jĆ15C14x1 y5 D Vj4y5 cos4kjx5and â24jĆ15C24x1 y5 D Vj4y5 sin4kjx5 for j D 11 : : : 1 J , eval-uated at the coordinates of the n prediction grid locations.Figure 3 shows the structure of two of these basis functions,4l1 p5 D 40115 and 4l1p5 D 421 15. A similar model, uE
tD ĂŞau
t ,is also used.
Small-Scale Wind Components. The small-scale windcomponents Qvt and Qut represent scales and types of dynam-ical processes not explained by the thin-ďż˝ uid approximation
110 120 130 140 150 160 170
â 20
â 15
â 10
â 5
0
5
10
15
20
0.010.01 0.01
0.01
0.010.010.01
0.01
0.02 0.020.02
0.02
0.020.02
0.02
0.03 0.03
0.03
0.030.03
0.040.04
0.04
0.04
East Longitude (deg)
Latit
ude
(deg
)
â0.01
â0.0
1
â0.01
â0.0
1
â0.02
â0.02
â0.03
â0.03
â0.04
110 120 130 140 150 160 170
â20
â15
â10
â5
0
5
10
15
20
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.02 0.02
0.02
0.020.02
0.03
0.03
0.03
0.03
0.04
0.04
East Longitude (deg)
Latit
ude
(deg
)
â0.01
â0.01 â0.01
â0.01
â0.01â0.01
â0.02
â 0.02
â0.02
â0.02
â0.03
â0.03
0.03
â0.04
(a)
(b)
Figure 3. Examples of Shallow-Water Equatorial Normal Mode BasisFunctions Used in the Analysis. (a) Northâsouth Hermite mode l D 0;Eastâwest Fourier mode domain wavenumber p D 1. (b) l D 2, p D 1.
near the equator. We would like these processes to representthe scales that are resolved in the NSCAT sampling and arecommonly thought to display multiresolution spatial behaviorassociated with fractal processes. We chose to represent themin terms of wavelet basis functions with compact support,
QvtD ĂŤbv
t 1 (12)
where bvt is an n-vector of temporally evolving random coefďż˝ -
cients and ĂŤ is an nďż˝ n matrix containing Daubechies waveletbasis functions of order two (evaluated on the prediction grid),modiďż˝ ed for closed domains (e.g., Cohen, Daubechies, andVial 1993); the âorderâ is the number of vanishing momentsof the wavelets. A similar decomposition is speciďż˝ ed for Qut .
Our use of wavelets is motivated by the observation thatthese small-scale processes are typically localized in spaceand time. The speciďż˝ c choice of the foregoing multiresolu-tion wavelet basis is based on its ability to represent fractalprocesses (e.g., Wornell 1993). This is critical in attemptingto explain the multiscale turbulence structure of wind ďż˝ elds(see Sec. 3.4). Moreover, this wavelet basis has advantages interms of computational efďż˝ ciency (see Sec. 4).
Spatial Mean. The spatial mean processes Ĺv and Ĺu
account for the climatological mean wind structure. In thetropical western Paciďż˝ c Ocean, the climatological winds areeasterly (i.e., out of the east, toward the west). Note thatour domain contains land areas (see Fig. 1). Given that near-surface wind behaves differently over land and sea (e.g., sur-face heating and/or frictional differences), the spatial meanďż˝ eld should include a dichotomous variable to delineatewhether a prediction grid location is over land or sea. Finally,although the climatological wind structure can change withseason and horizontal extent, our spatial domain is smallenough and our temporal domain short enough (approximately2 weeks) that we need not consider more complicated spatialor time-varying mean ďż˝ elds in this analysis.
3.3.2 Process Model Speci�cation. The decomposi-tions (3) and (4) and subsequent modeling lead to the statisti-cal models
utD Ĺu
C ĂŞaut
C ĂŤ but (13)
andvt
D ĹvC ĂŞav
tC ĂŤ bv
t 0 (14)
Our hierarchical Bayesian model at this stage requires speci-ďż˝ cation of a parameterized joint distribution
6Ĺu1 Ĺv1 8aut 9T
0 1 8avt 9
T0 1 8bu
t 9T0 1 8bv
t 9T0âË71 (15)
where 8aut 9T
0 represents the collection 8aut 2 t D 01 : : : 1 T 9, and
so on, and Ë generically denotes a collection of parametersto be speciďż˝ ed. The crucial point is that the dynamic aspectof our modeling is through time series models for the a andb vectors. We use autoregressive models for these evolutions.Hence we have appended their initial states to the collectionof unknowns. Priors for these initial states are discussed at theend of this section.
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Wikle et al.: Spaciotemporal Hierarchical Bayesian Modeling 389
As noted earlier, a critical modeling assumption is that allsix components of the gridded winds in (15) are mutuallyconditionally independent; that is, (15) is factored as
6Ĺu1ĹvâË768au
t 9T0âË768av
t 9T0âË768bu
t 9T0âË768bv
t 9T0âË70 (16)
Our justiďż˝ cation of the priori independence assumption isbased primarily on physical grounds. The theory of nondi-vergent two-dimensional turbulence implies that the velocitycomponents are uncorrelated across all spatial scales (e.g.,Freilich and Chelton 1986). As discussed in Section 3.4.2, werely strongly on theoretical and empirical results suggestingthat tropical surface wind ďż˝ elds behave like turbulent ďż˝ elds. Inaddition, Freilich and Chelton (1986) showed that the empir-ical cross-spectral densities of tropical surface wind compo-nents are very small, justifying the general prior modelingassumption of independence. Of course, dependence can arisea posteriori, especially in the presence of physically mean-ingful structures (e.g., storms). For example, for our 2-weekstudy period, posterior analysis yields a correlation betweenwind components, averaged over both time and space, of .3.
We next describe the prior distributions in (16). For econ-omy in presentation, we describe in detail only the models forthe v-components and hence suppress dependence on v. Themodels for the u-components were developed similarly andare summarized at the end of this section.
Spatial Mean. We chose a simple spatial regression modelfor Ĺ,
Ĺ D PĆ1 (17)
where P is a speciďż˝ ed design matrix. In our analysis, thisincludes an overall intercept term and a land/sea indicator vari-able (1, land; 0, sea). The regression coefďż˝ cient vector Ć isthen length 2 and is assigned a bivariate normal prior dis-tribution, Ć Gau4Ćo1 èĆ5. The hyperparameters of this dis-tribution were speciďż˝ ed based on an ordinary least squaresregression of NCEP data from a 4-month period roughly cen-tered around, but excluding, our study period. Speciďż˝ cally, forthe v and u components the prior means were 4Ć041 0025 and4Ć20711095. We assumed that the prior variance-covariancematrices were diagonal with relatively small variances. Weused preliminary data analysis in developing these speciďż˝ ca-tions. Because our study period is only 2 weeks long, genuineclimatological means (even seasonal means) would not servewell in centering the model. Further, these mean parameterswere not of interest in and of themselves; they merely offereda simple method for adjusting for a land-versus-sea effect.
Dynamic Models. One of the key features of our approachis that we seek to model empirically the atmospheric dynam-ics, so that wind information observed at time t can in prin-ciple propagate to nearby locations at time t C 1, where theremay be fewer observations (e.g., see Fig. 2). Thus we assumethat the coefďż˝ cient vectors (aâs and bâs) are conditionally inde-pendent and follow ďż˝ rst-order Markov vector autoregression(VAR) models: for t D 11 : : : 1 T ,
atâHa1atĆ11 èâĄa
Gau4HaatĆ11èâĄa5 (18)
and
btâHb1 btĆ11 èâĄb
Gau4HbbtĆ11èâĄb51 (19)
where Ha and Hb are VAR parameter matrices for the ENMand wavelet coefďż˝ cients and èâĄa
and èâĄbare the associated
VAR innovation covariance matrices.To initialize these VAR models, we assumed that a0
Gau4Ĺa1èa5 and b0 Gau4Ĺb1èb5. The hyperparameters Ĺa,èa, Ĺb, and èb were speciďż˝ ed based on an assumption ofmean 0 and diagonal covariance matrices with large variances.Speciďż˝ cally, èa was assumed to have variance 100 and èb
was given prior variance corresponding to the multiresolutionscaling discussed in Section 3.4.2.
3.4 Stage 3: Priors on Parameters
We assume that the parametersâ 2I 1â 2
B1â 21Ha1Hb1 èâĄa, and
èâĄbare mutually independent. Similar formulations are used
for the parameters relevant to the u-component model.
3.4.1 Autoregressive Parameter Matrices. As suggestedby the derivations in Section 3.3.1, to describe wave struc-tures that propagate in time, each pair of coefďż˝ cients al1p31 andal1p3 2 must be dependent. A simple model for such evolutionis a ďż˝ rst-order vector autoregression,
"al1p3 14t5
al1 p324t5
#D Ha
l1p
"al1p3 14t Ć ât5
al1p3 24t Ć ât5
#C âĄa
l1 p4t51 (20)
where Hal1p is a 2ďż˝ 2 propagator matrix, the âĄa
l1p4t5 are vectorsof random innovations, and ât is some time interval (.25 daysin our case). Application of simple trigonometric identitiesfor cos4âl1p4t C ât55 and sin4âl1p4t Cât55 suggests physicallybased prior information for the structure of Ha
l1p:
Hal1 p
D"
cos4âl1pât5 Ć sin4âl1pât5
sin4âl1pât5 cos4âl1pât5
#0 (21)
Given an equivalent depth he, âl1p can be determined fromdata analysis. For the v-wind, we used the values suggestedby Wheeler and Kiladis (1999) as prior means, namely âl1p
D2ďż˝ 6Ć01331Ć0181Ć0081 Ć005, .67, .59, .75, .75] for (l, p)D 6401 15, (0, 2), (1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (3,2)].Our prior knowledge regarding the last two modes is com-paratively uninformed. Note that in some 4l1p5 combinations,wave modes with the identical horizontal structure (i.e., basisfunction) can have different propagation characteristics. Forsimplicity, we chose for our prior the âdominantâ wave modesuggested by the data analysis of Wheeler and Kiladis (1999).Thus vec4Ha
l1 p5 is speciďż˝ ed to be Gaussian with means givenby (21) and diagonal covariance structure with relatively largeprior variances all set to 100. Sensitivity analysis showed thatthe posterior wind ďż˝ elds were not sensitive to these speciďż˝ ca-tions. Similar priors were developed for the u-components.
Our speciďż˝ cation for the prior on the VAR matrix Hb isbased more on a subjective sense of the dynamics. We expectsmall-scale features to have some persistence over the 6-hourtime intervals considered in this model. However, it is notclear from theory what the prior means and variances shouldbe or whether we should allow spectral interaction. Interaction
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390 Journal of the American Statistical Association, June 2001
of the spectral modes would be implied if we allowed nonzerooff-diagonal elements in Hb . The added level of complexityrequired to implement such a formulation was not justi� edin the current application. Instead, an effective interaction ofscales is parameterized by a fractal innovation variance struc-ture, as described in Section 3.4.2. We assume that the ele-ments of Hb ² diag46hb4151 : : : 1 hb4n5705 are distributed asindependent Gaussians,
hb4j5â N 4Ĺhb4j51â 2
hb4j55 2 j D 11 : : : 1 ka0 (22)
For the hyperparameters, we chose Ĺhb4j5 D 04 and â 2
hb4j5 D
001 for all j. These values reďż˝ ect our subjective physical priorof persistence in small-scale modes. Sensitivity analyses onthese hyperparameters showed that the posterior wind ďż˝ eldswere not extremely sensitive to the speciďż˝ cation.
3.4.2 Autoregressive Innovation Covariance Matrices.The VAR conditional covariance matrix èâĄa
is assumed to beblock diagonal, with J 2ďż˝ 2 covariance matrices, èâĄa
4l1 p5 onthe diagonal. For each l D 01 : : : 1L and p D 11 : : : 1P , thesecovariance matrices are assumed to be mutually independentand distributed as
èâĄa4l1p5Ć1 W 44Ĺ aSâĄa
4l1p55Ć11Ĺ a51 (23)
where W45 is a Wishart distribution with degrees of freedomĹ a and expectation SâĄa
4l1p5Ć1. For the v-component, thesehyperparameters were speciďż˝ ed to be Ĺ a
D 2 and SâĄa4l1p5 D
â 2âĄa
4l1 p5I, where â 2âĄa
4l1p5 D 4s24l1p5=2561Ć 4cos4âl1pât5527.
In this case, s24l1 p5 are climatological variances for each wavemode as observed by Wheeler and Kiladis (1999); that is,s24l1p5 D (2,133, 2,681, 3,047, 7,922, 305, 335, 200, 200), forthe eight ENMs used here. The posterior wind ďż˝ elds are notoverly sensitive to the choice of these hyperparameters. A sim-ilar speciďż˝ cation was developed for the u-component portionof the model.
For the wavelet coefďż˝ cient innovation covariances, weassumed that
èâĄb² diag4â 2
âĄb4151 : : : 1â 2
âĄb4n550 (24)
The choice of the hyperparameters were based on physicalideas. The spatial energy spectrum of tropical surface windshas been shown to behave like a self-similar random fractalprocess (Freilich and Chelton 1986; Wikle et al. 1999), inwhich the energy spectrum is proportional to the inverse ofthe spatial frequency taken to some power,
Sv4k5 / â 2v
âkâd1 (25)
where Sv4k5 is the spatial energy spectrum of v at spatialfrequency k, â 2
v is the wind component variance, and d isthe decay rate (e.g., Wornell 1993). In the tropical surfacewind case, d has been shown to be approximately equalto 5=3 over a broad range (1â1000 km) of spatial scales(Wikle et al. 1999). This spectral decay rate is consistent withfamous results from turbulence theory (Kolmogorov 1941a,b;see also Rose and Sulem 1978). It is also a robust empir-ical result in that recent observational studies of surface
winds (Freilich and Chelton 1986; Milliff et al. 1999; Wikleet al. 1999) and winds aloft (Lindborg 1999; Nastrom andGage 1985) demonstrate a similar power law relation withoutthe conditions for two-dimensional isotropic turbulence andan inertial subrange required by the theory of Kolmogorov.Wornell (1993) derived the relationship for variances of such afractal process in terms of scales of a wavelet multiresolutionanalysis. Chin, Milliff, and Large (1998) extended this resultto the two-dimensional case by assuming identical distributionof the âdiagonal,â âhorizontal,â and âverticalâ multiresolutionwavelet coefďż˝ cients. They showed that the variance of two-dimensional wavelet coefďż˝ cients is proportional to 2Ćl41Cd5Ć1,where l is the level of the multiresolution decomposition(l D 11 : : : 1Nl). We use these results, along with the result thatthe innovation variance for a ďż˝ rst-order autoregressive processcan be written in terms of the autoregressive coefďż˝ cient andmarginal variance (e.g., â 2
âĄbD 61 Ć h2
b7â 2b ), to derive the prior
variances for each multiresolution level in the âĄb process,
â 2âĄb
4l5 / 61Ć h2b4l5762Ćl41Cd5Ć171 (26)
where we substitute the prior mean ĹhbD 04 for hb4l5 and
let d D 5=3. We use this relationship to determine the inversegamma ( IG) priors,
â 2âĄb
4j5âqâĄb4j51 râĄb
4j5
IG4qâĄb4j51 râĄb
4j55 2 j D 11 : : : 1 kb0 (27)
That is, we deďż˝ ne all spectral indices within a givenmultiresolution scale (l) to have independent inverse gammadistributions with parameters qâĄb
4l5 and râĄb4l5 determined by
assuming a mean given in (26) and a suitable variance. Forinstance, we give a large variance to the largest wavelet scales,which overlap with the large-scale equatorial modes and canadequately be determined by the data. Alternatively, we assignsmall (inverse gamma) prior variances for small and mediumwavelet scales where observational data are less abundant.This is the most critical prior assumption in the Bayesian anal-ysis! Sensitivity analysis has shown that if we do not givenarrow priors on the small- and medium-scale wavelet modes,the posterior spectrum will not follow the 5/3 slope over allspatial scales, as is necessary for realistic wind ďż˝ elds. This issimply because some large spatial regions are not sampled bythe scatterometer. Thus, by using the narrow priors, we are ineffect constraining the posterior to physical reality, but in sucha way that it can be informed by the data, if available.
3.4.3 Measurement Error Variances. The measurementerror variances for the data model were assigned inversegamma distributions â 2 IG4q1 r5, â 2
I IG4qI 1 rI5, andâ 2
B IG4qB1 rB5. As noted in Section 3.2, Freilich and Dunbar(1999) showed that the NSCAT measurement error varianceis approximately 1.7 (m/s)2. Because we ignored âgriddingerrorâ in both space and time, we inďż˝ ated this value to aprior mean of 2.0 (m/s)2 and assumed a prior variance of .1.Hence we set q D 42 and r D 00122. There is little infor-mation in the literature concerning NCEP measurement errorvariances. We have partially accounted for the overly smoothnature of NCEP winds via the Ka matrix, and so suggest thatthe measurement error variance should be about the same as
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Wikle et al.: Spaciotemporal Hierarchical Bayesian Modeling 391
found for NSCAT [1.7 (m/s)2] at interior NCEP locations andtwice that [3.4 (m/s)2] at boundary grid locations. This latterassumption follows because fewer prediction grid locations areavailable for the change of support averaging (see Sec. 3.2).However, to reďż˝ ect our lack of certainty, we assigned largerprior variances (.3) than for the NSCAT variance. Thus we setqI
D 110631 rID 005531 qB
D 40053, and qBD 00074. Our poste-
rior wind ďż˝ elds were not extremely sensitive to these choices.
4. BAYESIAN ANALYSIS
The fundamental product of a Bayesian analysis is the pos-terior distribution of all unknowns. Explicit formulas for theposterior distribution for large complicated hierarchical mod-els such as those presented here are intractable. Hence we usea Markov chain Monte Carlo (MCMC) methodâspeciďż˝ cally,a Gibbs sampler.
4.1 Computation
Our example analysis involve 64 ďż˝ 48 ďż˝ 54 1661000prediction locations in space (i.e., 64 ďż˝ 48) and time (i.e.,54), and we have a large amount of data to ingest intothe model ( 2001000 observations over 14 days). Deriva-tion of the full conditional distributions used in a basicGibbs sampler implementation is straightforward; the rel-evant full conditionals are available on the Web athttp://www.stat.missouri.edu/ wikle/trop_wind_pap.html. Butthe high dimensionality of some of these distributions pre-cludes the use of traditional sampling algorithms. For instance,consider the full conditional distribution for the waveletcoefďż˝ cients,
bt⢠Gau6AĆ1
t gt1AĆ1t 71 (28)
for t D 11 : : : 1 T , where
At ² 4Í 0K0tè
Ć1t KtĂŤ C èĆ1
âĄbC H0
bèĆ1âĄb
Hb5Ć1 (29)
and
gt ² 44VtĆ KtĹv
Ć KtĂŞavt 5
0èĆ1t KtĂŤ
C bv 0
tĆ1Hv0
b èĆ1âĄb
C bv 0
tC1èĆ1âĄb
Hb500 (30)
Each At is a 31072 ďż˝ 31072 matrix, and many of the matri-ces from which it is computed are huge (e.g., Kt can be aslarge as 31072ďż˝ 61481). Standard methods for the generationof very high-dimensional multivariate normal random variates(see, e.g., Ripley 1987) are impractical, because we must sam-ple from such high-dimensional distributions for each time tand over many Gibbs iterations. Fortunately, the sparse spec-iďż˝ cation of Kt can be exploited computationally (e.g., Press,Flannery, Teukolsky, and Vetterling 1986, sec. 2.10). Simi-larly, the models for temporal evolution parameters (e.g., Hb
and èâĄb) involve sparse (e.g., diagonal) matrices. Further,
computations for the multiresolution wavelet transform are fast(order-n operations). The net result is that matrix multiplica-tions of the form Atw, for any n-vector w, can be performedin order-n operations.
To make sampling from such a distribution practical on ahigh-end workstation, we use iterative linear methods. Specif-ically, we use a conjugate gradient solver (e.g., Golub and
van Loan 1996, sec. 10.2). Details of this sampling approachare given in the Appendix. A key strength of the conjugategradient approach is that the sparse operations described in theprevious paragraph can be exploited. The iterative solver ter-minates after a preselected convergence criterion is met. Thesample obtained is an approximate sample from the true full-conditional distribution. We can control the degree of approx-imation by selecting a more or less rigorous convergencecriterion. For the results presented here, we have prescribeda rather rigorous convergence criterion (see the Appendix).If larger spatiotemporal domains are of interest, then trade-off between computation time and the degree of convergencebecomes important.
4.2 Gibbs Sampler Convergence
The Gibbs sampler was implemented separately on both theeastâwest (u) and northâsouth (v) wind components. (This isvalid under all of the conditional independence assumptionsdescribed earlier.) Strategies to assess the convergence of aGibbs sampler in high-dimensional models (e.g., 105 param-eters) such as presented here are not well developed. We baseour convergence diagnosis on visual assessment of randomlyand subjectively chosen model parameters obtained from pilotsimulations with varying starting values. Along with perform-ing a visual assessment, we examined the Gelman and Rubin(1992) convergence monitor. These assessments suggested noreason to reject convergence after about 700 iterations. Wethen ran a single chain (2,400 iterations) and discarded theďż˝ rst 800 iterations. We based inference on the remaining1,600 samples.
4.3 Posterior Wind Process
A particularly interesting time period in our data cen-ters on the mature phase of tropical cyclone Dale. In par-ticular, consider the u-component posterior mean wind ďż˝ eldfor 0000 UTC on November 7, 1996 Figure 4(a) shows theNCEP weather center u-wind component ďż˝ eld for this period,Figure 4(b) shows our (estimated) posterior mean for the u-wind component, Figure 4(c) shows the ďż˝ eld of posteriormeans for the u-wind spatial mean plus the equatorial wavemodes (i.e., Ĺu
C ĂŞaut ), and Figure 4(d) shows the associ-
ated wavelet mode posterior mean component (i.e., ĂŤ but ). The
posterior wind ďż˝ eld has signiďż˝ cantly more small-scale spatialstructure than the NCEP ďż˝ eld. Recalling the NSCAT samplingfor this period (see Fig. 2), it is clear that there is small-scalestructure in regions for which small-scale observations werenot available. This is a crucial and desirable feature of ourmodeling strategy.
4.4 Sensitivity
Assessing sensitivity to our prior/model speciďż˝ cations isextremely difďż˝ cult because of the modelâs size and complex-ity. We performed some sensitivity analyses one parameter at atime, by rerunning the Gibbs sampler with different values foreach parameter, albeit with fewer iterations. We would expectinteractions among sensitivities of various models and pri-ors on parameters at various levels, but performing âcompletefactorialâ sensitivity experiments is not feasible. We inves-tigated sensitivities mainly by visually inspecting the wind
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392 Journal of the American Statistical Association, June 2001
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Figure 4. East-West (u) Component of Wind at 0000 UTC on November 7, 1996 (in ms 1). (a) NCEP u-wind component; (b) posterior mean oftotal âblendedâ u-wind [i.e., sum of components shown in (c) and (d)]; (c) posterior mean of u-wind spatial mean component (Ĺ) plus equatorialmode components (ĂŞat ); (d) posterior mean of wavelet mode u-wind components (ĂŤbt ).
ďż˝ elds and examining the empirical spatial spectrum of theposterior winds to see how it compared to the desired 5/3slope discussed in Section 3.4.2. The posterior wind ďż˝ elds arenot sensitive to reasonable choices of the equivalent depth he,NCEP weighting scheme (Ka), and hyperparameters on mea-surement error variances. Similarly, the posterior wind ďż˝ eldsare not overly sensitive to the hyperparameters for Ć, Ha
l1p ,èâĄa
4l1 p5, and Hb . But as mentioned in Section 3.4.2, theposterior wind ďż˝ elds are very sensitive to the priors on èâĄb
,which must be narrowly centered around the required fractalvariances that give the desired 5/3 spatial spectra. This is nec-essary to ensure proper variability in the posterior winds overareas and time periods where NSCAT sampling is absent.
5. INFERENCE AND MODEL ASSESSMENT
Though again limited by model size and complexity,we considered three âvalidationsâ: external/physical, inter-nal/physical, and NSCAT data hold out/resample.
5.1 External Physical Veriâ cation and Inference
As stated in Section 1, to understand convective processesin the tropical atmosphere, one needs a detailed view of the
surface wind ďż˝ eld and its horizontal derivatives. Speciďż˝ cally,we consider the divergence of the surface wind ďż˝ eld. Thedivergence, deďż˝ ned at a point as ÂĄu=ÂĄx C ÂĄv=ÂĄy, measures theoverall rate at which air is being transported away from thatpoint. Conversely, if the sign of the divergence at a locationis negative, then air is converging on the point. Convergenceat the surface can be related, through a continuity equation, toupward vertical motion. If sufďż˝ cient moisture is available inthe atmosphere, then this rising motion leads to the formationof clouds and, through nonlinear dynamic and thermodynamicprocesses, the possibility of a tropical storm associated withdeep convection. This suggests that external veriďż˝ cation of ourmodel would involve comparing cloud imagery with diver-gence ďż˝ elds calculated from our posterior wind ďż˝ elds.
Figure 5(a) shows wind vectors and gridded estimates ofdivergence for a subset of the spatial domain at 0000 UTCon November 7, 1996 based on the low-resolution NCEPdata only. This period corresponds to the mature phase oftropical cyclone Dale. The NCEP ďż˝ eld represents the state-of-the-art wind and divergence ďż˝ elds currently available.Figure 5(b) shows a cloud top (or âbrightnessâ) temperatureimage for the same period as observed from the Japanese GMS
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Wikle et al.: Spaciotemporal Hierarchical Bayesian Modeling 393
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Figure 5. (a) NCEP Divergence (s 1) and Wind Fields (direction of arrows correspond to wind direction, and length corresponds to magnitude)for a Subregion of the Prediction Grid at 0000 UTC on November 7, 1996; (b) Cloud Top Temperature (deg K) Satellite Imagery for the SamePeriod; (c) Corresponding Blended Posterior Mean Divergence and Wind Fields.
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394 Journal of the American Statistical Association, June 2001
satellite. Colder cloud top temperatures on this plot generallycorrespond to higher clouds, which in turn are indicative ofdeep convection and tropical storm activity. Thus areas ofclouds in Figure 5(b) should be associated with darker blueareas (convergence) in Figure 5(a). The comparison betweenthe NCEP divergence ďż˝ eld and this cloud imagery clearlyshows that the NCEP ďż˝ eld does not capture the conver-gence associated with the cloud structures and bands of deepconvection associated with the tropical storm. Alternatively,Figure 5(c) shows the posterior mean wind vectors and sur-face divergence for the same period from our analysis. Theuse of NSCAT winds and a model capable of space-time prop-agation have added detail not present in the NCEP analysis.In particular, note the substantial agreement between areas ofconvergence in the wind ďż˝ eld and cloud bands in the tropi-cal cyclone. The physical agreement between convergence andcloud imagery shown here provides very strong physical evi-dence that the model is performing well.
5.2 Internal Physical Veriâ cation
An important check on our model is obtained by exam-ining realizations from the posterior distribution. Figure 6shows divergence/wind plots for two Gibbs-sampled realiza-tions (widely separated in âGibbs timeâ) for the cyclone Daleperiod shown in Figure 5. These realizations are physicallyrealistic, suggesting no reasons for questioning the plausibilityof the posterior distribution. Furthermore, Figure 6(c) showsthe posterior standard deviation for divergence at this sametime. Note that, as expected, the âtracksâ of low standard devi-ation correspond to the satellite sampling paths (see Fig. 2).
5.3 Hold Out/Resample Veriâ cation
Although it would be useful to inspect residuals from ourmodel, we do not have residuals in the traditional sense. Ourdata sources reďż˝ ect winds at either coarser (NCEP) or muchďż˝ ner (NSCAT) spatial scales. The modeled wind process isnever observed! Nonetheless, we investigated the modelâs abil-ity to generate plausible observational data.
Consider the time period represented in Figure 5. We rana separate Gibbs sampler but left out the NSCAT data forthis period. We then compared NSCAT observations to poste-rior means (and realizations) at the NSCAT locations by map-ping the posterior output to those locations via the appropriateKs4t
05. Figure 7(a) shows the relationship when all NSCATdata are included in the analysis. Figure 7(b) shows the resultwhen the NSCAT data for this time period are excluded.Similarly, Figures 7(c) and 7(d) show the same plots, butfor a realization from the posterior distribution. Given theamount of data removed (over 5 ďż˝ 103 observations), the lin-ear associations shown in these ďż˝ gures suggest that the modelis reasonable.
6. DISCUSSION
The wind ďż˝ elds from these analysis are currently being usedin studies of tropical cyclone development and its relation-ship to intraseasonal and interseasonal phenomena such as theMaddenâJulian oscillation and El NiĂąo, and the seasonal pre-diction of El NiĂąo. Additional studies of this kind will be pos-sible when the methodology is extended to cover the entire
tropical region. We are currently âportingâ this model to asupercomputing environment, which will allow such calcula-tions for larger domains. Because the posterior wind ďż˝ eldsgenerated by the current model show realistic small- andmedium-scale variability, the results from these analyses canthen be used to provide distributional forcing to tropical oceangeneral circulation models.
APPENDIX: HIGH-DIMENSIONAL MULTIVARIATENORMAL SAMPLING
Consider the full conditional distribution for some n ďż˝ 1 vector x,
x⢠N 4QĆ1g1 QĆ151 (A.1)
where Q ² Í 0K0KÍ C D is known and has dimensions n � n andg is a known n � 1 vector. De� ne n � 1 random vectors e11 e2
iid N 401 I5 and let
f ² Í 0K0e1 CD1=2e20 (A.2)
Consider the linear system
Qx D g C f0 (A.3)
Because Q is invertible by hypothesis and, with probability 1, g 6D Ćf ,the unique (with probability 1) solution to (A.3) is Qx D QĆ14g C f5.It can be easily shown that E4f5 D 0 and var4f5 D Q, and henceE4Qx5 D QĆ1g and var4Qx5 D QĆ1. Thus, with simulated e1 and e2 , thecorresponding solution to (A.3) is a sample from (A.1).
For n very large, we use iterative approaches to solve (A.3), ratherthan attempt the indicated matrix inversion directly. Speciďż˝ cally, weuse the conjugate gradient algorithm (e.g., Golub and Van Loan1996, sec. 10.2). Especially in the case of sparse systems as arisingin our model, this approach has computational advantages related tostorage and efďż˝ ciency. The basis of the algorithm is that the solutionto (A.3) coincides with the minimizer of the expression
minx
12
x0Qx C x04g C f5 0 (A.4)
Improvements over direct iteration, such as Newtonâs method orsteepest descent, come to mind. The conjugate gradient method issimilar, but has the property that all successive differences, xiC1 Ćxi ,between iterates are mutually Q-orthogonal (or âconjugateâ); that is,4xiC1 Ćxi50Q4xjC1 Ćxj5 D 0.
As with most iterative procedures, a key computational issue isthe rapid computation of powers of Q. Indeed, we can write (A.3) as
4ĂŤ 0K0KĂŤ C D5x D g C ĂŤ 0K0e1 CD1=2e21 (A.5)
where D1=2 is sparse for our models. Thus we do not have to store Q,and must only perform a series of vector multiplications. By makinguse of sparseness from our hierarchical implementation and spectraland multiresolution representations, we can do these multiplicationsvery efďż˝ ciently (e.g., in our case ĂŤ x corresponds to the inverse dis-crete wavelet transform).
With an iterative approach, a choice must be made as to startingvalues. We typically use the value for the previous Gibbs iteration orthe one-step-ahead âpredictionâ from the appropriate Markov model.Furthermore, although the conjugate gradient algorithm is known toconverge to the solution in at most n steps, n is far too large for thealgorithm to be run convergence for each MCMC iteration. Henceone must choose an approximate-convergencecriterion. In our imple-mentation, this criterion is speciďż˝ ed to be ËËgC fË, where Ë D 00005;this criterion is usually met after 15â30 iterations.
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Wikle et al.: Spaciotemporal Hierarchical Bayesian Modeling 395
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Figure 6. (a), (b) Wind and Divergence Field Realizations From the Posterior Distribution at 0000 UTC on November 7, 1996; (c) PosteriorStandard Deviation for Divergence (s 1) at the Same Time.
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396 Journal of the American Statistical Association, June 2001
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Figure 7. NSCAT u-wind Component at 0000 UTC November 7, 1996 Versus the Posterior u-Wind Downscaled to NSCAT Locations; (a) DataVersus Posterior Mean With NSCAT Data Included for This Period; (b) Data Versus Posterior Mean with NSCAT Data Deleted for This Period; (c)Same as (a), Except That a Realization From the Posterior Is Used; (d) same as (b), Except That a Realization From the Posterior Is Used. All dataare in m/s.
[Received July 1998. Revised September 2000.]
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