卫星高度计与地球科学：技术与应用手册

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"Satellite Altimetry and Earth Sciences-Lee.Lueng.Fu" 《卫星测高与地球科学》是国际地球物理学系列中的第69卷，由李-卢恩·傅（Lee-Lueng Fu）和安妮·卡森纳维（Anne Cazenave）编辑。这本书详细介绍了卫星测高技术及其在地球科学中的应用，涵盖了海洋学、大地测量学和地球物理学等领域。卫星测高技术起源于20世纪60年代的人造卫星时代，它通过在太空中的雷达测高仪对全球海面形状进行频繁测量。这种技术具有革命性的意义，因为它能够快速获取全球海洋的地形信息，而这是传统船舶测量无法实现的。海洋的环流、温度和盐度时刻在变化，船只需要数周或数月才能横渡大洋进行测量，因此无法实时全面地观测全球海洋。卫星测高技术的出现为海洋学家提供了一种独特工具，可以绘制全球海洋地形图，从而研究海洋环流及其随时间的变化。它对于理解洋流模式、海平面变化、气候变化以及地球动力学过程至关重要。此外，卫星测高数据还能用于监测冰川消融、海岸线变化、台风和飓风活动，以及评估全球变暖对海洋生态系统的影响。在大地测量学方面，卫星测高技术有助于精确测定地球的形状和地壳运动，这对于地震学、板块构造研究以及地壳均衡理论有重要贡献。同时，它在地球物理学中也有广泛应用，例如研究海洋重力场、地磁场和地球内部结构。该书详尽阐述了卫星测高的原理、数据处理方法和各种应用技术，为科研人员、工程师和学生提供了宝贵的参考资料。书中可能包括了数据校正、误差分析、时间序列分析、数据融合等关键技术的讨论，以及如何将这些数据应用于气候模型、海洋环流模型的构建和改进。《卫星测高与地球科学》是一部综合性的手册，不仅详细介绍了卫星测高技术，还展示了其在地球科学研究中的广泛和深远影响。通过这本书，读者可以深入了解这一领域的最新进展和技术挑战，以及它如何塑造我们对地球系统动态的理解。

1. SATELLITE ALTIMETRY 3

measurement imprecision can be reduced to less than 1 cm

by averaging the measurements over along-track distances

of ~ 100 km. Other sources of measurement error such as

corrections for atmospheric refraction and sea-state biases in

the range estimates are of much greater concern since they

typically have length scales on the order of 100 km or longer.

Through a concerted effort devoted to algorithm improve-

ment, the root-sum-of-squares of these various measurement

errors has been reduced to about 2.7 cm (see Table 11 in

Section 8.3). Equally important, there has been a parallel im-

provement in precision orbit determination that has reduced

root-mean-square errors of estimated orbit height H from

10 m for GEOS-3 to 2.5 cm for TOPEX/POSEIDON (see

Table 1). The overall root-sum-of-squares measurement ac-

curacy for the TOPEX/POSEIDON dual-frequency altime-

ter estimates of sea-surface height is therefore about 4 cm

(see Section 8.3).

One of the purposes of this chapter is to pay tribute to the

success of the intensive effort that has been devoted to the

technical details of altimetry over the past 2 decades. The

present ~4 cm state-of-the-art overall accuracy of the sea-

surface height estimates h has been achieved through ma-

jor technological advancements in precision orbit determi-

nation and a dedicated effort to improve each of more than

40 sensor and geophysical algorithms. This attention to algo-

rithm improvements has transformed altimetry from a semi-

quantitative measurement of the sea-surface height for which

the distinction between measurement errors and geophysical

signals was sometimes difficult to discern, to a highly quan-

titative measure of sea-surface height variability that is pro-

viding insight into the wide range of dynamical processes

summarized in later chapters. Moreover, a major benefit of

the high degree of accuracy that has been achieved with the

TOPEX/POSEIDON dual-frequency altimeter is that it is no

longer essential for users to be deeply versed in all of the id-

iosyncrasies of satellite altimetry. Altimetry has thus become

a standard tool for oceanographic research.

The interest in ever more subtle signals with smaller-

amplitude sea-surface height signatures in the altimeter data

demands a thorough understanding of the errors of each of

the corrections applied to altimeter data. As various algo-

rithms have improved, errors in other algorithms that were

once considered of secondary concern have become the fo-

cus of attention. Each of the corrections that must be applied

for accurate determination of the sea-surface height by satel-

lite altimetry is described from basic principles in this chap-

ter. Although full wavenumber-frequency descriptions of the

various measurement errors are not yet available, the magni-

tudes of the errors of each of the corrections can be estimated

as described herein. If not taken into consideration, some of

these errors might be mistakenly interpreted as variations of

the sea surface height.

As summarized above and shown schematically in Fig-

ure 1, accurate determination of the orbit height H is an

integral part of altimetric determination of the sea-surface

height h. Indeed, the improvement in orbit accuracy by more

than two orders of magnitude over the past decade is the pri-

mary reason for the improved accuracy and utility of altime-

ter data. A detailed discussion of the evolution of precision

orbit determination is therefore included as Section 4 in this

chapter.

Accurate estimates of R and H are not sufficient for

oceanographic applications of altimeter range measure-

ments. The sea-surface height given by Eq. (2) relative to

the reference ellipsoid is the superposition of a number of

geophysical effects. In addition to the dynamic effects of

geostrophic ocean currents that are of primary interest for

oceanographic applications (see Chapters 2, 3, and 4), h

is affected by undulations of the geoid

h g

about the ellip-

soidal approximation (Chapter 10), tidal height variations

h r (Chapter 6), and the ocean surface response

at-

mospheric pressure loading (Chapter 2). These effects on

the sea-surface height must be modeled and removed from

h in order to investigate the effects of geostrophic currents

on the sea surface height field. The geoid undulations, tidal

height and atmospheric pressure loading contributions to h

are briefly described here in Section 5; more detailed discus-

sions are given in later chapters. The dynamic sea-surface

height is thus estimated as

hd = h - hg - hr -- ha

= H-R+ZARj-hg-hr-ha.

(3)

While complicating altimetric estimation of the range R,

the alteration of the incident radar pulse by the rough sea

surface can be utilized to extract other geophysical informa-

tion from the radar returns. In particular, the significant wave

height 3 can be estimated from the shape of the returned sig-

nal as described in Sections 2.4.2 and 6. In addition, since the

sea-surface roughness is highly correlated with near-surface

winds, the wind speed can be estimated from the power of

the returned signal as described in Section 7.

The emphasis throughout this chapter is on the correction

algorithms applied to the dual-frequency altimeter onboard

the TOPEX/POSEIDON satellite (referred to hereafter as

T/P). This state-of-the-art altimeter sets the standard for fu-

ture altimeter missions since it is significantly more accurate

than any of the other altimeters that have been launched to

date. After discussions of the various elements of satellite

estimation of the sea surface height as outlined above, the

chapter concludes with a summary of the T/P mission de-

3As discussed in Sections 2.4.1 and 6, the significant wave height is a

traditional characterization of the wave height field that is approximately

equal to four times the standard deviation of the wave heights within the

altimeter footprint.

SATELLITE ALTIMETRY AND EARTH SCIENCES

sign and an assessment of the performance of the T/P dual-

frequency altimeter in Section 8. An overview of future al-

timeter missions is given in Section 9.

2. RADAR MEASUREMENT PRINCIPLES

Because of a fortuitous combination of physical prop-

erties of the atmosphere and the sea surface, the fre-

quencies most well suited to satellite altimetry fall within

the microwave frequency range of 2-18 GHz. Accord-

ing to the frequency band allocations defined in Sec-

tion 1-3.2 of Ulaby

et al.

(1981), this encompasses S-band

(1.55-4.20 GHz), C-band (4.20-5.75 GHz), X-band (5.75-

10.9 GHz), and Ku-band (10.9-22.0 GHz) radar frequen-

cies. Graybody emission of electromagnetic radiation from

the sea surface is very weak and the reflectivity of water

is high in this frequency band, thus allowing easy distinc-

tion between radar return and natural emission. At frequen-

cies higher than 18 GHz, atmospheric attenuation rapidly

increases, thus decreasing the power of the transmitted sig-

nal that reaches the sea surface and the reflected signal that

is received by the altimeter. At lower frequencies, Faraday

rotation and refraction of electromagnetic radiation by the

ionosphere increase and interference increases from ground-

based civilian and military sources of electromagnetic radia-

tion related to communications, navigation, and radar. In ad-

dition, practical design constraints on the size of spaceborne

antennas set a lower limit on the frequencies useful for satel-

lite altimetry. The antenna footprint size on the sea surface is

proportional to the wavelength of the electromagnetic radia-

tion and inversely proportional to the antenna size. A beam-

limited footprint diameter of 5 km, for example, would re-

quire an impractically large antenna diameter of about 7.7 m

for the Ku-band microwave frequency of 13.6 GHz at the

T/P orbit height of 1336 km [see Eq. (22) in Section 2.4].

Spatial resolution of altimetric measurements from smaller

antennas is enhanced by a technique known as pulse com-

pression. These and other principles of radar altimetry are

summarized in this section.

measured by the radar thus depends upon the scattering char-

acter of the sea surface, the parameters of the radar system,

and two-way attenuation by the intervening atmosphere.

The physical factors that affect the signal received by the

altimeter can be understood by considering a radar antenna

illuminating a footprint area A f on the sea surface from a

height R above the sea surface at a pointing angle 0 with

corresponding slant range

Ro,

as shown in Figure 2. Be-

cause of the curvature of the sea surface owing to the el-

lipsoidal shape of the earth and the presence of local geoid

undulations, there is a distinction between the pointing angle

and the antenna incidence angle (see Figure 2). The antenna

pointing angle 0 is the angle between the boresight of the

antenna (i.e., the peak of the antenna gain) and the line from

the satellite center of mass to the point on the earth's surface

that is vertically beneath the satellite (referred to as the nadir

point). The incidence angle 0' is the angle between the line

of propagation and the normal to the sea surface at the point

of incidence. For the small incidence angles less than ,~ 1 o

relevant to altimetry and the small geoid slopes that only

very rarely exceed 10 -4 (Brenner

et al.,

1990; Minster

et al.,

1993), the two angles are very nearly identical. Likewise, R

and

are very nearly identical for altimetry.

The parameters of the radar system are the wavelength

~. of the transmitted and received electromagnetic radiation,

the transmitted power

in watts, and the transmitting and

receiving antenna gains

and

Gr.

Since altimeter systems

use the same antenna to transmit and receive the radar pulses,

and

are

essentially the same and both can therefore

be denoted simply by G. The backscattered power from a

2.1. Normalized Radar Cross Section

The relationship between the power of the signal trans-

mitted by a radar altimeter and the backscattered power that

is received by the altimeter is fundamentally important to

altimetry. The electromagnetic radiation that is transmitted

toward the sea surface is attenuated by the intervening at-

mosphere. The signal that reaches the sea surface is partly

absorbed by seawater and partly scattered from the rough

sea surface over a wide range of directions. The power that

is reflected back to the altimeter is then attenuated by the

intervening atmosphere. The power of the returned signal

FIGURE 2 Schematic representation of satellite measurements of the

radar return from the approximately ellipsoidal sea surface with large-scale

geoid undulations. The angles 0 and 0 t are the antenna pointing angle and

incidence angle, respectively; R is the satellite altitude above the nadir

point;

is the slant range of the radar measurement at pointing angle 0;

and A

is the antenna footprint area.

1. SATELLITE ALTIMETRY 5

differential area

d A

within the antenna footprint on the sea

surface is related to X, Pt and the range R from the satellite

to the target area by

G2xZ P t

d Pr -- t 2 (47r)3

R 4

O', (4)

where the proportionality factor cr has units of area and

t)~ (R, 0) is the atmospheric transmittance, defined to be the

fraction of electromagnetic radiation at wavelength X that

is transmitted through the atmosphere from an altitude R

at off-nadir angle 0. The two-way transmittance is t 2 (R, 0).

A detailed derivation of Eq. (4) can be found in Section 7.1

of Ulaby

et al. (1986a).

This so-called radar equation is

equivalent to characterizing the backscattered power from

the location

d A

as having been isotropically scattered by a

fictitious sphere with cross-sectional area or. The proportion-

ality factor cr is thus referred to as the radar cross section of

the differential target area dA.

For oceanographic applications, the total returned power

received by the radar antenna is backscattered from a dis-

tributed target (the rough sea surface) over the area

A f

that

is illuminated by the antenna, rather than from a point-source

target within the antenna footprint. In this case,

dPr

and o-

are meaningful only when they are considered to be the av-

erage quantities over all of the differential target areas

within the field of view. The scattering properties of the

sea surface within the antenna footprint are characterized

by the differential radar cross section per unit area, which

is denoted as o0 and is related to the radar cross section by

cr = crod A.

In general, the dimensionless quantity or0 (re-

ferred to as the normalized radar cross section and expressed

in decibels) varies spatially over the antenna footprint. The

total returned power is obtained by integrating over the re-

turns from each differential area illuminated by the antenna,

X2 pt s

Pr -- t 2 (47r)BR4

G2ryodA.

(5)

This expression assumes that the antenna footprint area is

small enough that R and t;~ can be considered constant over

the footprint. The antenna gain is retained inside of the in-

tegral since G depends on off-axis angle and, to a lesser ex-

tent, azimuthal angle and therefore varies across the antenna

footprint area.

With the exception of o0, each quantity on the right side

of Eq. (5) is either a known parameter of the radar system

(G,)~, and

Pt)

or a physical parameter of the medium be-

tween the altimeter and the target (R and &). If the normal-

ized radar cross section o0 is spatially homogeneous over the

antenna footprint or is considered to be the average differen-

tial cross section per unit area over the antenna footprint,

then it can be passed through the integral on the right side

of Eq. (5). Then Eq. (5) can be rearranged to express or0 in

terms of the measured returned power

and the parameters

of the radar system. The radar equation can then be written

in the more useful form

(47r)3R 4

Pr,

(6)

rYO - t2G2X2Aeff p t

where Go is the boresight antenna gain and Aeff is the

"effective footprint area" defined to be the area integral

G2(O)/G 2

over the angles 0 subtended by the antenna

beamwidth. In Section 2.4.2, a distinction is made between

the footprint area within the beamwidth of the antenna and

the area within this footprint that is actually illuminated by

the radar.

Since all of the quantities in the multiplicative factor on

the right side of Eq. (6) are known parameters of the radar

system or can be determined from the measurement geome-

try, the returned power

and hence cr0 depend only on the

radar scattering characteristics (the "roughness") of the tar-

get area. The sea-surface roughness increases with increas-

ing wind speed. At the small incidence angles relevant to

satellite altimetry, Pr and therefore or0 decrease monotoni-

cally with increasing wind speed. Near-surface wind speed

can therefore be inferred from altimetric measurements of

the radar return. Expressed in decibels, a typical altimetric

measurement of o0 is about 11 dB. It is shown in Section 7

that or0 decreases from 20 dB at very low wind speed to about

5 dB at a wind speed of 30 m sec -1 (see Figure 56).

An important property of or0 can be deduced from Eq. (6).

If the scattering characteristics of the sea surface are statis-

tically homogeneous over the antenna footprint, then o0 is

insensitive to the distance between the radar antenna and the

sea surface. This is easily seen by noting that the radiant flux

density of electromagnetic radiation emanating from a point

source falls off as the square of the distance from the source.

The transmitted power that reaches the sea surface there-

fore decreases as R -2. Likewise, the backscattered power

that reaches the satellite also falls off as R -2. The power

received by the radar therefore decreases as R -4 from

spreading loss, in accord with Eq. (5). The resulting R -4 de-

pendence of

is exactly offset by the R -4 dependence in

the numerator of Eq. (6). Thus, assuming that proper cor-

rections for atmospheric transmittance & (R, 0) are applied,

measurements of or0 are independent of the altitude of the

measurement. Aircraft measurements of o0 are therefore the

same as those made from a satellite at any altitude, 4 as long

as the scattering properties of the sea surface are equivalent

over the footprints of the different radars.

4As noted by Rodrfguez (1988) and Chelton

et al.

(1989), cr 0 measured

by a satellite altimeter must be adjusted to account for the approximate

spherical shape of the earth. Depending on the orbit height R, this results in

a 10-20% reduction of cr 0 compared with the flat-earth approximation (see

Section 2.4.1).

SATELLITE ALTIMETRY AND EARTH SCIENCES

2.2. Ocean Surface Reflectivity

Thorough discussions of the theory of radar backscatter

from the sea surface can be found in Ulaby

et al. (1986a,

1986b). A brief overview is given here. The phenomena re-

sponsible for radar return from a rough sea surface are rea-

sonably well understood theoretically. The solution for the

radar backscatter can be obtained based on the principles

of physical optics, for which the scattered electromagnetic

fields are determined from the electromagnetic current dis-

tribution induced on the sea surface by the incident radar

signal. Alternatively, the solution can be obtained based on

geometrical optics, for which the wavelength of the radar

signal is considered to vanish and the electromagnetic radi-

ation is treated as a bundle of rays. Both approaches lead

to essentially the same result. At the small incidence angles

relevant to altimetry, Barrick (1968) and Barrick and Peake

(1968) showed that the radar return measured at the satellite

consists of the total specular return from all of the mirror-like

facets oriented perpendicular to the incident radiation within

the antenna footprint. The normalized radar cross section for

small pointing angle 0 is given by

cro(O ,~ O) -- 7rp2(O~

(y)lfx=fy=O,

(7)

where ( is the sea-surface height relative to mean sea level,

p(0 ~ is the Fresnel reflectivity for normal incidence angle

(defined to be the fraction of incident radiation that is re-

flected) and

p(gx, (y)

is the joint probability density func-

tion of long-wave sea surface slopes (x and (y in two or-

thogonal directions. Here "long wave" means the portion of

the ocean wave spectrum with wavelengths longer than that

of the radar signal (approximately 2.2 cm for the 13.6 GHz

primary frequency of the T/P dual-frequency altimeter). The

joint probability density function

p(gx, (y)

in Eq. (7) is eval-

uated at (x = (y = 0 for the specular scatterers.

The radar return at small incidence angles thus depends

explicitly on the spectral characteristics of long waves on

the sea surface and on the reflectivity of sea water. It also

depends on the radar frequency through the implicit depen-

dence of p(0 ~ on frequency. The reflectivity for normal in-

cidence angle is shown in Figure 3 as a function of frequency

for fresh water. (The presence of dissolved salts introduces

minor changes to the reflectivity only at frequencies lower

than about 1 GHz.) It is apparent that water is about an or-

der of magnitude more reflective at microwave frequencies

than at visible or infrared frequencies. For a given antenna

beamwidth, the much lower power requirement for a radar

transmitter at microwave frequencies than at infrared or vis-

ible frequencies is a distinct advantage for radar measure-

ments at small incidence angles.

To a first degree of approximation, the long-wave sea-

surface height distribution can be modeled as isotropic with

a Gaussian probability density function. Then Eq. (7) be-

comes

p2(0~ sec40 exp (

tan2 0)

o-0(0 ~0)-- 2s 2 9 2s 2 ,

(8)

where

S 2

is the mean square of the sea surface slopes. The

radar return at small incidence angles thus decreases with

increasing roughness of the sea surface (i.e., increasing s2).

Physically, a greater fraction of the incident radiation re-

flects specularly in directions away from the radar receiver.

Since the sea surface roughness increases with increasing

wind speed, or0 at small incidence angles is inversely re-

lated to wind speed. For a specified spectral density of the

sea-surface elevation, or0 can be computed accurately from

theory. What is required in order to determine wind speed

from measurements of or0 is a theoretical relation between

the wind field and the ocean wave spectrum.

1.00

t--

"o 0.10

.--- 0.01

Frequency (GHz)

106 105 104 103 102 10 1

Pure

Wat~

~inf Visible

Near Thermal Middle Far

rared Infrared Infrared . !nfrarecl .4* Microwave

200nm 1 gm 10gm 100gm 1 mm 1 cm 10cm 1 rn

Wavelength

FIGURE 3 The reflectivity at normal incidence angle for fresh water at 25~ for a smooth surface as a function of

frequency and the corresponding wavelength of the electromagnetic radiation. (From Maul, 1985. With permission.)

1. SATELLITE ALTIMETRY

At present, the relation between wind speed and the

ocean wave spectrum is not well understood. Indeed, the

mechanism for transfer of energy from the wind to ocean

waves is one of the important unsolved problems in phys-

ical oceanography. The input of wind energy to short cap-

illary and gravity-capillary waves depends on the velocity

profile near the sea surface and on wind-induced turbulence

in the air. The surface-wave amplitude is thus presumably re-

lated in some way to the friction velocity or the wind stress.

Although significant progress has been made in relating the

wave spectrum to the wind (e.g., Plant, 1986; Donelan and

Pierson, 1987), the most accurate models relating or0 to the

near-surface wind are still purely empirical. These models

express o0 as a function of the wind speed at a reference

height of 10 m above the sea surface (see Section 7).

2.3. Atmospheric Attenuation

2.3.1. Clear-Sky Attenuation

The transmittance tk for cloud-free subpolar, midlatitude,

and tropical atmospheres is shown in Figure 4 as a func-

1.0

, ..... I ' I' ' I ' I '

......,.. .. ........ ,o.

0.4-

' I t

0.2

0.0 , I , I ,, ,

0 40

80 120 160

F-

I ~ I ' I i

..........

GG:

..........

q/ ,

..I-. ~,,..-. J_._ , I

200 240 280

Frequency (GHz)

,-- -' ......... J ....... ' I ' I ' I

-- ~'~

..... Polar |

~\~. ."" ..........

9

0.9

""'"~

~ iv ,,------4

E= o8-

- ~ /

! Tropical

- ~ i

10 20 30 40

Frequency (GHz)

0.7

0.6

FIGURE 4 The transmittance at normal incidence angle for cloud-free

subpolar (dotted line), midlatitude (solid line) and tropical (dashed line)

atmospheres as a function of frequency. The frequency range 0 to 300 GHz

(after Grody, 1976) is shown in the upper panel and an enlargement for

the frequency range 0 to 40 GHz (data courtesy of E Wentz, based on the

atmospheric model of Liebe, 1985) is shown in the lower panel.

tion of frequency at incidence angle 0' = 0 ~ (generally re-

ferred to as normal incidence) for microwave frequencies

between 1 and 300 GHz. The one-way attenuation is cor-

respondingly defined as (1 - tk). The salient features of Fig-

ure 4 are a moderately strong water vapor absorption line

centered at 22.235 GHz, a strong oxygen absorption band

between 50 and 70 GHz, a strong oxygen absorption line at

118.75 GHz, and a strong water vapor absorption band cen-

tered at 183.31 GHz. The oxygen and water vapor molecules

that contribute to the attenuation are almost entirely con-

fined to the troposphere, which extends to altitudes of less

than 10 km at midlatitudes and about 18 km in the tropics

(see, for example, Figure 1.10 of Wallace and Hobbs, 1977).

From the differences between the curves in Figure 4 for a dry

subpolar atmosphere and a moist tropical atmosphere, it is

apparent that atmospheric transmittance generally decreases

with increasing frequency, owing primarily to the presence

of water vapor.

At the Ku-band frequency of 13.6 GHz that is the primary

frequency of the dual-frequency T/P altimeter, the clear-sky

one-way transmittance at normal incidence angle is seldom

less than 0.96 (see Figure 4), even in a moist, tropical at-

mosphere. From the radar Eq. (5), the power of the received

signal is determined by the two-way transmittance t 2. The

corresponding two-way attenuation (1 - t 2) at 13.6 GHz is

therefore generally less than 8%. At the secondary C-band

frequency of 5.3 GHz used to correct for ionospheric re-

fraction of the T/P range estimates (see Section 3.1.3), the

clear-sky one-way transmittance exceeds 0.98 (see Figure 4).

The clear-sky atmospheric attenuation at 5.3 GHz is thus less

than half that at 13.6 GHz.

The total attenuation of electromagnetic radiation with

wavelength k is characterized by the opacity (also referred

to as the optical thickness) of the atmosphere along the prop-

agation path. The opacity v~ is related to the transmittance

& by

tk - e -Tx (9)

Since the transmittance is the fraction of electromagnetic ra-

diation that is transmitted through the atmosphere, it is ap-

parent from Eq. (9) that the transmittance decreases expo-

nentially with increasing opacity.

The water vapor and dry-air (primarily oxygen) attenua-

tions of the returned power from which or0 is calculated by

the radar Eq. (6) are easily corrected. Since atmospheric gas

molecules are much smaller than the radar wavelength, at-

tenuation of or0 through a cloud-free atmosphere is governed

by Rayleigh scattering. The attenuation at any point z along

the path of propagation is therefore proportional to the air

density at that location. The total opacity of the cloud-free

atmosphere is thus proportional to the integrals of dry-air

gas and water vapor densities Pdry and Pvap along the propa-

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