TIME DEPENDENT WIND FIELDS 

FINAL REPORT 

Dudley B. Chelton 

College of Oceanography 

Oregon State University 

Corvallis, Oregon 97331 

(NASA-CR-173959) TIME DE€ENE;EN'I: MIND EXELDS 

CSCL 08C 

Einal ReFort (Oregon State Uriv.) 67 p 

c/m 43 

Ne7-13839 

Uaclas 

63/43 43639 

January 27, 1986 

JPL Contract No. 956567 

This work was performed for the Jet Propulsion Laboratory, California 

Institute of Technology, sponsored by the National Aeronautics and 

Space Administration under Contract NAS7-100. 
. 

This document contains information prepared by Oregon State University 

under JPL sub-contract. Its content is not necessarily endorsed by 

the Jet Propulsion Laboratory, California Institute of Technology, or 

its sponsors. 
ABS TRACT 

The work performed under this contract consisted of two basic 

tasks: 1) determination of the accuracy of Seasat scatterometer, 

altimeter, and scanning multichannel microwave radiometer measurements 

of wind speed; and 2) application of Seasat altimeter measurements of 

sea level to study the spatial and temporal variability of geostrophic 

flow in the Antarctic Circumpolar Current. 

The results of the first task have identified systematic errors 

in wind speeds estimated by all three satellite sensors. However, in 

all cases the errors are correctable and corrected wind speeds agree 

between the three sensors to better than 1 ms in 96-day 2" latitude 

by 6" longitude averages. 

-1 

The second task has resulted in development of a new technique 

for using altimeter sea level measurements to study the temporal 

variability of large-scale sea level variations. Application of the 

technique to the Antarctic Circumpolar Current yielded new information 

about the ocean circulation in this region of the ocean that is poorly 

sampled by conventional ship-based measurements. 
 

TABLE OF CONTENTS 

Page 

1. Introduction ...................................................... 1 

2. SASS wind speed accuracy .......................................... 3 

3. ALT wind speed accuracy ........................................... 5 

4. SMMR wind speed accuracy .......................................... 9 

5. ALT sea level variability in the Antarctic ........................ 11 

Circumpolar Current 

6. Conclusions and Recommendations ................................... 14 

7. List of publications supported by this Research Contract .......... 16 

References ........................................................ 17 

Appendix 1: A review of satellite altimeter measurement of ......: 19 

sea surface wind speed with a proposed new algorithm. 

Appendix 2: Comments on: Seasonal variation in wind speed ........ 33 

and sea state from global satellite measurements. 

Appendix 3: Temporal variability of the Antarctic Circum- ........ 41 

polar Current observed from satellite altimetry. 

Appendix 4: Observing large-scale temporal variability of ........ 45 

ocean currents by satellite altimetry: with application 

to the Antarctic Circumpolar Current. 
1 

measured by all three satellite sensors. 

ALT and SASS are both active radars which beam a pulse of 

microwave radiation at the sea surface and measure the power of the 

returned signal. The ALT radar points at satellite nadir while the 

SASS radar is a fan beam which measures from nadir to approximately 

60" incidence angle on both sides of the satellite (with a gap between 

10" and 25" incidence angles). The power of the backscattered signal 

is sensitive to the roughness of the sea surface. As the sea surface 

roughness increases with increasing wind speed, the power of the 

backscattered signal decreases at satellite nadir through specular 

reflection away from the receiving antenna and increases at incidence 

angles greater than 10" due to Bragg scattering. 

SMMR is a passive radiometer which measures the natural microwave 

radiation emitted from the sea surface. The "brightness temperature" 

of the sea surface from horizontal polarization is correlated with sea 

surface roughness and foam coverage. 

increase with increasing wind speed, the sea surface becomes a more 

efficient radiator :and the brightness temperature increases. 

As the roughness and foam cover 

Among the three satellite microwave wind speed estimates, 

off-nadir SASS measurements have received by far the greatest 

attention. 

calibration data used to derive sea surface wind speed from SASS 

Summaries of the evolution of the algorithm and the 

~ backscatter measurements can be found in the works by Boggs [1981] and 

1. Introduction 

Analysis of Seasat data has demonstrated that there are three 

microwave sensors capable of measuring sea surface wind speed from 

satellites. These are the scatterometer (SASS), altimeter (ALT) and 

scanning multichannel microwave radiometer (SMMR). 

measure the wind speed indirectly by measuring different physical 

aspects of the sea surface roughness. 

the sensor-measured sea surface parameters to wind speed have been 

derived from very limited data sets and then applied to the full 

96-day Seasat data set. 

under this contract was to evaluate the accuracy of wind speeds 

All three sensors 

The algorithms for converting 

One of the purposes of the research funded 
2 

Schroeder et al. [1982]. 

based on extensive aircraft scatterometer measurements prior to the 

launch of Seasat and aircraft underflights during the Seasat mission. 

In addition, the off-nadir SASS wind speed algorithm incorporated 

orders of magnitude more direct comparisons between satellite and in 

situ measurements than either the ALT or SMMR algorithms. 

therefore reasonable to assume that SASS wind speed estimates are the 

most accurate satellite measurement of wind speed. 

The off-nadir SASS wind speed algorithm is 

It is 

In Sec. 2, the results of an evaluation of the overall accuracy 

of SASS wind speeds by comparison with NDBO buoy winds is summarized. 

It was found that SASS wind speeds are in error by approximately a 

1 ms-' bias. 

detail. 

The probable cause of this simple error is discussed in 

In the research performed under this contract, off-nadir SASS 

wind speeds were used to evaluate the accuracy of both ALT and SMMR 

wind speeds. It was found that winds speeds from both ALT and SMMR 

differed significantly from SASS wind speeds. Investigations into the 

cause for these differences identified errors in both the ALT and SMMR 

wind speed algorithms. These errors are described in Secs. 3 and 4. 

When the algorithms are corrected, wind speed estimates from ALT and 

SMMR agree quite well with SASS estimates. 

The ultimate goal of these thorough investigations of the 

accuracy of satellite wind speeds is to use the data to study wind 

forced ocean circulation. 

not included in the research supported under this contract. 

An additional research project funded under this contract was 

However, application of the wind data was 

analysis of Seasat ALT measurements of sea level to study the spatial 

and temporal variability of geostrophic flow in the Southern Ocean. 

This work resulted in the development of a new technique for 

extracting sea level variability time series from ALT data and 

revealed new information on the structure of variations in the flow of 

the Antarctic Circumpolar Current. The results of this study are 

summarized in Sec. 5. 
3 

2. SASS wind meed accuracv 

For reasons discussed in the Introduction, vertically polarized, 

off-nadir SASS measurements are presently the most reliable remotely 

sensed estimate of sea surface wind speed. It is therefore reasonable 

to use these data as a basis for comparison with ALT data. In any 

event, there is no alternative high-quality wind data base which could 

be used to derive wind speed model functions for ALT and SMMR. In 

order to evaluate the accuracy of the SASS winds, we compared them 

with in situ measurements from 19 National Data Buoy Office (NDBO) 

buoys scattered around the coast of North America. During 1978, the 

winds were measured by NDBO buoys at l-s intervals for 8.5 min. The 

average over the 8.5-min. interval was then reported every 3 hours 

(with approximately 25% of the buoys reporting hourly). 

For purposes of comparison with vertically polarized, off-nadir 

SASS wind speed estimates, the NDBO buoy and SASS data sets were 

searched for matches within 100 km and 1 hour. 

comparison of the two measures of the wind speed is shown in Fig. 1. 

This figure suggests that there is a 1 ms” bias in the SASS wind 

speed estimates. 

et al. [1984]. 

final SASS wind speed algorithm was heavily tuned to a particular buoy 

wind recorder in the JASIN experiment. 

this recorder showed that it was overestimating the wind speed by 

about 10% [Weller et al., 19831, corresponding to a 1 ms error for 

the 10 ms’l winds typically observed in JASIN. 

confirmed until after the GDR processing of SASS data. kide from the 

simple 1 ms’l bias in Fig. 1, the agreement between SASS and NDBO buoy 

winds is quite good. 

wind speed (after removing the 1 ms’l bias) is 2.03 ms 

scatter is undoubtedly due to the general difficulty in making 

accurate wind measurements from a buoy [see k’eller et al, 19831 and to 

the differences between an instantaneous satellite measurement over a 

A scatter plot 

This bias has been independently discovered by Wentz 

It can probably be attributed to the fact that the 

Postexperiment calibration of 

-1 

This error was not 

The rms difference between the two estimates of 

-1 . Much of the 

finite footprint and a temporally averaged buoy measurement at a 

point. 
4 

n 

v; 

\ 

E 

n 

W 

W 

w 

[L 

(A 

D 

Z 

5 

m 

m 

c 

m 

SUOY WIND SPEED (m/s) 

Fig. 1. Scatter plot comparison of coincident 19.5 m verrically 

polarized, off-nadir SASS estimared ad h330 buoy measured vinC 

speeds. For a given orbiz, 211 verriczlly poiarized, off-nadir 

s,@Ss obsern-ations wirhin 100 'mu axd 1 hoc of B buoy measuremenr 

have been averaged. 

wind speeds wit%in the 100 kn ~-indor exceeded 3 ms 

excluded. 

826 observations and cne nns error abo~r -&e bias is 2.03 ms 

Dr=a from orbirs =here the razge of SASS 

-1 . 

nave been 

-1 . SASS rind s?eeb are birsed 1.06 cs nigh over rnese -1 

. 
Although it is admittedly an ad hoc "correction", we feel there 

is strong justification for removal of a 1 ms-' bias from the SASS 

GDR wind speeds. The range of uncertainty about this 1 ms bias is 

difficult to assess given such such a limited surface "truth" 

comparison data set, but a 1 ms-' bias removal is certainly an 

improvement over using the raw GDR SASS wind speeds with no bias 

adjustment. 

SASS wind speeds used in the analysis that follows. 

-1 

Therefore, a 1 ms'l bias has been removed from all of the 

3. ALT wind speed accuracy 

A complete summary of investigation of the accuracy of ALT wind 

speeds is given in Chelton and McCabe (1985) and Chelton (1985), 

included here as Appendices 1 and 2. 

examine the accuracy of the ALT wind speed algorithm as with SASS by a 

direct comparison with high-quality in situ wind speed measurements 

such as the NDBO buoy winds. However, a search for NDBO buoy 

observations within 100 km and 1 hour of ALT measurements resulted in 

only 14 independent buoy observations. This small number of matches 

is due to the small ALT footprint size and the fact that ALT measures 

only at satellite nadir (as opposed to SASS which measures over two 

500 km swaths in the off-nadir regime). Clearly, 14 independent 

measurements is too few for any meaningful comparison. 

Ideally, we would like to 

This lack of coincident satellite and buoy measurements is a 

long-standing problem in satellite wind speed algorithm development. 

To circumvent this problem, a new method of calibrating ALT wind 

measurements was proposed. Rather than comparing with in situ 

measurements, ALT measurements can be compared with vertically 

polarized, off-nadir SASS measurements (corrected for the 1 ms-' bias 

discussed above). 

nearest off-nadir SASS samples, it is clearly not possible to compare 

instantaneous measurements by the two sensors. 

spatial and temporal averages of ALT and SASS wind speeds were 

compared. Because of their different sampling characteristics, a 

Since the ALT nadir samples are 200 km from the 

In this research, 

5 
6 

given geographical region is sampled at different times by ALT and 

off-nadir SASS. 

geographical region, this time separation causes no problems. 

However, for variable winds, the samples by each sensor must be 

temporally averaged. The temporal average must be long enough so that 

the variable winds are sampled frequently enough to obtain a reliable 

estimate of the mean wind speed over the averaging period. 

Seasat mission, the maximum possible averaging period is 96 days. 

After experimenting with different spatial averages, 2" of latitude by 

6" of longitude was subjectively chosen as a trade-off between 

reliable spatial averages (limited by the small footprint size of 

individual ALT measurements) and number of ALT and SASS 

If the winds are steady and constant over the 

For the 

intercomparisons. 

Use of this proposed method of calibration is obviously dependent 

on the accuracy of SASS wind speed estimates. 

rather serious limitation since the accuracy of SASS winds has not yet 

been demonstrated over a broad range of conditions. 

1 ms 

in view of known problems with JASIN data used to calibrate the SASS 

wind speed algorithm). 

errors in SASS wind speed estimates are no worse than errors in wind 

speed estimates by conventional measurement techniques. 

wind speed comparison is based on spatial and temporal averages of a 

large number of individual observations, random errors in individual 

SASS wind speed estimates will be either eliminated or greatly 

reduced. Systematic errors, such as cross-track biases in SASS wind 

speed estimates, will also be greatly reduced since the spatial and 

temporal averages include individual wind speed estimates over the 

full SASS incidence angle range. 

This is clearly a 

In addition, the 

ad hoc correction is a little disturbing (although justifiable -1 

However, it is almost certainly true that the 

Since the 

ALT wind speed:retrieval is a two-step procedure. In the first 

step, the normalized radar cross-section 4" is computed from receiver 

gain (AGC), satellite attitude angle and satellite height. In the 

second step, the wind speed is computed from uo. 

wind speed estimation from Seasat, problems were identified in both 

steps of the procedure. 

In this study of ALT 
7 

The most fundamental problem is an error in the 0" algorithm 

implemented in Seasat ALT GDR processing which resulted in a 

discontinuous dependence of o0 on AGC. When this algorithm is 

corrected to remove all discontinuities, the resulting values of 6' 

are much more reasonable than those produced in the Seasat GDR's, 

corrected algorithm (suggested by D. Hancock (personal communication, 

1983)) is much simpler than the Seasat GDR 6" algorithm. 

The 

The wind speeds computed from corrected (IO still exhibit rather 

peculiar behavior. 

adopted from the GEOS 3 algorithm derived by Brown et al. [1981]. In 

this algorithm, the wind speed is first estimated from a three-branch 

model function and then corrected with a fifth-order polynomial 

adjustment. The peculiar behavior of computed wind speeds was traced 

to discontinuous derivatives at the two branch points of the 

three-branch model function. 

The Seasat GDR wind speed model function was 

The end result of the ALT wind speed research funded through this 

contract has been development of a new model function for wind speed 

estimation from corrected values of 0'. The proposed model function 

has the same functional form as that used on Seasat SASS data (both 

nadir and off-nadir). If 0" is expressed in decibels, the new model 

function estimates the wind speed at 19.5 m by 

where 

G = 1.502 

H -0.468 

The parameters G and H were estimated by least squares from a 

comparison of global 96-day, 2" by 6" averages of ALT 6" with 

vertically polarized, off-nadir SASS wind speed. The SASS wind speeds 

were corrected for a 1 ms-' bias which was evidently introduced into 

the Seasat SASS GDR processing due to calibration against an erroneous 

wind recorder in JASIN as discussed in Section 2. 
8 

The proposed model function of course produces ALT wind speeds 

which are consistent with SASS wind speeds. 

by 6" averages, the correlation hetween ALT and SASS wind speed is 

0.94, and the rms difference is 0.82 ms . A distinct advantage of 

this technique for ALT and SASS wind speed comparison is that the 

averaging removes any random component of error in individual 

estimates of wind speed. It should be born in mind that the validity 

of the proposed algorithm has only been demonstrated on spatially and 

temporally averaged data. 

For the global 96-day, 2" 

-1 

A natural question that arises is whether the proposed algorithm 

adequately describes instantaneous ALT estimates of wind speed. A 

rigorous test of performance on individual measurements of u" requires 

an extensive high-quality in situ data base for comparison. 

discussed previously, a search of the NDBO buoy data base identified 

only 14 independent buoy observations of wind speed within 100 km and 

1 hour of Seasat ALT oo measurements. 

meaningful intercomparison. 

planned for Geosat will correct this situation and allow an 

independent assessment of the proposed wind speed model function. 

As 

This is clearly too few for a 

The extensive in situ measurement program 

An alternative approach which yields some information on model 

function performance on instantaneous measurements is to determine the 

rms error of ALT oo measurements. It was found that the 

point-to-point rms error was 0.3 dB. 

large errors in estimated wind speed. For example, the error in an 

estimate of 20 ms wind speed is about 4 ms . 

This noise can lead to very 

-1 -1 

It was concluded that Seasat ALT uo measurement error is too 

large to provide reliable instantaneous estimates of wind speed. ALT 

wind speed estimates will only be useful when the data are temporally 

and spatially averaged. In this study, 96-day, 2" by 6" averages were 

used. The lower limits of averaging are the subject of further 

investigation. This problem is unfortunately addressable to only a 

limited degree from Seasat data because of intermittent sampling by 

ALT during the 96-day mission. 

One final caveat in the proposed model function is that it is 

-1 based on very few comparisons at wind speeds higher than 12 ms . 
9 

Thus the validity of the model function is suspect at these high wind 

speeds. 

situ measurement program such as that planned for Geosat. 

This can only be resolved from a carefully coordinated in 

4. SMMR wind speed accuracv 

The accuracy of SMMR wind speeds was evaluated using the same 

technique used for ALT wind speeds. Temporally and spatially averaged 

SMMR wind speeds were compared with similarly averaged SASS wind 

speeds, 

A careful investigation of the Seasat SMMR wind speed algorithms 

developed by T. Chester determined that they cannot be used in their 

present form because they contain errors. 

for the various geophysical parameters measured by SMMR, Chester 

applied an incorrect incidence angle correction to brightness 

temperatures. The Seasat SMMR was intended to measure microwave 

radiation at an incidence angle of 48.8'. However, because of 

spacecraft roll, pitch and yaw, the actual incidence angle varied 

about the average of 48.8' with an rms value of about 0.25'. The 

brightness temperature should decrease with increasing incidence angle 

but the Chester algorithm imposes the opposite effect. 

In all of the algorithms 

There are also problems with some of the individual algorithms 

for the various geophysical parameters. For example, there are two 

problems with the wind speed algorithm. 

fundamental weakness in the form of the model function. The Chester 

wind speed model function is a two-branch model function with a branch 

point at about 7 ms'l (see Fig. 2). The previously discussed analysis 

of ALT data showed that multiple-branch model functions can produce 

very bad results. 

the model function at the branch point result in different 

probabilities of obtaining wind speeds above or below the branch 

point. 

modified to be smoothly varying. 

The first of these is a 

The problem is that discontinuities in the slope of 

Thus, the Chester wind speed model function for SMMR must be 
10 

i 

c 

Fig. 2. Tne Chester 2-branch dgorithm for wind speed from the Seasat 

S.W. The quanciry x is e funczion of horizontally polarized 

10.7 Gtiz radia=ion, SST, sarer vapor, z~nos?heric lio&d varer 

and incidence angle. 
11 

The second problem with the Chester wind speed algorithm is an 

erroneous correction for SST effects. In the derivation of the Seasat 

algorithms, Chester compared SMKR estimated wind speed with SASS wind 

speeds along one orbit in the Pacific Ocean. Based on the results of 

the limited comparison, Chester applied an ad hoc SST dependent 

adjustment in the final Seasat geophysical processing. 

comparison of SMMR and SASS winds found that these "corrected" SMMR 

wind speeds differed from the SASS wind speeds by exactly the amount 

of the ad hoc SST dependent adjustment (Fig. 3). In other words, the 

Chester SST adjustment to wind speed should not have been applied. 

After removing the Chester SST adjustment (and accounting for a simple 

constant relative bias), the Seasat SMMR and SASS wind speeds compare 

very well over 2" latitude and 10" longitude 96-day averages (Fig. 4). 

The rms difference is 0.72 ms with a correlation of 0.93. 

A global 

-1 

5. ALT sea level variabilitv in the Antarctic CircumDolar Current 

A thorough discussion of this component of the research supported 

under this contract is given in Fu and Chelton (1984; 1985), included 

here as Appendices 3 and 4. 

constructing sea level time series from crossover differences of 

satellite altimetric measurements at geographical locations with high 

density of ascending and descending orbit ground track intersections 

Briefly, a new method was developed for 

(cluster 

employed 

removing 

of a few 

computed 

residual 

areas). In this method, an orbit bias adjustment is first 

to reduce the dominant, long-wavelength orbit error by 

from altimetric measurements a constant bias over a distance 

thousand kilometers. Residual crossover differences are then 

for each cluster location. 

errors, the sequence of crossover differences at each 

To further reduce high-frequency 

particular cluster location is smoothed using a two-dimensional 

running average window in the two-time domain spanned by the ascending 

and descending orbit times of the crossovers. The resulting crossover 

differences are then treated as measurements of time-lagged sea level 

differences, from which an irregularly spaced time series of sea level 
12 

-2 

2 -4 

-6 

t -I 

1 

Fig. 3. Difference between globally distributed 2' latitude by 10' 

longitude 96-aay average Seasac SMXR and SASS wind speeds as a 

function of SST. 

18 

f 16 

n 

\ rl 

b 

vr 

l2 

2 ID 

0 

t I 

c 

I 

t 

e7 r ,/-- 

Fig. 4. Comparison between Seaszt SEE and SASS 2' latitude by 10' 

longitude 96-day average vir15 speeds with the Cnester a6 hoc SST 

2tjusfment removed. 
13 

variation is estimated by a least squares technique. Finally, an 

evenly spaced sea level time series is constructed at each cluster 

location from the irregularly spaced samples using optimal 

interpolation. 

The method was applied to Seasat data over the Southern Ocean to 

study large-scale temporal variability of the ACC during the Seasat 

mission (July 7, 1978, to October 10, 1978). Due to the fact that the 

Seasat orbits were nonrepeating (except for the last 30 days of the 

mission), clusters of crossover differences were sampled in a finite 

area of 200 x 200 km to assure adequate temporal coverage. 

cluster area, the crossover differences are smoothed over a 20 x 20 

day running window to reduce the magnitude of measurement errors to a 

marginally useful value of 5 cm (rms). 

altimetrically measured sea level time series for each cluster 

location represents a spatial and temporal average of the true sea 

level variations. 

In a given 

Hence the resulting 

The altimetric results were compared with in situ bottom pressure 

measurements made by pressure gauges deployed at a depth of 500 m on 

the continental slope on either side of Drake Passage. The comparison 

shows fair agreement between the two measurements at the southern side 

of Drake Passage, but substantial disagreement is found at the 

northern side. It is speculated that this is due to a greater 

presence on the northern side of the ACC of baroclinic variability 

which causes the differences between sea level and bottom pressure 

measurements. 

A total of 229 time series were computed from the Seasat 

altimeter data, covering the entire ACC region from 40"s to 65's. 

empirical orthogonal function (EOF) analysis of the time series 

indicates that 73.5% of the total variability during the 96-day Seasat 

mission can be accounted for by the first EOF, which is characterized 

by a general decrease in sea level around the southern side of the ACC 

and a general increase in sea level to the north. 

change in surface geostrophic velocity indicates a generally eastward 

acceleration of the ACC. 

An 

The corresponding 

Substantial meridional components of the 

acceleration are observed over major topographic features. The small 

~~ 
14 

magnitude of the large-scale meridional velocity change (about 2 cm/s 

over the 3-month mission) indicates that the extent of meridional 

migrations of the ACC axis during the Seasat mission are, at most, a 

few hundred kilometers. 

The results of this analysis constitute the first direct 

obsenrational evidence for large-scale coherence in the temporal 

variability of the ACC. 

significance of the results are questionable due to the short duration 

and substantial measurement errors of the Seasat altimeter, the 

results have demonstrated the great potential of the methodology in 

application to more accurate and longer duration altimetric missions 

expected in the near future. 

for the altimeter proposed for the TOPEX mission [see TOPEX Science 

Working Group, 19811 is 14 cm, an order of magnitude improvement over 

Seasat. The TOPEX mission design calls for global altimeter 

measurements over a 3-year period. 

it will be possible to obtain from TOPEX altimetry a global network of 

sea level series with much improved accuracy for studying temporal 

variability of the world ocean circulation. 

Although the accuracy and statistical 

For example, the projected ms accuracy 

Using the method introduced here, 

6. Conclusions and Recommendations 

The conclusions from the investigation of satellite-measured wind 

speed accuracy supported under this contract are that the algorithms 

for all three satellite sensors contain systematic errors. 

the error in vertically polarized estimates of wind speed appears to 

be a simple bias of about 1 ms . For ALT and SMMR the errors are 

more complex and due to a combination of several errors in the 

respective algoritbs. These errors are discussed in detail in Secs. 

3 and 4. When the errors are corrected, ALT and SMMR wind speeds 

For SASS, 

-1 

agree with SASS wind speeds to better than 1 ms-' in 96-day, 2" 

latitude by 6" longitude averages. 

The recommendations from this analysis of wind speeds are: 
15 

The SASS V-pol wind speed model function should be modified to 

-I 

remove the 1 ms bias in wind speed. 

The full 96-day Seasat SASS data set should be reprocessed using 

the modified model function. 

A thorough investigation of SASS H-pol wind speeds should be 

carried out to determine whether any systematic errors exist. 

(This work is being done by P. Woiceshyn and colleagues at JPL 

and F. Wentz). 

The ALT uo algorithm should be corrected and the wind speed model 

function used to process the Seasat data should be modified to be 

continuously differentiable as discussed in Sec. 3. 

Seasat ALT wind speeds should be reprocessed using the corrected 

uo algorithm and modified wind speed model function. 

Errors in the T. Chester SMMR wind speed algorithm should be 

corrected and the model function modified to be continuously 

differentiable. 

The Seasat SMMR wind speeds should be reprocessed using the 

corrected algorithm and modified wind speed model function. 

The conclusions from the ALT sea level analysis supported under 

this contract are that altimeter data are indeed useful for studies of 

large-scale, low frequency variability of geostrophic currents. The 

results of this research represent the first application of altimeter 

data for studies of this nature. These conclusions have important 

implications for use of ALT data from the proposed TOPEX mission. 
7. List of Dublications suDDorted bv this Research Contract 

The following publications were supported by this Research 

Contract. These publications are included here as Appendices. 

Fu, L.-L., and D. B. Chelton , 1984: Temporal variability of the 

Antarctic Circumpolar Current observed from satellite altimetry. 

Science, 226, 343-346. 

Fu, L.-L., and D. B. Chelton, 1985: Observing large-scale temporal 

variability of ocean currents by satellite altimetry; with 

application to the Antarctic Circumpolar Current. 

GeoDhvsical Research, 90, 4721-4739. 

Journal of 

Chelton, D. B., and P. J. McCabe, 1985: A review of satellite 

altimeter measurement of sea surface wind speed; with a proposed 

new algorithm. Journal of GeoDhvsical Research, 90, 4707-4720. 

Chelton, D. B., 1985: Comments on: Seasonal variation in wind speed 

and sea state from global satellite measurements. 

GeoDhvs ical Research, 90, 5001-5008. 

Journal of 
17 

References 

Boggs, D. H., 1981: The Seasat scatterometer model function: the 

genesis of SASS-1. Rep. 622-230, 30 pp., Jet Propul. Lab., 

Pasadena, California. 

Brown, G. S., H. R. Stanley, and N. A. Roy, 1981: The wind speed 

measurement capability of spaceborne radar altimetry. 

Oceanic Ene., m, 59-63. 

IEEE J. 

Chelton, D. B., 1985: Comments on: Seasonal variation in wind speed 

and sea state from global satellite measurements. J. GeoDhvs. 

Res. , 90, 5001-5008. 

Chelton, D. B., and P. J. McCabe, 1985: A review of satellite 

altimeter measurement of sea surface wind speed; with a proposed 

new algorithm. J. Geophvs. Res., 90, 4707-4720. 

Fu, L.-L., and D. B. Chelton , 1984: Temporal variability of the 

Antarctic Circumpolar Current observed from satellite altimetry. 

Science, 226, 343-346. 

Fu, L.-L., and D. B. Chelton, 1985: Observing large-scale temporal 

variability of ocean currents by satellite altimetry; with 

application to the Antarctic Circumpolar Current. J. Geophvs. 

Res., 90, 4721-4739. 

Schroeder, L. C., D. H. Boggs, G. Dome, I. 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. GeoDhvs. Res., 87, 

3318-3336. 

TOPEX Science Working Group, 1981: Satellite altimetric measurements 

of the ocean. Doc. 400-111, Jet Propul. Lab., Pasadena, Calif. 
18 

Weller, R. A., R. E. Payne, W. G. Large, and W. Zenk, 1983: Wind 

measurements from an array of oceanographic moorings and from F/S 

Meteor during JASIN 1978. J. GeoDhvs. Res,, 88, 9689-9705. 

Wentz, F. J., S. Peteherych, and L. A. Thomas, 1984: A model function 

for ocean radar cross sections at 14.6 GHz, J. GeoDhvs. Res., 

- 89(C3), 3689-3704. 
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 90. NO. C3. PAGES 37074720. MAL X. 1985 

19 

A Review of Satellite Altimeter Measurement of Sea Surface Wind Speed : 

With a Proposed New Algorithm 

DUDLEY B. CHELTON 

College of Oreanopraphy. Oregon State Uniwrsity. Corrallis 

PATRICK J. MCCABE 

Jet Propulsion Lahoratorj, Pasadena, Calijornia 

The scheduled February 1985 launch of a radar altimeter aboard the U.S. Navy satellite Geosat has 

motivated an in-depth investigation of wind speed retrieval from satellite altimeters. The accuracy of sea 

surface wind speed estimated by the Seasat altimeter is examined by comparison with wind speed 

estimated by the Seasat scatterometer. The intercomparison is based on globally distributed spatial and 

temporal averages of the estimated wind speed. 11 is shown that there are sysiematic differences between 

altimeter and scatterometer wind speed estimates. These differences are traced to errors in the Seasat 

altimeter geophysical data record wind speed algorithm. A new algorithm is proposed which yields 

consistent estimates from the two satellite sensors. Using this new algorithm, the rms dinerence between 

spatial and temporal averages of the two wind speed estimates is less than 1 m/s, and their correlation is 

greater than 0.9. 

1. IKTRODUCTION 

Satellite radar altimeters were developed principally for 

measurement of the sea surface elevation from which the general ocean circuiation and temporally varying ocean currents 

can be studied (see Fu 119831 for a recent review). Altimeter 

data can also be used to infer the ocean surface significant 

wave height and wind speed from an examination of the shape 

and amplitude of the return radar signal [see Fedor arid 

Brown, 19821. For wind measurements, a satellite scatterometer is generally more useful since it mezsures the wind speed 

more accurately and also provides information about the wind 

direction. Wowever. it will be at least 5 years before the next 

sateliit- scatterometer is launched. Since an altimeter is scheduied for February 1985 launch on board the US. Xavy satellite Geosat it is useful to examine the accuracy of wind speeds 

estimated from radar altimerry. 

To date, three satellite altimeters have orbited the earth. 

Tne first was camed on board Skylab in 1973. The noise level 

of this firs: altimeter was too hi@ for the data to be of any 

vaiue for wind speed estimation. GEOS 3. launched in April 

1975, camed the first altimeter capable of estimating sea suriace wind speeds. The GEOS 3 altimeter observed the ocean 

suriace for 3 1.1 years until December 1978. An improved 

zlrimtter was launched on S:asat in July 1978. The Seasat 

zlrimeter operated until October 1978 when a power failure 

‘orought the mission to an unfortunate premature end. In this 

?EFT. we reviex wind speed remeval from the Seasat alrimeer and evaiuate the aigorithms for possible implementation 

on Geosat 

Seasat camed three microwave sensors capable of remotely 

r;.,:asurinp v;inds at the sea surface. Two of these, the scatterometer (S.4SS) and altimeter (ALT), were active radars which 

beamed a pulse of microwave radiation at the sea surface and 

ntvured the power of the backscattered radiation. Wind 

speed estimation from SASS and ALT is based on the prinCopyright 1985 by the American Geophysical Union. 

Paper number 4C1478. 

01 4P-Ol~ijSj/Ou?C-l?7S~Oj.~ 

ciple that backscattered power depends almost exclusively on 

short-wavelength roughness of the sea surface, which itself is 

dependent on wind speed. The third sensor, the scanning multichannel microwave radiometer (SMMR), was a passive radiometer which measured the power of the natural radiation 

emitted from the sea surface and intervening atmosphere. By 

correcting for atmospheric effects. the power of the residual 

radiation can be converted to the “brightness temperature” of 

the sea surface which is, in part. a function of wind speed (see 

Swifr [1980] for a summary). The data from these three sensors have been processed to geophysical data records (GDR’s) 

by the Jet Propulsion Laboratory. The wind speed algorithms 

are purely empirical and were developed using entirely different “calibration” data for each sensor. If the various algorithms have been derived appropriately, there should be no 

systematic differences in the wind speed estimates from all 

three sensors. In this paper, we compare ALT and SASS wind 

speed estimates in order to assess the accuracy of the corresponding algorithms. 

Most oceanographic and atmospheric applications of sea 

surface winds require knowledge of the vector wind field (both 

speed 2nd direction). For these applications, wind speed estimates from altimeters may not be useful. However, for studies 

of latent heat fiux, knowledge of only the wind speed is sufficient [Liu and Niiler, 19841, and ALT estimates of wind 

speed could be very useful. Recent evidence [Halpern and 

Knox, 19831 that cloud mo::ons can be used to infer the wind 

direction at the sea surface (21 least in tropical regions) suggests another potential application of ALT data. Vector surface Hinds could be determined by comjining altimeter estimates of surface wind speed with surface wind direction inferred from cloud motions. It is therefore useful to examine 

ALT wind speed estimates in some detail in order to evaluate 

the accuracy of present algorithms. This is particularly important in view of the fact that, at least for the next several years, 

the Geosat altimeter is the only hope for obtaining global sea 

surface wind data. 

There are a number of methods of determining the accuracy 

of satellite estimates of wind speed. The approach generaliy 

.. , .., 

I. . -.> 

4707 
20 CHELTOX AND MCCABE: hATELLlTE ALTIMETER MEASUREMENT OF SEA SURFACE ’H.1h.D SPEED 

- 

. 93 

E 

0 

ul 

P 

5 

Fig. 1. 

Antm 

10 

v w 180’ 27v 360. 

Wind Dnrclmn Rebtivc lo Forward Bean Antenna 

Schematic diagram showing the possible vector -. wind solulions from a forward andaft antenna pair with pointing angles xparated by 90’. For noise-free measurements. the true wind vector is one of the four intersections of the two curves. 

taken in the past has been to compare satellite backscatter 

measurements with high-quality surface “truth” measurements. Since exactly coincident observations (in both space 

and time) seldom occur, measurements within 100 km and 1 

hour are usually considered “coincident.” A fundamental dimculty is that satellites and in situ instruments sample very 

difierent characteristics of the same wind field. Satellites measure wind speed averaged over a finite spatial footprint at an 

instant in time. The size of the footprint varies, ranging from 

less than 10 km for ALT to 50 km for SASS. In comparison, 

conventional anemometers measure the turbulent wind field at 

a single point in space. Much of the inherent turbulent variability (which is generally not of interest to large-scale studies) 

can be removed by averaging over time. For example, to compare a satellite wind speed measurement of 8 m s-’ over a 

50-km footprint with an anemometer measurement of the 

same wind field, the anemometer record must be averaged 

over something like lo0 min (the time required for a Lagrangian particle in a typical 8 m s-l synoptic wind field to 

travel 50 km). Except in carefully conducted experiments, this 

atering of anemometer data is rarely done. 

Wenrr et a!. [1982] proposed another method of determining the accuracy of satellite estimates of wind speed. They 

compared instantaneous estimates of wind speed from different satellite sensors. To obtain coincident measurements, the 

analysis was restricted to satellite nadir (the point directly 

beneath the orbiting satellite). The results suggested that ALT 

wind spuds were biased low relative to .nadir SASS wind 

speeds. However, nadir SASS winds are not nearly as reliable 

as off-nadir SASS winds. In pan this is because the algorithm 

for nadir SASS wind speeds was developed by a circuitous 

method that did not involve any direct comparisons with in 

situ data (see Went: et ai. [19S2] for a summary). Since offnadir SASS wind speeds are 200 &:or more from sateliite 

nadir, coincident observations with ALT measuremenu do 

not exist 

An alternative method of determining the accuracy (or at 

least the consistency) of wind speeds measured from sateliites 

is to compare spatial and temporal averages (maps) of wind 

speed from the different sensors. This is the method used in 

this paper. The advantage of this approach is that it does not 

require coincident observations and therefore yields 2 far 

greater number of comparisons. The obvious iimitation is that 

if the winds are highly variable, this method requires a uniform sampling of the spatial area over the averaging time 

interval to be meaningful. 

Among the various satellite microwave wind speed estimates, 05-nadir SASS measurements have received by far the 

greatest attention. Summaries of the evolution of the alporithm and the calibration data used to derive sea surface wind 

speed from SASS backscatter measurements can be found in 

the works by Boggs [1981] and Schroeder et a!. [19S2]. The 

off-nadir SASS wind speed algorithm is based on extensive 

aircraft scatterometer measurements prior to the launch of 

Seasat and aircraft underfiights during the Seasat mission. In 

addition, the off-nadir SASS wind speed algorithm incorporated orders of magnitude more direct comparisons between 

satellite and in situ measurements than either the ALT or 

SMMR algorithms. It is therefore reasonable to assume that 

SASS wind spccd estimates are the most accurate satellite 

measurement of wind speed. In this study, we use off-nadir 

SASS wind speeds as calibration data to examine the accuracy 

of ALT wind speed estimates (section 3). The two estimates of 

wind speed are found to differ significantly. To determine 

whether the differenas can be attributed to errors in ALT 

estimates, the Seasat ALT GDR wind speed algorithm is 

examined in detail in section 4. It is found that errors and 

weaknesses exist in the present algorithms. The errors are corrected and an improved algorithm is proposed in section 5. 

Wind speed estimates using the new algorithm agree quite 

well with SASS wind speeds. 

2. RADAR MEASUREMENTS OF SEA SURFACE WIND SPEED 

In order to understand the similarities and differences between ALT and SASS wind speed measurements, we include 

here a brief summary of wind speed estimation from satelliteborne radars. More detailed discussions can be found in the 

works by Moore and Fung [1979] and Barrick and Swift 

[1980]. Both ALT and SASS operated in approximately the 

same microwave frequency range (13.5 GHz for ALT and 14.6 

GHz for SASS). The nature of returned radar power depends 

strongly on the incidence angle of the radiation, defined to be 

the angle measured in a vertical plane between satellite nadir 

and the pointing angle of the radar. At small incidence angles 

(less than !Go) the backscattered radiation results piimarily 

from specular reflection from ocean waves having wavelengths 

longer than the incident radiation (about 2 cm for ALT and 

SASS). As the wind speed increases, the sea surface roughness 

increases and a greater fraction of the incident radiation is 

reflected away from the satellite. Thus, at incidence angia 

near nadir, the power of the backscattered radiation is inversely related to wind speed but independent of wind direction. 

Both ALT and SASS provided measurements of wind speed 

in rnis small incidence angle regime. Wind speed measurements in this near-nadir regime are attractive from the point 

of view that a long, continuous 3 1;’2 year record of global 

6C.N 

40. 

20. 

160- w 140. 120’ 100. 

Fig. 2. Map of NDBO buoy locations during summer 1978. 

-. 

.. 

,‘ 

. : 

,- 
21 

55.5 TO 65.N 

0.E TO 360.E 

CHELTO"; AND MCCABE: SATELLITE ALTIMETER MEASUREME~T OF SEA SL~RFACE ~IND SPEED 

nadir wind speed is available from the GEOS 3 altimeter 

(April 1975 to December 19781. In addition, altimeter wind 

speds will soon be available from Geosat. The limitations are 

that (1) the sensitivity of near-nadir radar backscatter to wind 

speed is relatively weak so that small errors in backscatter 

measurement can lead to sizable errors in wind speed, (2) no 

information on wind direction can be obtained from nearnadir backscatter measurements, and (3) the spatial coverage 

of near-nadir measurements is small. For Seasat orbiting at an 

altitude of 800-km, f8' of incidence angle covers an area 140 

km wide centered on the satellite ground track. For the scatterometer this area is broken up into approximately 50 km 

cells by using doppler filtering of the backscattered radiation. 

For the altimeter, which measures backscatter from only a 

single cell centered at satellite nadir, the cross-track footprint 

size ranges from 2 to 10 km, depending on sea state. 

At incidence angles ranging from 10" to 15", there is no 

strong dependence of backscattered power on either wind 

speed or wind direction. Thus no wind speeds can be retrieved 

from satellite radar measurements in this range of incidence 

an g 1 e s . 

At incidence angles greater than lY, few of the longer wave 

faces are favorably oriented for specular reflection. and the 

shorter ocean wave crests act analogous to a diffraction grating. The radar selectively samples resonant backscatter from 

ocean waves with wavelength governed by the Bragg scattering equation. At 30" incidence angle. the Bragg resonant ocean 

wavelengths are approximately 2 cm for the 14.6-GHz SASS 

radar. The amplitudes of these short capillary-gravity waves 

are found to increase with increasing-wind speed, and the sea 

surface becomes more effective at scattering incident radiation. 

Thus the power of the backscattered radiation in this off-nadir 

regime increases with increasing wind speed. The sensitivity of 

backscattered power to wind speed improves with increasing 

incidence angle so that wind speed estimation is less sensitive 

25 

20 

h 

n 

\ 

E 15 " 

0 

w 

w 

L 

ln 

10 2 

f 

3 

ul 

v) 

<5 

ln 

.. ,! , .,. ..(.. I.,,. 

0 5 10 15 20 25 

BUOY WIN3 SPEED (m/s) 

Fig. 3. Scatter plot comparison of coincident 19.5 m vertically 

poianzed, ofi-nadir SASS esrimzted and NDBO buoy measured wind 

speeds. For a given orbit, all vertically poianzed, off-nadir SASS observations within 100 km and 1 hour of a buoy measurement have 

been averaged. Data from orbits where the range of SASS wind 

speeds within the 100 km window exceeded 3 m s-l have been excluded. SASS wind speeds are biased 1.06 m s-' high over these 826 

observations and the rms error aboui the bias is 2.03 m s-'. 

GLOBAL OCEAN 

1947 085 

16 

to errors in backscatter measurement than at satellite nadir. 

However, the total returned power decreases with increasing 

incidence angle. As a consequence. wind speeds in excess of 10 

m s-' are required at large incidence angles (greater than 55') 

in order to achieve adequate signal to noise ratio with the 

Seasat SASS. 

In the off-nadir regime, the backscattered power is dependent also on wind direction. The returned power is maximum 

when the antenna is pointed upwind or downwind and minimum when the antenna is pointed crosswind. This anisotropic 

behavior varies approximately as the cosine of twice the angle 

measured in a horizontal plane between the wind direction 

and the antenna pointing angle (see, for example. Moore and 

Fung [1979]). There is also a small contribution which varies 

as the cosine of the angle between wind direction and antenna 

pointing angle (which leads to a slight asymmetry between 

upwind and downwind backscatter for a gven wind sped). 

For a single backscatter measurement, the possible vector 

wind solutions thus lie along a quasi-harmonic curve in wind 

speed-Kind direction space. Tiis is sbown schematically in 

Figure 1. A second antenna measurement from a difier:nt 

viewing angle (optimally 90" separation from the first antenza) 

defines a second quasi-harmonic curve of possible vector wind 

solutions. For noise-free measurements, the true wind vector 

must lie at one of the intersections of the two curves 

The Seasat SASS carried forward and aft antennas oriented 

45' and 13Y, respectively, relative to the satellite ground 

track. A pair of such antennas was mounted on each side 0: 

the spacecraft. A fan beam antenna design was used to obtain 

off-nadir backscatter measurements over an incidence angle 

range from 25" to 55" (corresponding to a 500-km-wide swath 

on each side of the spacecraft, ceniered 450 km from satellite 

nadir). In each of these off-nadir regimes (port and starboard 

side of the spacecraft), 12 Doppler filters and range gares were 

used to subdivide electronically the illuminated SASS footprint into separate resolution cells and to exclude interfering 

signals from side lobes of the antenna. The doppler filter 

bandwidths were designed to give approximately 50-km crosstrack cell spacing, and the measurement timing was set to give 
22 CHELTO\ ASD hIcC~.ur: SATELLITE ALTIMETER MEASUREMEST OF Sw SCRFACE Wiw SPEED 

I 3. SEASAT ALT AND SASS WIND SPEED COMPARlSOh 

For reasons discussed previously. vertically polarized, offnadir SASS measurements are presently the most reliable remotely sensed estimate of sea surface wind speed. It is therefore reasonable to use these data as a basis for comparison 

with ALT data. In an> event, there is no alternative highquality wind data base which could be used to derive a wind 

speed model function for ALT. In order to evaluate the accuracy of the SASS winds, we compared them with in situ 

measurements from 19 National Data Buoy Oflice (NDBO) 

0 

0 5 IO 15 20 

195m Wind Speed (m/d 

Fig. 5. Histogram of 19.5 m wind speeds computed from approximakly 3 million observations during the full 96day Seasat mission 

using the Seasat GDR algorithms. Bin size is 0.25 m s-'. 

approximately 50 km along-track cell spacing. A forward antenna measurement was combined with an aft antenna 

measurement at the same location to form a paired antenna 

multiple wind vector solution like that shown in Figure 1. The 

time separation between forward and aft antenna measurements at the same location varied from about a minute at 25" 

incidence angle to about 4 min at 55' incidence angle. 

The schematic in Figure 1 shows the possible wind direction 

solutions relative to one of the antenna pointing angles (in this 

case, the forward antenna). Note that the vector wind solution 

from the two antennas is not unique; there are four possible 

solutions (the four intersections of the two quasi-harmonic 

curves). For more general orientation of wind direction relative to antenna pointing angle, the number of possible vector 

wind solutions varies from two to four. 

The ambiguity in vector wind solution is a serious impediment to most applications of SASS winds. A methodology has 

been proposed for subjective selection of the true solution 

[JVurtele et nl., 19821. The technique is very labor intensive 

and nonautomated. To date it has been applied only to 2 

weeks of global SASS data in September 1978 (P. M. Woiceshyn, personal communication, 1984). However, for purposes of comparison with ALT wind speed estimates, the ambiguity in SASS solution is not a problem. Although the 

multiple solutions differ widely in direction, they differ very 

little in speed (see Figure 1). Thus the speeds corresponding to 

the multiple solutions can be averaged to obtain an estimate 

of the true wind speed. This procedure has been applied to the 

SASS winds used in this paper. 

One final characterisric of SASS data should be noted. The 

microwave radiation trmmitted and received by the SASS 

antenna can have either vertical or horizontal polarization. 

However, in the off-nadir regime, the power of the measured 

backscatter for a fixed wind speed is greater for vertical than 

for horizontal polarization. The difierence increases with increasing incidence angle [see Jones e! al., 19773. Thus vertical 

polarization gives a somewhat greater signal to noise ratio 

and consequently a gentrally more reliable estimate of wind 

speed. (This is not always true since given an adequate signal 

to noise ratio, horizontally polarized backscatter is more sensitive to changes in wind speed than vertically polarized backscatter.) In addition, 75% of all Seasat SASS off-nadir Kinds 

were constructed by pairing vertically polarized forward and 

aft antenna measurements. Since the SASS wind speed algorithm was based predominantly on comparisons between vertically polarized SASS measurements of backscatter and in 

situ measurements of wind speed, vertically polarized SASS 

winds are more reliable than other polarization combinations. 

R'e therefore restrict the SASS winds used in this study to 

vertically polarized, of-nadir data. 

buoys scattered around the coast of North America (Figure 2). 

During 1978, the winds were measured by NDBO buoys at 1-s 

intervals for 8.5 min. The average over the 8.5-min interval 

was then reported every 3 hours (with approximately 25% of 

the buoys reporting hourly). 

For purposes of comparison with vertically polarized, offnadir SASS wind speed estimates, the NDBO buoy and SASS 

data sets were searched for matches within 100 km and 1 

hour. A scatter plot comparison of the two measures of the 

wind speed is shown in Figure 3. Figure 3 suggests that there 

is a 1 m s-' bias in the SASS wind speed estimates. This bias 

has been independently discovered by Wentz et al. [1984]. It 

can probably be attributed to the fact that the final SASS 

wind speed algorithm was heavily tuned to a particular buoy 

wind recorder in the JASIN experiment. Postexperiment Calibration of this recorder showed that it was overestimating the 

wind speed by about 10% [Weller et al., 19831, corresponding 

to a 1 m s-l error for the 10 m s-' winds typically observed 

in JASIN. This error was not confirmed until after the GDR 

processing of SASS data. Aside from the simple I m s-' bias, 

in Figure 3, the agreement between SASS and NDBO buoy 

winds is quite good. Much of the scatter is undoubtedly due to 

the general difliculty in making accurate wind measurements 

from a buoy [see Weller er al., 19831 and to the differences 

between an instantaneous measurement over a finite footprint 

and a temporally averaged measurement at a point. 

.4lthough it is admittedly an ad hoc "correction," we feel 

there is strong justification for removal of a 1 m s-' bias from 

the SASS GDR wind speeds. The range of uncertainty about 

this 1 m s-l bias is dillicult to assess pien such a limited 

suriace 'truth" comparison data set, but a 1 m s-' bias removal is certaintly an improvement over using the raw GDR 

SASS wind speeds with no bias removed. Therefore, a 1 m s-l 

bias has been removed from all of the SASS vrind speeds used 

in the analysis that follows. 

Ideally, we would like to examine the accuracy of the ALT 

wind speed algorithm by a similar direct comparison with 

high-quality in situ wind speed measurements such as the 

NDBO buoy winds. However, a search for NDBO buoy observations within 100 km and 1 hour of ALT measurements 

resulted in only 14 independent buoy observations. This small 

number of matches is due to the small ALT footprint size and 

.- 

.- 

I. 

: - L 

I . .. 

p 21 

Orbit 145, 7 July 1984 

I, a,,, 

Oi8W ' 190 ' 2w ' 21. A' ' ' ;4- ' 250 

LATITUDE 

Fig. 6. Wind speeds measured over a 2-min period during descending orbit 145 over the central Pacific (approximately 16O'Wj on July 7, 1978. 

._ 
23 

CHELTOK AND MCCABE: S4TELLITE ALTIMFTEK kiEASUREME9 OF SEA SURFACE WIND SPFXI) 

the fact that ALT measures only b; satellite nadir (as opposed 

to SASS which measures over two 500 km swaths in the offnadir regime). Clear]). 14 independent measurements is too 

few for any meaningiul comparison. 

This lack of coincident satellite and buoy measurements is a 

long-standing problem in satellite wind speed algorithm development. To circumvent this probiem, we propose a neu 

method of calibrating ALT wind measurements. Rather than 

comparing with in situ measurements, we suggest comparing 

ALT measurements with vertically polarized, OK-nadir SASS 

measurements (corrected for the 1 m s-' bias discussed 

above). Since the ALT nadir samples are 200 km from the 

nearest off-nadir SASS samples, we clearly cannot compare 

instantaneous measurements by the two sensors. We propose 

comparing spatial and temporal averages of ALT and SASS 

wind speeds. Because of their different sampling characteristics, a given geographical region is sampled at different times 

by ALT and off-nadir SASS. If the winds are steady and constant over the geographical region. this time separation causes 

no problems. However, for variable winds, the samples by 

each sensor must be temporally averaged. The temporal 

average must be long enough so that the variable winds are 

sampled frequently enough to obtain a reliable estimate of the 

mean wind speed over the averaging period. For the Seasat 

mission. the maximum possible averaging period is 96 days. 

After experimenting with different spatial averages, we subjectively chose 2' of latitude by 6" of longitude as a trade-off 

between reliable spatial averages (limited by the small footprint size of individual ALT measurements) and number of 

ALT and SASS intercomparisons. 

Use of this proposed method of calibration is obviously 

dependent on the accuracy of SASS wind speed estimates. 

This is clearly a rather serious limitation since the accuracy of 

SASS winds has not yet been demonstrated over a broad 

range of conditions. In addition, our 1 m s-' ad hoc correction is a little disturbing (although justifiable in view of known 

problems with JASIN data used to calibrate the SASS wind 

speed algorithm). However, it is almost certainly true that the 

errors in SASS wind speed estimates zre no worse than errors 

in wind speed estimates by conventional measurement techniques. Since the wind speed comparison is based on spatial 

and temporal averages of a large number of individual observations, random errors in indi\idual SASS wind speed estimates v;iil be either eliminated or greatly reduced. Systematic 

TABLE 1. Loss in Backscattered Powei Due to honzero Attitude 

h.@e of Spacecraft 

Attitude, dep , L,,,. dB 

0.00 

0.05 

0.10 

O.! 5 

0.20 

0.25 

0.30 

0.35 

0.4 

0.45 

0.50 

0.55 

0.0161 

0.0391 

0.10E; 

c.23 1 

0.31(;2 

0.5914 

0.8449 

1.1445 

1.4904 

1.8826 

2.213 

2.8066 

0.60 3.3386 

0.65 3.9 178 

0.70 4.5430 

0.75 5.2 158 

TABLE 2. Calibrated AGC as a Function of Attenuator Step lor 

Calculation of crc ~~ ~ 

Attenuator Step k A,. dR 

8 

9 

10 

11 

12 

13 

14 

15 

16.55 

21.15 

30.30 

35.67 

32.27 

48.07 

54.52 

60.58 

errors, such as cross-track biases in SASS wind speed estimates, will also be greatly reduced since the spatial and temporal averages include individual wind speed estimates over 

the full SASS incidence angle range. 

In any event, the proposed calibration method will produce 

wind speed estimates which are at least consistent between 

sensors. It will be seen later than the new ALT wind speed 

algorithm proposed in this paper is an improvement on the 

existing Seasat GDR algorithm. It should be born in mind, 

however, that any changes in the SASS wind speed algorithm 

at a later time may require adjustments to the new ALT wind 

speed algorithm. A rigorous test of the proposed algorithm 

will require an independent comparison with a large number 

of high-quality in situ measurements. This will be possible 

with the Geosat altimeter, as a carefully coordinated surface 

measurement program is being planned to provide coincident 

satellite and in situ measurements. 

A scatter plot of 96-day averages of ALT versus SASS wind 

speeds over nonoverlapping 2" by 6" areas is shown in Figure 

4. ?nese 2" by 6' areas are globally distributed from latitude 

6Yh to 55"s. The southern boundary of the averaging region 

was chosen to eliminate any measurements over ice. For wind 

speeds greater than 5 m s-', Figure 4 suggests a systematic 

underestimate of wind speed by ALT. On the surface, this 

underestimate appears to be a very simple error, and one 

might be inclined to apply an ad hoc correction to the ALT 

wind speeds to make them agree with SASS wind speeds. 

Howeve;. a more detailed investigation reveals that this apparently simple discrepancy between ALT and SASS Hind 

speeds is actually symptomtic of rather serious problems with 

the Seasat ALT GDR wind speed algorithm. 

A histogram of globally distributed individual ALT wind 

speed estimates is shown in Figure 5. The histogram shows a 

peculiar bimodal distribution of Rind speeds. This bimodal 

structure has previously been noted by Chelron er nl. [19Sl], 

who speculated that it could be a true geophysical phenomenon attributed either to geographical inhomogeneities in the 

wind field or to temporal evolution of the wind field over the 

96-day Seasat mission. However, subsequent analysis showed 

that the bimodal structure was present in all geographical 

regions and during all periods of the Seasat mission. In fac: 

examination of raw data along satellite ground tracks revealed 

abrupt toggiing between about 4.5 and 8. m s-l (Figure 6). 

These abrupt changes cannot be geophysical and provide 

rather clear evidence for problems somewhere in the algorithms used to retrieve wind speed from ALT. In tht next 

section. the ALT wind speed algorithm is examined in detail 

to identify the source of this behavior. 

4. SEASAT ALT GDR WIND SPEED ALGORITHMS 

As discussed previously, ALT is an active radar which 

transmits I3.5-GHz microwave radiation and measures the 
24 

TABLE 3. Calibration Attenuator Value for Calculation of bo 

Using Either the Seasat GDR or Fedor Algorithms 

Seasat GDR Fedor 

x. C.. dB C'. dB 

F 

9 

10 

11 

12 

13 

14 

15 

43.7 

36.1 

30.3 

24.2 

18.2 

12.2 

6.2 

0 

43.9 

36.3 

30.15 

24.78 

18.18 

12.38 

5.93 

-0.13 

power of the radiation backscattered from the sea surface. In 

order to account for variations in the power of the transmitted 

signal, the power of the return signal is normalized by the 

transmitted power. This normalized return power is called the 

normalized radar cross section which is usually denoted by d'. 

For ALT measurements at satellite nadir, no is inversely related to wind speed (see section 2). Estimation of wind speed 

from ALT is thus a two-step procedure. First, no is computed 

from parameters measured by the ALT receiver and from the 

measured height and attitude angle of the satellite. Then the 

wind speed is computed from (I' using an empirically derived 

model function. Errors in ALT wind speed estimates could 

therefore be due to problems in either the uc algorithm or the 

wind speed model function (or both). These two steps in the 

retrieval of wind speed are examined separately in this section 

to identify the causes for errors in ALT wind speed estimates. 

4.1. The GO Algorithm 

In order to maintain a constant output level from the Seasat 

ALT receiver and to operate the electronics within the linear 

response region of all receiver stages, an automatic gain control (AGC) loop was implemented in the electronics package. 

The power of the return radar signal was thus directly related 

to the AGC setting required to maintain the receiver output at 

the desired level. The algorithm used to convert Seasat ALT 

AGC to uc is britfiy summarized in this section. A more detailed technical discussion can be found in the work by Townsend [1980]. 

The AGC was controlled by a digital step attenuator which 

provided gain variation over a 0-63 dB range in I-dB steps. 

For uc values ranging from 10 to 15 dB (a range which in96-day Seasat mission), the AGC setting ranged from 28 to 33 

dB. .4GC measurements were recorded at 0.1-s intervals. Each 

AGC value was scaled and added to the previous AGC value 

to obtain an averagc AGC which was stored on the telemetry 

data tape for later transmission to a :ground-based receiving 

station. Thus, while the .4GC gates were separated by I-dB 

steps. the smoothed values received at the ground station had 

finer resolution. This find AGC was stored on tape with a 

least count of 1/16 dB. The integer and fractional parts of 

AGC were stored as separate data words in the telemetry 

string. These AGC values were further smoothed when 10 

consecutive measurements were block averaged before converting to ue (see discussion below). 

The normalized radar cross section G' was computed by a 

rather complex algorithm. The primary contribution to 0' 

was, of course, determined by the value of AGC. However, the 

AGC contribution must be corrected for loss from variations 

in the pointing angle of the antenna due to pitch, roll. and yaw 

of the sateliite. The reduction of o0 is given as a function of 

cl.4 ruues 80% of sl! GO va!ua meuured globally during the 

satellite attitude angle in Table 1. A change in attitude of only 

0.5' translates to a G' loss of 2.3 dB. which is quite substantial. 

A search of the data determined that the average attitude 

angle during the 96-day Seasat mission was 0.28". In addition 

to the correction for antenna pointing angle, G' must also be 

corrected for variations in the height of the satellite above the 

sca surface. These height variations are due to a small eccentricity about a circular orbit. 

The Seasat ALT periodically executed an internal calibration during which a calibrated portion of the transmitted 

pulse was diverted into the antenna receiver for the purpose of 

AGC calibration and stability monitoring. It was found that 

incorporating internal calibration mode AGC values into 

computation of uo eliminated any residual effects of transmission power changes and variations in receiver gain from 

changes in ambient temperature in the electronics package. 

Briefly, the internal calibration data were incorporated in the 

G' algorithm as follows. First, the 0.1-s AGC values were 

block averaged over 1 s. This smoothed AGC value (call it A,) 

was compared with the values A, in Table 2 which correspond 

to eight of the stepped calibration values fed to the receiver 

during internal calibration mode. The operating region corresponding to the ocean AGC measurement was determined by 

identifying the value A, for which the absolute value (A, - A,) 

was minimum. Define this value of k to be k. Then uo was 

computed by 

' 

. 

G' = B - Ci - AL + L,,, + 30 log,, (hjh,) 

where 

B bias, determined from prelaunch calibration measurements to be 39.93 dB for the GEOS 3 ALT: for the 

Seasat ALT, this bias was estimated to be 38.33 dB (see 

section 4.2); 

Cf attenuator value for calibration step 1;. obtained from 

the second column of Table 3; 

Ac =Ai - A, where AG is obtained from Table 2; 

correction for loss in G' due to nonnadir antenna pointing angle; value was linearly interpolated from Table 1 

based on known satellite attitude angle; 

h height of satellite above sea surface. km; 

h, reference height of satellite above sea surface (= 796.44 

km). 

La,, 

The authors were unable to determine from the published or 

"gray" literature the motivation for or history of this complicated G' algorithm. This is particularly puzzling in view of the 

fact that fixing f;. = 10 in Tables 2 .and 3 yields an improved 

algorithm (see discussion below). 

To determine whether the G' algorithm above could be responsible for the bimodal wind speed distribution in Figure 5, 

we constructed a histogram of Seasat to values computed 

t 

1 ,I 

Oi 11 

8 'Io"i 12 14 16 

uo (dB) 

Fig. 7. Histogram of G~ computed from approximately 3 million 

observations during the full 96-dag Seasal mission using the Seasat 

CDR algorithm. Bin size is 0.1 dB. 
CHELTOh AND MCCABE: SATELLITE ALTIMEER MEASUREME~T OF SEA SURFACE WIND SPEED 

using the algorithm described above. Figure 7 shows a pronounced bimodal structure. suggesting that the d algorithm is 

a likely candidate for the bimodal behavior of the wind 

speeds. A more detailed investigation found that the bimodal 

u' distribution was evident in all geographical regions and at 

all times during the Seasat mission. 

The cause for the bimodal oC distribution becomes immediately apparent when the AGC contribution to oC is converted 

from tabular to graphical form (Figure 8). To isolate the AGC 

contribution, the attitude angle and satellite height were fixed 

at 0.28" and 800 km. The functional relationship between crc 

and AGC is discontinuous at 27.3, 33.0. and 39.0 dB values of 

AGC. Note that the AGC contribution cannot produce any u' 

values between 11.2 and 12.0 dB because of the discontinuity 

at 33.0 dB AGC. This coincides with the valley in the go 

distribution in Figure 7. The sharp cutoffs in uo values as 

implied from the AGC contribution get smoothed by the satellite height and attitude angle corrections discussed above. 

Clearly, these discontinuities are undesirable attributes of 

the u' algorithm used in Seasat GDR processing. These discontinuities have previously been discovered independently by 

D. Hancock (personal communication, 1983) and L. Fedor 

(personal communication, 1983). However, the effects of these 

discontinuities on wind speed retrievals have not heretofore 

been fully appreciated. Both Hancock and Fedor modified the 

0' algorithm to be a continuous funtion of AGC. The Fedor 

algorithm is exactly the same as the Seasat GDR algorithm 

except that the attenuator values C, in column 2 of Table 3 

are replaced with the values given in column 3. The Hancock 

uc algorithm fixes L = 10 and uses the Seasat values of Af and 

Ck in Tables 2 and 3. These two modified uo algorithms are 

shown together in Figure 9; they differ only by a relative bias 

of 0.15 dB. Because of their similarity, there is no strong preference for using one of the algorithms over the other. We 

prefer the Hancock algorithm because it is simpler. The alporithm for G' then becomes 

(ic = B - 60.6 + .4, + La,, + 30 log,, (hlh,) 

The Seasat G' values were recomputed from AGC, satellite 

height, and attitude angle using the Hancock algorithm. A 

histogram of the resulting corrected G' values is shown in 

Figure 10. The bimodal distribution is no longer present and 

the G' values are approximately Gaussian. We have no way of 

proving that this distribution is "correct," but it certainly 

2o c 

26 28 30 32 34 36 38 

AGC (dB) 

Flg. 8. Automatic gain control (AGC) contribution to G' in the 

Seasat GDR algorithm. 

t 

25 

AGC (dB) 

Fig. 9. Hancock (solid line) and Fedor (dashed line) algorithm for 

AGC contribution to 0' 

seems more reasonable than the bimodal distribution in 

Figure 7. 

Using corrected uc values, we recomputed wind speeds 

Using the Seasat ALT GDR wind speed model function (see 

section 4.2). A histogram of the resultant wind speeds is shown 

in Figure 11. It is evident from Figure 11 that the distribution 

of ALT wind speeds still exhibits rather peculiar behavior. 

There are abrupt increases in the wind speed distribution at 

about 7.8 and 10.0 m s-'. These features cannot be due to 

problems with the corrected uc algorithm since the GO distribution is so smoothly varying (Figure 10). The problems must 

therefore lie in the wind speed model function. 

4.2. Wind Speed Model Ftrnction 

The roots of the ALT wind speed model function used in 

the Seasat GDR processing can be traced back to early studies 

of the relation between the mean square sea surface slope (s2) 

and the wind speed. Cox ond Munk [1954] suggested a simple 

linear relationship based on 23 observations. Wu [I9723 reanalyzed the Cox and Munk data and suggested a two-branch 

logarithmic relation between (s2) and wind speed. One loparithmic relation was proposed for Kind speeds less than 7 m 

s-', and a second logarithmic relation was proposed for wind 

speeds greater than 7 m s-'. This two-branch fit was morivated by the dimensionally derived wave number spectrum 

proposed by Phillips [1966] for a fully developed sea state. 

From this assumed equilibrium spectrum, Wu derived an expression for (?) in terms of a contribution from capillary 

waves and a contribution from longer gravity waves. He conjectured that at low wind speeds, only the gravity waves contribute to (?). while at higher wind speeds, both gravity and 

capillary waves contribute. Wave components in the capillary 

Q' (dB1 

Fig. 10. Histogram of uc computed from approximately 3 million 

observations durins the full 96-day Seasat mission using the Hancock 

modified algorithm. Bin size is 0.1 dB. 

.. 
26 

CHELTOk AND MCCABE: SATLLLITE ALTIMETER IMEASUREMEVT OF StA SUREACE WIND SPEEV 

0 5 10 I5 20 

19.5m Wind Speed h/s) 

Fig. 11. Histogram of 19.5 m wind speeds computed from ap 

proximately 3 million observations dunng the full 96day Seasat mission using the Hancock modified uo algorithm and the Scasat GDR 

Brown el a/. [1981] wind speed model function. Bin size is 0.25 m s-'. 

range have a larger mean square slope than gravity waves, 

and (s2) of capillary waves is much more sensitive to changes 

in wind speed than (s2) of longer gravity waves. Justification 

for the sudden change in regimes at 7 m s-' is not apparent 

from Wu [1972]. It is also noteworthy that the two-branch 

logarithmic fit to the 23 Cox and Munk observations of (s') 

does not appear to be a statistically significant improvement 

over the simpler single branch linear fit. 

Brown [1978] showed that the normalized radar crosssection GO (see section 4.1) for 0" incidence angle microwave 

backscattering IS inversely related to the mean square sea surface slope (?). He proposed a two-branch algorithm of the 

form 

0.617 

n In ul0 + b Qo = 

where ul, is the neutral stability wind speed at 10 m above the 

sea surface. Expressed in terms of decibeis, 

c"(dB) = -2.1 - 10 log,, (n In uIo + b) 

Brown [1979) applied this algorithm to ALT measurements of 

G' from GEOS 3 which collected data from April 1975 to 

December 1978. The GEOS 3 ALT operated at 13.9 GHz but 

was othenise very similar to the Seasat ALT in terms of uD 

measurements. The constants n and b (see Table 4) in the 

expression above were evaluated by least squares analysis of 

39 ship observations and 19 buoy observations of wind speed. 

The 9.2 m s-' branch point was selected after conducting 

numerous numencal expenments. The resulting correlation 

between measured and estimated wind speed was 0.84 and the 

rms difference was 2.6 m s- '. 

Although we have not conducted such an analysis (since we 

do not have the in situ data base used by Brown [19?9]), we 

are confident that it would be difficult to justify the twobranch fit using the usual analysis of variance arguments [e.g., 

Draper and Smirh. 198!). A single-branch algorithm could certainly fit the observations equally well within the bounds of 

statistical uncertainty from the scatter in the data. Brown's 

motive for a two-branch algorithm was evidently the theoretiTABLE 4. Constants in Brown's [1979] Two-Branch Logarithmic 

Wind Speed Model Function 

Wind. Speed. m s-' a b 

< 9.2 0.02098 0.01075 

> 9.2 0.08289 -0.12664 

cal work of Wu [1972]. Note, however, the differences in 

branch points between the Brown and Wu models. 

The reliability of the Brown [ 19793 two-branch algorithm 

was clearly very suspect because of the questionable quality 

and limited quantity of observations used to estimate the parameters n and b. To test further this early model. Brown el al. 

C198l-J compared GEOS 3 measurements of uo with additional high-quality buoy observations of wind speed. A total 

of 184 matches were identified within 110 km and 1.5 hours. 

They found that the parameters in the original two-branch 

model did not adequately fit the expanded in situ data set. In 

particular, in the vicinity of the branch point (9.2 m s- '), wind 

speeds estimated from the two-branch algorithm were somewhat low. Brown et al. therefore conducted another series of 

numerical experiments to obrain a model which yielded lower 

mean square error. They ultimately settled on a three-branch 

algorithm of the same form as the earlier two-branch algorithm. The coeficients n and b evaluated by least squares 

analysis of the 184 buoy observations are given in Table 5. 

As with the earlier two-branch algorithm, it is our opinion 

that the three-branch algorithm is dificuit to justify in view of 

the large scatter in the observations. For example, the range of 

Q' values for wind speeds around 8 m s- was 3 dB. Again, we 

do not have the in situ data base used by Brown et nl. [1981] 

to conduct such an analysis, but we are confident that an 

analysis of variance would conclude that the error variance for 

the three-branch algorithm is, at best, only marginaliy improved over a simple single branch algorithm. 

In order to estimate wind speed from a given measurement 

of go, the three-branch algorithm must be inverted. Using the 

values of n and b in Table 5, the estimated 10 m wind speed is 

u^lo = CXP [(S - b)/a] . 

where 

s = ~0-IsWB)+2.11/10 

Over the 184 data points, the correlation between PI, and the 

observed ul0 was 0.89 with an rms error of 1.86 m s-'. 

Brown et al. [1981] constructed a histogram of the errors 

and found them to be somewhat skewed from a Gaussian 

distribution. They therefore introduced a second stage to the 

wind speed model function which applied a fifth-order polynomial correction to the wind speed estimated from the threebianch model function, 

This polynomial correction was applied only for wind speeds 

less than 16 rn s-'. The coeficients c, determined by minimizing the mean square error are given in Table 6. This polynomial correction succeeded in achie\<ng errors distributed 

mare symmetrically about ztio but did not improvo the corrclation between estimated and observed wind speeds. 

The wind speed model function proposed for GEOS 3 ALT 

TABLE 5. Constants in Brown er aL's 119811 Three-Branch 

Logarithmic Wind Speed Model Function 

Wind Speed, m s-' 8, dB 

c 7.87 > 10.9 0.01 595 0.017215 

7.87-10.03 10.12-10.9 0.039893 -0.031996 

> 10.03 c10.12 0.080074 -0.124651 

, .- 
27 

. 

was thus a two-stage procedure. The wind speed was first 

estimated using a three-branch logarithmic model function. A 

fifth-order polvnomial correction was then applied to obtain 

an improved estimate of wind speed. 

Since the basic functions of the Seasat and GEOS 3 altimeters were the same, Fedor and Brown [ 19821 proposed applying the Brown er al. [1981] GEOS 3 wind speed algorithm to 

the Seasat uC measurements. This required a calibration of uo 

measurements by the two instruments in order to eliminate 

any relative bias. The GEOS 3 mission (April 1975 to December 1978) entirely overlapped the 96-day Seasat mission (July 

7. 1978, to October 10, 1978). The two data sets were searched 

to identify points where Seasat and GEOS 3 ground tracks 

intersected within 1 hour. Owing to data storage limitations, 

GEOS 3 essentially sampled only the northeast Pacific, the 

western Atlantic, and the Gulf of Mexico during the time 

period of the Seasat mission [see Chelron, this issue] so the 

range of environmental conditions available for comparison 

with Seasat data was quite limited. A total of 20 ground track 

intersections was found. One of these was eliminated because 

of an anomalously large discrepancy between Seasat and 

GEOS 3 measurements of 6'. The remaining 19 observations 

covered only a range from 10 to 13 dB (with one observation 

at 15 dB). Over these 19 pairs of GO measurements, Seasat was 

found to be biased high by 1.6 dB with a standard deviation of 

0.37 dB. 

In spite of the questionable reliability of this bias estimate 

because it was based on so few observations and over such a 

limited range of G@, the 1.6-dB bias was incorporated in the 

Seasat ALT wind speed algorithm. The Brown er al. [1981] 

GEOS 3 three-branch logarithmic algorithm with fifth-order 

polynomial correction was then applied directly to the corrected Seasat data using the parameters defined in Tables 5 

and 6. To be consistent with the other wind sensors on Seasat 

(SASS and SMMR), the 10-m wind speeds estimated by the 

GEOS 3 algorithm were converted to a height of 19.5 m. 

Using a neutral stability Mind profile, this results in 

Thus the ALT wind speed algorithm used in the Seasat GDR 

processing was not based on any direct comparison between 

Seasat ALT G' and in situ m:asurements of wind speed. Surprisingly, a comparison of ALT estimated wind speed with 

buoy measurements yielded good results [Fedor nnd Brown. 

1952). This is rathei pu&ig iii vie% of the fact tha: the G' 

values used in the GDR processing are now known to have 

been in error (see section 4.1). 

Ha\+ing established the hisrorical development of the Seasat 

-4LT wind speed aigorithm, we cow wish to determine nhether the algorithm can account for the peculiar behzvior of the 

wind speed histogram in Figure 11. The noteworthy features 

of Figure 11 are sharp peaks at about 8 and 10 m s-' and 

valleys at about 7.5 and 9.5 m s-'. The slopes of the two 

TABLE 6. Coefficients in Fifth-Order Polynomial Correction to 

Brown el al.'s [198l] Three-Branch hgarithmic Model Funciion 

n c, 

1 LO87799 

2 -0.3649928 

3 4.062421 x IO-' 

5 3.258189 x lo-' 

4 -1.9049~2 10-3 

1 I! ! I I I, ! 

8 IO 12 14 16 

uo (dB) 

Fig. 12. The Brown et al. [I9811 three-branch wind speed model 

function with fifth-order polynomial correction. This model function was implemented in the Seasat GDR processing. Insert shows en- largement ol region in vicinity of branch points. 

peaks are much steeper on the left side than on the right side. 

That is, the transitions from valley to peak as wind speed 

increases are very abrupt. 

These abrupt jumps in the wind speed distribution coincide 

exactly with the two branch points in the three-branch logarithmic wind speed algorithm. This algorithm (with fifth-order 

polynomial correction). inverted IO give the 19.5 m wind speed 

in terms of GO, is shown graphically in Figure 12. Note the 

discontinuous derivatives at the two branch points. The probability density function for u19.5 can be expressed in terms of 

the probability density function for GO by 

[see Bendnr and Piersol, 19713. Thus, as long HG') is smoothly 

var5ing (which it is, see Figure IO), then the probability den- 

! @I 55.S TO 65.N 

O'E TO 56C.E 

0 4 B 12 16 20 

SASS WIND SPEED (m/s> 

Fig. 13. Scatter plot of global 96-day. nonoverlapping 2' by 6" 

average G" (computed using the Hancock modified alponthm) as a 

function of SASS wind speed (corrected for a 1 m S-' bias). Conrinu- ous curve shonfs least squares fit to the data (see Table 71. 
28 

CHfLTOh ASV ls4CCABE: SATELLITE ALTIMETER k1LASUREMEST OF SEA SURk4CE WIND SPEED 

TABLE 7. Least Squares Fit Parameters (With 95% Confidence Limits) to z Model Function of the 

Form U' (dB) = 10[C + H log,, ulP.,] 

Number of 

Region 2" hy 6' Averages G H ~ 

Global Ocean 1947 1.502 t 0.020 -0.468 k 0.023 

Pacific Ocean 1062 1.503 5 0.026 -0.473 5 0.032 

Atlantic/lndian oceans 885 1.507 -+ 0.030 -0.471 0.034 

Northern hemisphere 75 1 1.523 5 0.031 -0.499 0.041 

Southern hemisDhcrc 1196 1.505 0.027 - 0.470 k 0.029 

sity of u19,5 is inversely related to the magnitude of the slope 

of the wind speed model function. This accounts for the discontinuities in the wind speed distribution in Figure 11. For 

each branch point uB, the slope of the model function is steeper to the right than to the left of up Hence there is greater 

probability of obtaining u19,5(uB-) than ~19.5(~B *). The larger 

jump in the histogram at 8 m s-' (corresponding to the 10.9- 

dB branch point) is explained by the fact that the change in 

model function slope is larger than at the 10.12-dB branch 

point (see Figure 12). 

We conclude that it is imperative that any model function 

used to compute wind speed from uo must have continuous 

slope everywhere. Thus the Brown et al. three-branch algorithm must be modified. One possibility would be to modify 

the existing algorithm using a smooth polynomial fit to the 

Brown et al. curve. However, since the Brown et al. algorithm 

was derived for the GEOS 3 altimeter and simply implemented on Seasat data (see earlier discussion), there is merit in 

deriving a new wind speed model function independent of past 

models. 

5. A PRoPosm NEW WIND SPEED MODEL FUNCTION 

During the 1960's and early 1970's a number of radar 

measurements of the sea surface were made from aircraft by 

?he Nava! R-esesach Laboratory. These measurements were 

made over a broad range of incidence angles, wind speeds. and 

wind directions. From off-nadir measurements (incidence 

angles greater than IS"), the data suggested a power law relationship berween the normalized radar cross-section GO (in 

real units) and the wind speed [Guinard er a!., 19713. More 

extensive aircraft measurements made by NASA Johnson 

Space Center and NASA Langley Research Center supported 

the empirical power law relationship [Jones and Schroeder, 

19783. A power law relation was therefore adopted to estimate 

wind speed in the ofi-nadir regime from the Seasat satellite 

scatterometer (SASS). 

In the near-nadir regime (incidence angles less than 10'). GO 

also appeared to obey a power 1aw.relation to the wind speed. 

This was appartntly first noted bj Daley er al. [I9733 from 

data collected from the NRL flight program. Barrick [1971] 

found additional support for a power law relation from an 

analysis of aircraft nadir radar measurements made by Raytheon in 1969 and 1970. Consequently, a power law relation 

was also adopted for nadir-regime SASS backscatter measurements. If G' is expressed in decibels the power law relation 

becomes 

G'(dB) = lO[G + H log10 ~19.53 

where u19.5 is the wind speed at 19.5 m above the sea surface. 

For SASS GO, the parameters G and H were estimated from 

least squares analysis of an extensive in situ data base [see 

Boggs, 1981; Scliroeder ef a!., 19823. 

From the discussion in section 4.2, we have shown that 

whatever form is chosen for the ALT wind speed model function, it is important that it be continuously differentiable. We 

propose adopting a model function for ALT with the same 

form as that used on SASS data. Besides being continuously 

differentiable, an advantage of this model function is its simplicity compared with the Brown et a!. [1981] three-branch 

algorithm with fifth-order polynomial correction. 

As discussed previously, the quantity of high-quality buoy 

data is too small to estimate the parameters G and H by direct 

comparison with in situ wind speed measurements (only 14 

independent NDBO buoy measurements within 100 km and 1 

hour of ALT no measurements). We have therefore used the 

spatial and temporal averaging technique of section 3 to estimate G and H for ALT by comparison with vertically polarized off-nadir SASS wind speed estimates. Based on the results 

of section 3, we removed a 1 m s- bias from the SASS data 

prior to comparison with ALT measurements of GO. While we 

are very aware of the limitations of this method of ALT \;ind 

speed model function calibration, we cannot find any viable 

alternative. given the limited in situ data base available for 

intercomparison. Tuning to future altimeter missions will 

clearly require a well-planned in situ measurement program in 

conjunction with the satellite measurements. Such a measurement program is being coordinated for Geosat. 

As in section 3, we chose 2" latitude by 6" longitude and 

96-day averages for comparison of GO from ALT and wind 

speed from SASS. If we use angle brackets to denote temporal 

and spatial average, the proposed form for the wind speed 

model function can be expressed in terms of averages by 

. 

Note that the wind speed enters the expression as (log,, 

Is i 

i 

; .. 

Wind Speed (m/d 

Fig. 14. Comparison of Brown er aL's [19Sl] three-branch wind 

speed model function (thin curve) and the new model function proposed here (heavy curve). The heavy curve is the same as that snow in Figure 13. 
29 

1"'i'''l"' 

NORTHERN UEMISPHERE C 7S1 08s. - 

CHELX)..; AND VCCABE: SATELLITE ALTIMETER h!tASUREMEhT OF SU SURtACE WIND SPEED 

14 m 

m 

U 

0 

v) 

Y 

U 

0 

4 

I 

v 

c, 12, 

: 10. 

1 :\ 

8- 

1 

- 55.S TO 65.N 

120.E TO Zl3O.E 

IIIIIIIII IIII 

\. I 

14 r\ 

m 

D 

0 

v, 

Y 

U 

0 

< 

I 

v 

0 12 

: 10- 

- 

- 

55.5 TO 65.N 

210.E TO 120.E I. 

0 4 8 12 16 

SASS WIND SPEED (m/s) 

16 

14 

12 

10 

8 

16 

14 

12 

10 

8 

1 0.N TO 65.N 

0.E TO 360.E 

0 4 8 12 16 20 

I~~!,~I~,TStI37T 

d SOUTHERN HEMISPHERE 

1 196 08s. 

\ \ 

55.5 TO 0.N 

OWE TO 360.E 

,I I .I 

SASS WIND SPEED (m/s) 

Fig 15. Same as Fig. !3 except dam have been scpaiated inKO four geographical regions: (a) Pacific Ocean (both 

hemispheres): (b) Arlantic and Indian oceans (both hemispheres); (c) northern hemisphere (Pacific and Atianric oceans); (6) 

southem hemisphere (Pacific, Aliantic. and lnaian occans). Continuous curves represent ieast squares fit to the global data 

(see Figure 13). 

u19.5) and not log,, (u,~.~); the constants G and H can be 

camed through the averaging operation, but the logarithm 

cannot since it is a nonlinear operation. 

In our intercomparison, all data within 200 km of land 

boundaries were excluded to eliminate the possibility of land 

contamination in the footprint or side lobes of the radar 

measurements. This 200-km land mask also eliminates much 

of the data over ice (primarily in the southern hemisphere) 

which can also resulr ii. spurious GO measurements. As in section 3, only data between latitudes 65"N and 55"s were included. 

A scatter plot of global 96-day nonoverlapping 2" by 6" 

average ALT co and SASS u19.5 is shown in Figure 13. Thc 

solid line in Figure 13 corresponds to our proposed model 

function. This curve is based on the least squares parameters 

estimated from the global data (see Table 7). The error bars in 

Table 7 correspond to the 95% confidence regions about the 

estimated parameters (computed using standard methods. see, 

for example, Draper nnd Smith [19Sl]). It is evident that the 

proposed model function fits the data quite well. The rms 

error about the least squares fit line is 0.26 dB, and the correlation between cC and the wind speed is 0.92. 

For comparison, the new proposed model function is shown 

in Figure 14 together with the Brown et al. [1981] threebranch model function. The two algorithms are very nearly 

the same for wind speeds less than 10 m sL1. However, for 

higher wind speeds (corresponding to the third branch of rne 

Brown et al. algorithm), the two model functions diverge rap- 

.. -. 

. ,.. 1. 
20 I" 

CLOBXi OCEAN 

1947 OBS. 

h 

n 

\ 

E 

v 

0 

W 

W 

m 

O 

a 

f 

t2 

0 4 B 12 16 20 

SASS WIND SPEED (m/s) 

Scatter plot comparison of 96-day, nonoverlapping 2" by 

6 average wind speeds estimated using the proposed new algorithm 

and SASS wind speeds (corrected for a 1 m s-' bias). Straight line 

represents perfect fit. 

Fig. 16. 

idly. That the two model functions agree at the lower wind 

speeds is comforting since the preponderence of data (over 

90% in both cases) used to derive both model functions fell 

within the 3-12 m s-l range. The descrepancy at higher wind 

speeds is disturbing. However, because of the paucity of observations in this regime, both model functions must be considered suspect at wind speeds greater than 12 m s-'. In 

defense of the model function proposed here, we are aware of 

no physical basis for a sudden increase in sensitivitiy of GO to 

wind speed at around 10 m s-'. We therefore feel that our 

model function is preferable to the Brown et al. three-branch 

model function. This hypothesis will be testable with the extensive high-quality in situ measurement program planned for 

Geosat. 

It is useful to investigate the stability of the proposed new 

wind speed model function to gain some feeling for its reliability. One method of achieving this is to examine the performance of the model function in different geographical regions. In Figure 15. the global 96-aay, 2" by 6' average data 

have been separated into four geographical regions: Figure 

15~. Pacific Ocean (both northem and southern hemispheres): 

Figure 15b, Atlantic and Indian oceans (both northern and 

southern hemispheres); Figure 15c, northern hemisphere (Pacific and Atlantic oceans); and Fi&re 15d, southern hemisphere (Pacific, Atlantic. and Indian oceans). In all four plots, 

the solid line corresponds to the model function derived for 

the global data set (Figure 13). It is apparent that the single 

model function fits the data in all four regions very well. 

For comparison, the least squares fit of parameters G and H 

estimated separately for each region are given in Table 7. In 

aI1 cases, the C 2nd H values computed from the global data 

fall easily within the error bars of the G and H values computed separately for each of the four geographical regions. The 

worst case is the northern hemisphere region. However, this is 

the least reliable of the four regions because of the very limited 

range of the observed wind speeds (4-8 m s- '). This is reflected in the larger error bars on the H coefficient (see Table 7). In 

the other three regions, the observations cover a broader 

range of wnd speeds due to strong austral winter winds in the 

southern hemisphere during the Seasat mission. The estimated 

coefficients G and H are consequently more reliable 

We conclude that the G and H values derived for the global 

data set are an adequate representation in all geographical 

reeions. Inverting the proposed algorithm to obtain the 

19.5-m wind speed from measurements of uc gives 

A scatter plot comparison of 96-day, 2" by 6" average SASS 

wind speed (corrected for a 1 m s-' bias) and ALT wind speed 

(computed from G' corrected using the Hancock algorithm) is 

shown in Figure 16. The correlation between the two estimates of wind speed is 0.94 over a range from 4 to 15 m s-'. 

The slope of the least squares fit straight line is 1.005 2 0.048 

with an offset of (0.084 0.409) m s-l and an rms error of 

0.82 m s-'. 

From Figure 16, there is a suggestion that the proposed 

model function may be overestimating wind speeds in excess 

of 12 m s-'. Alternatively, SASS could be underestimating 

these high winds speeds. Since the number of comparisons at 

these high wind speeds is small, the statistical significance of 

the discrepancy is questionable. The discrepancy at high wind 

speeds could be eliminated with a polynomial correction of 

the sort used in the Brown et al. [1981] wind speed model 

function. However, since it is not yet known whether the error 

lies in ALT or SASS wind speed estimates, we prefer not to 

apply a polynomial correction at this time. The Geosat in situ 

measurement program should resolve this apparent discrepancy at high wind speeds. 

As a final point, the efTect of the new model function on the 

distribution of wind speeds is shown in Figure 17. The peculiarities evident in Figures 5 and 11 are no lonp present; the 

distribution varies smoothly over the full range of wind 

speeds. The peak in the distribution is at about 7.5 m s- I, and 

the overall average value and standard deviation are 8.5 and 

4.6 m s- ', respectively. 

' 

6. Dacussio~ AXD Co~crusross 

The launch of the U. S. Navy altimetric satellite Geosat 

scheduled foi February 1985 has motivated an in-depth investigation of wind speed estimation from ALT measurements of 

radar backscatter. ALT wind speed retrieval is a two-step procedure. In the first step (see secrion 4.1). the normalized radar 

cross-section G' is computed from receiver gain (AGC). satellite attitude angle and satellite height. In tht second step (see 

section 4.2), the wind speed is computed from G'. In our study 

I I 

4t 

0 5 10 15 20 

19.5m Wind Speed (rn/s) 

Fig. 17. Histogram of 19.5-m wind speeds computed from approximately 3 million observations during the full 96-day Seasat mission using the Hancoc; modified algorithm and the new proposed 

wind speed model function. Bin size is 0.3 m s-'. 
31 

~HELTOS ASV MCCABE: SATELLIE ALnMETER MEASLIREME~T OF SEA SURFACE ~IND SPEED 

of ALT wind soeed estimation from Seasat, we identified 

probiems with both steps of the procedure. 

The most fundamental probiem is an error in the G' algorithm imnlemented in Seasat ALT geophysical data record 

GDR) processing which resulted in a discontinuous dependence of uc on-AGC. When this algorithm is corrected to 

remove all discontinuities, the resulting values of ui are much 

more reasonable than those produced in the Seasat GDR's. 

The corrected algorithm (suggested by D. Hancock (personal 

communication, 1983)) is much simpler than the Seasat GDR 

u' algorithm (section 4.1). 

The wind speeds computed from corrected uo still exhibit 

rather peculiar behavior. The Seasat GDR wind speed model 

function was adopted from the GEOS 3 aigorithm derived by 

Brown et al. [ 1981 3. In this algorithm, the wind speed is first 

estimated from a three-branch model function and then corrected with a fifth-order polynomial adjustment. We traced 

the peculiar behavior of computed wind speeds to discontinuous derivatives at the two branch points of the three-branch 

model function. 

In this paper, we have proposed a new model function for 

wind speed estimation from corrected values of uc. The proposed model function has the same functional form as that 

used on Seasat SASS data. If uc is expressed in decibels, the 

new model function estimates the wind speed at 19.5 m by 

;,9.5 = ~~l~u"ldB)/iO-G~/Hl 

where 

G = 1.502 

H = -0,468 

Tnc parameters G and H were estimated by least squares from 

a comparison of global 96-day, 2" by 6" averages of ALT uD 

with vertically polarized, off-nadir SASS wind speed. The 

SASS wind speeds were corrected for a 1 m s-l bias which 

was evidently introduced into the Seasat SASS GDR processing due to calibration to an erroneous wind recorder in 

JASIN. 

The proposed model function of course produces ALT wind 

speeds which are consistent with SASS wind speeds. For the 

dobal 96-aay. 2" by 6" averages. the correlation between ALT 

and S.ASS wind speed is 0.94. and the rms difference is 0.82 m 

s- '. A distinct advantage of this technique for XLT and SASS 

wind speed comparison is that the averaging removes any 

random component of error in individual estimates of wind 

speed. It should be born in mind that the validity of the proposed algorithm has only been demonstrated on spatially and 

temporally ayeraeed data. 

A natural question that arises is whether the proposed algorithm adequately describes instantaneous ALT estimates of 

wind speed. A rigorous test of performance on individual 

measurements of uo requires an extensive high-quality in situ 

data base for comparison. Using the NDBO buoy data base, 

we were able to identify only 14 independent buoy observations of wind speed within 100 km and 1 hour of Seasat ALT 

uo measurements. This is clearly too few for a meaningful 

intercomparison. The extensive in situ measurement program 

planned for Geosat will correct this situation and allow an 

independent assessment of the proposed wind speed model 

function. 

An alternative approach which yields some information on 

model function performance on instantaneous measurements 

is to determine the rms error of ALT ui measurements. We 

examined the global Seasat ALT data base and determined 

that the point-to-point rms error was 0.3 dB. The effects of 

this measurement noise are shown in Figure 18. The dashed 

lines show the wind speed estimated with a 0.3-dB error in the 

measurement of 0'. Retrieval of low wind speeds is relatively 

insensitive to errors in measurement of 0'. However, at high 

wind speeds, errors in measurement of uo result in very poor 

estimates of wind speed. For example, a +0.3-dB error in uo 

at 20 m s-' results in an estimated wind speed of 24 m s-'. 

Even at moderate wind speeds of 8 m s-', a i0.3-dB error in 

u" yields an estimated wind speed of 9.5 m s-'. 

We conclude that Seasat ALT uc measurement error is too 

large to provide reliable instantaneous estimates of wind 

speed. It seems to us that ALT wind speed estimates will only 

be useful when the data are temporaliy and spatially averaged. 

We have used 96-day. 2' by 6" averages in this study. The 

lower limits of averaging are the subject of further investigation. This problem is unfortunately addressable to only a 

limited degree from Seasat data because of intermittent sampling by ALT during the 96-day mission. 

One find cave~t in the proposed model function is that it is 

based on very few comparisons at nind speeds higher than 11 

m s-'. Thus the validity of the model function is suspect at 

these high wind speeds. This can only be resolved from a 

carefully coordinated in situ measurement program such as p-. pkinned for Geosat. 

Go (dB) 

Fig. 18. The proposed new ALT wind speed model function (coniinuous curve). The dashed lines show the efiects on estimated wind 

rpeed when 6' is in error by 2 0.3 dB. 

Acknorledprnents. We would like to acknowledge F. Wentz for 

helpful discussions throughou: this project and D. Hancock for the UK of his unpublished G' algorithm. M. Freiiich and an anonymous 

reviewer contribured valuable criticism and comments on an early 

version of this manuscript. We thank F. Kuykendall for writing and 

implementing a wonden'ull?. eficient computer program for identifying matches between sarellite and buoy wind speed measurements. 

Finally. we would also liie to thank Pam Wegncr for typing tne 

manuscript. This work was carried out at Oregon State University 

and the Jet Propulsion Laboratory under KASA contract NASi-100. 

The Jet Propulsion Laboratory-s Pilot Ocean Data System provided 

the use of their compater for accessing the Seasat data archieves and 

for Ihe data analysis presented in this pzper. 

.. 

.: 

REFERENCES 

Barrick, D. E., Wind dependence of quasi-specular microwave sea 

Barrick. D. E.. and C. T. Swift. The Seasat microwave instruments in 

Benaat. J. S. and .4. G. Piersol. Random Dura: Analysis and Measurescatter, IEEE Trans. Anrennas Propag., AP22. 13-Sl36. 1974. 

hisiorical perspective. ]€€E J. Oceanic Eng.. OES. 74-79, 1950. 

niem Procedures, 407 pp.. Q'dey-Interscience, kw \iork. 197 1. 
CHELTOV AVV MCC4BE: SATELLITE ALTIMETER hlEASUREME\T Of %A SURFACE h'lVD SPEED 

Boggs. D. H.. The Seasat scatterometer model function: The genesis 

of SASS-1. Rep. 622-230. 30 pp.. Jet Propul. Lab., Pasadena. Calif., 

1981. 

Brown. G. S., Backscattering from a Gaussian distributed, perfectly 

conducting. rough surface, I€€€ Trans. Antennas Propag.. AP26, 

472-482, 1978. 

Brown, G. S., Estimation of surface winds spceds using satellire-borne 

rddar measurements at normal incidence, J. Geopnvs. ke5.. 84, 

3974-3978.1979. 

Brown, G. S., H. R. Stanley, and N. A. Roy, The wind speed measurement capability of spaceborne radar altimetry, IEEE J. Oceanic 

Eng., OE6. 59-63, 1981. 

Chchon, D. B., Comments on "Seasonal variation in wind spced and sea state from global satellite measurements," J. Geophys. Res., this 

issue. 

Chelton, D. B.. K. J. Hussey, and M. E. Parke, Global satellite 

measurements of water vapor, wind speed and wave height, Nature, 

Cox, C. S., and W. H. Munk. Statistics of the sea surface derived from 

sun glitter, J. Mar. Res.. 13, 198-227, 1954. 

Daley, J. C.. J. T. Ransone, and W. T. Davis, Radar sea return from 

JOSS-11, Rep. 7534, Nav. Res. Lab., Washington, D. C., Feb. 21, 

1973. 

Draper, N. R., and H. Smith, Applied Regression Andvsis, 709 pp., 

John Wile)., New York, 1981. 

Fedor, L. S., and G. S. Brown, Wave heieht and wind speed mcasurements from the Seasat altimeter, J. Geophys. Res., 87, 3254-3260, 

1982. 

Fu, L.-L., Recent progress in the application of satellite altimetry to 

observing the mesoscale variability and general circulation of the 

oceans, Ret). Geophvs. Space Phys., 21. 1657-1666.1983. 

Guinard, N. W, J. T. Ransone, and J. C. Daley. Variation of the 

NRCS of the sea with increasing roughness, J. Geophys. Res., 76, 

153-1538, 1971. 

Halpern. D.. and R. A. Knox, Coherence between low-level cloud 

motion vectors and surface wind measurements near 0'. 152"W 

from April 1979 to February 1980, Atmos. Ocean, 21,82-93, 1983. 

Jones, W. L., and L. C. Schrocder, Radar backscatter from the ocean: Dependence on surface friction velocity, Boundnry hyer Mereorol., 

13, 133-149, 1978. 

Jones, W. L., L. C. Schroeder, and J. L. Mitchell, Aircraft measure2Y4,529-532, 1981. 

ments of the microwave scattering signature of the ocean, I€€€ J. 

Oceanic hng.. O€t. 5241, 1977. 

Liu. W. T., and P. P. Niiler, 1984: Determination of monthly mean humidity in the atmospheric surface laver over oceans from satellite 

data. J. Phss. Oceanog., 14. 1451-1457, 1984. 

Moore. R. I;.. and A. I(. Fung. Radar derermination of winds at sea, 

hoc. I€€€. 67. 1504-1521. 1979. 

Phillips, 0. M.. The Dvnamics o/ the Upper Ocean, Cambridge University Press. hew Y ork. 1966. 

Schroeder. 1. C., D. H. Boggs, G. Dome, I. M. Halberstam. W. L. 

Jones, W. J. Pierson, and F. J. Wcntz, The relationship between 

wind vector and normalizcd radar cross section used to derive ' 

Seasat-A satellite scatterometer winds, J. Geophys. Res., 87, 3318- 

3336, 1982. 

Swift, C. W., Passive microwave remote sensing of the ocean-A 

review, Boundary Layer Meteorol. 18,25-54, 1980. 

Townsend, W. F., An initial assessment of the performance achieved 

by the Seasat-1 radar altimeter, IEEE J. Oceanic €ng., OES, 8G92, 

1980. 

Weller, R. A., R. E. Payne, W. G. Large, and W. Zenk, Wind measure- ments from an array of oceanographic moorings and from F/S 

Meteor during JASIN 1978, J. Geophvs. Res., 88,9689-9705, 1983. 

Wen& F. J., V. J. Cardonc. and L. S. Fedor, lntercomparison of wind 

speeds inkrred by the SASS, altimeter, and SMMR, J. Geophys. 

Res., 87,3378-3384, 1982. 

Went& F. J., S. Peteherych, and L. A. Thomas, A model function for 

ocean radar cross sections at 14.6 GHq J. Geophys. Res., 89, 3689- 

3704, 1984. 

Wu, J., Sea surface slope and equilibrium wind-wave spectra, Phys. 

Wurtele, M. G., P. M. Woiceshyn, S. Peteherych, M. Borowski, and 

W. S. Appieby, Wind direction alias removal studies of Seasat 

scatterometer-derived wind fields, J. Geophys. Res., 87, 336-%3377, 

1982. 

D. B. Chelton, College of Oceanography, Oregon State University, 

P. J. McCabe, Jet Propulsion Laboratory, Pasadena, CA 91109. 

. 

Fluids, 15, 741-747. 1972. 

Corvallis, OR 97331. 

(Received August 24, 1984 ; 

accepted October 10. 1984.) 

.. 
33 

JOURNAL OF GEOPHYSICAL RESEARCH. VOL. 90, NO. C3, PAGES 5001-500S, MAY 20, 198.Z 

Comment on "Seasonal Variation in Wind Speed and Sea State 

From Global Satellite hleasurements" by D. Sandwell and R. A, oreen 

DUDLEY B.CHELTOK 

College of Oceanography, Oregon State University. Corvallis 

In a recent paper, Sandwell and Agreen C1984; hereafter SA] 

presented figures of global seasonal wind speed and sea state 

as measured by the GEOS 3 satellite altimeter. Since that 

time, Chelton and McCabe [1985; hereafter CM] have found 

that problems exist in the algorithms used to retrieve wind 

speed from altimeter measurements of radar backscatter. 

These problems were discovered too late to be of use in the 

analysis of SA. However, because they have a significant 

impact on the accuracy of wind speed estimation from altimeters. it is important that they be pointed out now so that the 

results of SA are not misused. Although the results presented 

here do not alter many of the conclusions of SA in a qualitative sense, they do become important for any quantitative 

interpretation of the seasonal winds presented by SA. In addition, the data distribution maps presented here (Figures 3a-31) 

are useful for pointing out limitations in other applications of 

GEOS 3 data (e.g., use of the altimeter sea level measurements 

to study surface geostrophic currents). 

A detailed description of sea surface wind speed estimation 

from radar altimeters is gven in CM. Briefly, wind speed retrieval is a two-step procedure. In the first step, the power of 

microwave radiation backscattered from the sea surface is determined from parameters measured by the altimeter receiver. 

To account for variations in transmitted power, the return 

power is normalized by the transmitted power. This normalized radar cross section is usually referred to as oo. The (I' 

measurements must be corrected for variations in the attitude 

angle of the satellite and variations in the height of the satellite above the sea surface. 

The second step in wind speed retrieval is to estimate wind 

speed from GO. This estimation is based on the principle that 

the roughness of the sea surface increases with increasing wind 

speed. The backscattered power measured by the alumeter 

receiver consists of microwave radiation reflected specularly 

from the sea surface over an approximate IO-km footprint 

directly beneath the satellite. As the sea surface roughens, 

much of the transmitted radiation is specularly scattered away 

from the radar antenna Thus mind speed is inversely relatcd 

to GO. The algorithms used to estimate wind speed from G' are 

purely empirical, based on comparisons with coincident 

measurements from buoys. 

In a detailed investigation of the performance of wind speed 

estimation from the Seasat altimet:r. CM identified problems 

with both steps of the procedure for wind speed retrieval. For 

Seasat, the power received by the altimeter antenna was converted to a constant output level for other receiver stages 

using a digital step attenuator automatic gain control (AGC). 

Thus G' can be computed directly irom AGC [with the aiorementioned satellite attitude angle and height corrections). CM 

found that the tables used to compute G' resulted in a duCopyright 1985 by the American Geophysical Union. 

Paper number 4C1495. 

0 I &0227/85/OO4C- 149SO?.OO 

continuous dependence on AGC. This algorithm was incapable of producing uo values in the range from 11.2 to 12.0 

dB, which corresponds to wind speeds from about 5 to 7 

ms-'. This error is clearly serious for wind speed estimation, 

since this wind speed range is very near the peak in a distribution of global wind speeds. 

After correcting for the error in the Seasat oo algorithm, 

CM still found peculiarities in the estimated wind speeds. 

These peculiarities were traced to discontinuities in the slope 

of the model function used to estimate wind speed from oo. 

For Seasat, wind speeds were computed from uc using a threebranch logarithmic model function developed by Brown et al. 

[1981]. The change in the slope of the model function at the 

two branch points (10.12 and 10.9 dB values of go) resulted in 

abrupt changes in the distribution of wind speeds computed 

from 6'. CM proposed a new single-branch model function 

which is very nearly the same as the Brown et al. model 

function for wind speeds less than 11 ms-I (corresponding to 

the two lower branches of the three-branch algorithm). For 

higher wind speeds, the two model functions diverge rapidly. 

Wind speed retrieval from the GEOS 3 ALT differed from 

the Seasat ALT both in the hardward configuration and in the 

algorithm used to estimate wind speed from GO. On GEOS 3. 

the backscatter was measured directly using an analog receiver. (This was changed on Seasat to a digtal system because of 

calibration problems inherent in analog systems.) Since G' was 

measured using an analog system, wind retrieval from GEOS 

3 should be immune from errors of the type found in the 

Seasat G' algorithm (aside from calibration problems with the 

analog system). 

However, the nature of the algorithm used to compute wind 

speed from GEOS 3 measuremena of G= does introduce errors 

in the v;ind speed. Wind speed esrimation was based on a 

two-branch algorithm proposed by Brown [I9791 which had 

the form 

o'(dB) = -2.1 - 10 loglo (a In ul0 - b) 

where G' 1s measured in dB and u,~ is the wind speed at 10 m 

above the sea surface. The parameters a and b were evaluated 

by least squares analysis of a very limited ship and buoy data 

base. The result was a two-branch algorithm with a branch 

point at 9.2 ms-'. For the lower wind speed branch, 

u = 0.02098 and b=0.01075. For the upper wind speed 

branch u = 0.08289 and b = -0.12664. 

In order to compute wind speed, this algorithm must De 

inverted, which gives 

11 = exp CCS - b)bI 

where 

S = ~O-luoldB)+2.11110 

This model function for wind speed is shown graphically in 

Figure 1. Note the discontinuous slope of the model function 

,. . .. : 

.. 

.. 

r. .. 

':. . . 

:. 

.. .- 

5001 
34 

CHFLTOK ' COMMLYTARY 

(To(dB) 

Fig. 1. The Brown [1979] two-branch wind specd modal function 

implemented in geophysical processing of GEOS 3 ALT data. 

at the branch point at 9.2 ms-' (corresponding to a uo value 

of 10.3 dB). 

(Note: Brown et aI. [I9811 later developed a three-branch 

algorithm for computing wind speed from uo for the GEOS 3 

ALT. This aigorithm had exactly the same form as that given 

above (with branch points at 7.87 and 10.03 ms-') and was 

based on an expanded data base consisting of high-quality 

measurements from National Data Buoy Ofice buoys. The 

three-branch algorithm was implemented in geophysical processing of the Seasat ALT data but was not used in the final 

geophysical processing of GEOS 3 ALT data.) 

Problems with the Brown [I9793 two-branch wind speed 

model function are easily identified with a simple histogram of 

wind speeds from the GEOS 3 ALT (Figure 2). There is an 

abrupt change at 9.2 ms-' in the global distribution of wind 

speeds. Based on the results of CM, this abrupt change is due 

to the discontinuous slope at the 9.2 ms-' branch point. The 

approximate factor of 2 change in slope at 92 ms-' results in 

a factor of 2 increase in the number of Hiind speeds observed 

in the 9.2-9.4 ms-' bin over the 9.b9.1 ms-' bin. 

Thus as noted in CM, it is essential that any model function 

used to compute wind speed from GO be continuously difierenuabie. Ciearly, the GEOS 3 ALT ciara must be reprocessed 

using a new model function for wind speed in order to obtain 

reliabie estimates of global winds. A form for the new model 

function is proposed in CM. 

4. 1 

lorn Wind Speed (m/s) 

Fig. 2. Histopam of 10-m wind speeds computed from approximately 3.3 million observations during the full 3j year GEOS 3 mission using the Brown [1979] two-branch wind speed modal function. 

Wind speed measurements less than 0.01 ms-' have been exciuded. 

Bin size is 0.2 ms - l. 

I I 

Fig. 3a. GEOS 5 pound tracks during all umes when the ALT was 

turned on during January. 

Since a large number of individual wind speed estimates are 

averaged in each 2' square in the maps presented by SA, a 

histogam of the average uind spetds ail1 of course not exhibit the abrupt change at 92 rns-'. Indetd, according to the 

Central Limit Tntorem, a histogam of averaged winds must 

be very nearly Gaussian. However, the peak in this distribution will be biased toward a lower wind speed than for 

averages of individual winds computed using a more accurate 

model functioo. (This is not in general true but is true for the 

case considered here.) Therefore the seasonal wind speeds presented by SA are generally lower than would be obtained if a 

more accurate model function were used (see Figures 4 and 16 

of CM). 

Before any steps are taken to reprocess the data, it is useful 

to examine tne spatial and temporal distribution of the GEOS 

f 

.. ... 

.- . .. 1. 

. 
35 

CHELTQS: COMMEXTARY 

I I 

FEBRUARY I S78 

~ 

Fig. 36. As in Figure 3n exapt for February. 

3 ALT cC measurements in some detail. This information is 

given to a very limited degree by SA in their global summaries 

of the number of observations by 2” latitude and lonptude 

regions. They show the total number of boreal %+inter (which 

they define to be Dectmber, January, February, and March) 

and summer (defined as June: July, Aupusr and Sepiember) 

observations over the % year GEOS 3 mission (April 1975 to 

November 1978). A more detailed examination reveals serious 

limitations in the seasonal wind speeds presented by SA. The 

GEOS 3 ground tracks during all times when the ALT was 

turned on are shown in Figure 3 for each individual month 

during the 34 year mission. 

Some of the salient features of the boreal winter (December, 

January, February, and March) sampling by the GEOS 3 

ALT are: 

1. Only the western North Atlantic and eastern North PaI l 

~ ~~~~~ 

Fig. 3c. As in Figure 3a except ior March. 

cific oceans were sampled every month during the 3 year 

GEOS 3 mission. 

2. There were very few measurements south of 30% during 

any month. 

3. The entire Indian Ocean was sampled only during the 

month of March 1976. 

4. The western Xorth Pacific Ocean w2s sampled by only 

a very small number of ground tracks in January, February, 

and Mzrch of 1978. 

The eastern Korth Atlantic Ocean ~2: sampled by only 

a very smali number of ground tracks In February 1976, December 1977. and February and March 1978. 

6. The entire South Atlaniic Ocean was essentially unsampled. (SA block out much of this region in their figures, but the 

reliability of the seasonal mean in regions not blocked oct 

remains questionable.) 

5. 

. .. . .. . 

-I. . 

.. . - .. - .. .. .. 

.. 

.. 
36 

CHELTOX : COMMEWTARY 

Fig. 3d. As in Figure 3a except for April. 

. -_ 

- 

Fig. 3e. As in Figure 3a except Tor May. 
37 

CHELTON: COMMEXTAR'I 

\ _h ,-.- 7 -.-- '- c'r ---I /-. /-\J--- 

Fig. 3.f. As in Figure 3a except for June. 

- .ULY I971 

,:-? 

Fiz. 39. As in Figure 30 except for July. 

.. . 
Fig. 3h. As in Figure 30 except for August. Fig. 3i. As in Figure 3a except for September 

._. - 
39 

Fig. 3j. As in Figure 30 except for October. 

w-7 -,-..r-d 4 i 

Fig. 3k. As in Figure 3, except for November. 

-. 

. , . .. 
40 

Fig. 31. As in Figure 30 except for Dccembcr. 

Some of the salient features of boreal summer (June, July, 

AU~USL and September) sampling by the GEOS 3 ALT are: 

I. Tht western Nod Atlanuc and eastern liorth Pacific 

Oceans were sampled nearly every month during the GEOS 3 

mission. 

2. Again, the number of observations south of 30"s was 

relatively small. 

3. The central and South Indian Ocean was never sampled. (SA block out the southern Indian Ocean but show contours in the central Indian Ocean.) 

4. The eastern half of the equatorial Pacific Ocean was 

essentially unsampled (except for a few ground tracks in 

August and September 19771. 

Sampling of the South Pacific Ocean was essentially 

limited KO August 1977 (with a very lew ground tracks in June, 

July. and September 1977). 

6. The western South Atlantic Ocean was sampled only in 

June 1976 (with a few pound tracks in August 1976). 

From these figures, it is clear that, except for the western 

ral and spatial sampling of the global Ocean by the GEOS 3 

ALT was very irregular. In many areas, the seasonal mean 

values presented by SA are based on measurements taken in 

different years. Since it is known that there is strong interannual variability in global winds, the representativeness of the 

seasonal winds in SA is questionable. This is true even in a 

5. 

t 

North Atlantic and eastern North Pacific Oceans, the tempo- I 

relative sense where winds in one geographical region are 

compared with those in another (since the winds in different 

regions may have been sampled by GEOS 3 in different years). 

The conclusion is that patterns of winds presented by SA 

are, at best, only qualitatively correct because of weaknesses in 

the two-branch model function for wind sped and irregular 

temporal and spatial sampling by GEOS 3. It is questionable 

whether reprocessing GEOS 3 ALT data for wind speed 

would be useful scientifically for more quantitative examination of global wind speeds. The maps in Figure 3 help identify 

geographic regions where the mean wind speeds presented by 

SA are least trustworthy. Although reliable statistics on seasonal winds could probably be computed from GEOS 3 data 

in a few selected regions (using a corrected single-branch wind 

speed model function), SKatistics on global winds would probably not be very meaningful. 

Acknowledgmenrs. I would like to thank the Jet Propulsion Laboratory's Pilot Oaan Data System for use of their computer for 

accessing the GEOS 3 data archive and for the data analysis prcscnted in this paper. Th~s work was carried out at Oregon Stare University under NASA contract NASi-100. 

REFERENCES 

Brown, G. S, Esumauon or surface wind spetds using satellite-borne 

radar measurements at normal mddena, J. Geopnys. Res. 84, 

3974-3978, 1979. 

Brown, G. S.. H. R. Stanley. and N. A. Roy, The wind sp&d measuremat cauabilxty of space-borne radar ?.!*metry, I€€€ J. Oceanrc 

Lng, OEd. 5963,1981. 

Chelton. D. B, and J. P. McCabe, A review of satellite alwetcr 

measurements of sea surface amd speed, with a proposed new 

algorithm J. Geophvs. Res, in press. 1985. 

Sandwell, D. T, and R W. Aprren, Seasonal variatjon in wind spud 

and sea state from global satellite measurements, J. Geophys. Res, 

89,2011-2051, 1984. 

D. B. Chelton, College or Oaanography, Ore_pon Srate University, 

Conally OR 97331. 

(Received September 21 1981; 

accepted Dtccmber 14,1984.) 

. revised December 1, 1984; 

, 

..- . 

.. :. . 

. --_ 
41 

Reprint Series 

19 October 1984, Volume 226, pp. 343-346 SCIENCE 

Temporal Variability of the Antarctic Circumpolar Current 

Observed from Satellite Altimetry 

Abstract. Sea level measurements by the Seasar alrimerer were used to study rhe 

temporal variability of rhe Anrarcric Circumpolar Current between July and October 

1978. Large-scale zonal coherence in zhe cross-stream sea level diference was 

observed, i’ndicaring a general increase in ihe surface geostrophic velocity of rhe 

current around rhe Sourhern Ocean. The result demonsrrares the power of sarellire 

alrimerry 10 monitor ihe variabiliry of large-scale ocean currenrs. 

Recent observations (1, 2) have revealed that there are substantial temporal variations in the mass transport of the 

Antarctic Circumpolar Current (-4CC). 

The magnitude of the variation (peak-topeak range) has been estimated to be 40 

sverdrups (1 Sv = IO9 kdsec), about 30 

percent of the mean transport. ACC 

transport variations of this magnitude 

have profound efiects on the circulation 

.of the world’s oceans, which are important determinants of the global climate 

(3). 

Because synoptic global observations 

of the ACC are lacking, very little is 

known about the spatial structure of its 

temporal variability. For example, what 

is the zonal scale of the variabiiiry? DOCS 
42 

2400 

Fig. 1. Long-term averaged 

. sea-surface dynamic topography of the Southern Ocean 

(11). expressed in dynamic 

meters relative to the 1000- 

dbar level. The dots indicate 

the locations of six pairs of 

altimeter crossover clusters (A 

through F). and the X'S indicate the locations of two bottom pressure gauges. 

3300 150° 

the transport of the ACC vary in phase 

around the Southern Ocean? Answers to 

these questions can come only from 

global observing systems. The cost of 

using a network of ships to observe the 

ACC for an extended period of time 

would be prohibitive. One of the most 

promising means of monitoring ocean 

circulation over large spatial scales is 

with a satellite radar altimeter system 

(4). 

A radar altimeter measures the altitude of a spacecraft above the sea surface by radio pulse ranging. With an 

independent knowledge of the height of 

the spacecraft (the orbit height) referenced to an earth-fixed coordinate system, the height of the sea surface referenced to the same coordinate can be 

readily obtained. Reviews of the applications of altimetric measurement of the 

sea surface to the study of ocean currents can be found in the literature (5). 

W-e present here some results of an attempt to use the data collected by the 

Seasat altimeter (6) to study the largescale temporal variability of the ACC 

during the 3-month lifetime of Seasat. 

The approach used is a variant of the 

so-called crossover-dserence technique 

(7). A full description of the technique 

will appear elccwhere (8). A crossover 

difference refers to the diEerence between the altimeter measurements made 

at the intersection of an ascending 

(nonhbound) and a descending (southbound) ground track. It comprises three 

components: orbit error, altimeter measurement error, and true sea level variations. With both orbit and altimeter measurement errors properly corrected, the 

crossover differences have been used to 

map the mesoscale variability of ocean 

currents (7). However, as far as we 

know, the information contained in the 

crossover dserences has not yet been 

exploited to infer the large-scale temporal variability of ocean currents. 

Of the various errors contained in the 

crossover differences, orbit error is by 

far the most significant. As a first step to 

reduce orbit error, we applied a crossover adjustment (9) to the Seasat geophysical data records (10) in overlapping 

Jul. Aug. Sep. Oct. 

,I , !< I II , ,I I 

190 220 250 280 

Day in 1978 

Fig. 2. Time series of the north-south sea level 

diKerences (in centimeters) across the six 

altimeter crossover cluster pairs shown in 

Fig. 1. The nippled region around each curve 

represents the r.rn.s. error. The dashed line in 

(.4) shows a 20-day running average of the 

north-south pressure difference (in millibars) 

across the two bortorn pressure gauges (deployed ar a depth of 500 m) sho\r.n in Fig. l. 

areas of approximately 2500 km by 2500 

km around the Southern Ocean between 

40"s and 65"s. This procedure reduced 

the root-mean-square (r.m.s.) crossover 

difference from 146 to 32 cm. After deleting anomalous values greater than 60 cm, 

the residual r.m.s. crossover difference 

was reduced to 24 cm, which was still 

dominated by residual errors from various sources (orbit, sea-state bias, range 

delays in the media, and ocean tides). 

Since the time scales of the errors are 

generally less than a week whereas those 

of the large-scale oceanic variability are 

generally greater than a month, the residual crossover differences can be 

smoothed over a 20-day window to further reduce the errors. This smoothing 

procedure reduced the r.m.s. residual 

crossover difference from 24 to 9 cm 

with an error of 6 cm (8). The resultant 

smoothed residual crossover differences 

constitute the data base for the results 

presented here. 

Because of the near-repeat orbit of 

Seasat, there is a regularly spaced grid of 

points with a high density of crossovers 

(referred to here as clusters). We have 

identified a total of 229 clusters around 

the ACC, each measuring approximately 

200 km by 200 km and containing about 

200 crossovers during the 3-month Seasat mission. For each cluster area the 

crossover adjustment was applied again 

to remove a constant bias from each 

track so as to minimize the residual 

crossover dxerences. Each resultant 

bias then effectively represents an alongtrack average of the time-varying sea 

level over a distance of -200 km and 

hence reflects predominantly the largescale variability of ocean currents. We 

thus generated a sea level time series at 

each cluster by ordering all the resultant 

biases by their corresponding altimeter 

overpass times. 

To examine the zonal structure of the 

ACC variability, we have selected six 

pairs of crossover clusters along the 

main axis of the fiow and computed the 

sea level difference across the current. 

The sea level difierence between two 

clusters is proportional to the average 

surface geostrophic velocity perpendicular to the line segment connecting the 

two clusters. Displayed in Fig. 1 are the 

positions of the six pairs of clusters 

(labeled from A to F) superimposed on a 

map produced by Gordon el al. (11) of 

the long-term averaged sea-surface dynamic height relative to the 1000-dhar 

level. The dynamic height is proportional 

to the stream function of the surface 

geostrophic flow relative to the 1OOOdbar level. The directions of this average 

relative flow are indicated by the arrows 
43 

on the figure. The two clusters of pair A 

are in the vicinity of two bottom pressure 

gauges (denoted by Xs on Fig. 1). which 

were deployed at a depth of 500 m on the 

continental slope as part of the International Southern Ocean Study (ISOS) program to monitor the variability of the 

4CC in the Drake Passage (I). The sea 

?vel difference across pair A will be 

compared with the pressure difference 

across the two gauges. 

The sampling interval of the sea level 

time series generated from altimeter 

crossover differences is irregular as a 

result of the irregular overpass times of 

the altimeter. To compute the difference 

between two time series, we interpolated 

the time series to a common time grid at 

daily intervals, using the optimai interpolation scheme discussed by Bretherton et 

al. (12). The resultant sea level differences across the six cluster pairs (northem cluster minus southern cluster) for 

the period from day 193 (12 July) to day 

284 (11 October) of 1978 are shown in 

Fig. 2. The r.m.s. error is indicated by 

the stippled region around each curve. A 

rise in the sea level difference implies an 

increase in the average surface geostrophic velocity between the two clusters. Because each sea level time series 

is a solution to a difference equation, 

there is an arbitrary constant associated 

with each series. This constant was determined here such that each series started from zero at day 193. 

The dashed line superimposed on 

curve A represents a 20-day running 

mean of the north-south pressure diflerence (in millibars) between the two bottom pressure gauges (deployed at a 

depth of 500 m) in the Drake Passage. If 

the flow variability in the region were 

due entirely to barotropic motions, then 

the two curves would match each other 

to the extent of measurement errors (13). 

However, the flow variability in the 

Drake Passage has z substantial baroclinic component (14, which can account for part of the discrepancy between the two curves. In fact, the discrepancy found here is due primarily to 

the dserence between sea level and bottom pressure at the northern side of the 

Passage, where the flow variability is 

known to be higher. Nevertheless, the 

general increase in the pressure difference (about 20 mbar) across the Drake 

Passage indeed hzs a counterpart in the 

sea level difference measured by the 

aitimeter (about 10 cm). 

Figure 2 shows that there is some 

zonal coherence in the increase in the 

sea level difierence across the ACC during the Seasat mission. The net increase 

in the sea level difTrrence across the 

Fig. 3. Color-coded map of the 

low-frequency (period longer 

than 20 days) sea Level 

changes measured by the Seasat altimeter over the Southern Ocean between 403 and 

65"s from 12 July to 11 Octe 

ber 1978 (October minus July). 

The directions of the comesponding change in surface 

geostrophic velocity arc indicated by the arrows. The dots 

designate the locations of the 

altimeter measurements used 

to construct the map. 

other five sections is 20 to 30 cm. This 

change in sea level slope is indicative of 

an eastward acceleration of the ACC. 

The variations of the sea level difference 

across the ACC do not appear to be 

exactly in phase around the Southern 

Ocean. Owing to the limited duration of 

the time series, however, it is not possible to draw any quantitative conclusions 

about the coherence and phase characteristics of the variability. 

Figure 3 is a color-coded map showing 

the net sea level changes (day 284 minus 

day 193) at the 229 clusters (indicated by 

dots) around the Southern Ocean (15). 

Because the observed sea level varistions are basically characterized by linear trends over the 3-month period (16), 

such a diEerence map is an eEective 

representation of the variability. The 

large spatial scales of the variability are 

clearly evident. During this period, sea 

level was decreasing over the green areas around Antarctica and increasing 

over the yellow to brown areas to the 

north, indicating a general eastward acceieration of the ACC. The directions of 

the corresponding change in surface geostrophic velocity are indicated by the 

arrows, showing substantial meridional 

components in three repons over major 

topographic features (17): 240" to 270". 

330" to 30", and 60" to 90". At longitudes 

0" and 90" there are exceprions to the 

generally eastward acceleration. The 

large, localized increzsc in sea level 

southeast of South America and south of 

Australia is probably due to local phenomena unrelated to the ACC. 

To the best of our knowledge, the 

results presented here provide the first 

direct observsiional evidence for zonal 

coherence in :he temporal Vanability of 

the ACC. Moreover, they demonstrate 

that even an altimeter with only 1-m 

accuracy (for the measurement of the 

sea-surface height) is able to detect 

large-scale sea level variability with decimeter magnitudes. The shortness of the 

Seasat data set allows only a glimpse of a 

phenomenon whose temporal scales are 

apparently longer than the 3-month data 

record. Future altimetric missions such 

as TOPEX (Ocean Topography Experiment) (18), on satellites equipped with 

improved instruments, hold the promise 

of providing a more accurate, multiyear 

view of the large-scale temporal variability of ocean currents such as the ACC. 

LEE-LUENG Fu 

Jet Propulsion Laboratory, 

California institute of Technology, 

Pasadena 91 109 

College of Oceanography, 

Oregon State University, 

Corvallis 97331 

DUDLEY B. CHELTON 

References urd Sots 

1. R. B. Weam. Jr.. and D. J. Baker, DccpSea R~S. 27.4,~~ ~8o). 

2. T. Whitworth 111, J. Phys. Oceunogr. 13, 2045 110171 ,.,..-,. 3. National Research Council, Occun Rcscarchjor Lindersranding Climuric Vuriarions Wational 

Academy hss. Washinplon, D.C., 1983). 

4. W. Munk and C. Wunsch. Philos. Trans. R. 

SOC. London Ser. A 3U7. 4?9 (1982). 

5. C. Wunsch and E. M. Gaposchkin. Rev. 

Gcophys. Space Phys. 18.M (1980); L.-L. Fu. 

ibid. 21. 1657 (1983). 

6. B. D. Tapley, G. H. Born, M. E. Parkc, 1. 

Gcophvs. Res. 87. 3179 (1982). 

7. R. E. Cncney and J. G. Marsh, €os (Trans. Am. 

Geo hys Union) 62. 713 (1981): A. L. Gordon cr af. J.'Geophys. Res. 88, 755 09S3). 

8. L.-L. Fu and D. B. Chcllon. in preparation. 

9. In the crossover adjustment we subtracred a bias from each track such that the crossover differences were minimized -in a least-squares sense over a given gcogiaphlcal area. Owing to 

its large spatial scales (-1O.OOO km). orbit error can be substantially reduced by this adjustment withour compromising oceanic SIFWIS. 

IO. Most of the altimcrer mcilsuremcni errors have 

been panially corrected for in the data [see (6)]. 

11. A. L. Gordon. E. Molinelli. T. Baker, J. 

Cruph\,s. RES. 83. 2043 097R). 
12. F. P. Bretherton. R. E. Davis. C. B. Fandry, 

DcrpSra Rrs. 23. 559 0976). 

13. If the density field remains unchanged as for 

barotropic motions. a change in sea level of I cm causes a cnange in bottom pressure of -1 mbar. 

14. W. D. Nowlin. R. D. F’illsbury, J. Bottero. 

Drcp-Sra Rrs. =A. I (1981). 

15. Before contounng, the data were interpolated to 

a grid 400 km by 400 km and smoothed over adjacent grid points. The resultant rms error for 

the map is estimated lo be less than 4 cm. 16. The correlation coefficient (sea level versus time) averaged over the 229 clusters (weighted 

by variance) is 0.7 2 0.3. 

17. See (11) for a bathymetry map. 

18. The rms accuracv of the TOPEX altimeter is 

expected to be 14 cm [see TOPEX Science 

Working Group. Snrellrrr Alrrmerrrr Mrasurrmrnrs of rhr Orean (Publication 400-11 1. Jet 

Propulsion Laboratory. Pasadena, 1981) 

19 be thank M. Parke for useful discussions. The 

lSOS nressure record used in the study was provided by R. B beam This research was performed at the Jet Propulsion Laboratory, 

California lnstitute of Technology, and at Oregon State University unaer contract uith the 

National Aeronautics and Space Administration. 

:O April 1984: accented 11 June 1984 
JOURNAL OF GEOPHYSICAL RESEARCH. VOL. 90, NO. C3, PAGES 4721-4739. MAY 20. 1985 

45 

Observing Large-Scale Temporal Variability of Ocean Currents 

by Satellite Altimetry: With Application to the Antarctic Circumpolar Current 

LEE-LUENG Fu 

Jet Propulston Luboratory. Caljfornio institute oJ Technology, Pasadena 

DUDLEY B.CHELTON 

College of Oceanography, Oregon State University, Coruallis 

A new method is developed for studying large-scale temporal variability of oman currents from 

satellite alttmetric sea Level measurements at intersections (crossovers) of ascending and descending orbit 

ground tracks. Using this method. sea level time series can be constructed from crossover sea level 

difierences in small sample areas where altimetric crossovers are clustered. The method is applied to 

Seasat altimeter data to study the temporal evolution of the Antarctic Circumpolar Current (ACC) over 

the 3-month Seasat mission (July-October 1978). The results reveal a generally eastward acceleration of 

the ACC around the Southern Ocean with meridional disturbances which appear to be associated with 

bottom topographic features. This is the first direct observational evidence for large-scale coherence in 

the temporal variability of the ACC. It demonstrates the great potential of satellite altimetry for svnoptic 

observation of temporal variability of the world Ocean circulation. 

1. INTRODUCTTON 

It has been increasingly recognized that global measurement of the sea surface topography by satellite altimetry is a 

viable technique for obtaining quantitatively useful information on the general circulation and variability of surface 

geostrophic currents of the world ocean [e.g., Munk and 

Wunsch, 19821. General descriptions of satellite altimetry can 

be found in Wunsch and Gaposchkin [I9801 and in Stewart 

[ 19831. Progress in application to observations of the general 

circulation and mesoscale variability of the ocean has recently 

been reviewed by Fu [1983a]. To obtain absolute geostrophic 

velocity from altimetrically measured sea surface topogaphy, 

adequate knowledge of the geoid (the earth's equigeopotential 

surface) is required. Because the geoid is not accurately known 

over most of the world ocean, altimetric observations have not 

yet improved our knowledge of the oceznic general circulation 

[see Tai and Wunsch, 1984J. However, the use of repeated 

altimetric observations whereby the unknown time-invariant 

geoid can be removed has provided a global view of the statistics of mesoscale variability of he oceans [e.& Cheney et d, 

19s;; Fu 19S3bJ. Due to the presence of substantial, iongwavelength orbit errors in alktric measurements, the scales 

of variability studied have been confined primarily to mesoscaies (with a dominant iengh scale of about 100 kmi. Thus 

far, there have been no attempts to use satellite altimeter data 

to study large-scale (scales greater than mesoscales) temporal 

variability of the ocean. 

In :his paper we dernonstrzte a new method by which altimetric measurements can 'be used to construct sea level time 

series for the study of large-scale temporal variability of oean 

currents. A number of techniques are empioyed to reduce 

long-waveiength orbit errors plus other errors without compromising loa-frequency, large-scale oceanic signals. The essence of the technique is the separation of sign21 2nd error by 

utilizing fundamental differences in their spatial and temporal 

characteristics. The method is appiied here to Seasat altimeter 

Copyright 1985 by the American Geophysical Union. 

Paoer number 4C1494. 

0 146-0227~85/001C-149~S05.00 

data [see Tapfey et ai., 1982aJ to study temporal variability of 

the Antarctic Circumpolar Current (hereafter referred to as 

ACC) around the Southern Ocean. Some preliminary results 

of this study have been briefly reported in Fu and Cheiron 

[1984]. A more complete and detailed description is given 

here. 

Recent in situ observations in Drake Passage [e.g., Whitworth, 1983; Wearn and Baker, 19801 have revealed rather 

substantial temporal variations in the mass transport of the 

ACC. Due to a lack of synoptic observations around the 

Southern Ocean, very little is known about the spatial structure of this temporal variability. With its global coverage, 

altimeter data can be used to examine the spatial coherence of 

changes in the fiow over spatial scales this large. Because of 

the short duration of the Seasat mission and errors in the 

altimeter data, the accuracy and statistical significance of the 

results presented here are not very high. Emphasis in this 

paper is placed more on demonstration of the methodology 

and its potential with more accurate and longer durztion altimetric missions expected in the near future. 

We begin with a discussion in section 2 of a new method for 

generating sea level time series from dtimetnc measurements. 

Data characteristics and error reduction techniques are described in sections 3 and 4, respectively. Testing of the method 

using Seasat altimeter measurements in the Drake Passage 

area and comparisons of the results with in situ measurements 

are discussed in section 5. Application to the entire ACC area 

is presented in section 6. and the results are discussed m terms 

of variations in geostrophic velocity. Finally, a summary and 

conclusion are given in section 7. 

2. THEMETHOD 

The essence of the approach is to construct sea level time 

series from altimeter crossover difierences, Le., the difierence 

between altimetric measurements made at the intersection (2 

crossover) of an ascending (northbound) and a descending 

(southbound) ground track. A crossover difference comprises 

three basic components: orbit error, altimeter measurement 

error. and true sea level variation. After reducing the two error 

components, altimeter crossover differences have previously 

472 1 
been used to map the statistics of mesoscale sea level variability [eg, Cheney and Marsh, 1981; Gordon et ul., 19831. 

However, information contained in the crossover differences 

has not yet been exploited to examine the temporal history of 

sea level variations. 

The method is most easily explained if .we assume for the 

moment that the crossover differences are error-free. Effects of 

orbit error and measurement error will be discussed in later 

sections. Consider a small control area traversed by M 

ascending and N descending orbit tracks and assign to each 

track an unknown constant to represent an along-track 

average of sea level across the control area. We wish to make 

an estimate of the M + N unknown average sea levels (one for 

each track) from the M x N crossover differences. The problem is thus an overdetermined one and can be solved by a 

least squares procedure. To pose the problem mathematically, 

denote the M + N unknown along-track average sea levels by 

a,,, (m = 1, , M) and d, (n = 1, ... , N), where a,,, and d, are 

the sea levels for ascending and descending tracks, rcspectively. Define rmn to be the altimeter-measured sea level difference at the crossover point corresponding to the mth ascending and nth descending track (ascending minus descending). 

The sum of square differences between sea level changes measured by the altimeter at the crossover points and changes in 

true sea level averaged across the control area is then 

MN 

= 1 2 (am - dn - rmnY (1) nI=1 "=I 

The average sea ievels a, and d, can be determined by minimizing Z. The resulting equations for um and d, can be obtained by differentiating with respect to a, and d, to get 

N N 

Nan- Edj= Ermj (rn=l;..,M) (2) 

xai-Md,= Irk (n=l;-.,N) (3) 

J-1 j-I 

M M 

i= 1 i- 1 

Since the sum of the M equations of (2) is equal to the sum of 

the N equations of (3), (2) and (3) represent only (h' + M - 1) 

independent equations. This simply means that an arbitrary 

constant exists in the solution. This constant can be fixed by 

setting 

a, =o (4) 

Then the solution to (21, (3. and (4) is unique and represents 

the difiercnccs of a,,, and d, from a,: the unknown initial value 

of the sea level. By arranpng a, and d, in order of increasing 

time, a time series of sea level in the control area is then 

obtained. Note that if sea level is truly constant across the 

control area but temporally vaqkg between ground tracks, 

then a,,, - d, = r,,,,, with error-free measurements. 

It should be apparent that a good control area must met 

rwo requirements. FirsL the number of crossovers in the area 

must be high to ensure adequate temporal resolution for the 

resulting time series. The number of crossovers increases with 

the size of the control area. However, the second requirement 

is that the dimension of the control area be small compared 

with the typical scale of oceanic variability so that averapc sea 

level over the area is a meaningful quanrity. In application to 

real data, some compromise must be made between these two 

conflicting requirements. 

3. SEASAT ALTIMETER DATA AND CROSSOVERS 

To examine the ACC around the entire Southern Ocean, all 

Seasat 1-s average altimeter data from the Geophysical Data 

Records (GDR. see Lorell et ol. [1980]) between 40"s and 

65"s were used in this study. Most of the major errors had 

already been corrected in the GDR. These errors include tropospheric water vapor and ionospheric free electron range 

delays, tides. and the inverse barometer effect of atmospheric 

pressure loading (see Tapiey ef 01. [19S2a] for a detailed discussion). Before computing crossover differences. we applied 

three additional levels of processing to the data. First, a 

"glitch editing" procedure was performed to replace anomalous data values with their local medians across seven successive measurements. (The criterion for anomaly was a deviation of more than 2 m from the local median.) Second, a 

correciion for sea-state bias was made by adding 7% of the 

altimeter-measured H (significant wave height) to the sea 

surface height measurement [Born et al., 19823. Third, all data 

flagged for anomalous scatter in the I-s averages of returned 

radar power were eliminated. Such scatter occurs, for example, 

when there is ice or land in the footprint or side lobes of the 

altimeter antenna pattern. 

To facilitate computation of crossover differences; the data 

set was divided into 12 overlapping subsets, each covering 25" 

of latitude and 40" of longitude. Within each subset, crossover 

points were located using the method described by M. E. 

Parke and L. R. Stavert (unpublished manuscript, 1985). and 

the corresponding crossover differences were then computed. 

To minimize the effects of short tracks and anomalous data on 

the orbit error reduction procedure to be discussed in section 

4.1, TWO screening tests were applied to the computation: first, 

individual tracks with less than 10 crossovers were deleted 

before computing crossover differences; second, crossover dilferences exceeding 6 m were eliminated. A total of 125,222 

crossover points were thus located, resulting in a rms crossover difference of 146 cm over the Southern Ocean. This value 

is significantly lower than the global value of 180 cm (M. 

Parke, personal communication, 1984), primarily due to the 

relatively low orbit error over the Southern Ocean (M. E. 

Parke and L. R. Stavert, unpublished manuscript, 1985). 

Because Seasat was locked in a repeat orbit during the last 

month of the mission, the density of crossover points is highest 

at the intersections of the repeat tracks. These intersections 

are thus ideal locations for constructing se2 level time series in 

a control area as discussed in section 2. Displayed in Figure 1 

are the Seasat tracks in the vicinity of Drake Passage. The 

repeat-track intersections are indicated by open circles. After 

some experimentation with different sizes for the control area, 

we settled on a 200 x 200 km square around each repeattrack intersection. The exact location of the center of the control area was determined so as to maximize the number of 

crossovers in the square. We thus restrict attention to sea level 

variations with wavelengths longer than 400 km. Each control 

area thus determined contains a ~oup of crossovers and is 

hereafter referred to as a cluster area. A total of 229 cluster 

areas which yielded good quality data were identified around 

the South:m Ocean. Two sample cluster areas are shown by 

the open squares in Figure 1. The two solid circles indicate the 

locations of bottom pressure gauges that were deployed in 

early 1976 as part of the International Southern Ocean Studies (ISOS! program to monitor the variability of the ACC 

through Drake Passage [see Wearn and Baker, 19803. The two 

pressure gauges measured the bottom pressure at a depth of 

500 m on the continental slope through 1980, thus bracketing 

the Seasat mission from July 7, 1978, to October 10, 1978. The 

altimeter-derived sea level rime series in the two cluster areas 

will later be compared with the pressure gauge data. Before 

proceeding to the construction of time series in the cluster 

areas, we first discuss treatment of errors in the data. 

.- _I . .-. ~ 

.. . 

.. 

, .. 7. 

.. . . .. . .. !. 

.. 

.. 

.. .. 

.. 
45 

280 285 295 300 305 310 

-50 

-55 

-60 

-65 

the open squares 

4. ERROR REDUCTION AhD ANALYSIS 

Successful application of the method described in section 2 

to real data depends on the size of orbit error and measurement error in the crossover differences rM. Among the various 

errors, orbit error is by far the most signifcant The global 

rms orbit error in the Seasat GDR has been estimated to be 

150 cm [Topley et ai., 1982uJ. Since orbit errors are approximately uncorrelated from track to track this corresponds to a 

rms crossover Werene of 210 cm. These orbit errors would 

clearly obscure any oceanogaphic si-pals likely to be observed. Some method is thus required to redxe the size 0: 

orbit error. .4 method that has been successiullg used in a 

nuaber of past studies is discussed in detail below. This is 

then followed by a discussion of residual mevurement errors 

and their reduction by a smoothing scheme. 

4.1. Reducrion of Orbit Error 

Since its spatial scaie is large (the dominant wavelength is 

the circumference of the earth; see Topiey er oI. [198ta]). orbit 

error can be greatly reduced by a simple modeling efforr: Over 

a distance of a few thousand kilometers (the meridional dimension of the Southern Ocean), the orbit error along track i 

can be expressed as [see Ropp, 19791 

ei(t) = zi + Bit + yjr2 + higher order terms (9 

where t is time, ui, fli, and yi are parameters whose values are 

determined for each track so as to minimize the resulting 

crossover differences in a least squares sense over a given 

geographic ares. The least squares solution procedure used 

here is exactly the same as the one discussed in section 2 

except that the control area is now very large (larger than the 

scales of true sea level variability). The estimated orbit error is 

then subtracted from each track of data. This is now a standard technique for reducing orbit errors in altimetric measure- , 

ment. 

For studies of mesoscale varizbility, the first two or three 

terms in (5) have been used to model the orbit error [Cheney 

ond Morsh. 1981; Gordon er oi., 19831. Since the focus of thc 

present study is large-scale variability, only the first term, a 

constant, is used in (5). Inclusion of the linear and quadratic 

terms would remove any me sez level slope and curvature 

along track and hence eiiminate iarpe-scale geostrophic currents. which are the si-mals of interest in this study. It should 

be kept in mind, however, thzt the residual orbit error after 

removing only a constant bias (hereafter referred to as an 

orbit bias adjustment) from each track is higher than that 

resulting from also removing the higher-order terms in 151. It 

vi11 be shown later that residual orbit error can be further 

reduced by temporally smoothing the crossover difierences 

within a ciuster are& 

Applicatiom of the orbit bias adjustment to the 12 subsets 

of data (see section 3) reduced-the rms crossover diference 

from 146 to 32 cm averaged around the entire Am. Shown in 

Figure 2 are histograms of the crossover differences before and 

afier the orbit bias adjustment in one of the subset regions 

(4OoS-65"S, 180"-220"). The rms crossover difference in this 

particular re~on is146 cm before the adjustment and 34 cm 

after the adjustment. It is unlikely that the large residua! 

crossover difierences at the tails of the histogram (Figure 2. 

lower panel) were caused by large-scale sea level changes. 

Since these outliers would carry a large weight in determining 

.. 

. . .. 

. .. 

-50 

ORIGNAL @'AGE -85 

OF POOR QuAW 

-55 

-60 

-65 

280 285 290 265 30[1 305 31 0 

Fig. 1. Seasat altimeter data coverage in the Drake Passage area. Open circles represent intersections of 3-day repeat 

ground tracks. Solid circles denoic the locations of lSOS botiom pressure gauges. Two sample ciuster areas are shown b) 
12 

L 

2 

CR(?SS[?\IES D!FFERENCE (MI 

Fig. 2 Histograms of the crossover differences in the region 4o0S-65"S, 180'-220" before (upper panel) and after (lower 

panel) an orbit bias adjusrment. 

sea level from least squares solutions, we deleted all the residual crossover differences exceeding 60 cm (about two standard deviations). This resulted in deletion of about 6% of the 

data reducing the rms crossover differenct to 24 cm around 

the entire ACC. 

4.2. 

To determine si@ content of the residual crossover difierences, we investigated the residual errors contained in the 24 

cm rms variability. From the error budget for the Seasat altimerer data archived in the GDR [Tapley er a[., 1982aJ, the 

follouing major error souras for residual crossover differences were identified: resodual orbit erroi, instrument noise, 

sea-state related bias, range delay by ionospheric free electrons 

and tropospheric water vapor, sea level variations caused by 

ocean tides and atmospheric loading (inverse barometer eifects). The ma-mitude of each of these errors after applying the 

orbit bias adjustment of section 4.1 is discussed below. 

By applying the orbit bias adjustment to a simulated 

orbit error data set, we estimate the residual rms orbit error to 

be 10 cm. A detailed discussion of the method used to obtain 

this estimate is given in Appendix A. 

For I-s average data in the GDR, the rms instrument 

noise is about 5 cm [Tapley et al., 1982~1. Because this is 

A Budgerfor rhe Residual Errors 

1. 

2. 

essentially a white noise (i.e., uncorrelated from point to 

point). it is not reduced by the orbit bias adjustment, which 

only affects errors with spatial scales on the order of 1500 km 

(the size of the orbit bias adjustment control area). 

3. After the 7% correction for the sea-state bias, Born 

et al. [I9821 estimated the nns residual sea-state bias error to 

be 2% of HI,,,. The average measured by the Seasat 

altimeter over the Southern Ocean is about 4 m [see Chelron 

er a/., 19813. Therefore the rms sea-state bias error is estimated 

to be 8 cm. However, it should be kept in mind that the result 

of Born el al. [1982] was based on the average of a wide range 

of sea-state conditions at a number of geographical locations 

and might not be representative of the wLlter high sea-state 

conditions in the Southern Ocezn during the Seasat period. 

Therefore 8 cm may be an'underestimate, and the eiininated 

outliers of Figure 2 (lower panel) could in fact be due IO 

sea-state bias errors. However, we will assume that the 8-cm 

estimate is approximately correct. The scale of the sea-state 

bias is 500-1000 km, wxch is too short to be sipi5:antly 

reduced by the orbit bias adjustment. 

The ionospheric free electron range delay has been corrected in the GDR with an estimated rms residual error of 3 

cm [Lorell et al., 19823. Because this error is distributed over a 

wide range of spatial scales, it is not clear how much it can be 

reduced by the orbit bias adjustment. We therefore conserva4. 

48 

.. . 

., 

,... . .. . 

.s' 

. .. _- 

.. . 
49 

TABLE 1. Aliimetrv Error Budgets 

rms Magnitude. cm 

Before Afier 

Error Source Adjustmen; Adjustment 

Instrument noise 

Orbir 

Sea-slate bias 

Ionosphere free electrons 

Troposphere water vapor 

Atmospheric pressure loading 

Ocean tides 

Root-sum-squares 

5 

103 

6 

3 

4 

3 

10 

104 

5 

10 

8 

3 

4 

3 

4 

16 

tively estimate the residual error due to ionospheric effects to 

be 3 cm. 

The tropospheric water vapor range delay has been corrected in the GDR using (when available) water vapor 

measurements from a scanning multichannel microwave radiometer (SMMR) on board Seasat. The rms error of this correction is estimated to be about 3 cm [Tapley et nl., 198261. 

However. SMMR data are available for only about 80% of 

the altimeter measurements. The water vapor analyses from 

the Fleet Numerical Oceanography Center (FNOC) were used 

when SMMR data were missing. The correction using FNOC 

data has an estimated rms error of 6 cm [Tapley et ol., 198261. 

Therefore the combined rms error for the water vapor effect is 

approximately 4 cm. Because the scale of this error is only 

50-500 km. it is not significantly reduced by the orbit bias 

adjustment. 

Sea level variations due to atmospheric pressure loading 

have been corrected in the GDR by using FNOC sea level 

pressure analyses. The rms error of this correction is estimated 

to be 3 cm [Tapley et a/., 1982~1. Part of this error might have 

been removed by the orbit bias adjustment, since the scales of 

atmospheric pressure variations are of the order of several 

thousand kilometers. However, we have not attempted to estimate the reduction of error since the atmospneric mass field 

over the Southern Ocean is so poorly known. We therefore 

conservatively leave the residual error due to pressure efiects 

at 3 cm after the orbit bias adjustment. 

Tie global tide model of Schwiderski [1980] has been 

used to remove the tide-induced sea level changes from the 

GDR. The global rms error of this model is reputed to be of 

the order of 10 an Beczse the sdes of :he bzronopic a;ez~-. 

tides are of the order of several thousand kilometers, part of 

the tidal error is removed by the orbit bias adjustment. .4s B 

rough estimate of the amount by which tidal error can be 

reduced, we used the difierence between the model of Schsiderski and the model of Parke and Hehershort [1980] along 

Seasat tracks as 2 simulated tidal error data set We then 

appiied to this signal the bias adjustment discussed in section 

4.1. Tnis resulted in a reduction of the ms tidal error from 11 

to 4 cm. 

The error budgets before and after the orbit bias adjustment 

are tabulated in Table 1. The root-sum-squared total error 

after adjustment is about 16 cm. Since these errors are approximately uncorrelated from track to track, tl-’ ,is corresponds to an rms error of 22 cm for crossover difierences and 

accounts for 80% of the total variance of residual crossover 

diflerences. Assuming signal and noise are uncorrelated the 

signal-to-noise ratio for the residual crossover difierence is 

:nus only 0.2. Clearly, further processing is required to reduce 

5. 

6. 

7. 

errors to a level where temporally varying oceanic signals can 

be detected. 

4.3. A Smoothing Sclienie 

The errors listed in Table 1 have time scales ranging from a 

few hours to a few day. This is generaliy much shorter than 

the time scales of large-scale oceanic variability (1 month or 

greater). It is therefore possibie to temporally smooth those 

crossover differences that lie in a given cluster area to further 

reduce errors. In this section we present a smoothing technique which is analogous to a running average filter. 

A particular crossover difference is a function of two times 

and can thus be expressed as Di(ti, ii‘), where ii and ti’ are the 

times of ascending and descending tracks. respectively. When 

the two times of one crossover difference are close to those of 

another, the two crossover differences can be averaged to 

reduce random errors. In a given cluster area, we replaced the 

crossover diflerence Di by a smoothed value di defined by 

(6) 

The summation is over the Ni individual crossover differences 

Dj satisfying either of the following two conditions: 

Condition A 

Ilj - < T If,’ - ti’( < T 

Condition B 

Ifj - li’l < T 11,’ - fit < 7 

The coeflicients sj in (6) are either + 1 or - 1. The sign is 

determined as follows. The individual crossover differences 

were defined to be ascending minus descending altimeter 

measurements (see section 2). A temporal evolution of sea level 

is thus implicit in each crossover difference. The signs of the sj 

in (6) are chosen to preserve the temporal sense of the crossover difference Di. For example. if the ascending time of crossover i occurs after the descending time (Le.. ri > ti‘)! then all of 

the terms in the summation (6) must correspond to increasing 

time. For crossovers Dj for which tj > rj‘, the si-en of sj is thus 

positive. However. for crossovers Dj for wnich tj -= rj‘, tne sign 

of sj is negative. 

After experimenting with difierent values for T, we settled 

on a value of 10 days as a compromise between temporal 

resolution and error reduction. The smoothing scheme is then 

eqaivaien: to averaging over a two-dimensional (20 x 20 day) 

running window as shown schematically in Figure 3, which 

displzys the position in the t, I’ plane of the crossovers in a 

sample cluster area (595, lOO’W). For a given crossover denoted by the asterisk, the unbroken square represents the 

represents the window corresponding to condition B. 

Shown in Figure 4 is a histogram of Ni for all of the crossovers that fall in the $29 cluster 2reas around the Southern 

Ocean (a total of 65,989 crosscwrs). The average number of 

individual crossovers used to oorain a smoothed value is 24. 

These crossover differences are not all independent. When the 

number of crossovers is Izrge, ,Vi crossover points are (to a 

close approximation) the result of the intersections of h’”” 

ascending tracks and descending tracks. Thus the 

number of independent samples is approximately 2h’1’2, or 9.8 

on the average. One then expects that the smoothing scheme 

would reduce the rms error by a factor of (9.8)1’2 = 3.1, resulting in an rms error of 7 cm for the smoothed crossover 

window corresponding to condition A, and the dashed square ... 

.. 
290 

27@ 

$ 

& 

I 

t 

250 

0 

z 

0 

3 

230 

0 

In 

LL: - 

i 

210 

I90 

xxx 

xxx 

xxx 

XXX 

xxx 

XI* 

XXX 

xxx 

xxx 

xx 

xx 

xxx 

xx 

xx 

xxx 

xx 

xxx 

xx 

FU AYI) CHELTOS: ALTIMETRIC OBSERVATIONS OF OCEAN CURREhTS 

xx xxx xx X X X 

xx xx x xxx xxxx xx xx xxx xx 

xx xx x xxx xxxx xx xx xxx xx 

xxx xxxx xx xx xxx xx 

xI(xxxxxx 

xxxxxxxx 

xxxxxxxx 

xxxxxxxx 

XYXXXXXX 

xxxxxxxx 

xxxxxxxx 

xxxxxxxx 

xxxxxxxx 

x-x-x-x-x-x-x* 

?xxxxxxy 

~xxxxxxk 

kxxxxxxk 

L------' 

?xxxxxxy 

xxxxxxxx 

xxxxxxxx 

xxxxxxxx 

xxxxxxxx 

xxxxxxxx 

I I,,,, I, 

190 210 230 250 270 290 

QSCENDING TIME 

Fig. 3. Locations of crossovers in the plane spanned by their 

ascending and descending times (in day numbers of 1978) in a sample 

cluster area (59"s. 100"W). The two squares represent the two 

smoothing windows for the crossover denoted by the asterisk. 

differences. This corresponds to an rms error of 5 cm for 

altimetric measurement of sea level. A more rigorous error 

estimate gives a value of 4.6 cm (see Appendix B). 

Note from Figure 1 that within a cluster area a piven 

ascending track does not always intersect all the descending 

tracks (and vice versa). In other words, the intersections of 

some of the ascending and descending tracks traversing the 

same cluster area fall outside its 200 x 200 km boundary. 

Therefore the number of crossovers along an individual track 

differs from track to track in a cluster area, making the sea 

level estimate for each track differently constrained in the least 

squares solutions given by (2) and (3). The estimated sea level 

would be more reliable for those tracks with the greatest 

number of crossovers. Computation of a smoothed crossover 

difference di by (6), however, can be performed for any desired 

pair of ascending and descending times z and t'. regardless of 

whether such a crossover actually exists. From the set of 

ascending and descending ground track times in a cluster area, 

we computed a smoothed crossover difference between all possible pairs of ascending and descending ground track times, 

even when the corresponding intersection fell outside the cluster boundary. Consequently, the number of smoothed crossover differences used as inputs to (2) and (3) is the same for all 

tracks. resulting in equal reliability for all sea level estimates in 

a cluster area. The total number of smoothed crossover differences in a cluster area is thus always somewhat greater than 

that of the actual raw crossover differences. 

By applying this smoothing scheme to all 229 cluster areas, 

the rms residual crossover difference was reduced from 24 to 

10 cm. Since the residual error has been estimated to be 7 cm, 

the signal-to-noise ratio is increased to about 1 (assuming 

signal and noise are uncorrelated). The effect of smoothing on 

the crossover differences is illustrated in Figure 5, which shows 

sample scatter plots of crossover difference versus time difference between ascending and descending tracks both before 

and after smoothing. The sample cluster area chosen is the 

one shown in Figure 1 by the square near the southern side of 

Drake Passage. Before smoothing, the crossover differences 

are dominated by errors and appear to be random with no 

systematic dependence on time difference. After smoothing, a 

general increase of crossover difference with time difference 

can be clearly seen, suggesting that a low-frequency signal has 

emerged from the noise. The smoothed residual crossover dirferences at the 229 cluster areas formed the data base for 

constructing sea level time series using the method described 

in section 2. Results at the two cluster areas shown in Figure 

1, and their comparisons with the bottom pressure gauge data 

are discussed in the next section. 

, 

5. SEA LEVEL TIME SERIES 

The ability to extract true sea level signal from noisy 

measurements using the methods described in sections 3 and 

4.3 is investigated in this section using both altimeter and in 

situ data in the vicinity of Drake Passage. We first examine 

the performance of the methods by applying them to errorfree known signals. As known signals, we have chosen the 

Wearn and Baker pressure gauge data on the north and south 

sides of Drake Passage as shown in Figure 1. After demonstrating the success of the methods on known signals, we 

apply the methods to actual Seasat altimeter data at cluster 

areas near the bottom pressuie gauges and cornpait the results with the bottom pressure time series. 

NUMBER OF CROSSOVERS 

I 

1 .. 

. .- 

, .- 

Fig. 4. Histogram of the number of crossovers averaged in each smoothed crossover difierence computed over all the 

cluster areas. 
W io W 2 15 I 

X R 

X 

xx X 

X xx 

"X 

xxx 

X X 

X xX 

X 

X X 

x xxxx ,?(x 

X xx *x xx x xx xx xx 

xx~x" xx X xx 

EX *x* xI(y X * I I I Y IXXt I 

10 20 30 40 50 60 70 80 

TIME DIFFERENCE [DAYS] 

B 

xx xx 

rx xx 

V 

I 

30 40 50 60 70 BO 0 

TIME OiFfEilENCE [DRYS) 

SC2tKer plots of crossover difierence versus the corrcsponding tCmDOr2l separation beiort (Upper panel) and after (lower panel) 

smoothing. Xote the difierence in vertical scale between the two 

Fig 5. 

panels. 

5.1. Test Solutions 

It is useful to denonstrate the reconstruction of a known 

time series from knowledge of its time-lagged difierences (analogous to noise-free altimeter crossover differences). As two 

examples -of known signals, we used the Wearn and Baker 

pressure data which are shown in Figure 6 by the thin lines 

(with the mean values removed). If the bottom pressure variations are indicative of bzrotropic ocean signals, 1 mbar 

change in bottom pressure is equivalent to approximately 1 

cm change in sea level. It is not important here whether or not 

the bottom pressure data represent true sea level changes or 

whether the bottom pressure measurements are noisy. We 

simply wish to determine how well a specified signal (with 

- 

time scales similar to those likely to exist in the altimeter data) 

can be recovered using the crossover difference technique described in section 2 and the smoothing technique described in 

section 4.3. 

A set of "crossover differences" was first generated by sampling tne raw pressure data at the times when Seasat fiew over 

the enter of each track in the two cluster areas indicated in 

Figure 1. These crossover differences were then smoothed 

using the 20-day smoothing technique described in section 4.3 

and used as inputs to (2) and (3). The resulting estimated 

bottom pressure time series are shown in Figure 6 by the open 

circles. The sampling intervals of the time series are irregular 

due to the irregular overpass times of the altimeter. The 

20-day smoothed "altimeter sampled" data can be compared 

with the true 20-day running mean of the raw data (denoted 

by the heavy lines in Figure 6). For each test case, the single 

arbitrary constant in the solution was determined by minimizing the squares of the differences between the altimeter 

sampled data and the low-passed raw data. The results are 

quite satisfying. As expected, the reconstructed time series 

closely resemble the true 20-day running mean of the original 

time series. 

5.2. 

The same technique was applied to the actual altimeter 

measurements in the two cluster areas shown by the open 

squares in Figure 1. The estimated sea level time series obtained by applying (2) and (3) to the smoothed crossover diL 

rerences are shown by the crosses in Figure 7. Comparisons 

with the low-passed bottom pressure data (solid lines) will be 

discussed in section 5.3. 

The error bars in Figure 7 denote an rms error of 54.6 cm, 

which was obtained from a study in which the effects of the 

errors listed in Table 1 were simulated in the reconstruction of 

a given known time series (see Appendix B for details). Note 

that the expected error estimated from the more rigorous 

analysis in Appendix B differs little from the crude 5-cm estimate given in section 4.3. 

The presence of a small-amplitude bimodal osciliation with 

approximately a 3-day period (e.g., the early part of the record 

at the northern cluster and the latter part of the record at the 

southern cluster) is caused by the effects of the smoothing 

scheme on aitimeter measurement errors and can be explained 

as follows. Over a given cluster area, the altimeter data are 

characterized by an irregularly spaced series of ascending and 

descending orbit pairs separated in time by about li2 day. 

Successive pairs of passes are about 3 days apan exccpt for 

the existence of data gaps. Thus the closely spaced points 

constituting the bimodal oscillation in Figure 7 alternate between ascending and descending orbits. For crossovers separated by less than a day, the true sea level differene should be 

sma:' However, the crossovc~ differences are sometimes large 

becz-se of altimeter measurement errors. The crossover differences associated with successive pairs of samples satisfy condirion A in section 4.3 and are therefore averaged together to 

produce smoothed crossover difierences from which the sea 

level estimates are computed. Thus the effect of an anomalous 

crossover difference spreads over the 20-day smoothing 

window. This can result in a finite and slowly varying offset 

between sea level estimates from ascending and descending 

orbits, which accounts for the bimodal oscillatory behavior 

found for some of the clusters. In section 6.2, gappy time series 

like those shown in Figure 7 are interpolated to a uniform 

time grid using an objective technique which eliminares the 

erroneous oscillatory behavior (see also Appendix C). 

Altimetrically Derived Sea Level Time Series 

I 

.- 

I. 

i.. - I .' .- . 

.. 
c 

Fk ANI) CHELTO\. ALTMETRIC OISFR~ATIOSS OF OCEA~ CURRE\TS 

52 

-I2 ILO 140 2bD 2'1D 2;D 2;D 2;D 2;D 2kD 2;O 2'aO 2'90 

YEAR DRY OF 1978 

I 

J 

W 

W '0 

* -6 

I 

I 

I 

-I2 IbO I90 20C 2;O 2;O 230 240 250 260 270 260 240 

YERR DRY OF 1978 

Fig. 6. Thin lines represent daily bottom pressure (in centimeters of its equivalent barotropic sea level change) at the 

southern (lower panel) and northern (upper panel) side of Drake Passage. Heavy lines represent a 20-day running average 

of the daily pressure record. Open circles show the reconstructed pressure record at altimeter overpass times using the 

method described in the text. 

53. Comparison With Bottom Pressure hf easuremenfs 

Ideally, we would like to compare the sea level time series 

derived from altimetry with in situ sea level measurements. 

Unfortunately. such measurements do not exist. To our 

knowledge. the only time series measurements which can be 

readily compared with the Seasat altimeter results in the 

Southern Ocean were made by the two aforementioned 

bottom pressure gauges deployed at a depth of 500 m on 

either side of Drake Passage (see Figure 1). As noted earlier, 

variation in bottom pressure is proportional to variation in 

sea level when the water movement is barotropic. However, in 

the presence of baroclinic motions, these two measurements 

can difier considerably. Sea level and bottom pressure can also 

difier by the simple fact that the bottom pressure gauges and 

altimeter cluster areas are not exactly collocated (see Figure 

1). Nevertheless, we feel that a comparison between the two 

measurements can still shed some light on the vzlidity of the 

altimetric results. 

As shown in Figure 7, there is fair agreement between the 

sea level and low-passed bottom pressure time series in the 

southern region of Drake Passage (lower panel), whereas the 

two time series difier substantially in the northern region 

(upper panel). This geographic difference can be rationalized 

in terms of baroclinic variability. The currents in Drake Passage are concentrated along several baroclinic fronts [e.g., 

Kodin and Clifford, 19821. The cross-stream migration of 

these fronts can cause large baroclinic si-gnals in sea level measured at a fixed location. In addition, the energetic eddy field 

in this region 2!so has a si-pificmt barochic component In 

general, the total fiow variability (frontal migration plus 

eddies) in Drake Passage has been shown from in situ 

measurements to increase northward [Kowlin et al, 1981). 

resulting in a larger discrqancy between sea level and bottom 

pressure in the northern region. In addition, the greater distance between the cluster area and the pressure gauge in the 

nonnern region could also amomi for part of the observed 

discrepancy. 

The better agreement between observed sea level and 

bottom pressure in the southern region suggests that the Aow 

variability there is predominantly barotropic and that the altimetric sea level time series is a vaiid measure of true sea level 

variations to the extent of esrimated measurement errors. 

6. TEMPORAL VARIABILITY OF THE 

ANTARCTIC CIRCUMPOLAR cURREh7 

The crossover difference technique described in section 2 

was applied to the smoothed crossover difierences at each of 

the 129 cluster areas around the Southern Ocean to obtain 
129 estimated sea level time series. The geographical distribution of the 229 cluster areas is shown in Figure 8 along with 

the average sea surface dynamic topography relative to lo00 

dbar assembled from 50 years of snip data [Gordori et ai.. 

I%'l]. The iarge-scale variability of the ACC is well sampled 

by the altimetric ciuster areas. The sea level variabiiity at these 

219 cluster areas over the 3-month Seasat mission is descrimd 

in this section in terms of empirical orthogonal functions 

(EOF's). EOFs provide an effcient summary of the covariability over a large array of time series. Their use in data 

analysis is now a standard practice in oceanography (see Davis 

[1976] for a thorough discussion). A very brief summary of 

the formalism is given in section 6.1. The technique is applied 

to the 229 sea level time series in section 6.2, and the results 

are then discussed in section 6.3 in terms of geostrophic velocitY. 

6.1. Empirical Orthogonal Functions 

Let qm(t) represent the altimetrically measured sea level at 

time t in cluster area m. We wish to express q,(t) at the cluster 

locations x,. rn = 1, . . . , M, in terms of M orthonormal functions Fk(x,) by 

M 

qm(t) = ak(t)Fk(x,) rn = 1, '' ' 9 M (7) I;= 1 

xxx 

I T *X x xx 

:< xxx 

xx x 

- 

X6 

u 

J 

J 

W 

A 

'0 

a 

W 

* -6 

-12 

For this study, M = 229. EOFS are uniquely defined among 

the large number of possible orthonormal functions by the 

constraint that the corresoonding amplitudes a, satisfy 

(a,ft)a,(r)) = A,6, (8) 

where the braces are used to denote sample mean value 

averaged over t, A,, is a constant, and 6, is the Kronecker 

delta. It can be shown that EOFs are the eigenvectors of the 

M x M mean cross product matrix C with elements 

Cij = {qi(t)qj(t)J (9) 

As a consequence of the orthonormality of the EOFs, the 

amplitude time series associated with the kth EOF is computed from the sample observations by 

The total mean square variability. summed over the M cluster 

points, is apportioned among the M modes such that the 

fraction of total mean square variability accounted for by 

mode k is 

xxxxx 

X 

YERR DqY Gf 1978 

Fig. 7. Estimaies (denoted by crosses) of sea level variation from Seasat altimetric crossover differences at the southern 

cluster area (lower panel) and northern cluster area (upper panel) in Drake Passage ISEX Figure 1 for IocaLions). Solid lines 

represent the 20-day running average of the correspondme bottom pressure record. 
EU AXl) CHELTQt: ALTIMETRIC OBSERVATIONS OF 0CkAn.i CURRENT5 54 

.- 

240' 

150" 330" 

60' 

long-term average dynamic height (in dynamic meters) relative to lo00 dbar [from Gordon er ai., 19821. 

Fig. 8. Locations of the 229 Seasat altimeter crossover cluster areas in the Southern Ocean. Contours represent the 

r. 

The virtue of EOFs over other possible orthonormal expansions is their eficiency of representation. Suppose the 

EOFs are ordered by decreasing {ak'([)}. Then no other expansion of the ofrm 

x 

~~'(r) = bdt)Gdx,) k=1 

in terms of K < M functions G,(q,) gives lower total mean 

square error 

M 

,E = 1 iCtl,(t) - Vm'(0I2} m= 1 

than is obtained when the G,kJ are the EOFs [Duck, 19761. 

Moreover, when the variability is dominated by large-scale 

coherent signals, only a small number of modes are requred 

to represent most of the variability. 

63. Sea Level Vuriabiliry 

To carry out an EOF analysis of the sea level time series, 

the objective analysis technique described by Brerhmron et a/. 

[1976) was used to interpolate the irregularly spaced qm(r) to a 

uniformly spaced time -mid. With uniform sampling, the mean 

cross-product matrix C- (see (9)) can easily be computed. Details of the objective interpolation are discussed in Appendix 

C. The interpolated time series were constructed at daily intervals beginning July 7, 1978 (day 188), the first day of Seasat 

altimeter data reception, and ending October 10, 197s (day 

283), the last day of Seasat operation. The sea level time series 

at each cluster point was constructed to yield changes in sea 

level relative to day 188. The EOF computation reveals that 

approximately 99% of the total mean square variability of the 

229 time series can be represented by the first three EOFs 

which account Tor 73.5%, 16.3%, and 8.7%, respectively, of 

the total mean square variability. Only the first EOF is discussed here; the spatial scales of the second and third EOFs 

are smaller and are considered to be dominated by errors. 

The first EOF of sea level variability is contoured in Figure 

9. Sea level rises and falls together in regions where the sign of 

the EOF is the same. The magnitude of the EOF value indicates the intensity of variability. Large spatial scales of the 

variability are clearly evident. The amplitude time series of the 

first EOF (Figure 10) is characterized by a secular increase 

over the 3-month observation period. It is apparent from 

Figure 9 that mulitplication of the first EOF by its amplitude 

time series indicates a general decrease in sea level around the 

southern side of the Southern Ocean (negative areas near Antarctica) and a general increase in sea level to the north. The 

total sea level change in centimeters from the beginning to the 

end of the Seasat mission can be obtained by multiplying the 

EOF value by 7, the total chage in the amplitude time series. 

Since gradients of sea level reflect geostrophic surface velocities (see section 6.3). this pattern of sea level variation implies 

a general eastward acceleration of the ACC around Antarctica. This is represented in Figure 9 by the generally eastward 

arrows south of 50"s in the core of the ACC. EOF analysis 

thus suggests that the increase in ACC mass transport observed at Drake Passage [Weurn und Baker, 19803 is a local 

manifestation of a large-scale, eastward acceleration of the 

ACC. The eastward acceleiation is notably disrupted in the 

three regions defined by longitude sectors 230"-270", 330"-0)"- 

20", and 60"-80". Flow variations over these regions are discussed in section 6.3 in terms of possible topographic influences. 

North of the core of the ACC. the scales of variability are 

smaller, refiecting. the possible influence of the subtropical 

. 
55 

150' 

240' 

I 

330" 

90" 30" 

60" 

I 

Fig. 9. The first EOF of sea level variability. Areas of negative values are stippled. Directions of the corresponding 

change in surface geostrophic velocity are indicated by the arrows. 

convergence zones and the southern boundaries of subtropical 

ewes. As an example, the large localized sea level increase 

southeast of South America (300"-330") is probably caused by 

an intensification of the southeastward extension of the Bradl 

Current [see Reid et 01, 19773. Similarly, the large localized 

sea level increase south of Australia (11Oo-13O0) probably refiects an intensification of the Leeuwin Current as it turns 

eastward around the southwest corner of Australia [Cresswell 

and Golding, 1980; Legeckis and Cresswell, 19811. 

Since the nature of sea level variability is so simple (essentially a secular trend over the 96-day Seasat mission). the 

character of variability indicated by EOF analysis can be determined by an alternative simple method. Differencing sea 

level from day 188 to day 283 at the 229 ciuster points yields a 

pattern of total sea level change that is virtually identical to 

the spatial EOF shown in Figure 9 [see Fu and Chelron, 

19843. The advantage of the EOF analysis is that it provides 

an amplitude time series (Figure 10) from which the temporal 

evolution of large-scale sea ievel variability can be examined 

in greater detail. 

Although the first EOF accounts for a large fraction 

(73.5%) of the total mean square variability, it IS useful to 

examine the local representativeness of the EOF. The percentage of variability accounted for in each cluster area by the first 

-2 I I I I I I I I I I 

180 190 200 210 220 230 240 250 260 270 280 290 

YERR DRY OF 1978 

Fig. 10. Amplitude time series of the first EOF of sea level variability. 
56 

150' 

30" 

I 

60" 

Fig. 11. Contour plot of the percentage of sea level variability accounted for ai each cluster region by the fil rst EOF. 

EOF is contoured in Figure 11. Using the notation of section 

6.1, the function plotted is 

P(xd = {a, 2(r)Jr, 2(X,)/{s,2(t)} 

which can be considered a measure of representativeness of 

the EOF; the EOF is most representative of the sea level 

variability in regions where the value of P(x,) is high. As the 

map clearly illustrates, the first EOF is a good representation 

of sea level variability in the core region of the ACC (south of 

50"S), accounting for &80% of the sipd in each cluster 

area 

6.3. Geostrophic Velocity Variability 

Over periods longer than 20 days, sea level variations (after 

inverse barometric correction) essentially refiect variations in 

surface geostrophic currents [Wunsch, 19721. Sea level 

measurements can thus be used to compute directly the variability of surface geostrophic velocity. Denote the net 96-day 

. change in surface geostrophic vel~city associated with the first 

EOF by 6u (eastward) and 61: (norihward). Tnen 

--- (6H) f CY 

gz 6t: = - - (6H) f zx 

where bH is the net 96-day change in sea level associated with 

the first EOF, g is the gravitational constant, f is the Coriolis 

parameter (negative in the Southern Hemisphere), and x and y 

are the east-west and north-south coordinates, respectively. To 

compute 6u and 60, 6H was first interpolated to a uniform 

4" x 4" grid using a two-dimensional cubic spline fit to the sea 

level changes associared with the first EOF at the 229 cluster 

points. Then 6u and 6u were evaluated at the grid point from 

the resulting spline coeflicients. Figure 12 shows the resulting 

net 96-day vector change in surface geostrophic velocity associated with the first EOF. The 4" x 4" grid points are located at the tails of each vector. Superimposed on the figure 

are the 0°C and 5°C isotherms at 100-m depth (shown by 

dashed and solid lines, respectively) taken from Gordon et ol. 

C1982, plate 11). These subsurface isotherms represent ap 

proximate boundaries of the polar frontal zone which contains 

the high-speed core of the ACC (T. Whitworth, personal communication, 1981). Areas with iepth las than 3000 m are 

stippled in Figure 12. The 3000-m contours were subjectively 

smoothed from Gordon et d. [1982, plate 2). 

The generally eastward acceleration of the ACC discussed 

in section 6.2 is clearly evident in Figure 12. As noted earlier, 

there are three regions where the eastward acceleration is disrupted These disruptions are apparently associated sith topographic features. In the sector oT longitudes 330"-0"-20', the 

acceleration becoma southward at M, downstream from a 

region of rough topopaphy comprising the meridionally oncnted South Sandwich Island Chain and Trench to the east of 

Drake Passage. This southward acceleration is probably a . , 

manifestation of a lee wave generated by the topogTaphic fezchange is westward, indicating a deceleration of the ACC over 

the 96-day Seasat mission. This deceleration may be related to 

the upstream southwind acceleration at 340". The southward 

acceieration occurs near the regon where the Weddell Sea 

-- nvre derives its inflow [e.g., Gordon and Huber, 19841, indicating a possible intensification of the gyre during the Seasat 

observation period. 

Between longitudes 240" and 270", a change from southward acceleration to northward acceleration is found along 

the 0°C isotherm, and a generally westward acceleration (deceleration of the ACC) occurs along the S'C isotherm. These 

.. 

.. 

tures. Between longitudes 350" and 20', the net velocity .. 
57 

240" 

. . . 

150 330' 

2.0 

- 

emlrec 

Fig. 12. Net vector change in surface geostrophic velocity associated with the first EOF of sea level variability. Also 

shown are the 0°C (dashed line) and 5°C (solid line) isotherms at 100-m depth [from Gordon er al., 19823. Areas shallower 

than 3OOO m are stippied. 

features also may be manifestations of a lee wave generated by 

the upstream Pacific-Antarctic Ridge. 

The third disruption of the eastward acceleration of the core 

of the ACC is in the region between 60" and 80". In this region 

there is a large area of northward acceleration apparently 

caused by the obstruction of the ACC by the Kerguelen Plateau 

Since the flow variability (Figure 9 and 12) has substantial 

meridional component in some regions, it is useful to aetermine whether the extent of the meridional migration of the 

ACC axis as suggested from the altimeter data is reasonable. 

From Figure E, the magnitude of the altimetrically measured 

velociry change is generally less than 2 ems. much smaller 

than the 20-40 cmis backgound surface velocity typically observed in this region [Whinvorrh er aZ., 19821. From the ratio 

of the velocity change to the mean backgound velocity and 

the zonal spatial scale of the variabilitg, we can roughly estimate the meridional scale of cross-stream deflection due to the 

flow variability. A schematic diapam of initial (straight line) 

and final (curved iine) paths of a streamline is shown in Figure 

13. The cross-stream deflection scale (denoted by d) is thus 

related to the along-stream scale of the variability (denoted by 

L) by the following relation: 

I L------l 

Fig. 13. Schematic diagram showing the relationship between the 

sparial scale of cross-stream defiecuon (d) and the magnitudes (c. c', 

and 6~) and along-stream spaIial scaie (L) of velocity vanabihty. 

d 6v 

Lv 

-2- 

where 6t. and t are the magnitudes of the velocity change and 

the mean background velocity, respectively. For example, consider a streamline roughly along the 5°C isotherm in the 

sector of longitudes 320'4'-20'. The cross-strezq conponent 

of the velocity difierence changes from northward to southward over a distance of about 4700 km (60' in longitude at 

45"s). so the corresponding L is about 2300 km. If we assume 

I: = 20 cm/s and 6r = 2 cmk then d 2 230 km. which is conparable to the findings of Bowen and Sromrnel [I9713 based on 

1 gear's repeated Discoverj I1 sections along 0" and 20'E 

longitudes. We conclude that the meandering of the core of 

the ACC deduced from the altimeter data is not unreasonable. 

7. SUMhIARY AhD COSCLUSIOS 

A method has been developed to consma sea ievel time 

series from crossover difierences of satellite altimetric 

measurements at geographical locations with high density of 

ascending and descending orbit ground track intersections 

(cluster areas). An orbit bias adjustment is first employed to 

reduce the dominant. long-wavelength orbit error by removing from altimetric measurements a constant bias over a distance of a few thousand kilometers. Residual crossover difierences are then computed for each cluster location. To further 

reduce high-frequency residual errors, the sequence of crossover difierences at each particular cluster location are 

smoothed using a two-dimensional running window in the 

two-time domain spanned by the ascending and descending 

_. 

_. 
58 

2 

4c 

CT 

1@ 

C?(?S5(?\/ER G!FFESEtqCE IVj 

Fig. 14. Histograms of the crossover differences resultins from simulated orbit error before (upper panel) and after 

(lower panel) an orbit bias adjustment. 

orbit times of the crossovers. The resulting crossover differences are then treated as measurements of time-lagged sea 

level difierences, from which an irregularly spaad time series 

of s:a level variation is estimated by a least squares technique. 

Finally, an evenly spaced sea level time series is constructed at 

each cluster location from the irregularly spaced samples using 

optimal interpolation. 

Tie method is applied to Seasat data over the Southern 

Ocean to study large-scale temporal variability of the ACC 

during the Seasat mission (July 7, 1978, to October 10, 1978). 

Due to the fact that the Seasat orbits were nonrepeating 

(except for the last 30 days of the mission), clusters of crossover differences are sampled in a finite area of 200 x 200 km 

to assure adequate temporal coverage. In a given cluster area, 

the crossover differences are smoothed over a 20 x 20 day 

running window to reduce the ma-mitude of measurement 

errors to-a marginally useful value of 5 cm (rms). Hence the 

resulting altimetrically measured sea level time series for each 

cluster location represents a spatial and temporal average of 

the true sea level variations. 

The altimetric results were compared with in situ bottom 

pressure measurements made by pressure gauges deployed at 

a depth of 500 m on the continental slope on either side of 

Drake Passage. The comparison shows fair agreement between the two measurements at the southern side of Drake 

Passage, but substantial disagreement is found at the northern 

- 

side. We believe that this is due to a greater presence on the 

northern side of the ACC of barociinic variability which 

causes the differences between sea level and bottom pressure 

measurements. 

A total of 229 time series were computed from the Seasat 

altimeter data, covering the entire ACC regon from 40"s to 

65"s. An EOF analysis of the time series indicates that 73.5% 

of the total variability during the 96-day Seasat mission can 

be accounted foi by the first EOF, which is characterized by a 

general decrease in sea level around the southern side of the 

ACC and a general increase in sea level to the north. The 

corresponding change in surface geostrophic velocity indicates 

a generally eastward acceleration of the ACC. Substantial meridional components of the acceleration are obstrved over 

major topopraphic features. The small magnitude of the largescale meridional velocity change (about 2 cm/s over the 3- 

month mission) indicates that the extent of meridional migrations of the ACC axis during the Seasat mission are, at most, a 

few hundred kilometers. 

The results presented here constitute the first direct observational evidence for large-scale coherence in the temporal 

variability or the ACC. Although the accuracy and statistical 

significance of the results are questionable due to the short 

duration and substantial measurement errors of the Seasat 

altimeter, the results have demonstrated the great potential of 

the methodology in application to more accurate and longer 

;- , . 

.. 

.. 

__ .- 

. .. . 

.~ 

.._ 
59 

FU AND CHELTON : ALTIMETRIC cMSERVATlONS OF OCEAN CURREh' 

K-5 t 

X 

-lo llg0 140 2bo 2;o 2;o 2;o 2h0 2kO 260 2'lo 2'So 2'90 

YERR DRY OF 1978 

Fig. 15. An example showing reconstruction of the Wcarn and Baker pressure record on the north side of Drake 

Passage using error-lree (circles) and contaminated (crosses) crossover differences sampled at altimeter overpass times. 

duration altimetric missions expected in the near future. For 

example, the projected rms accuracy for the altimeter proposed for the TOPEX mission [see TOPEX Science Working 

Group, 19811 is 14 cm, an order of magnitude improvement 

over Seasat. The TOPEX mission design calls for global altimeter measurements over a 3-year period. Using the method 

introduced here, we will be able to obtain from TOPEX altimetry a global network of sea level time series with much 

improved accuracy for studying temporal variability of the 

world ocean circulation. 

8. APPENDIX A: REDUCTION OF ORBIT ERROR 

BY A BIAS ADJUSTMENT 

As discussed in section 4.1, the rms crossover difference of 

the Seaszt altimeter data between 40"s to 65"s is 146 cm 

before the application of an orbit bias adjustment. To a close 

approximation (within a few centimeters), the crossover difierence is attributable to orbit error. In this appendix we investigzte the extent to which an orbit error of the given ma-enitudc 

can be reduced by the bias adjustment discussed in section 4.1. 

The approach is to apply the bias adjustment to a simulated 

orbit error field. 

Since the dominant frequency of orbit error is once per 

revolution [Tapley er al., 1982a], the orbit error along a given 

track can be modeled as A sin (For + P). where A is an amplitude, Fo is the once-per-revolution frequency (1/101 min-' for 

Seasat), f is time, and P is a phase angle. Since orbit error is 

essentially random from track to track, A and P can be approximately modeled as random numbers with specified probability distribution functions (pdf). The pdf of P was simply 

assumed to be uniform between 0 and 2~. After experimenting 

with different pdfs for A, we found that a uniform pdf between 

0 and a maximum value A, resulted in a crossover difference 

hstogam with shape resembling that shown in Figure 2. To 

make the simulated rms crossover difference close to the observed value of 146 cm, A, was chosen to be 253 cm. We 

applied this orbit error model to the same sample region used 

in section 4.1. The histogam of the resulting crossover difierI I I I I 

8 16 24 32 40 -10 

LAG [DRYS) 

Fig. 16. Sample erroi autocovariance function estimated from simulations. The solid line represents a least squares fit by 

an analytical function (see text). 
FL AND CHELTOY: ALTlHCTRlC OBSERVATlO\S OF OCEAh CI'RRE'ZTS 

i 

60 

I00 

75 

50 

25 

0 

-25 

I I I f E, 

e 16 24 32 40 -50 

LRG [DRYS] 

Fig. 17. Sample sea-level autocovariance functions estimated from altimetrically derived sea level time series with 

variance in the range 0-20 cm2 (circles). 2WO cm2 (crosses), 40-60 cm2 (triangles), and greater than 60 cm2 (asterisks). 

Solid lines represent least squares fits by analytical functions (see text). 

ences is shown in Figure 14 (upper panel). The standard deviation is 145 cm, the skewness is 0.02, and the kurtosis is 0.99. 

For comparison, the corresponding parameters for the real 

data in Figure 2 (upper panel) are 146 cm. -0.02, and 0.89, 

respectively. 

After application of the orbit bias adjustment (as discussed 

in section 4.1) to the simulated orbit error, the rms crossover 

difference was reduced from 145 to 14.2 cm. A histogram of 

the residual crossover differences is shown in the lower panel 

o! Figure 14. Since the orbit biases of the two tracks in the 

crossover difierence are essentially uncorrelated. an estimate of 

the residual orbit error is thus 14.2 cm divided by 21'2, or 10 

an. 

9. APPEXDIX B: ERROR ESTJMATE FOR THE ALTIMETRICALLY 

DERIVED SEA LEVEL TIME SERIES 

In section 4.3. the rrns error of the smoothed crossover 

difierences wa crudely estimated to be 7 cm. This estimate, 

when divided by 2'" (Le., 5 cm), can be considered a rougb 

estimate for the rms error of the altimetrically derived sea level 

time series obtained through (2) and f3). In this appendix a 

more rigorous error estimare is made from a simulation study 

in which the various eirors listed in Table ! are izrlude:! in 

the generation of simulated time series. In addition to an ms 

error estimate. an esrimate of the error autocovariance function is also obtained. This error autocovariance is required for 

obtaining interpolated sea level time series by optimal estimation scheme (see Appendix C). 

The residual errors after the orbit bias adjustment of section 

4.1 (set Table 1) can be grouped into thret categories: (1) 

poini-to-point random: this category includes only instrument 

noise and has an rms value of 5 cm. (2) Track-to-track 

random: this category includes residual orbit error! sea-srate 

bias, ionosphere and tropospheric range delay, and atmospheric pressure loading. Although not all of these - .r rors are 

strictly random from track to track, their decorrelation time 

scales are believed to be shorter than a week. The root-sumsquares error from these sources is 14.1 cm. (3) Track-to-track 

systematic; this category includes only the tidal error, which is 

estimated to be 4 cm after orbit bias adjustment (see section 

4.2). 

Tie method used to estimate the statistical characteristics of 

: 

the errors can be summarized as follows. First, an error-free 

time series was obtained for a given cluster area by solving (2) 

and (3) using smoothed crossover differences sampled at altimeter overpass times from the Wearn and Baker bottom pressure record on the north side of Drake Passage. The procedure is exactly the same as that used to generate the simulated time series in Figure 6. Then a "contaminated" time 

series was generated by adding simulated errors. The northern 

Wearn and Baker pressure record was first resampled with 

added errors from all the three categories defined above. To 

simulate type 1 errors. a random number with a Gaussian pdf 

with a standard deviation of 5 cm was added to each sampled 

value. To stimulate type 2 errors, a random number with a 

Gaussian pdl with a standard deviation of 14.1 cm was added 

to each sampled value (sampled at the center of each track). 

Finally, to simulate type 3 errors (tidal error), a sinusoidal 

signal with an amplitude of 5.7 cm (corresponding to an rms 

signal of 4 cm), a frequ:ncy of 1j12.49 h-' (the Mz tidal 

frequency), and an arbitrary (but fixed for all sample times) 

phase was added to each sampied value. With these errors 

added, crossover differences were computed as before. After 

smoothing using the method described in section 4.3, these 

contaminated crossover differences were then used to g. Oneiate 

a Contaminated time series for the cluster area using (2) and 

(3). The difierence between the two time series is thus a simulared error time series. An example showing error free and 

contaminated time series are displayed in Figure 15. 

Such simulations were performed at 110 clusters to obtain 

an ensemble of simulated error time series from which a 

sample erro: autocovariance function was obtained (Figure 

16). The autocovariance funcrion was computed at lags wirh 

daily intervals. Because the error time series were irregularly 

spaced, the autocovariance was computed through a binning 

process. The value at each daily lag was an average within a 

1-day bin centered at the lag. As shown in Figure 16 the 

zero-lag autocovariance (i.e., the error vzriance) was 20.7 cm2 

and the first zero-crossing of the autocovariance function was 

at about 16 days. Thus the rms error for the altimetric time 

series is 4.6 cm. in close agreement with the rough estimate of 

5 cm in section 4.3. This corresponds to an rms error of 6.5 cm 

in crossover difierences. The finite lag zero crossing is primarily a result of the smoothing of a random noise. 
61 

Fu AND CHLLMK: ALTIMETRIC OBSERVAT~ONS OF OCWN CURREWS 

. 

TABLE C1. Parameters for R(7) 

Variance Ejumber of 

Category Range. an2 Clusters Q.m2 P.d-' 

I a- 20 99 9 0.166 

I1 Xc40 63 26 0.137 

I11 MI-60 39 46 0.121 

I v 60- 28 80 0.116 

To be useful for the optimal interpolation discussed in Appendix C, the error autocovariance function was modeled by 

the four-parameter, positive definite (see Appendix C for definition) analytical function 

E(T) = Db(7) + A exp (-ET) cos (CT) (B1) 

where 7 is the lag and b(~) represents a special function for the 

contribution from random errors and is defined to be unity at 

7 = 0 and zero elsewhere. The parameters A, E, and C were 

obtained by fitting the exponential-damped cosine function, 

represented by the second term of (Bl), to the sample error 

autocovariance function through a nonlinear least squares 

procedure. Then D was obtained as the difierence between the 

variance and A. The resulting estimates of the parameters 

were D = 2.4 an2, A = 18.3 an2, B = 0.0389 d-', C = 0.102 

d-'. E(7) is shown by the solid line in Figure 16. 

* 

' 

10. APPENDIX c: OPTIMAL INTERPOLATION OF AN 

IRREGULARLY SPACED TIME SERIES 

There are many methods which can be used to interpolate 

an irregularly spaced time series to a regular time grid. All 

methods should produce similar results; if the interpolation 

method makes a great deal of difference in the final time series, 

then the process under investigation is probably undersampled temporally. The method we adopt is an objective scheme 

based on optimal estimation theory. It is preferable over other 

techniques because it provides a measure of the accuracy of 

each interpolated estimate based on statistical information 

about the si-mal and measurement error. 

Optimal estimation was first used by meteorologists to generate synoptic maps of atmospheric variability [see Gandin, 

1965; Alaka and Elcander, 19721. The method has been ap 

plied to oceanographic data by Bretherton er ai. [1976). who 

have also thorou-phly reviewed the subject. Only a brief account of the method is given here with specific emphasis on 

application to the altimetric sea level time series. 

Consider a set of 5 imperfect measurements of sea level at 

cluster area m. Use h,,,(fi) to denote the true sea level and qmitJ 

the corresponding imperfect measurement at time ti, i = 1, N. 

Then 

(C1) 

where &,(ti) is the measurement erroi. For simplicity in the 

notation that follows, we will drop the subscript rn with the 

understanding that an interpolated time series must be generated for each of the 129 cluster points. An estimate of sea level 

at time t can be constructed from all A' imperfect measurements by 

%(ti) = uti) + t,(rJ 

N 

h(t) = ZiV(ti) 

i= 1 

Note that the estimate A(t) is biased unless both the true value 

h(r) and the measurements V(ti) have zero mean value. This is 

easily seen by noting that the bias of the estimate is 

N 

(h(t) - Mt)) = 1 Zi<q(ti)> - (h(i)) is1 

Here angle brackets are used to denote the true expected value 

over a hypothetical infinite ensemble of realizations. The true 

ensemble averages (h(t)) and (q(ii)) are not, in general, 

known. To minimize this bias, the sample mean of the 

measurements q(ri) was removed prior to constructing the OFtimal estimate by (C2r. This sample mean was then added 

back to the estimate h(t) to obtain the final interpolated time 

series. 

in general, the estimate (C2) should be constructed from 

only those measurements within a specified time window 

about the estimation time r in order to iimit the size of the 

matrix to be inverted in forming the optimal estimate (see 

below). For the application here, the number of irregularly 

spaced measurements was small enough that all of the 

measurements at a particular cluster point were used to construct the estimates. 

The expected square error of the estimate (C2) is 

N NN 

<Ch(t) - h(t)12> = (h2(t)> - 2 

where Ai and Dij, defined by 

aiAi + 1 1 aiDi,aj (C3) in 1 1-1 1-1 

are the elements of an N x 1 vector (denoted by A), and an 

N x N matrix (denoted by D), respectively. The optimal estimate h(t) is obtained when the expected square error is minimized with respect to the parameters z,. Thic. minimum is 

obtained by setting the derivative of (C3) with respect to ai 

equal to zero which gives the N equations (one Cor each i) 

A' 

Ai= Cz,Dij 

J= 1 

Solving these equations for the N parameters ai gives 

N 

si = 1 Dij-'Aj (C6) J'1 

where Dij-' is the i, jth element of the inverse of matrix D. 

Note that (C5) simplifies the expression (C3) for the expected 

square error of the estimate h(r) to 

N 

([h(r) - i;(t)]2) = <i?(t)) - ZiAi (C7) 

,=I 

It is apparent from (C2), (C61, and (0) th? the sez level 

estimate h(t) and its expected square error require knowledge 

of the vector A and matrix D. If we aSsume that the measurement errors &(til are uncorrelated with the signal h(t) and that 

the statistics are stationary, then 

where R(7) and €IT) are the signal and error autocovariance 

functions defintd by 

R(r) = (h(t)h(r + T)) 

E(7) = <E(t)E(1 - 7)) 

The true expected values within angle brackets in (C9) must be 

estimated from sample statistics. Since the sample measurements are irregularly spaced, it is necessary to approximate 

R(7) and €(r) by continuous analytical functions in order to 

compute Ai for general samplc time r and interpolation time 

I,. These analytical representations must satisfy the very im- 

.. 
62 

1 I I 

180 196 20G 210 220 230 24b 25C 260 270 280 290 .IO I 

TERR DRY OF 1978 

-30 IbO I90 200 2;c' 220 230 2aO 25C 260 2;O 2bG 230 

YERR G2T OF 1978 

** j I 

i2G 1SO 2C3 2lC 220 23: 240 250 260 270 280 290 -1 0 

YEQR DRY OF 1978 

Fig. 18. Interpolated sea level time series at three sample cluster areas. The venical bars represent error estimates 

obraincd from the optimal interpolation technique (the square root of (C7)). 

portant constraint that they be positive definite functions (i.e.. 

the corresponding autocovariance matrix be positive definite 

for any arbitrary set of sampling lags) in order to assure that 

the mean square estimate 

Kh 

<C2(r)> = 1 1 aiDi,aj i-1 j-1 

h" NN 

= C SriR(ri - rj)zj + C 2 Sr,E(t, - tj)zj 

is nonnegative. Failure to satisfy this constraint can lead to 

very bad estimates by (C2). A function of lag T is assured of 

J'1 1-1 J'1 

being positive definite if its Fourier transform is everywhere 

positive. This corresponds to positive spectral density at all 

frequencies. We used this criterion for positive definiteness in 

selecting proper functional representations for R(7) and E(T). 

The functional form for E(7) has already been given in Appendix B. The analytical form for R(7) was determined as 

follows. We used the 229 altimetric sea level time series to 

estimate R(r). First, a sample autocovariance function was 

computed at daily lags for each irregularly spaced time series 

using the same binning procedure as that used for estimating 

E(T). Because the statistics of the sea level time series were not 

spatially homogeneous, we divided the 229 time series into 

.. 
63 

four categories according to their mean square variability he., 

the variance) and computed an average autocovariance function for each category. The results are shown by different 

svmbols in Figure 17. The definition of the four categories and 

the number of their corresponding samples are given in Table 

C1. Note from Figure 17, that the sample autocovariances for 

categories I and I1 are much less noisy than those of catepories 111 and IV due to the fact that the statistics were compiled 

over a larger number of clusters. Each sample autocovariance 

function was fitted (through a nonlinear least squares procedure) by a positive-definite analytical function of the form # 

R(r) = Q sin (Pr)/(Pr) (C10) 

The coeficients P and Q for each category are also given in 

Table C1. The resulting R(r) are shown by continuous lines in 

Figure 17. 

Using R(T) and E(7) represented by (ClO) and (Bl), respectively, optimal estimates of sea level along with corresponding 

error estimates were constructed for the 229 time series using 

(C2), (C6), and (C7) at daily intervals from July 7, 1978 (day 

188), to October 10, 1978 (day 283). Each interpolated series 

was computed relative to a sea level value of zero at day 188. 

Three examples are shown in Figure 18. The upper two panels 

represent typical cases, while the bottom panel shows an extreme case in which the measurement errors are exceptionally 

large. The interpolation performs fairly well even for the extreme case. At sample observation times, the estimated value 

is exactly equal to the measured value when the measurement 

error E is zero. However, for nonzero measurement error, this 

is not true in general. The deviation of interpolated values 

from sample observations is thus due to the effect of finite 

measurement errors. For the typical cases (upper two panels 

in Figure 18), the deviation is consistent with the estimated 

error bars. 

Acknowledpmenrs. We benefited from many useful discussions 

with M. E. Parite of the Jet Propulsion Laboratory. Comments by C. 

N. K. Mooers of the Naval Postgraduate School on the manuscript are also appreciated. The ISOS pressure record used in the study was 

provided by R. B. Wearn of the University of Washington. The re- search described in this paper was carried out bv the Jet Propulsion 

Laboratory, California lnstitute of Technology, and by Oregon State 

.Giversity under contract with the National Aeronautics and Space 

Administration. 

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D. B. Chelton, College of Oceanography, Oregon State University, 

L.-L. Fu, Jet Propulsion Laboratory, California Institute of TechCorvallis, OR 97331. 

nology, 4800 Oak Grove Drive, Pasadena. CA 91 109. 

(Received September 18, 1984; 

accepted October 27, 1984.) 

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