84 MONTHLY WEATHER REVIEW Vol.uMell3

A Simple Scheme for Objective Analysis in Curved Flow '- STANLEY G. BENJAMIN

Program for Regional Observing and Forecasting Services, NOAA, Environmental Research Laboratories, Boulder, C0 80303and National Centerfor Atmospheric Research*, Boulder. C0 80307

NELSON L; SEAMANDepartment ofMeteorology, The Pennsylvania State University, University Park, PA 16802

(Manuscript received 21 February 1984, in nal form l4 March 1985)

ABSTRACT

An objective analysis scheme has been developed which combines use of three different weighting functions,two of which are anisotropic (elliptical and banana-shaped). The effective distance between a grid pointand an observation point used for the anisotropic functions may be applied in any objective analysis schemewhich uses distance to calculate weights or correlations, but a successive-correction scheme is used here as avehicle for testing. This relatively simple and computationally inexpensive scheme produces wind andmoisture analyses in which along-ow autocorrelation is accentuated, especially in regions of curved ow,and thus simulates conventional subjective analysis procedures. Sample analyses from a case study arepresented which demonstrate the improvement which may result from using this scheme rather than onewith the circular weighting function alone. ,

In tests with an analytically dened, curving jet stream, the multiple weighting function scheme with thebanana tnction was superior to schemes using the circular function either alone or with an ellipticalfunction for all of the error statistics considered, including a 30% reduction in rms vector error.

This objective analysis scheme also includes an alternative method for calculating corrections at individualgrid points which is designed to eliminate discontinuities which may occur when more common correctionmethods are applied. Additional analytical tests and sample analyses conrm that the new correction methoddecreases noise in gradients (e.g., vorticity, divergence) of analyzed elds which result with the use of othercorrection methods in data-sparse regions or over the entire domain when the ratio between the grid spaceand the mean station separation is small (S-10%). The analytical tests also indicate that the new correctionmethod performs slightly better than other methods for the analyzed variable itself (as well as the gradient)regardless of the scale.

1. IntroductionAn eicient scheme is presented for objective anal-

ysis in situations with curved ow. A hierarchy ofthree weighting functions is used, including two an-isotropic functions, banana-shaped and elliptical, andthe isotropic circular function. Although these func-tions are incorporated into a successive-correctionscheme in this paper, the principles on which theyare based may be used in any objective analysisscheme which uses observation point-to-grid pointdistance to calculate weights or correlations.

The resulting scheme accentuates along-ow auto-correlation of analyzed variables, similar to that fre-quently observed in the atmosphere, while only min-imally increasing computational expense. It also vir-tually eliminates gradient discontinuity problems oftenfound in regions of low or sharply changing datadensity, encountered with some objective analysistechniques.

"' The National Center for Atmospheric Research is sponsoredby the National Science Foundation.

ti-1-rig-i

1985 American Meteorological Society

The importance of objective analysis of meteoro-logical variables, especially for accurate and compu-tationally efficient initialization of numerical forecastmodels, has been manifested in the development ofincreasingly sophisticated techniques during the past30 years. Some of the earliest schemes for meteoro-logical objective analysis used polynomial tting tech-niques (Panofsky, 1949; Gilchrist and Cressman,1954; Johnson, 1957). Later, distance-dependentweighting functions were introduced (Bergthorssenand Diiiis, 1955) and simplied by employing themethod of successive corrections (Cressman, 1959).In Cressmans scheme, a rst-guess eld (from aprevious forecast or analysis) is modied by obser-vations using a weighting function ranging from 0.0to 1.0 which produces a circular area of inuencearound each observation point.

More recently, Endlich and Mancuso (1968), Inman(1970), and Ceselski and Sapp (1975) incorporatedinto their objective analysis schemes elliptical weight-ing functions oriented with the major axis along thewind direction. A number of other approaches to the

JULY 1985 STANLEY G. BENJAMIN

objective analysis problem have been suggested, suchas spectral least squares ts (Flattery, 1967), thevariational method of incorporating various dynamicconstraints (Sasaki, 1958), techniques designed toobtain any desired level of detail (Barnes, 1964), andoptimum (statistical) interpolation (Gandin, 1963;Schlatter, 1975). Of these recent methods, optimuminterpolation has become the most widely employedat facilities charged with operational numerical fore-cast responsibilities and is now used at nationalcenters in the United States, the Soviet Union, Can-ada, France, Great Britain, Australia, and at theEuropean Centre for Medium Range Weather Fore-casts (R. McPherson, personal communication, 1984).ln part because of the computational expense ofoptimum interpolation, especially in its multivariateform, variations of the relatively simple successive-correction method continue to be widely applied formeteorological objective analysis.

Satellite imagery has conrmed the characteristicstreakiness of meteorological variables, particularlywind and moisture, which has been taken into accountfor decades in conventional subjective analysis pro-cedures. Spatial autocorrelation is generally higheralong-ow than cross-ow for these variables, partic-ularly in such crucial features as jets, moist and drytongues, conveyor belts (Browning, 1971), andother subsynoptic scale features. For wind maxima,this is generally due to the elongated nature of frontalzones. Moisture streakiness, on the other hand, isusually associated with patterns of horizontal advoc-tion. Rasmussen (1982) has demonstrated using sta-tistical methods that anisotropic functions orientedalong the wind are signicantly superior to isotropicfunctions as correlation models for the relative hu-midity eld at all levels above 1000 m. Moreover, healso found that the optimum degree of anisotropy isproportional to wind speed. The National Meteoro-logical Center currently incorporates anisotropy incorrelation functions for moisture analysis using op-timum interpolation (Dey and Morone, 1985).

Although elliptical weighting functions produceimproved analyses for straight ow, especially on themesoscale (Sasaki, 1971), satellite observations alsofrequently indicate considerable curvature in streakedfeatures. Deciencies were noted by the authors insuccessive-correction analyses using the circular andelliptical weighting functions in areas of high windspeed and strong curvature in numerical studies ofmesoscale severe storm development (Anthes et al.,1982). This demonstrated the need for a simplecurved ellipse or banana-shaped weighting functionfor use in curved ow.

Such a banana-shaped function is described inSection 2 along with an elliptical function used forstraight flow situations. A description of correctionsto the rst-guess variable elds by the successive-correction objective analysis scheme combining use

AND NELSON L. SEAMAN 1185

of the three weighting functions and their transitioncriteria is given in Section 3. Results of tests with thisscheme are shown in Section 4, followed by a sum-mary in Section 5.

2. Formulation of anisotropic weighting functionselongated along straight and curved streamlines

Several requirements governed the development ofanisotropic weighting functions in this study. It wasconsidered important that the functions maintainsimilar shape for dicrent isopleths (from 0.0 to 1.0),and that the banana (curved) weighting functionmathematically approach the elliptical weightingfunction as the curvature approached zero. The axisof maximum elongation should not deviate signi-cantly from the observed streamline curvature. Asimple method to vary elongation with wind speedwas needed. We also required a functional formsimilar to that of Cressman (1959), which smoothlyapproaches a value of zero (without a zero-orderdiscontinuity) as distance from grid point to obser-vation point increases to some critical maximumdistance. The Cressman (1959) isotropic circularweighting function has the form

R2 - d?-kWfjk = fOI' dijk < R

Wijk = 0 fOI' dgjk .2 R

where R is an arbitrary but constant radius of inuenceand djjk the distance from the grid point (i, j) to anobservation point k. Finally, the function must notincrease greatly the computation effort required forthe objective analysis scheme.

a. Straight owInman (1970) suggested a weighting function similar

to Cressmans to enhance along-ow spatial autocor-relation of objectively analyzed meteorological vari-ables. His function has the form

R3 - d%,,,.~~ = -- 2W R3 + d,?,-,, ( )where Rf = R2(1 + BIV,-J-I cos20), ,8 is a constant ofelongation, normally 0.02 to 0.20 [s m"] and

C059 = (Dijk'vij)/(|vij|dijk)-Here V,-J, = wind at a grid point (i, j) [m s"], andD,-J-k is the position vector from the grid point (i, j)to observation point k and other variables are asdened in (1). This function is of the same form asCressmans isotropic function, but is elongated in thedirection of the wind according to the factor

Eij(iVl) = (1 + 6lviji)l/2- (3)However, the (1 + cos20) form of the Inman function

FelipeHighlight

FelipeHighlight

FelipeHighlight

felipeHighlight

1136 MONTHLY WEATHER REVIEW VOLUMEII3produces a peanut-shaped (i.e., lobed) area of inuencefor which the lobes increase in size as the product46|V,-J,-I increases. The peanut-shaped region might beuseful at a point experiencing upstream conuenceand downstream diuence. However, since this cannotbe assumed to be the general case, a simple ellipticalshape should be preferred.

To improve general applicability, a true ellipticalweighting function was developed in the form

R2 d,,,2 . '_ --iz2 , if dmz < R2Wijk R + dm ' (4)

.0, if d,,,2 2 R2,where '

dmz = (_x_'2 + ya) iEk2(|V|)

X = (Dijk ' V1

JULY 1985 STANLEY G. BENJ.AMlN AND NELSON 1.. SEAMAN 1187

Northt

Q_-II;

rnhh Piilqrid point)""'Ilu_.

\\ I _

-"L-.

-2"

////"-.

//u-

./._' ar\' '5"\bservotion point)'14 \ curved streamline

(passing through Pk

I11 Q @ -0c lelik .............center of 3 _,curvature

----rw

FIG. 1. Conguration for variables used inbanana weighting function.

(4) functions in this paper. We emphasize that theseeffective distances are not onlyuseful for successive-correction schemes but may also be used to incor-porate anisotropy in curved or straight ow in anyobjective analysis scheme which calculates weights orcorrelations as a function of distance.

The banana function degenerates to the ellipticalfunction (4) if there is no curvature, and to thecircular function (1) if there is also no elongation.Thus, only one weighting function (the banana func-tion) need be used in our scheme, but its limitingidealizations (the elliptical and circular functions) areapplied in regions of weak wind speed and curvatureto minimize computations. The inuence area of abanana function with elongation Ek(lV|) = 2.25 isshown in Fig. 2.

The banana weighting function requires calculationof the streamline radius of curvature, rk, at eachobservation station before the successive correctionprocedure is initiated. This calculation may be simplyaccomplished using the rst guess wind eld. Theexpression for relative vorticity, ',, in natural coor-dinates can be solved for rk, the streamline radius ofcurvature | I

V'* s+-nvwan (6)

where n is the distance along a perpendicular to theleft of the ow.

In the event that a rst-guess wind eld is notinitially available or is known to be of poor quality,there are a few alternatives. One may assume thatcurvature of the geostrophic wind is approximatelythat of the actual wind and calculate curvature usinga rst-guess geopotential eld. This alternative pro-

cedure has been tested, and while it produced reason-able curvature elds, the rst (relative vorticity)method is recommended if a rst-guess wind eldexists, due to its simplicity and usually greater accu-racy. If rst guess elds of both winds and heightsare unavailable, a preliminary wind analysis may beproduced with the circular weighting function fromwhich rk may then be computed.

In one experiment (not shown), the radius ofcurvature rk, was recalculated after each of threesuccessive scans. Although this improved the accuracyof rk, there was little change in the resultant meteo-rological analyses (e.g., analyzed winds changed lessthan 1 m s*). This test, plus numerous applicationsin which only rk of the rst guess eld was used,suggest that the analysis scheme is fairly stable withrespect to modest curvature errors such as are antic-ipated in a 12 h forecast from an operational model(e.g., National Meteorological Center or EuropeanCentre for Medium Range Weather Forecasts). Useof curvatures from less reliable longer-range forecastsfor the rst guess, with no successive recalculation ofrk, may result in serious errors.

3. Objective analysis scheme incorporating a hierar-chy of three weighting functions

The general objective analysis scheme describedhere utilizes the method of successive correction scanswith a decreasing radius of inuence for each scan(Cressman, 1959). The computation of grid pointvalues of a variable or uses the relation

oz = ao + Aa,-j (7)where

KZ (Wr2jkAak)Au" A; -A (8)

2 wijkand -0:0 rst guess for variable or at grid point (i, j)a corrected variable oz at grid point (i, j)Auk dierence at observation point k (observed value

minus rst guess value)

F ___ _

FIG. 2. Weight isotachs (0.1-0.9) for typical banana weightingfunction. The X is an observation point and the curved arrowrepresents a horizontal streamline.

felipeHighlight

felipeHighlight

1188 MONTHLY WEATHER REVIEW VOLUMEII3

w,-J.-k weight, 0.0-1.0, from banana, elliptical or cir-cular function

K number of observations with positive weighting' values for grid point (i, j).

The application of weighting functions in (8) differsfrom the two most commonly reported methods,given by Cressman (1959) in the form

K2 (Wr'jkAak)

Aer,-j " (9)

and by Haltiner and Williams (1980) in the formK .

Z (Wr'jkA1k)Aoz,-I - K ' (10)

E Wijk

An initial radius of inuence (before elongation) of1.6 times the mean radiosonde station separation wasused, as suggested by Stephens and Stitt (1970). Thisradius was decreased by a factor of 0.7 in each oftwo successive scans.

The form of the correction calculation given inEq. (8) was chosen to reduce spurious discontinuitiesin the resultant objectively analyzed elds that canbe introduced when either Eq. (9) or (10) is used.Such discontinuities have long been known to occur(e.g., Cressman, 1959) and are particularly serious indata-sparse regions or when cross-scale analyses arenecessary (e.g., analysis on mesoscale grids when onlysynoptic-scale observations are available). Cressman(1959) recommended application of a spatial lter toremove the resultant spurious features. However, thisoperation inevitably removes some potentially usefulinformation as well. Ogura and Chen (1977) proposeda two-part technique to solve this problem, rstanalyzing the data on a "coarse grid via the Cressmanscheme (10), and then interpolating to a ne meshwith cubic splines. Equation (8) allows the analysesto be made in a one-part scheme without introductionof substantial discontinuities, even in data-sparseregions, and thus generally eliminates the necessityfor a separate smoothing operation.

The reason for the discontinuities introduced by(9), which calculates the simple average of the weighteddifferences, w,-J-,

JULY 1985 STANLEY G. BENJAMIN

their use will yield a more accurate analysis. In thisresultant hierarchy, the banana weighting function isapplied to regions of strong, curved ow. An ellipti-cally shaped weighting function is used in regionswith strong ow but without signicant curvature.Finally, the circular weighting function is applied toareas of relatively light winds. Regions of signicantcurvature were dened as having radius of curvatureless than 3R0, where R0 is the radius of inuence forthe initial scan. The critical wind speed for use of theanisotropic weighting functions was chosen as

25, (0.U20)P for P> 500mbV, --{I Hms] 15, for P>0, esta diferena no desprezvel em relao ao angulo azimutal entre os vetores de vento do fg (Vfg) e da observao (Vobs). Isso intuitivo pois um local muito distante do outro tem grandes chances de ter ventos diferentes.

felipeHighlight

1190 MONTHLY WEATHER REVIEW VOLUMEII3

ow gradients. Therefore, two additional changesweremade to minimize unintended smoothing duringobjective analysis of relative humidity. First, an ad-ditional fourth scan with a yet smaller radius ofinuence was performed. -This tends to bring themaximum radius of inuence used in the anisotropicfunctions closer to the mean station separation in thenal scan (as suggested by Stephens and Stitt, 1970).Since moisture is a passive variable (not dynamicallyinterrelated with the mass or momentum elds exceptin areas of condensation or evaporation), the additionof smaller-scale RH variations was not considered tobe detrimental. Second, before the objective analysiswas performed, RH -values were transformed to an-other variable designed to conserve moisture maxima.Transformation of discontinuous meteorological pa-rameters such as ceiling height for objective analysiswas suggested by Penn et al. (1963) but they did notapply this technique to relative humidity. In thecurrent analysis scheme, the following transformationsuggested by Rasmussen (1982) was used .RH"_ = (1 RH)"2; RH between 0.0 and 1.0. (13)After objective analysis is completed, RH values arerecovered via the inverse transformation. Objectiveanalysis of the transformed relative humidity, RH*,exhibits greatest accuracy in moist areas (RH > 75%).This characteristic is advantageous for initializationof numerical models since inappropriate smoothingin regions of near-saturation can result in errors inthe forecast commencement of precipitation.

4. Resultsa. A case study comparison

The multi-weighting function scheme described inthis paper has been tested and introduced as theanalysis technique used to obtain initial conditionsfor the Penn State mesoscale model (Anthes andWarner, 1978). As of this writing, it has been usedsuccessfully in hundreds of analyses for 10-15 differentcases with grid increments from 2 to 220 km. Here,we report details of the initializations developed forthe 10-11 April 1979 case, which featured episodesof severe convective storms, including a destructivetornado outbreak in the Red River Valley. Prominentamong the atmospheric features of this case was astrong upper- to midtropospheric jet, strongly curvedin the base of a large-amplitude trough over thewestern United States, which propagated into Texasfrom northern Mexico.

The NMC hemispheric octagonal grid data wereused as the rst-guess elds for this study. Themesoscale grid spacing was As = 111 km on a 37X 37 mesh. The initial radius of inuence R0 for therst scan (without taking into account elongation inareas of high wind) was 555 km. Thus, areas of highcurvature where the banana function was applied met

the criterion, lr,-jl < 1670 km (3R0). Curvature valueswere calculated from Eq. (6) with An = 2As. Whendi"erent rst-guess analysis sources are used for whichthe winds are known to be more noisy than the NMCgridded analyses, the nite difference An should belarge enough to avoid spurious calculated values ofcurvature. Satisfactory values of curvature were ob-tained in tests with An as large as 6As (660 km).

The percentage of grid points with a wind speedexceeding the pressure-dependent critical value [Eq.(1 1)] increased sharply from about 35% at 850 mbto near 85% at 300 mb. On the otherhand, the meanabsolute curvature of the ow [1/r,-_,-"J, where r,-j isthe streamline radius of curvature at grid point (i, j)]decreased somewhat with height, as might be expectedsince ow generally becomes increasingly zonal withheight. When considering wind speed and curvaturejointly, it was found that the percentage of pointsmeeting criteria for use of the banana weightingfunction increased with height.

Objective analyses using the multi-weighting func-tion scheme were performed both with curvaturecalculated once (from the rst-guess wind eld) andalso with curvature recalculated after each scan. Onlyinsignicant differences resulted. However, the extracomputation may be quite important for an accurateanalysis if features of the rst-guess curvature eldare highly smoothed or misplaced. A

Analyses of wind and moisture at all levels as wellas those of diagnosed elds such as vorticity anddivergence were created to compare performances ofthe original Cressman scheme with circular weightingfunction, and the multi-weighting function schemewith anisotropic functions. The 400 mb wind obser-vations and analyzed elds from the 10-11 April1979 case are provided for comparison in Figs. 4a-c.The elds were objectively enhanced by the samestandard synoptic and special SESAME rawinsondeobservations as well as bogus (simulated) values overdata-void regions produced from subjective analyses.The maximum speeds in the jet extending .fromCalifornia into northern Mexico are about equal inthe two analyses. However, the anisotropic weightingfunctions used to produce the analysis shown in Fig.4c lled in areas between observations along the jet,resulting in a much smoother jet feature than thatdepicted in the analysis using only the circular func-tion (Fig. 4b). Since a maximum of curvature existedat the base of the-trough, the coherence of the jet inFig. 4c in that region may also be attributed to useof the banana function. The increased smoothness ofthe anisotropic analyses is also found in the divergenceeld (not shown).

A second comparison of objective analysis tech-niques, this time for a scalar eld, is presented inanalyses of 700 mb relative humidity for the samecase. To concentrate attention on the effect of theanisotropic weighting functions at this level, analyses

Q

JULY 1985 STANLEY G. BENJAMIN AND NELsoN L. SEAMAN 1191IT ---- - -- -~ - e r

A inH 5 cc - e, .

-r'z,Q\ ri~\\ .\\\ *;\\ \ =\ \-~. ',\ -.._/ .

P \ _- 3; __n~ . i K. g.i J

FIG. 7b. As in Fig. 7a but using Eq. (9). Points A and B arelocations of radiosonde observations 433 and 532, respectively.Solid arc lines are the radii of inuence for the successive scansabout A and B.

-%%/ w-4a.1; 5QA-+////Q2\ \////g //\;"r1; ///'/'/4?,-/"PH\ //2/C//C?/C/ //Z////8

/

"/i,/* /r._//lg?vvy //y/M

'\\

/@77% //g "~/"7 W??? 1/4*.Zmnna

n?K

--1'

1 \4o=

/Z "In

Equation (8) produced, on average, the smallest errorsin the wind gradient terms in this test and shows verylittle scale dependency. We again note that theseresults were obtained in tests without data-sparseregions which. would have handicapped the perfor-mance of conventional equations (9) and (10).

To demonstrate the effect of low data density whenusing the three methods of applying the weighting

l-\---- Pil t -J l r__ _ __ T-t ._.... \

W l

_ 1... .1

, -lV '

. f :-

@r*?"*@r ;a;?;ian2i?iAi sa-r-/zrrssfran/a//y/has,gj\;, /r414?//avQM4- ~@;~w@M@@ww?-Wqa/X"//y///>{/4/ /E/L/l\//;\g//E/>\.

JULY 1985 STANLEY G. BENJAMIN

functions, a sample situation was devised using realdata from two radiosonde observations at 1200 GMT25 July 1975 on a mesoscale grid at 850 mb. Thedomain of 41 >< 41 points had a mesh increment of15 km and was centered at St. Louis, Missouri(38.62N, 90.18W). A National Meteorological Cen-ter hemispheric analysis (381 km mesh) was used asa rst guess. The resulting wind elds, obtained withcorrections to the rst guess eld calculated accordingto (8)-(10), appear in Figs. 7a-c, respectively. Thelocations of the two observations from radiosondestations 433 (Salem, Illinois) and 532 (Peoria, Illinois)are shown with the radii of inuence for the successivescans about each station. The wind at Salem wasreported as 7.2 m s from 315 deg, and at Peoria as5.1 n1 s" from 330 deg. Because the wind is light,the areas of inuence are circles in this case withradii of 520 km, 364 km, and 255 km for the threescans. Figure 7b clearly shows the discontinuities inthe isotachs concentrated along the radii of inuenceof each scan that form when (9), Cressmans originalform of correction to the rst guess, is used in thisdata-sparse environment. The same discontinuousisotach pattern appears in Fig. 7c when (10) is used.Introduction of the revised application of weightsaccording to (8) causes the objectively analyzed windeld in Fig. 7a to have smooth features and the samegeneral pattern.

The amplied effect of such discontinuities insecondary gradient-dependent elds is shown in Fig.8 for the vorticity elds at 850 mb calculated fromthe winds in Fig. 7. The vorticity in Figs. 8b and 8c,calculated with (9) and (10) respectively, show extreme

____ ' _ if | _V if _ _ if mi

I : : : i. . . gI : : 0 :

_ 0 .u ' ,,,,,, --0 o o - . Q , ,,, , , _ , _ . H 1 . I H, ""--a--0Q-........... .. :

E . E . 5

: if 4 T5 E E '. \ .1 I .a n .- 1 >, .. .. . .. . .. . . .-1 - Q I Q n U 0 I I U I I . .'I 3 ..... .1........................................................... ~--1 --- '*. .. .. .. . _

- :l or =

I 1 I. . .. . .. . .. . .. . .; r.oa 1.12

H 2 e.5 : -in-E. . . . .. . - . . .. . - .. . . f . I. - 3 . IE *l.O~ = .-.-.!.....,.....,,,,,...,,:,__,,,_,_, ,,,,,____,;, + ____,,,,,, ._._.... -..-......... .

M lg 0 . i I

I =-151 = = - ; - 5 2 2 I

. 5--LQ '1 49 \

. -,aq7 g = 5 I 1 '=. . E * ~ E E

' ' A ' ',' I I I ' f-~-~-~-~--.-......, ..,.-....-.,..,,,,,..,,,,.,,,,_______, QIIIJIOIOIIuIAlllllII""""""'.' -----"" I-"1"-11"". - 1,; ' I . 1. ' IJ . - ! 2 _ 5 - '. __ ___ - __ ,_,_---- _ _- _- __ t 7}

FIG. 8a. Diagnosed 850 mb vorticity calculated from objectivelyanalyzed wind eld, using Eq. (8), for 1200 GMT 25 July 1975.Contours in units of s X 105.

AND NELSON L. SEAMAN 1197

....... ..; . . . . _ _ _ _ _ _ I _ _-._ - _.... .. l

...... 5 5

\=""l7'-\~5

l

I AxlB I _. L _

I IU

ElxLFIG. 8b. As in Fig. 7a but using Eq. (9). Points A and B are

locations of radiosonde observations 433 and 532, respectively.Solid arc lines are the radii of inuence for the successive scansabout A and B. -

distortions concentrated along the successive radii ofinuence. The relatively small distortions in the windshave caused the vorticity eld to become so chaoticthatits actual pattern can hardly be guessed. However,when the new method of application of the weightsaccording to Eq. (8) is used in this difcult testsituation, (Fig. 8a), the resulting vorticity pattern hasonly minor distortions and the pattern is easily un-

R l \\%_ _ -. _,...,_;__ . I 1, _ ,2 , '_ _ i pl

FIG. 8c. As in Fig. 8b but using Eq. (10).

1198 . MONTHLY WEATHER REVIEW - VOLUMEII3

derstood. Similar improvement was found in thegeostrophic wind, relative humidity and other elds(not shown). If necessary, the small remaining distor-tions could be removed by application of a simplespatial lter. However, for most situations, lteringshould be unnecessary. *

5. Summary and conclusionsQ

This paper describes a scheme for successive-cor-rection objective analysis which combines threeweighting functions, a banana-shaped anisotropicfunction and its two limiting forms as wind curvatureand speed decrease (elliptical and circular weightingfunctions). Transition criteria are specied whichlimit use of the more complicated weighting functionsto those areas where it is required.

As with earlier anisotropic weighting functions, thescheme presented here is designed for objective anal-ysis of meteorological variables which exhibit higherautocorrelation along-ow than cross-ow. Theuniqueness of this scheme is found in improvedanalyses in curved ow while economy is preservedthrough well-behaved but relatively simple anisotropicweighting functions and a consistent set of transitioncriteria. Analyses of wind and moisture for a samplecase study using the anisotropic scheme are presentedwhich show improvement over those using the circularscheme, particularly in regions with pronounced jetfeatures and curvature in the wind eld. Mancuso etal. (1981) have shown that further improvementsmay result from applying anisotropic functions toanalyses on isentropic rather than isobaric surfaces.The scheme presented here may also be effectivelyapplied to objective analysis of other scalar variablessuch as aerosol or pollutant concentration.

Further tests demonstrated that the analysis schemeis also valuable for minimizing spurious discontinuitiesin resultant elds in certain regions. These areasinclude data-sparse zones such as continental marginsor mesoscale grids located in regions for which syn-optic scale data observations are available. The schemeis especially helpful for reducing errors in secondarygradient-dependent variables. Thus, the successivescan objective analysis scheme presented here dem-onstrates a number of important improvements whileremaining relatively easy to implement in a varietyof situations. '

Acknowledgments. We acknowledge with thanksthe careful reviews given by Thomas Schlatter, Ying-Hwa Kuo, Gary Rasmussen, Stanley Barnes andToby Carlson, and suggestions on the calculation ofcurvature made by Richard Anthes.

This work was supported by the U.S. Air Forceunder Grant AFOS-79-0125 and by the NationalScience Foundation under Grant ATM-80-1 1295.The manuscript was capably typed by D. Cormanand N. Warner.

\|

REFERENCESAnthes, R. A., and T. T. Warner, 1978: Development of hydro-

dynamic models suitable for air pollution and other mesome-teorological studies. Mon. Wea. Rev., 106, 1045-1078.

--, Y. -H. Kuo, S. G. Benjamin and Y.-F. Li, 1982: Theevolution of the mesoscale environment of severe local storms:Preliminary modeling results. Mon. Wea. Rev., 110, 1187-1213.

Barnes, S. L., 1964: A technique for maximizing details in numericalweather map analysis. J. Appl. Meteor., 3, 369-409.

Bergman, K. H., and T. N. Carlson, 1975: Objective analysis ofaircraft data in tropical cyclones. Mon. Wea. Rev., 103, 431-444.

Bergthrssen, P., and B. R. Docs, 1955: Numerical weather mapanalysis. Tellus, 7, 329-340. _

Browning, K. A., 1971: Radar measurements of air motion nearfronts. Weather, 26, 320-340.

Ceselski, B. F., and L. L. Sapp, 1975: Objective wind eld analysisusing line integrals. Mon. Wea. Rev., 103, 89-100.

Cressman, G. P., 1959: An operational objective analysis system.Mon. Wea. Rev., 87, 367-374.

Dey, C. H., and L. L. Morone, 1985: Evolution and performanceof the National Meteorological Center Global Data AssimilationSystem: January-December, 1982. Mon. Wea. Rev., 113, 304-318. .

Endlich, R. M., and R. L. Mancuso, 1968: Objective analysis ofenvironmental conditions associated with severe thunderstormsand tornadoes. Mon. Wea. Rev., 96, 342-350.

Flattery, T. W., 1967: Hough functions. Tech. Rep. 21, Dept.Geophys. Sci., The University of Chicago, Chicago, IL 60637,175 pp.

Gandin, L. S., 1963: Objective Analysis of Meteorological Fields.Gidrometeorologicheskoe lzdatelstvo, Leningrad. Translatedfrom Russian, Israel Prog. Sci. Transl., Jerusalem, 1965, 242DP-

Gilchrist, B., and G. P. Cressman, 1954: An experiment in objectiveanalysis. Tellus, 6, 309-318. .

Goodin, W. R., G. J. McRae and J. H. Seinfeld, 1979: A comparisonofinterpolation methods for sparse data: Application to windand concentration elds, J. Appl. Meteor., 18, 761-771.

Haltiner, G. J., and R. T. Williams, 1980: Numerical Predictionand Dynamic Meteorology. Wiley and Sons, 477 pp. .

Inman, R. L., 1970: Operational objective analysis schemes at theNational Severe Storms Laboratory. Tech. Cir. No. 10, NSSL,ESSA, Norman, OK 73069, 91 pp.

Johnson, D, H., 1957: Preliminary research in objective analysis.Tellus, 9, 316-322.

Mancuso, R. L., R. M. Endlich and L. J. Ehernberger, 1981: Anobjective isobaric/isentropic technique for upper air analysis,Mon. Wea. Rev., 109, 1326-1334.

Ogura, Y., and Y.-L. Chen, 1977: A life history of an intensemesoscale convective storm in Oklahoma. J. Atmos. Sci., 34,1458-1476. -

Panofsky, H. A., 1949: Objective weather map analysis. J. Mete0r.,6, 386-392.

Penn, S., B. Kunkel and W. D. Mount, 1963: On objective analysisof continuous and discontinuous parameters. J. Appl. Mete0r.,2, 345-350.

Rasmussen, R. G., 1982: Some techniques for the objective analysisof humidity for regional scale numerical weather prediction.Ph.D. dissertation, Cooperative-thesis No. 67, Drexel Universityand National Center for Atmospheric Research, 366 pp.

Sasaki, Y., 1958: An objective analysis based on the variationalmethod. J. Meteor. Soc. Japan, 36, 77-88.

-, 1971: A theoretical interpretation of anisotropically weightedsmoothing on the basis of numerical variational analysis. Mon.Wea. Rev., 99, 698-707.

Schlatter, T. W., 1975: Some experiments with a multivariatestatistical objective analysis scheme. Mon. Wea. Rev., 103,246-257.

Stephens, J. J ., and J. M. Stitt, 1970: Optimum inuence radii forinterpolation with the method of successive corrections. Mon.Wea. Rev., 98, 680-687.