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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
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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
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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
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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.
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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-,
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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.
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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
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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/>\.
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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).
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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.
\|
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