unerical Simulation of a lew Correlation Function for
the Clinatic Statistical Structure of the Height Dynanic Field
Shing-Chung Onn
Kuang-Horng Wang, Jen-Ko Wei, and Tai-Hwa Hor
Computational Fluid Dynamics Laboratory
Chung Cheng Institute of Technology
Ta-Hsi, Tao-Yuan, Taiwan, R.0.C.
ABSTRACT
For the long-term peroid in the low-latitude region of the earth, the statis-
tical structure of the height dynamic field at 500 mb during the winter sea-
sons has been studied successfully by using a new correlation function {0 (r)
= A(r) * EXP [ B(r) ] } where A(r) and B(r) are two general polynomials. The
best selections of the degrees of these two polynomials can be found in the
least-squares sense. The results show that this new mixed-type correlation
function can yield more accurate fitting than Gandin's formula( 1963 ) {0 (r)
=a %* EXP (-b%r %* 2) }. The height dynamic fields at the regular nest
grids are then computed and compared with those obtained from the measured
data at irregular observational stations. The troughs of the resulting height
dynamic fields can be identified very clearly.
INTRODUCTION ‘:
For the study on macroscale turbulence of meteorology, the univariate and the
multivariate scheme are used to analyze the measured data of the upper air
observation points. With them, several techniques ( Gandin ( 1963 ) ; Kricak
( 1967 )s Panchey ( 1969 ); kluge ( 1970 ); Steinitz et al. ( 1970 )i Schlat-
ter (1975 ); Schlatter et al. ( 1976 )s Thiebaux et al.( 1986 )+ Mitchell et
al. (1990 ) ) of the optimum interpolation had been developed successfully
to study the correlation functions of the height dynamic fields. The computed
results based on the above two schemes were also compared with those of the
numerical weather forcast. ( Schlatter et al. ( 1976) ) It was concluded that
two univariate schemes ( one for height, one for wind ) can fit data as well
as the multivariate scheme. The present method uses the univariate scheme to
analyze the measured data. As an example, for the long-term peroid in the low
latitude region of the earth, the statistical structure of the height dynamic
field at 500 mb during the winter seasons has been studied successfully by
using a new mixed-type correlation function.
ANALYSIS
Correlation Function
f(r) , a meteorological variable, can be defined as the sum of its mean f(r)
and the deviation f'(r) where r is the distance of two geographic points.
Between two meteorological variables f and g, the cross-covariance function
C_ can then be defined as
fg
C (r,r)=f(r) + g(r) (1)
fg 1 2 1 2
where r and r are respectively the distances between one unknown and two
1 2
known geographic points shown as Fig. 1. Furthermore, the corresponging auto-
covariance function C (r,r) and variance functionC (r,r) can be
f 1 2 at
defined from Eq. (1). Between two meteorological variables f and g , the
crosscorrelation function can then be defined as
C (r,r)
fe 1 2 F
oe (rar)e (2)
fg 1 2 V2
{Cc (r,r)+C (r,r)]
ff 1 1 202
ee
Similarly, the corresponding autocorrelation function 9 (r,r) can be
defined from Eq. (2). Normally , it is homogeneous( position independent ) and
isotropic (direction independent ), i.e.,
o (r,r) = » (r,r) = » (d ) = o (d ). (3)
ff 1 2 ff 2 1 ff 12: ff 21
The above d can be computed by
ij
2 2 1/2
d = (r +r -2rrcs@ ) (4)
ij ij ij ij
and shown in Fig. 1 with i = 1, NT and j=1, NT where NT is the total of
observational stations. If different and nonzero d are indicated by d ,
iJ
the above p (r,r_) can further be represented by 0 (d) withk = 1,
ff i j ff ok
(NT+ CNT-1 )/ 2] .
Therefore, the above raw correlation data o (d_) can be computed from the
measured values of climatological variables during the long-period of time in
a specific region. Then, the corresponding autocorrelation function Boe d)
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can be found.
For a specific distance D between two geographic points, the distance-aver~
L
age autocorrelation data with homogeneity and isotropy can be defined as
B (dD) = ——_ ; (5)
L :
where (p> Gy sids Dew Dy with (i =1,M and L=1,N)
i L L
and 2 ‘ = NT+ (NT-1 )/2 if A D is provided and N is defined as the total
L=1
of the eee autocorrelation data. Also, 6 (D) withL=1, KN
L
can be employed to construct the distance-average autocorrelation function
B (D>) in the least-squares sense.
ff
The Least-Squares Method
The negative squared exponential form ( Gandin (1963) ), {6 (D)= a>
EXP (- b + D #2.) } , was first proposed to fit the distance-average auto-
correlation data 0 i. De } in the least-squares sense where L=1, N. The
present method uses the Univariate scheme to develop a new correlation func-
tion to analyze the distance-average correlation data in the least-squares
sense. It is a mixed-type function of the polynomial form and the exponential
form. Certainly, it is also general form of the Gandin's ( 1963 ) and Buell's
(1972 ) empirical correlation formulas. It can be expressed as
a (D)=fACD)] + [EXP (B(D))]
ma i mb i
={2a D1] + [EXP( Db D)] (6)
isd i is0 i
where ma and mb may be two arbitrary and different positive integers. The re-
sulting sum of the square errors between the correlation model and the corre~
lation data is
N 2 N _ nb i 2
es tind - [CEbD)+In( Sa DDI) (7)
jel oj Gl j isQ i j izO ij
vhere { (D, » )13=1, NJ represents the distance-average autocorrela~
jd
tion data. Differentiate Eq.(7) with respect to a and b where k = 0, ma
k
for a and k = 0, mb for b to obtain
k k
k
D
N 2 N j
F =gd ec. /ga = & { [———~——— ] + « } (8)
ko jel ij ko j=l ma i j
Zal( dD )
i=0 i J
and
N 2 N k
G =df ce /ab = =F (D € J : (9)
ko jel J ko gel oj
ae
From Eqs. (8) and (9), the (ma + mb + 2) nonlinear algebraic equations, {
F=0 | k=0, ma} and{G=0 | k= 0, mb}, can be solve simultaneously
k
k
using the Newton-Raphson method ( Chow ( 1979) and Gerald ( 1978) ) by
first guessing arbitrary initial values for {a| k= 0, ma} and {b1 k = 0
k k
» mb }. The final numerical solutions of a and b are considered satisfac-
k
tory when both
iter +1 iter “5
Max (1 a -a 1}<10 where k = 0, ma
k k
and
iter +1 iter -5
Max (1 b -b 1} < 10 where k = 0, mb
k k
The convergence processes of maximum errors for a and b are indicated in
Fig. 2.
Optimum Interpolation Based On Observational Stations
For an observational point k, the deviation f'(r ) can be found from
f(r) =f (r)-F Cr) (10)
k k k
f (r) and f (r) are respectively the measured meteorological and the cli-
k k
matical mean values. Therefore, the analyzed deviation of the meteorological
parameter f at an arbitrary geographic point G can be evaluated by
NT
f'(r)=E a f(r) qi)
G kl &k k
if weighting coefficients, { a | k= 1, NT), are provided. Actually, the
k
weighting coefficients are not given up to now. However, they can be found by
using the method of the optimum interpolation in the least-squares sense. The
resulting simultaneous equations for the weighting coefficients { a | k=1
k
» NT } are
NT
Le (r,r)azo (r,r) (12)
k=l ff m k k ff m @
for m= 1, XT.
Reliability of Distance-Average Autocorrelation Models
For the observational raw autocorrelation data as shown in Fig. 3, the dis-
tance-average autocorrelation data computed from Eq. (5) are then employed to
generate the present new mixed-type autocorrelation model in the least-sq
uares sense. This model is compared with the distance-average autocorrelation
data and the Gandin's model. The results are shown in Fig. 4.
From Fig. 4, it can be concluded that the present new mixed-type distance-
average autocorrelation model is much more accurate and reliable than Gan-
din's model for representing the distance-average autocorrelation data in the
low-latitude region.
f
SAMPLE_PROBLEH
In this study, the basic measured data are provided by 78 observational sta-
tions as shown in Fig. 5. Their height field on 500 mb and at 1200 GMT during
3 winter months ( December, January, and February ) of each year from 1977 to
1983 is then employed to generate a new mixed-type distance-average autocor-
relation model. This model is finally used to compute height fields on the 18
x 16 grid for two prescribed days. For these days, the height fields based on
the measured data of 78 observational stations are contoured as Figs. 6 and 7
for the following comparing work. Based on the above results of objective a~
nalysis, the meteorological parameter at the regular grid can then be employ-
ed to serve as the initial guess of the numerical weather prediction ( NWP ).
NUMBER OF OBSERVATIONAL STATIONS USED FOR OBJECTIVE ANALYSES
Between computed and measured deviations of the meteorological parameter at a
target point G, the error (E') in the least-squares sense depends on the num-
- 509 -
ber (NO) of observational stations used for objective analyses. ( Gandin (
1963 ) + Wang ( 1986 ) ) The relation between E’and NO is shown in Fig. 8. It
was concluded that the measured data of the nearest seven observational sta-
tions around a target point are enough to compute accurate meteorological pa-
rameters. ( Wang ( 1986) ) In this study, the number of observational sta-
tions used for objective analyses is eight.
RESULTS AND CONCLUSIONS
ly
Fig. 4 compares this new mixed-type model with the distance-average auto-
correlation data and the Gandin's model. As can be seen, the mixed-type
model is much more accurate and reliable than Gandin model for represent-
ing the distance-average autocorrelation raw data in the low-latitude re-
gion. Specially, the above phenomena are more obvious when the separation
distances between two geographic points are less than 370 KM and greater
than 2250 KH. Since the large areas in the low-latitude are the sea. The
observational stations on the sea are sparse, the separation distances be-
tween the target points on the sea and the nearest 8 obseryational sta-
tions may be greater than 2000 KM. Therefore, the present autocorrelation
model is much more reliable for computing the meteorological parameters in
the low-latitude region than the Gandin's model.
. Figs. 6 and 7 show the height fields based on the measured data of 78 ob-
servational stations. Several obvious troughs of height fields are found
in these contours and marked by arrows. For each figure, the high and low
of the height field are indicated and the gradient of the height field
across Japan in the S-N direction is also found to be higher than those of
the other regions in the figure.
. As mentioned above, Fig. 8 shows that the measured data of the nearest
seven observational stations around a target point are enough to compute
the accurate meteorological parameters. This result can also be found by
comparing Fig. 9 with Fig. 10. They are the interpolation-error contours
of height fields computed respectively from the nearest 8 and 20 observa-
tional stations. As can be seen, these two contours are very similar.
. Figs. 11 and 12 are the height fields computed respectively on January 7
and January 10 of 1983 from this new mixed-type autocorrelation model by
optimum interpolation method of objective analyses. They are respectively
compared with Figs. 6 and 7. The same analyzed results can be found.
From the above results and experiences, the following conclusions can be
drawn +
. For the low-latitude region, since the large areas are the sea, the accu-
racy of the autocorrelation model. is very important for the full range of
the separation distances between the target points and the observational
stations.
Because of the similarity of correlation data distributions for the mid
and high latitudes, it can be anticipated that the algorithm of the numer-
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ical simulation for this new mixed-type correlation function will also be
valid for different latitudes regions of the earth provided the measured
correlation raw data are given.
APPENDIX
In Fig. 4, the resulting coefficients of Eq. (6) for the ma=5 and mb =5
case are
a = 0.99998D-01 a= 0,23054D-02 a = -0.33989D-03
1
0 2
a = -0.11815D-02 a = -0.10476D-04 a= 0.46725D-04
3 4 5
b = 0.22635D+01 = b = -0.35200D+00 ~—sb_ = -0.68385D-01
0 1 2
b = -0.71690D-02 b= 0.12955D-02 b= 0,91964D-03
3 5
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Fig. 1. Definitions of distances among
three geographic points ( A, B.C).
CA:FOR POLY PART
(CB:FOR EXP PART
ERMAX
Pe ee eee
ime oreo e.tiben arena.
NUMBER OF ITERATION
Fig. 2. The convergence processes of nax-°
inun errors for coefficients (al k = 0,
5) and {bi k=0,5).
k
~- 512 -
vaMeur
0.s01e100 Ea vee cere ee Pe ae
CLUE O.IKEY Oem O.RLOLIOF 0.6008 0%
visitng
Fig. 3. The observational raw autocor-
relation data for the height field as a
function of the separation distance bet-
ween every pair of points for 78 radio~
sonde stations of the low latitude region
Fig. 5. Geographic Locations of 78 ob-
servational stations.
Fig. 7. The height field based on the
measured data of January 10, 1983 at 78
observational stations. ( From Central
Weather Bureau, R.0.C. )
O:DISTANCE-AVERAGE CORRELATION CATA
PACD FYPE FUNCTION HITE
EUS POLY mY § EGS EXP
TT" GISTANCER( 1Ome-3) KH
Fig. 4, The new aixed-type autocorrela~
tion model and its distance-average raw
autocorrelation data. ( Gandin's model
for the low latitude is presented togeth~
er as a reference. )
ALY
*% ‘e
Fig. 6. The height field based on the
measured data of January 7, 1983 at 78
observational stations. (From Central
Weather Bureau, R.0.C. )
og
el
oy cs %
Fig. 8. The relation between the inter-
polation error ( £') and the number (HK )
of observational stations used.
- 513 -
Fig. 9. The interpolation-error contour
of the height field computed from the
nearest 8 observational stations.
ite 1YPe MUTA 6? BE —uie. 7, 1909
Fig. 10. The interpolation-error contour
of the height field computed from the
nearest 20 observational stations.
é
ninco THE WIM 6 SE <n. 10,_ 1903
TREOPOTENTIR. EVENT OF WE FIA AT 500 MB
Fig. 11. The height field conputed by
optinue interpolation method of objective
analyses froa the new mixed-type autocor-
relation node. ( January 7, 1983 )
TGEOPOTENTIR, PEIGHT OF Ne ASIA AY S00 FO
Fig. 12. The height field computed by
optimum interpolation method of objective
analyses from the new mixed-type auto~
correlation model. ( January 10, 1983 )
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