fNASA-CR-176471 ) DEVELOPMEN T Of THE H86-166.7

VARIATIONA L SEASA T DAT A ANALYSI S TECHNIQUE

Final Report (Oklahoma Univ.) 8C p ^^ unclas

HC A05/M F A0 1 G3/U3 052U7

national Aeronautics and Space Administration

Goddard Space Flight Center

FINAL REPORT

Research Grant No. NAG 5-289

Development of the Variational SEASAT Data Analysis Technique

submitted by

" Dr. Yoshi K. Sasaki

Director, Cooperative Institute for Mesoscale Meteorological Studies

and

Dr. Lang-Ping Chang

Research Scientist,Cooperative Institute for Mesoscale Meteorological Studies

and

Dr. James S. Goerss

Research Scientist, Cooperative Institute for Mesoscale Meteorological Studies

from

The University of Oklahoma

Office of Reserach Administration

1000 Asp Avenue, Room 314

Norman, OK 73019

December 1985
TABLE OF CONTENTS

PAGE

I. Introduction 1

II. Retrieval of the Sea Surface Wind from the SEASAT-A

Satellite Observations 2

III. Variational SEASAT Data Analysis Technique 5

a. Estimate of the surface geostrophic wind 6

b. Variational adjustment technique 8

IV. Results of Numerical Experiments 12

a. Determination of weights 12

b. Influence of the boundary condition 15

c. Effect of data density of surface pressure 17

d. Effect of quality of SEAS AT-de rived wind data 19

e. Influence of the surface temperature field 20

V. Concluding Remarks 21

Tables 23

Figure Captions 27

Figures 30

References. 78
I. Introduction

Approximately 70% of earth's surface is covered by oceans. In terms of

conventional meteorological observations, oceans are data-sparse areas. With the

development of weather satellites in the 1960's, a vast amount of remotely sensed

asynoptic data has since become available. Among various satellite weather data,

the temperature soundings are of great potential use in determining the mass distribution of the entire atmosphere. In order to enter these satellite-derived

temperature soundings into an atmospheric prediction system, one needs to obtain

the geopotential height field for many different isobaric surfaces. The success

of this task, however, crucially depends upon the quality of the sea-level pressure analysis. It thus follows that an improved sea-level pressure analysis in

data-sparse areas over the oceans is vital for improved accuracy of the geopotential height information derived from the satellite soundings, which will in turn,

result in a better representation of the entire mass field in an atmospheric

prediction system. Since the surface wind is closely associated with the surface

pressure gradient, insertion of satellite-derived surface wind into the

conventional weather data is expected to considerably improve the sea-level

pressure analysis in the data-sparse ocean areas. The variational SEASAT data

analysis technique is specifically designed for this purpose.

Stated in a brief manner, the working principle of the SEASAT-A satellite

sensing of the surface wind over the sea is as follows: The microwave radar scatterometer carried by the SEASAT-A satellite is very sensitive to the backscatter

of the centimeter length ocean waves created by the action of the surface wind

over the oceans. As a result, one can derive the vector wind over the sea surface

by analyzing the backscatter of radar waves. Jones ^t_ aj_. (1982) compared the

SEASAT-derived surface wind data with observations from the Joint Air Sea

-1-
Interaction Experiment (JASIN) and concluded that the satellite-derived sea

surface wind has an accuracy of up to ±2 m/s in speed and ±20° in direction.

These numbers will be considered characteristic of the retrieved SEASAT wind

field. By combining the densely spaced SEASAT-derived wind data with the sparsely

distributed sea-level pressure observations via a variational adjustment technique

subject to some appropriate physical constraint(s), an improvement in the sealevel pressure analysis is expected. Such an improvement will certainly be very

helpful in upgrading the model forecasts of the atmospheric prediction systems in

view of the large areal coverage of the earth's surface by the oceans.

In the following sections, we shall demonstrate how a simple marine boundary

layer scheme in conjunction with a variational adjustment technique can be

developed to help improve the sea-level pressure analysis by using the SEASATderived wind of a limited-area domain in the ocean.

II. Retrieval of the Sea Surface Wind from the SEASAT-A Satellite Observation

The SEASAT-A satellite was launched on June 28, 1978 by the National Aeronautics and Space Administration. It was the first satellite dedicated to establishing the usefulness of microwave sensors for remote sensing of the oceans. After

just a little more than three months of operation, it failed to operate in its

orbit due to a massive short circuit in its electrical power system. In spite of

its short operational life, it was fortunate that enough data had been collected

before its operation came to an end. u

The most important microwave sensor the SEASAT-A satellite carried was a

radar scatterometer for detecting the surface wind over the ocean. The physical

basis of this lies in the sensitivity of the microwave radar backscatter to the

centimeter length ocean waves created by the action of the surface wind. This is

known as the Bragg scattering (Wright, 1966; Moore and Fung, 1979). The strength

-2-
of the radar backscattering cross section, 0°, is proportional to the capillary

wave amplitude which is assumed to be in equilibrium with the wind friction speed

u*. The wind direction can be determined because the radar backscatter is anisotropic. Once the friction speed u* is given, the wind at 19.5 m above the sea

level under a neutral stability condition can be obtained. This wind is defined

as the satellite-derived wind that would result if the atmosphere were neutrally

stratified with a dry adiabatic lapse rate.

The history and details of the development of a geophysical algorithm to

compute the 19.5 m neutral stability wind speed and direction from radar backscatter measurements are given by Jones et al. (1982). Three candidate algorithms

were developed by researchers from the University of Kansas (Dome et_ al_, 1977),

the City University of New York (Pierson et_ £l_, 1974), and by the Remote Sensing

System (Jones et_ a_l_, 1978). A review of their work is given by Schroeder et al.

(1982). The official wind vector algorithm known as SASS-I (J3_EASAT-A^ Jattellite

jJcatterometer), described in Jones ^t_ aj_. (1982), represents a cooperative effort

by the algorithm developers to incorporate the desirable features of its predecessors into a single algorithm.

The SASS-I model function is expressed by

F(9, x, e, w) = logir.a° = G(9, x, e) + H(9, x, e) log,ftw (2.1)

where w is the windspeed. The values of functions G and H are tabulated for incidence angles 9 from 0° to 70°, for relative azimuth angle x (antenna azimuth minus

wind direction) from 0° to 180°, and for horizontal and vertical polarization, e=

H or V. This lookup table, called the SASS-I Table, is given by Schroeder et al.

(1982).

Given the model function, estimates of the wind speed, W, and direction, a,

-3-
are found which produce local minima for the following sum of squares:

N

S(W, a) = E nog1Qa° - *(Q±t x^ e^ W)}2/£2 (2.2)

where

Xi

The sum is over the N observations in a data group. The instrument parameters

CT> 9.1 > <l>.i » and e. are the radar backscatter cross section, the incidence angle,

the antenna azimuth angle, and the polarization for the i-th observation, respec2

tively. The sum is weighted by 1/6. where 6. is the expected standard deviation

between the cr° measurement and the model function. Unfortunately, the number of

candidate wind vectors (called aliases) varies from two to four (with four being

most common). One alias will correspond to the solution for the true wind vector,

and the others will be false solutions. Given that the proper alias has been

determined, Jones et al. (1982) reported that comparisons of SASS-derived winds

with high quality withheld surface truth at the JASIN-SEASAT workshop show better

agreement than the SASS specification ± 2m/s for wind speed and ± 20° for wind

direction. We will assume that the alias removal problem will have been eliminated or greatly reduced before the next instrument is launched, and that these

figures are representative of the data quality one might expect from satellitederived surface wind observations.

Using a large-scale marine planetary boundary layer model (Brown and Liu,

1982), Brown et_ a±_. (1982) calculated synoptic scale wind fields for comparison

with surface data collected during JASIN and the Gulf of Alaska Experiment

-4-
(GOASEX). Synoptic scale pressure and temperature fields were input into the

model in order to obtain the wind field. These analyses did not use satellitederived wind observations. They found that the model-derived wind fields, when

compared with the JASIN and GOASEX platform wind values, yielded a maximum error

of ± 2m/s and ± 20°. This is only equal to the SASS specification accuracies.

Brown and Liu (1982) reported that at mid-latitudes, a 1 mb error in 400 km grid

pressures would contribute a 20% or 2m/s error to a lOm/s wind. The implication

of this is that we may expect (at least in the higher latitudes) to obtain very

accurate pressure gradient information from satellite-derived surface wind

measurements.

III. Variational SEASAT Data Analysis Technique

In this section, we will discuss how the SEASAT-derived wind data can be

utilized in a variational adjustment technique to help improve the sea-level pressure analysis.

On the average, the stability conditions for the marine boundary layer can be

assumed to be nearly neutral (Gray, 1972). It is therefore reasonable to assume

that the SEASAT-derived wind field well approximates the neutral stability vector

wind at 19.5 m above sea level with an accuracy up to ± 2m/s in speed and ± 20° in

direction. Thus, it is expected that by using a simple marine planetary boundary

layer scheme such as the geostrophic drag law, one may extract the surface pressure gradient information (equivalently but more conveniently the surface geostrophic wind) from the satellite-derived wind. After the surface geostrophic

wind components have been estimated from the satellite wind, then together with

the sparsely distributed observations of the surface pressure, a variational

adjustment technique using as a weak constraint the geostrophic relation may yield

the best estimate of the surface pressure and wind fields in the least squares

-5-
sense. In Section Ilia, we shall discuss the procedure of estimating the surface

geostrophic wind from the SEASAT-derived wind. Details of the variational

adjustment technique will be examined in Section Illb.

Ilia. Estimate of the surface geostrophic wind

The SEASAT-derived wind can be regarded as an approximation to the wind at

19.5 m above sea level under a neutral stability stratification. Consequently,

the simplest way to estimate the surface geostrophic wind from the SEASAT-derived

wind field is perhaps to make use of the geostrophic drag law together with an

empirically established relationship between the surface stress and the surface

wind. The geostrophic drag law for the neutrally stratified atmospheric boundary

layer is given by (Tennekes, 1973):

kua

 u*

IT

 = ^fT -

 B

»

 (3*1}

* o

kv

—* = -A. (3.2) u

*

In these equations, A=5 and B=2 (Blackadar and Tennekes, 1968), which are

experimentally determined similarity constants; f is the Coriolis parameter; u* is

the frictional speed, z is the roughness length; ug and vg are the components of

the surface geostrophic wind respectively parallel and perpendicular to the

direction of the surface stress (see Fig. I); and k is the von Karraan constant

0.35 (Businger jat_ a_l_., 1971).

By denoting the magnitude of the surface geostrophic wind as G and the angle

between the surface stress and the surface geostrophic wind as a (see Fig. 1),

(3.1) and (3.2) may be combined to give

r r \c r 9 2

— = B + £n -±- + (£-£- - A^) , (3.3)

Z U^ U^i

o *

-6-
u^A

sin a = •££- (3.4)

It is reasonable to use the well known logarithmic wind profile relation to describe the wind in the lowest say, 20 m of the atmospheric boundary layer:

ku = u.JIn — . (3.5)

z * z

o

In order to close our marine boundary layer scheme, we shall make use of the following empirical formula for the neutral drag coefficient

2

u

C

* T

DN H -^—^ = 10~J (0.75 + 0.067 UIQ) (3.6)

after Garratt (1977), where UIQ is the wind speed in m/s at 10 m above the sea

level.

Thus, with the help of (3.3) through (3.6), we may compute the vector surface

geostrophic wind from any given satellite-derived wind (with speed u^g^ in the

direction of the surface stress). The zonal component of this wind will be

denoted as ug and the meridional component denoted as vg in notation. Similarly,

(3.3) through (3.6) can also be used to find the wind at 19.5 m above sea level if

the vector surface geostrophic wind is given.

The purpose of the current study is to develop a useful variational technique

to help determine more accurately the sea-level pressure field by making use of

the SEASAT-derived wind data in data-sparse areas over the oceans. Consequently,

in order to assess the accuracy of this variational adjustment technique, we follow the procedures given below to create assimilated SEASAT wind data:

-7-
(i) For a given field of sea-level pressure of a selected target area

over the sea, compute the associated geostrophic wind components.

(li) Compute the vector wind field at 19.5 m using (3.3) through (3.6).

(iii) Add a normally distributed random error of zero mean and standard

deviation 2 ra/s to the wind speed and add another normally distributed random error of zero mean and standard deviation 20° to the

wind direction of the vector wind resulted in (ii). This new wind

field shall be regarded as the assimilated SEASAT-derived wind

field.

(iv) The assimilated SEASAT-derived wind at each grid point is converted

to the surface geostrophic wind using (3.3) through (3.6). After

the zonal and the meridional components of the surface geostrophic

wind have been obtained, they will be employed in the variational

adjustment procedure discussed in the next section.

Illb. Variational adjustment technique

The variation adjustment of the SEASAT data starts with the minimization of

the following functional

X Q

J s r

 x

2 / e/ (u

 -

 us)2 + (v

 - V2 + A (p-

p

s

)2

2

+ITB, r tf RT 1 9P»2 . / . RT 19P\ii,^j , /OT \ r(f v

 - — A0 + (-

fu

 ~— a ae} ] } dedx>

 (3

-

7)

where ug and vg represent the known surface geostrophic wind components derived

from the assimilated SEASAT wind; Pg is the sparsely located surface (sea-level)

pressure; a the earth's radius; X the longitude; 0 the latitude; A and B are the

weights; and all the other variables carry their conventional meaning.

-8-
Essentially (3.7) means that we request the wind and pressure fields to be as

close to the "observed" values as possible subject to the weak-form constraint of

the geostrophic relation.

By letting the first variation of J vanish and by dropping the boundary integral terms arisen from application of integration by parts, the following three

partial differential equations are yielded:

A,_ _ . . 1 3 r BRT RT 3£n P A(p-V

 + P aT t (fv

 -

u - u + f (fu + SI !*£_?_) = o , (3.9)

S ad o

v - v + f (fv - -SL_ 1|2_L) = 0 . (3.10)

s acosS 8X

Since we dropped the boundary terms in the above derivation, we have tacitly

assumed that the geostrophic wind relation holds on the boundary when solving the

P-equation.

In practice, Equations (3.8), (3.9) and (3.10) are solved in discretized

form. The computational mesh used is a staggered one as shown schematically in

Fig. 2. The outermost boundary grids are the pressure grids except for the four

corners where additional velocity grids are present. The grid interval is taken

to be 4° longitude and 4° latitude, the same grid resolution as the CIMMS Global

model.

-9-
The finite-difference equations for (3.8), (3.9), and (3.10) are

P AX a cos

B

i.;h

 Ti.J-H * Ti.;

< . i»J

.. 2 2fl (P, ._._. + P, . ,. 2 AX AXa cos 6 i

(Bi.J

(P. + P .) 2 AX ~ P

1

 » J 1 > J"1 ^ » J

R

2

 r

(Bi-i.j + "i.J^Vi.j + Ti,;h rP

J-i.J

2

 L

z T? ^P 5" *• 2 ' ^ ^ ;

T

i,j

 + Vi.j

 pi.j — n>- y 2 A6

f 1.1 1.1+1 1+1.1

u. . - u + (-LJ

 1>j+1

 4

 i+1

»J i±LJ±L) (f 2 1>r£2 1 ul f .

T4- T 4- T + T P 4- P P 4- P

' - ' p p y , 3 1 1 . n

2 2 '- •

-10-
V — V +1 '" '" '-* '"1 (f ^ If

v>

 • vsi,j

 l 4 } U2i; Ir2

T P + P P + i±!il±Lv ^>j +

i i+i,j+i i+i,J

j A o o acos6.AVP. .+ P. + P .+ p

(3.13)

For a P-grid point on the boundary, the pressure gradient in the direction normal

to the boundary is related to the u- or v-component through the finite-difference

form of the geostrophic relation. For a corner point on the boundary, the pressure is computed by averaging the two pressure values respectively obtained from

the u- and v-geostrophic components.

The finite-difference equations (3.11) through (3.13) are solved one at a

time in a cyclic manner. The initial guess for u and v is the surface geostrophic

wind components computed from the satellite-derived wind. The initial P field is

obtained by averaging the three pressure fields individually computed from the

geostrophic wind equations with the pressure observation at any three grid points

(assuming that at least three grid points will have observed surface pressure).

The iterative process is terminated when the difference in u and v between two

successive iterations is no greater than 0.1 cm/s and that in P is no greater than

0.01 mb. Different initial guesses for P should only slightly influence the rate

of convergence of the iterative procedure, not the final result.

IV. Results of Numerical Experiments

A number of numerical experiments were designed to investigate the following

aspects of the variational SEASAT data analysis technique:

(i) determination of the weights, (ii) influence of the boundary condition, (iii)

effect of data density of pressure, (iv) effect of quality of the SEASAT-derived

wind data, and (v) influence of the surface temperature field.

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In all the numerical experiments we ran, the target area was chosen to be the

region bounded by Latitudes 16°N and 32°N and by Longitudes 152°W and 168°E in the

Pacific Ocean. For each case of the numerical experiments, the true solution

against which results of the variational SEASAT data analysis technique was

compared was a certain FGGE IIIB surface pressure field and its associated

geostrophic wind field expressed in a mesh of 4° grid resolution.

IVa. Determination of weights

The first problem we face in developing the SEASAT variational adjustment

technique is to determine Weights A and B in (3.7). Even though we know of no

clearcut way to choose these weights without making numerical experiments, a close

look at Equations (3.8), (3.9), and (3.10) would more or less shed some light on

how to determine them.

To begin with, we notice that the density of surface pressure observations is

rather low over the ocean areas. Thus, values of p0 are known only at a few grid o

points in the entire domain. The surface pressure at all other grid points needs

to be determined by the satellite-derived surface wind data. In order for this to

hold, Weight A has to take on a very large value at those grid points where surface observations of pressure are available, and A has to be small enough (effectively zero) at all the other P-grids. For any given assignment of the A values

at all grid points, a vanishing value of B would mean no adjustment of the wind.

I.e., u and v would take on the surface geostrophic wind components u. and v0 s s

computed from the SEASAT wind data. As the value of B gradually increases, the

adjustment of wind is also allowed to increase. If a larger and larger value of B

is used, then we would suspect that since the adjusted surface geostrophic wind

will have less and less to do with the original surface geostrophic wind, the

adjustment may eventually become unsoundly large. When this takes place, the

-12-
f

resultant pressure field will become meaningless so that it may bear no resemblence to the true pressure field. Thus, by defining the target area to be the

smallest area bounded by two meridians and two parallels and containing all the

known P_ points, we may determine the best value of B that results in the smallest s

RMS error of the target area.

In order to obtain numerical values of A and B, we chose to use the mesh

shown in Fig. 3 as the computational domain. The dots in the mesh indicate that

observed surface pressure values at those points were given. This domain is certainly larger than the target area defined above. The presence of a buffer zone

surrounding the target area as shown in Fig. 3 is intended to prevent the boundary

condition from contaminating the solution in the target area. An extensive

discussion of this subject will be presented in Section VIb.

In making numerical experiments to determine values of A and B, we first ran

our variational code with the February 25, OOz, 1979 data in 4° grid resolution.

We found that by assigning an A value of 0.25 E-4 m^/g or larger to those grid

points with observed surface pressure, the resultant surface pressure values at

these points remained unchanged from the observations Pg. Thus, by fixing A to be

0.25 E-4 nr/g at the grid points of known Pg, we list in Table 1 the RMS errors of

the target area for three values of B versus six values of A at the grid points of

unknown Pg. The range expanded by those A and B values was chosen to be wide

enough to reveal a general trend. We remark in here that because the surface

pressure field is what we are interested in getting, when the terra RMS error is

used, it refers to the RMS error of surface pressure in the target area. This

convention will hold throughout the entire discussion of this study. It can be

seen from Table 1 that the smallest RMS error appears when A=0.25E-13 m3/g and

B=1.E7 s^, although other combinations of A and B may result in RMS errors that

-13-
are only slightly larger. Table 1 also reveals that a monotonic reduction of the

value of A does not always yield a smaller RMS error. Thus, we decided to choose

A to be 0.25 E-4 m3/g for the grid points of known Pg, 0.25 E-13 m3/g for all

other P-grids, and B equal to 1.E7 s2 as the most reasonable choice. In an attempt

f\

to find out whether or not B = 1.E7 s was also a reasonable value for other data

sets, we also included the March 01 and March 09, OOz, 1979 data sets in the

numerical study. The RMS errors computed for six B values using the three data

sets are presented in Table 2. It can be seen that when B falls in the range

between 1.0 E6 and 1.0 E8 s2, the RMS error of the target area reaches a general

low-value region. The fluctuation of the RMS error is small within this range of

B, indicating that any value of B in this range can be considered acceptable.

When B falls out of this range, however, the RMS error drastically increases.

Thus for all three data sets tested, A=0.25 E-4 nr/g or larger for known Pg grids,

A=0.25 E-13 m3/g or smaller for unknown Pg grids and B=1.0 E7 s2 for all grid

points is a reasonable choice. If a value other than 1.0 E7 s' is to be used for

B, then as along as B lies in the range between 1.0 E6 s2 and 1.0 E8 s2 the choice

will be considered acceptable, too. From this point on, all the numerical

experiments to be discussed were conducted with the specification of A and B

described above.

Plots of the true solution of P, u and v on February 25, OOz, 1979 are shown

in Fig. 4. Figs. 5a, 6a and 7a give plots of the computed P-, u- and v- fields

for B= 1.E5, I.E7 and 1.E10 s2. The difference fields of P, u and v between the

true solutions shown in Fig. 4 and the numerical solutions shown in Figs. 5a, 6a

and 7a are respectively plotted in Figs. 5b, 6b and 7b for easy comparison. In

all the pressure plots, two longitudes and two latitudes are drawn to mark out the

target area, and dots are used to denote locations of the input pressure

-14-
observations. It is obvious that the error is generally smaller inside the target

area, but it may become very large away from the target area. This point will be

discussed in detail in the next section. A comparison of Figs. 5b, 6b and 7b

verifies our previous conclusion drawn from examination of Table 1 that B=1.E7 s2

produces the least RMS error compared with B=1.E5 and 1.E10 s2. Similar plots for

data sets of March 01 and March 09, OOz, 1979 are respectively shown in Figs. 8,

9a,b, 10a,b, lla,b and Figs. 12, 13a,b, 14a,b, 15a,b. A close examination of

o

these figures shows that our previous choice of B in the range of 1.E6 and 1.E8 s

based on the February 25 data set is also supported by using the two additional

data sets.

VIb. Influence of the boundary condition

In the previous section, we discussed employment of a computational domain

larger than the target area in running numerical experiments. The reason for that

will be made clear in this section.

When surface pressure observations are located near or on the boundary of the

domain, one might suspect that discretization of the boundary condition could

create inaccuracies that would enter into the interior part of the domain in the

computation. And if this would indeed turn out to be true care must be taken to

alleviate this difficulty.

As was mentioned in the previous section, the target area Is the smallest

area bounded by two meridians and two parallels containing all the grid points

with surface pressure observations. Therefore, in order to determine the effect

of the boundary condition, computational domains of sizes the same as and larger

than the target area with identical surface pressure observations were used to

suit our purpose. We actually employed four different domains in running

numerical experiments. Domain A was identical with the target area. Domain B

-15-
exceeded the target area by one grid interval in each of the four directions.

Domain C was larger than Domain B by one grid distance in the four directions.

Domain D was an extension of one grid interval of Domain C in the longitudinal

direction only. Actually, Domain D was the same as the computational domain used

in the numerical experiments discussed in the previous section, and the same set

of the five surface pressure observations used in the computations in Section Via

was also used for Domains A, B and C. The February 25, OOz, 1979 data set was

used as input data to run numerical experiments on these three domains. The

resultant RMS errors of the target area for the four computational meshes A, B, C

and D are listed in Table 3. This table clearly shows that the closer the surface

pressure observations are located near the domain boundary, the larger is the RMS

error of the target area as we suspected earlier. It is also clear that an

expanded computational domain may advantageously be used to alleviate the

difficulty associated with the boundary condition. The RMS error decreases when

grid points with pressure observation are farther removed from the domain

boundary. Moreover, the RMS errors obtained on Mesh C and Mesh D are seen to be

not too different from each other, which indicates that a buffer zone at least two

grid intervals long and wide as shown in Mesh C will be required to prevent the

boundary condition from contaminating the interior domain. This will be used as a

general rule. The best way to determine the appropriate computational domain ,

however, is to conduct a series of numerical experiments with an increasing size

of the buffer zone. When there is no further change in RMS error due to change in

the size of buffer zone, the appropriate computational domain is thus decided.

The true solution of u, v and P is shown in Fig 4. The u-, v- and P-fields

computed on Meshes A, B, C, D and their difference from the true solutions are

shown in Figs. 16a,b ,17a,b, 18a,b and 6a,b, respectively. As before, two longi-

-16-
tudes and two latitudes were drawn in each of these plots to mark out the target

area. It can be seen by comparing Figs. 6b, 16b, 17b and 18b that when an

expanded computational domain is used, the error is extremly large outside the

target area, whereas inside the target area the error is significantly smaller.

This finding justifies our use of Grid D in conducting the numerical experiments

presented in Section Via.

Vic. Effect of data density of surface pressure

The essence of the current SEASAT data analysis technique is to use the dense

wind data obtained by the SEASAT observation to help improve the surface pressure

analysis in the data-sparse area over the oceans. Thus, it is straightforward to

expect that the denser the surface pressure observation is, the more accurate will

be the result of the variational SEASAT data anlysis. In order to verify this

speculation by making numerical experiments, we chose to analyze the February 25,

OOZ, 1979 data set on the D grid using A and B values found in Section Via. Three

cases were run: (i) Case 1 used only three of the five Pg grid points appeared in

the numerical experiments discussed in the previous two sections; (ii) Case 2

used the same five Pg grids as before; (iii) Case 3 used seven grid points of

known PS with five of these points identical with the Pg points in Case 2. Table

4 gives the RMS error of the target area for all three cases. As expected, the

RMS error for Case 3 is the smallest whereas the RMS error for Case 1 is the

largest. Plots of the computed P, u and v fields and their difference from the

true solution are respectively shown in Figs. 19a,b, 6a,b, and 20a,b. It is

informative to note that when too few surface pressure observations are available,

the result of the variational SEASAT data analysis is rather inaccurate as

evidenced by Figs. I9a,b. A comparison of the difference fields plotted in Figs.

19b, 6b and 20b shows a persistent increase in accuracy of the resultant pressure

-17-
analysis as a greater number of surface pressure observations became available.

However, this was the case simply because the input pressure data in Case 1 formed

a subset of the pressure data in Case 2, which themselves formed a subset of the

pressure data in Case 3. In general, a greater number of pressure observations do

not necessarily lead to a more accurate pressure analysis. It is expected that

when more observations of surface pressure are available in regions of strong

pressure gradient, will we get a better pressure analysis. In order to verify

this point, two additional runs were made on the same computational domain using

the same A and B values. Six pressure data points were used in both runs, but the

locations of those pressure data were different. In the first run, three pressure

observations were located equidistantly on the eastern boundary of the target

area, and another three pressure data were regularly spaced on the western

boundary of the target area. In the second run, however, the pressure data points

were chosen such that three regularly spaced data points appeared on each of the

northern and southern boundaries of the target area. RMS errors computed for the

target area are respectively 0.647 and 1.084 mb for the two cases, and contour

plots of P-, u-, v- and the difference fields are respectively shown in Fig. 21

and Fig. 22. It is obvious that a much better accuracy was obtained in the former

run when compared with the latter. The reason for this is not very difficult to

explain: An examination of the pressure field shown in Fig. 4 reveals that regions

of strong pressure gradient are located more close to the eastern and western

boundaries than the northern and southern boundaries of the target area.

Consequently, because more data points were available in regions of strong

pressure gradient, a better accuracy was thus resulted in the former run when

compared with the latter. By the same token, because regions of strong pressure

gradient in Fig. 4 are closer to the boundary than to the interior of the target

-18-
area, if the two pressure data points located in the middle of the eastern and

western boundaries of the target area in Fig. 21 were respectively shifted inward

for two grid intervals, then a decrease in accuracy would be expected. An

additional run was made to clarify this speculation. The resultant RMS error in

this case is 1.2 rab, much larger than the RMS error of 0.65 mb associated with

Fig. 21. The graphical results of this additional run are shown in Fig. 23 for

ease of comparison.

VId. Effect of quality of SEASAT-derived wind data

When a new SEASAT satellite is launched into space in the future to provide

sea surface wind data through remote sensing, it is expected to carry refined

sensing devices. Consequently, the wind data derived from the satellite

observation should more accurately represent the true surface wind over ocean

areas. A reasonable question to ask in this regard is, "If the speed and

direction errors associated with the SEASAT-derived wind data can be reduced by

introduction of new devices, what impact will it have on the accuracy of the

SEASAT pressure analysis?". In an attempt to assess the consequence of this, we

performed the following numerical experiment: We reduced the standard deviation

of the windspeed error from ±2 to ±1 m/s, and reduced the standard deviation of

the error in wind direction from ±20° to ±10° in assimilating the satellitederived wind data. The same data set and computational specification used in

obtaining Fig.6 were employed. The resultant RMS error of the target area is

found to be 0.86 mb which is less than the RMS error of 0.89 mb associated with

Fig. 6. A simple comparison of these two numbers does not reveal how the RMS

error gets a little smaller, so we examine the P-, u- and v-fields and their

difference from the true solution as plotted in Fig. 24. By comparing the two

pressure difference fields in Figs. 6b and 24b, we observe that the RMS error is

-19-
almost halved near the central part of the target area, although the reduction in

error quickly falls flat toward the boundary of the target area, where larger

errors appear and dominate the total RMS error. This result reflects two things

of different nature: (i)Effort to improve the quality of the SEASAT-derived wind

data is worth seeking, because it tends to significantly reduce the pressure error

near the central part of the target area; and (li)Except for the error imbedded in

the SEASAT-derived wind data, other factors such as the total number of surface

pressure observations, where they are located In relation to regions of strong

pressure gradient, and what computational domain Is in use, as have been discussed

in the previous sections may all influence the outcome of the SEASAT data analysis

technique. Thus without enough number of surface pressure observations In the

region of interest, there is an upper limit on the accuracy of the SEASAT pressure

analysis technique even with upgraded SEASAT-derived wind data.

Vie. Influence of the surface temperature field

In all the previous numerical experiments discussed, we have tacitly assumed

that the surface temperature field Is completely known at every grid point.

However, unless the sea surface temperature field is already obtained elsewhere,

say by blending the satellite temperature observations into the conventional data,

the temperature can only be known at those same grid points where surface pressure

is observed. When this is not the case, a convenient alternative is to use a

constant temperature field in the SEASAT data analysis. This constant temperature

can be taken as, for instance, a representative mean temperature of the entire

domain. Thus, we made three numerical runs to find out how much influence this

may have on the outcome of the SEASAT pressure analysis. The control case was

chosen to be the one shown In Fig. 6. In the three runs we made, the constant

temperature was respectively chosen to be 300, 283 and 290 °K, the last one being

-20-
the average of the five given surface temperature observations. The 300 and 283

°K temperatures were used to reprent more extreme cases. The resultant RMS errors

were found to be 0.91, 0.89 and 0.87 mb, respectively for the 283, 290 and 300 °K

temperatures. By comparinf it with the RMS error of 0.89 mb of the control case,

we see that the use of a representative mean temperature obtained by averaging the

given surface temperature observations almost produced an identical result to the

control case. Use of the two extreme temperatures yielded somewhat different

surface pressure fields from the control case, yet the difference is not too

distinctive. Graphical comparisons of these results can be made by examining

plots of the velocity and surface pressure fields and their difference from the

true solutions as shown in Figs. 25, 26 and 27. We consequently determined that a

fairly good surface pressure field can be obtained by the SEASAT data analysis

technique without knowing the complete surface temperature field.

V. Concluding Remarks

The SEASAT-derived wind can be considered as the neutral stability vector

wind at 19.5 m above sea level with an accuracy up to ±2 m/s and ±20° respectively

in speed and direction. These errors are considered characteristic and were used

in assimilating the SEASAT-derived wind data from the geostrophic wind field of a

chosen surface pressure field. With the help of the geostrophic drag law and an

empirical formula of the neutral stability drag coefficient, the assimilated

SEASAT-derived wind data could be converted to the corresponding surface

geostrophic wind at 10 m above the sea level. A variational problem with the

geostrophic wind relation as the weak constraint was then formed to combine the

sparse surface pressure data with the dense SEASAT-derived wind data together.

After this variational problem was solved numerically, the resultant adjusted

pressure field was compared with the input true pressure field. In conducting

-21-
this study, we have reached the following conclusions:

(1) Weight A that appears in the computation of the variational problem can be

reasonably chosen to be 0.25E-4 nr/g at those grid points where surface pressure

observation is available. For all other grid points without the pressure data, a

i

o

reasonable value of A is 0.25E-13 nr/g. Weight B can be assigned a uniform value

of 1.E7 s^ over the entire domain, however, any value between l.E6s^ and 1.E8 s

is also considered acceptable.

(2) In order to avoid possible contamination of the region of interest (target

area) by the boundary condition, it is recommended that a computational domain

larger than the target area by at least two grid intervals in each of the four

directions be employed. Any domain smaller than this will likely lead to very

inaccurate result. A more appropriate way to determine the size of the

computational domain is through conducting a series of numerical experiments with

increasing buffer zones. When no more change in RMS error with increasing size of

the buffer zone is observed, the proper computational domain is thus determined.

(3) The greater the number of surface pressure observations there is, the higher

the accuracy of the resultant pressure field will be. This is especially true in

regions of strong pressure gradient.

(4) An upgrade in quality of the SEASAT-derived wind data will result in an

improvement in accuracy of the SEASAT pressure analysis technique. However, the

extent to which the accuracy can be improved has an upper limit. Beyond that

limit, further improvement can be expected only if more surface pressure

observations become available.

(5) The surface temperature field is usually not available at each mesh grid.

However, in the current SEASAT data analysis technique, the employment of a

constant temperature can result in a fairly reasonable surface pressure field.

-22-
Table 1. List of pressure RMS errors In mb for determining Weights A and B.

s~*

t-4

V)

^— x

CO

t-l

f.

bl

1

i

1.0 E5

1.0 E7

1.0 E9

Weight A (nr/g) for grid points without

surface pressure observation

0.25 E-10

5.1

3.6

2.0

0.25 E-ll

4.8

I. A

1.786

0.25 E-12

3.5

0.9

1 .760

0.25 E-13

1.3

0.891

1.759

0.25 E-15

0.909

0.896

1.759

0.25 E-22

0.914

0.896

1.759

-23-
Table 2. List of pressure RMS errors In mb for determining

Weight B for three different data sets

-J

01

(S)

1C

.J

_/~^

feb 25 OO/

1979

Mnr ] OO/

1979

Mnr 9 OO/

1979

Weight B (s2)

1.0 E5

1.3

2.0

1.3

1.0 E6

0.9

0.9

1.0

1.0 E7

0.9

1 .0

l.n

1.0 E8

0.9

0.9

1.0

1.0 E9

1.8

1 .1

>*

1.4

1.0 E10

3.6

4.4

2.2

-24-
Table 3. List of pressure RMS errors in mh for determining the boundary effect

RMS error

Computat tonal Domain

A

2.6

B

1 .1

C

0.9

D

0.9

-25-
Table 4. List, of Pressure RMS error-, In mh for different data densities

of surface pressure observation

RMS

error

No. oC Pressure Observations In domain

3

Weight B (s2;

1.E6

2.0

1.E7

2.1

1.E8

1.9

1.K9

1.9

5

Weight B (s2)

I.E6 I.E7J I.E8

! |

0.9 ! 0.9 ! ().<)

!1

I.E9

1.8

7

Weight B (s2)

1.E6

0.7

1.E7

0.7

1.E8

0.8

I.E9

1.8

-26-
Figure Captions

Fig. 1 Schematic representation of the geostrophlc drag law. <* Is the angle

between the surface stress and the geostrophlc wind.

Fig. 2 Schematic of the staggered mesh used In conducting the SEASAT

varlatlonal data analysis. P^ . Is the grid point at which the (1,1)-

th surface pressure and weights A, B are located. (u,v)j . represents

the grid point where the (l.j)-th u and v are located. 'The Corlolls g itn

associated with the 1-th row of p

parameter foj is associated with the i-th row of u,v; ^21-1

wi

Fig. 3 Schematic Illustration of the computational domain, the target area,

and the buffer zone. The dots in the mesh Indicate locations of the

surface pressure observations.

Fig. 4 True solution of the surface pressure and goestrophlc wind fields on

February 25, OOz, 1979. From top to bottom: Contour plot of P In 2 mb

Intervals, contours of the field in 2 ro/s, and contours of v jn 2 m/s.

o

Fig. 5a Variatlonally adjusted fields of P, ug and VR in 2 mb and 2 m/s contour

intervals for A = 0.25E13m3/g and B = 1.E5 s2. Dots in the p-plot

indicate locations of given surface pressure observations.

Fig. 5b Contour plots of the difference between the fields shown In Figs. 4 and

5a In 0.5 mb and 2 m/s intervals. Dots in the ^pplot indicate

locations of given surface observations.

n

Fig. 6a Same as in Fig. 5a except B = 1.E7 s .

Fig. 6b Same as in Fig. 5b except B = 1.E7 s2.

Fig. 7a Same as In Fig. 5a except B - 1.E10 s2.

Fig. 7b Same as In Fig. 5b except B = 1.E10 s2.

Fig. 8 Same as in Fig. 4 except for using the March 1, OOz, 1979 data set.

Fig. 9a Same as in Fig. 5a except for the March 1, OOz, 1979 data set.

Fig. 9b Same as in Fig. 5b except for the March 1, OOZ, 1979 data set.

Fig. lOa Same as In Fig. 6a except for the March 1, OOZ, 1979 data set.

Fig. lOb Same as in Fig. 6b except for the March 1, OOZ, 1979 data set.

Fig. 11 a Same as In Fig. 7a except for the March 1, OOZ, 1979 data set.

Fig. lib Same as in Fig. 7b except for the March 1, OOZ, 1979 data set.

Fig. 12 Same as In Fig. 4 except for using the March 9, OOz, 1979 data set.

-27-
Fig. Ha Same as in Fig. 5a except for the March 9, OOZ, 1979 data set.

Fig. 13b Same as In Fig. 5b except for the March 9, OOZ, 1979 data set.

Fig. 14a Same as In Fig. 6a except for the March 9, OOZ, 1979 data set.

Fig. 14b Same as In Fig. 6h except for the March 9, OOZ, 1979 data set.

Fig. 15a Same as In Fig. 7a except for the March 9, OOZ, 1979 data set.

Fig. 15b Same as In Fig. 7b except for the March 9, OOZ, 1979 data set.

Fig. 16a Same as In Fig. 6a except for the use of Grid A (see text).

Fig. 16b Same as In Fig. 6b except for the use of Grid A (see text).

Fig. 17a Same as In Fig. 6a except for the use of Grid B (see text).

Fig. 17b Same as In Fig. 6b except for the use of Grid B (see text).

Fig. 18a Same as In Fig. 6a except for the use of Grid C-(see text).

Fig. 18b Same as In Fig. 6b except for the use of Grid C (see text).

Fig. I9a Same as In Fig. 6a except that only three of the five pressure

observations used In Fig. 6a were employed as the Input data.

Fig. 19b Same as In Fig. 6b except that only three of the five prebsure

observations used In Fig. 6a were employed as the Input data.

Fig. 20a Same as In Fig. 6a except two additional pressure observations were

Included In the Input data.

Fig. 20b Same as In Fig. 6b except two additional pressure observations were

included in the input data.

Fig. 21 a Same as in Fig. 6a except that the Input pressure observations used

were different.

Fig. 2lb Same as in Fig. 6b except that the input pressure observations used

pressure observations.

Fig. 22a Same as in Fig. 6a except that the input pressure observations used

pressure observations.

Fig. 22b Same as in Fig. 6b except that the input pressure observations used

pressure observations.

Fig. 23a Same as In Fig. 6a except that the input pressure observations used

pressure observations.

-28-
Fig. 23b Same as in Fig. 6b except that the input pressure observations used

pressure observations.

Fig. 24a Same as In Fig. 6a except that the errors in the assimilated SEASAT

wind data were reduced from ±2 m/s and 20° to ±1 m/s and 10°.

Fig. 24b Same as in Fig. 6b except that the errors in the assimilated SEASAT

wind data were reduced from ±2 m/s and 20° to ±1 m/s and 10°.

-29-
V

v

g

=0

direction of surface stress

*****

Kig 1

-30-
•w. v

in. v

K, V-

[

f

r

r

f

r

P

— 1M1I

5

•C -'

D

•u.

•*»<

•>

K

>

1>

p

ft

ir

;'-1

c

Iv

I''1

r

1

i/ '

f

P

-i/f'-'

v.

*-•(

7

c.;-.

•u

»•

>

;-.;-i

^

c

r

V

•r<

f

I

\f-

;'-<

fr

f

p

v

>

'•'•>

M

•<;-<

>

*•>

-r<

>

»>.,;

u

>

j/ -

i

/

V

i

r

r

. i/

7

p

r

V*

P

^/

5

<.-i ;M

w

•v-l

>

•/-'

K,

-•;

-r..;-,

1J

~>

\

\rV-

;'T'

t

V

'-r 1

\^

. V

Fig. 2

-31-
1

I

1 t 1' •(

1

1

r •

L _

1

i- -4

1

•i

J - ' J.I _ - ' ' 1

• i T i

; : ' ' ; : ki

 '

 - • — -

 ' -••( i ' • , m ' ^

' I I 1 ! , ' '

i i i i , _ . .! _ 1 1.0 . . _ 1 - i- -

1

 • JTARGE T j i '

; . 1 ARE A ' ' t

» , _ 1 - j _ T _

'

 ! i iii ,

' ,.(..._ . j ..... _ ... . .... , ...... 4

BUFFE R -

L

L _ _ --ZON E -' -

i i

1

 !

_ i . ,.

. I

I

I

f-

-32-
ORIGINAL PAGE IS

OF POOR QUALITY

-33-
Fig. 5a

-34-
Fig. 5b

-35-
Fig. 6a

-36-
ORIGINAL PAGE »S

OF POOR QUALTTY

Ml»' _ !i -I_7L./'L/V

/'" ^0 \x \

Fig. 6b

-37-
I I

A\

80

H

I I/ I

Fig. 7a

-38-
ORIGINAL PAGE IS

OF POOR QUALITY

Fig. 7b

-39-
Fig. 8

-40-
Hv\v////^^^<

' " ^<~S 'A." - \

• ~^A' ~-J l^

Fig. 9a

-41-
Fig. 9b

-42-
Fig. lOa

-43-
I 1- 1

Fig. lOb

-44-
1 /-r o 1 4 ; o

Fig. lla

-45-
Fig. lib

-46-
Fig. 12

-47-
Fig. I3a

-48-
Fig. I3b

-49-
Fig. 14a

-50-
Fig.

-51-
Fig. 15a

-52-
Fig. 15b

-53-
Fig. 16a

-54-
Fig. 16b

-55-
Fig. 17a

-56-
Fig. 17b

-57-
5sM/v/\y

r \\

//A\..'//i'. //A ,

Fig. 18a

-58-
Fig. 18b

-59-
l\ ! I I I I L,

Fig. 19a

-60-
ORIGINAL PAGE IS

OF POOR QUALITY

Fig. 19b

-61-
ORIGINAL PAGE IS

OF POOR QUALITY

Fig. 20a

-62-
Fig. 20b

-63-
Fig. 21a

-64-
v

 -•-

/I I I

Fig. 21b

-65-
Fig. 22a

-66-
Fig. 22b

-67-
Fig. 23a

-68-
--. 3 - ^ 1 O

Fig. 23h

-69-
J I

Fig 24a

-70-
">

Fig. 24b

-71-
I I I

Fig. 25a

-72-
Fig 25b

-73-
Fig. 26a

-74-
Fig. 26b

-75-
Fig. 27a

-76-
Fig. 27b

-77-
REFERENCES •

Blackadar, A.K., and H. Tennekes, 1968: Asymptotic similarity in neutral

barotropic atmospheric boundary layers. J. Atmos. Sci., 25, 1015-1020.

Brown, R.A., and W.T. Liu, 1982: An operational large-scale marine planetary

boundary layer model. J. Appl. Meteor., 21, 261-269.

, V.J. Cardone, T. Guymer, J. Hawkins, J.E. Overland, W.J. Pierson, S.

Peteherych, J.C. Wilkerson, P.M. Woiceshyn, and M. Wurtele, 1982: Surface

wind analyses for SEASAT. J. Geophys. Res., 8, 3355-3364.

Businger, J.A., et al., 1971: Flux-profile relationships in the atmospheric

surface layer. J. Atnos. Sci., 28, 181-189.

Dome, G.J., A.K. Fung, and R.K. Moore, 1977: An empirical model for ocean radar

backscatter and its application in inversion routine to estimate winds speed

and direction effects. Paper presented at Proceedings URSI Commission F., La

Baule, France, April 28 to May 6, 1977.

Garratt, J.R., 1977: Review of drag coefficients over oceans and continents. Mon.

Wea. Rev., 105, 915-929.

Gray, W.M., 1972: A diagnostic study of the planetary boundary layer over the

oceans. Atmos. Sci. Paper No. 179, Colorado State University, 95 pp.

Jones, W.L., F.J. Wentz, and L.C. Schroeder, 1978: Algorithm for inferring winds

stress from SEASAT-A. AIAA J. Spacecraft Rockets, 15, 368-374.

, L.C. Schroeder, D.H. Boggs, E.M. Bracalente, R.A. Brown, G.J. Dome, W.J.

Pierson, and F.J., Wentz, 1982: The SEASAT-A satellite scatterometer: The

geophysical evaluation of remotely sensed wind vectors over the ocean. J.

Geophys. Res.,87, 3297-3317.

Moore, R.K., and A.K. Fung, 1979: Radar determination of winds at sea, Proc.

IEEE,67, 1504-1521.

Pierson, W.J., V.J. Cardone, and J.A. Greenwood, 1974: The applications of

SEASAT-A to meteorology. Tech. Report, City University of New York.

Schroeder, L.C., D.H. Boggs, G. Dome, T.M. Halberstam, W.L. Jones, W.J. Pierson,

and F.J. Wentz, 1982: The relationship between wind vector and normalized

radar cross section used to derive SEASAT-A satellite scatterometer winds. J.

Geophys. Res., 87, 3318-3336.

Tennekes, H., 1973: Similarity laws and scale relations in planetary boundary

layers. Workshop on Microneteorology, A.M.S., Boston, 177-216.

Wright, J.W., 1966: Backscattering from capillary waves with application to sea

clutter. IEEE Trans. Antennas Propagat., AP-14, 749-754.