Doppler Velocity from Sea Surface on the Spaceborne ... - AMS Journals

372

JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY

VOLUME 20

Doppler Velocity from Sea Surface on the Spaceborne and Airborne Weather Radars SATORU KOBAYASHI

AND

HIROSHI KUMAGAI

Communication Research Laboratory, Cloud Profiling Radar Group, Tokyo, Japan (Manuscript received 20 February 2002, in final form 25 June 2002) ABSTRACT The Doppler operation applied to the sea surface is studied for the spaceborne and airborne weather radars. In the space mission, a small amount of beam misalignment can cause large contamination of a platform velocity into a measured Doppler velocity. In this paper, theoretical bases for retrieving water droplet velocity via the Doppler signal from a sea surface are considered for pulse-pair operation in the nadir direction. On a fast-moving platform, Doppler fading is so crucial that the finite beamwidth of an antenna is taken into account in the formalism based on a scalar Kirchhoff theory. Correlation signals are derived for both the coherent and incoherent scatterings that have substantial differences in Doppler velocity. Furthermore, the incoherent Doppler velocity is shown to include a path-difference term, which is characteristic of the incoherent surface scattering, set apart from the incoherent body scattering. Applying the retrieval methodology to the Doppler signal from a ground surface is also discussed.

1. Introduction The spaceborne weather radar with Doppler function is expected to provide resourceful information on the distribution of velocities of water droplets on a global scale. As a space mission, single-beam operation in the nadir direction rather than dual-beam operation (Amayenc et al. 1993) is being designed as a cloud profiling radar (European Space Agency 2001), in which detection of velocities to an accuracy of 0.1–0.5 m s 21 , not including systematic errors, is required. The largest portion of the systematic errors arises from a misaligned beam that may cause contamination of a platform velocity of the order of 10 m s 21, in addition to the lineof-sight velocity of clouds. To illustrate this effect, an ideal Doppler operation in the nadir direction shall be considered for spaceborne and airborne radars. In Fig. 1, the platform of a velocity vpl moves in parallel to a sea (geoid) surface, namely, with a null flight angle. A radar beam is transmitted exactly in the nadir direction toward clouds. Obviously no contamination of the platform velocity occurs on the measured Doppler velocity from the clouds. However, the operation is not always performed under the ideal configuration, but rather in configurations such as in Fig. 2. Although the beam in the figure is intended to be aligned in the direction normal to the flight direction, imperfect beam control causes a mispointing angle of u B from this direction. The Corresponding author address: Satoru Kobayashi, Communication Research Laboratory, Precipitation Radar Group, 4-2-1 Nukii-kita, Koganei, Tokyo 184-8795, Japan. E-mail: [email protected]

q 2003 American Meteorological Society

orders of u B can be controlled typically at 0.18 and 18 for the spaceborne and airborne radars, respectively. Furthermore the flight direction is not parallel to the surface, with a finite flight angle u A . In this configuration, the Doppler velocity from clouds can be measured as a body scattering quantity (e.g., Kobayashi 2002; Kobayashi et al. 2002) in the form of

y dop 5 2y pl sinu B cosw 0 1 y cld ,

(1)

in which w 0 is an azimuth angle of the mispointed beam, and y pl has been defined as | vpl | . In Eq. (1), the first term represents contamination of the platform velocity, and the second term y cld denotes the line-ofsight velocity of clouds. Once the contamination of the first term is removed from the equation, the velocity of clouds along the gravitational direction can be retrieved through the transformations of y cld /cos(u A 1 u B ). The above consideration on Eq. (1) allows us to determine the unknown mispointing angles of u B and w 0 for precise Doppler measurement. However, the contamination due to w 0 is a second-order small quantity so that the angle u B will be the main concern of the rest of the paper. A clue to the unknown u B has been considered to be provided by the Doppler signal from a sea surface (e.g., Testud et al. 1995; Bosart et al. 2002). This procedure would be trivial if the Doppler effect were measured as a direct frequency shift in the incoherent scattering. In this case, the Doppler frequency shift of vsea2v from the sea surface and the corresponding Doppler velocity of y sea2v could be given for the nonrelativistic limit (e.g., Nezlin 1993), respectively, in the forms of

MARCH 2003

KOBAYASHI AND KUMAGAI

FIG. 1. An ideal configuration of spaceborne and airborne Doppler operation. The platform of a velocity vpl moves in parallel to a sea (geoid) surface. A radar beam is transmitted exactly in the nadir direction toward clouds. No contamination of the platform velocity occurs on the measured Doppler velocity of the clouds.

v sea2v 5 2k · vpl

(2)

y sea2v 5 v sea2v /2|k|,

(3)

in which k is a radiation wave vector. Since the y sea2v for Fig. 2 gives the same value as the first term of Eq. (1), the subtraction of y sea2v from Eq. (1) would retrieve the line-of-sight velocity y cld . In actuality, the removal of a contaminated velocity is not so clear for the conventional Doppler radar system that involves pulse-pair and discrete Fourier transform (DFT) operations. Here notice that the latter Doppler information has no substantial difference from the former one, because sampled pulses on the DFT operation are identical to contiguous pulse pairs. In both the operations, the phase changes between samplings are measured, instead of the direct Doppler frequency shift vsea2v , in which there is no guarantee that the measured Doppler velocity should be given by Eqs. (2) and (3). For an airborne Doppler radar, Testud et al. (1995) derived an elaborate procedure to correct the Doppler velocity for navigation errors (pitch, roll, etc.) by using sea surface echo, theoretically based on an expression of Eq. (3). Fortunately their procedure will be proven, in this paper, to be legitimate for airborne missions involving a small flight angle u A , hence in most airborne experiments. However, the precise expression of the sea surface Doppler velocity from space is indispensable when applying the approach of Testud et al. (1995) to the spaceborne mission at a much higher velocity. Historically a scalar Kirchhoff approximation has been adopted to study the scattering intensity from a sea surface with large-scale roughness (Beckmann and Spizzichino 1963; Bass and Fuks 1979; Ishimaru 1978), followed by the advanced theory to take more realistic conditions such as vector theory into consideration (Bass and Fuks 1979; Fung et al. 2001; Stogryn 1967). However, Valenzuela (1978) and Furuhama et al. (1986) noted that the scalar Kirchhoff theory can give a sufficiently precise scattering intensity, as far as the nearly normal incidence onto the isotropic rough surface is

373

FIG. 2. A schematic configuration of Doppler operation from spaceborne and airborne radars. The platform of a velocity vpl moves with an flight angle of u A . The ideal alignments are to be set vertical to the flight direction. The mispointing angle of u B is caused by imperfectness of beam controlling ability.

concerned in the centimeter-wavelength regime. It indicates that the scalar theory can be adopted as a first step to calculate the correlation signal from a sea surface on pulse-pair operation. In this paper, formulation of Doppler velocity from the sea surface and its application to the Doppler weather radar will be mainly studied, rather than scattering cross sections and their wind dependences such that Valenzuela (1978) and Apel (1994) discussed on intensity measurement. Regarding Doppler operation from a fast moving platform, Doppler fading is so crucial that the beam-spread pattern and its associated phase delay to the center of a footprint (i.e., nonuniform illumination) will be taken into account in the formalism. Furthermore, an isotropic Gaussian rough surface will be assumed for simplicity throughout the paper, the validity of which will be discussed in section 6. The paper is organized in the following scheme. In section 2 the scattered wave from a random sea surface is formulated under the general Kirchhoff theory. In section 3, the time correlation signal is generally explained for the surface scattering. In section 4, the coherent correlation from a sea surface is calculated. In section 5, the incoherent correlation is derived for a coarsely random sea surface. The conclusions and discussion are found in section 6. Detailed derivations are found in appendices. 2. Scattered wave a. Coordinate system In this section, the amplitude of the scattered wave from a random sea surface is formulated for a radar with a finite beam width on a moving platform. Figure 3 is schematic diagrams of pulse-pair Doppler operation from the platform of a velocity vpl . Figure 3a is a side view for the case when the azimuth angle of a mispointed beam is set at the azimuth angle of w 0 5 0. Figure 3b is a top view for a general w 0 (±0). The angles u A and u B , defined in the same manner as in Fig. 2, are assumed to be constant during a timescale of the operation (ø100 ms). The points O(0) and O(t) on the averaged random surface denote the centers of the radar

374


VOLUME 20

r9 on the random surface will be represented by a Gaussian random amplitude f [x9; O(t)], along with the instant transverse coordinates x9 and y9 referenced to the origin O(t), in the form of r9 5 x9 1 f [x9; O(t)]zˆ.

(7)

In Eq. (7) the average and variance of the random amplitude f [x9; O(t)] are defined, respectively, as ^ f [x9; O(t)]& 5 0

(8)

^ f [x9; O(t)]& 5 s . 2

(9)

2 z

It is noted that the random amplitude f [x9; O(t)] at time t is related to that of t 5 0 through the transformation of f [x9; O(t)] 5 f [x9 1 vc t; O(0)] 1

E

t

y zs [t9, x9 1 vc t; O(0)] dt9,

(10)

0

FIG. 3. Schematic diagrams of pulse-pair Doppler operation from a platform moving at a velocity of vpl : (a) side view for the case of the mispointed beam of its azimuth angle of w 0 5 0, (b) top view for a general w 0 ± 0. The angles u A and u B are defined in the same manner as in Fig. 2. The points O(0) and O(t) on the averaged random surface are the centers of footprints at times 0 and t, respectively, and the points C(0) and C(t) are the bases of the vertical lines on the averaged surface from the platform points P(0) and P(t), respectively. The moving frame with the origin O(t) and its axes are indicated in the figure. (c). A position vector r9 on the random surface is represented by a Gaussian random function f (x9; t) via the instant transverse coordinates x9 and y9 at time t.

footprints at times 0 and t, respectively, and the points C(0) and C(t) are the bases of the vertical lines from the platform points P(0) and P(t) to the averaged surface, respectively (Figs. 3a,b). At an arbitrary time t, a coordinate frame of its origin comoving with the point O(t) is chosen to represent points on the random surface, the axes of which are indicated in the figures. Then the coordinate vector of the platform position P(t) from the origin O(t), referred to as r(t), can be represented in the form of

in which the vectors x9 on both sides are referenced to the origin O(t); y zs [t, x; O(0)] defines the vertical component of the random sea surface velocity, due to wave action at time t and position x in reference to the origin O(0). Here, v c designates the velocity of the moving center O(t) of the beam spot, given by vc [

d → C(0)O(t) dt

5 (2y pl cosuA 2 y pl sinuA tanu 0 cosw 0 )xˆ 2 y pl sinuA tanu 0 sinw 0 yˆ,

(11)

in which the following vector relation from Figs. 3a and 3b has been used: →

→

→

C(0)O(t) 5 C(0)C(t) 1 C(t)O(t)

5 2y pl t cosuA xˆ 2 r⊥ (t).

(12)

Furthermore, in Eq. (10), the transverse component y ⊥s of the sea surface has been ignored because of its small contribution to the measured velocity for the nearly nadir operation, characterized by | y ⊥s u 0 | K 1 K | y pl u 0 |

(13)

b. Incident and scattered waves

→

r(t) [ O(t)P(t ) 5 r⊥ (t) 1 (z 0 1 y pl t sinuA )zˆ,

(4)

in which the transverse vector r⊥ (t) has been defined as →

r⊥ (t) [ O(t)C(t ) 5 (z 0 1 y pl t sinuA )(tanu 0 cosw 0 xˆ 1 tanu 0 sinw 0 yˆ) (5) with the slant/incident angle of

u 0 [ uA 1 uB,

(6)

and z 0 in Eqs. (4) and (5) designates the initial altitude of the platform. As shown in Fig. 3c, a position vector

When an incident wave cin (r; t) is transmitted from the platform position r(t) at time t in Fig. 2 toward the sea surface along the direction of 2r(t), the scattered wave c s (r; t) can be represented as follows through Green’s theorem:

c s (r; t) [ c (r; t) 2 c in (r; t) 5

E[

g(r, r9)

s

]

]c (r, r9; t) ]g(r, r9) 2 c (r, r9; t) dS, ]n9 ]n9 (14)

MARCH 2003

in which c (r; t) is the resultant of the incident and scattered waves, and g(r, r9) denotes a Green function in free space. An explicit form of the incident wave at a point r9 on the surface S can be written for a small beamwidth of u d K 1:

c in (r, r9; t) 5

e ik|r2r9| 2|x9| 2 421s 22 r , e |r 2 r9|

(15)

where the last exponential term (Kobayashi 2002) represents the transverse amplitude pattern by a circular antenna with the parameter of

sr2 5

|r| 2 u d2 2.61 . 2p 2 ln2

(16)

For later use, we shall define the incident (k in ) and backscattered (k) wave vectors, respectively, as k in 5 2k

3 exp[i2ky pl sinuA t(cosu 0 )21 ] 3

1 k cosu 0 zˆ.

(17)

k |r9| 2 . 2r

(18)

Then, as shown in Eq. (A8), Eq. (14) can be calculated with the expansion of Eq. (18) into the form of

e i2kr 4pr 2

E

(21)

in which the amplitude reflectivity Ramp(r9) has been replaced with the averaged value of Ramp over the integral surface. 3. Time correlation signal The time correlation signal in pulse-pair operation is derived by taking an ensemble average of c *s (r1 ; t1 )c s (r 2 ; t 2 ) over the random function f [x; O(t)]: R(t1 , t 2 ) 5 ^c *(r s 1 ; t1 )c s (r 2 ; t 2 )&.

(22)

Decomposing c s (r; t) into the averaged term

c 0 (r; t) [ ^c s (r; t)&

(23)

c s (r; t) 5 c 0 (r; t) 1 dc s (r; t).

To account for Doppler fading, the phase terms in the incident wave cin [Eq. (15)] and the Green function g(r, r9) [Eq. (A7) in appendix A] are expanded up to the second order:

5i

e ik|x| 2r021 e 2|x| 2 221 sr22 e ik · r dx,

s

5 (k sinu 0 cosw 0 xˆ 1 k sinu 0 sinw 0 yˆ)

k|r 2 r9| ø kr 1 k in · r9 1

E

and the deviation term dc s (r; t) from c 0 (r; t), we can rewrite c s (r; t) as

k [ kr(t)/|r(t)| 5 k⊥ 1 k z zˆ

c s (r; t)

375


e ik · r9 e ik|x9| 2 r 21 e 2|x9| 2 421s r22 R amp (r9)(k · c n9) dS,

s

(24)

Equation (24) leads Eq. (22) to the form of R(t1, t 2 ) 5 c 0*(r1; t1 )c 0 (r2 ; t 2 ) 1 ^dc s*(r1; t1 )dc s (r2 ; t 2 )&.

(25)

The first term, called coherent correlation Rcoh , arises from averaged characteristics of the random surface. The second term, called incoherent correlation Rinc , reflects the deviation/fluctuation from the averaged characteristics. Notice that these two terms are interrelated through the existence of an averaged random surface that is constrained in the space as a boundary. This space–constraint feature plays an important role in the time correlation signals for the random surface scattering, set apart from the random body scattering such as the incoherent scattering from clouds.

(19) in which the negative of the scattering wave vector has been introduced as

k 5 k⊥ 1 k z zˆ [ kin 2 k 5 22k,

(20)

and R amp(r9) and (c n9) designate the amplitude reflectivity and the normal vector at the position r9 on the random surface respectively. Equation (19) is seen to be reduced to previous formulas in Beckmann and Spizzichino (1963), Bass and Fuks (1979), and Ogura (1990, chapter 5) when ignoring the terms of | x9 | 2 that is crucial to Doppler operation. Taking the maximum antenna gain G into account, the calculations in appendix A yield a simplified form of the scattered wave [Eq. (A17)]:

c s (r; t) 5 2i

!

A 0 e i2kr0 pG 2 |k | 2 k R 4pkr02 k 2 k z amp

4. Coherent correlation The coherent correlation Rcoh can be calculated from Eqs. (21) and (23) in the form of R coh (t1, t 2 ) 5 c 0*(r1; t1 )c 0 (r2 ; t 2 ) 5

|A 0| 2 pG 2 2 2 k k |R amp | 2 exp[i2ky pl sinuAt (cosu 0 )21 ] 4 2 p 2r 04 k 2 z 3

)E

)

2

e ik|x| 2r021 e 2|x| 2 221 s r22 e ik⊥ · x ê ik z f [x;O(t)] & dx . (26)

s

Using the stationary phase approximation, Eq. (26) can be represented asymptotically for a large value of k: R coh (t1, t 2 ) 5

Pt G 2 cos 2 u 0 |R amp | 2 e 25.2(u 0 /u d ) 2 4 2 k 2 r02

376


3 exp[2k z2 sz2 ]

3 exp[2(|x1 | 2 1 |x 2 | 2 )221 sr22 ]

3 exp[i2ky pl sinuA t (cosu 0 )21 ],

(27)

in which the transmitted power of P t 5 4p | A 0 | 2 and the time interval of t [ t 2 2 t1 have been inserted, and s 2z represents the variance of surface amplitude as defined in Eq. (9). In Eq. (27), the exponential decaying term of exp[2k z2s 2z ] represents the well-known diffusing effect by randomness, which leads to zero when the randomness s z is much larger than the radar wavelength. The other exponentially decaying term of e 25.2(u 0 /u d) 2 represents a specular reflection due to the effect of the finite beam width e 2|x|2 2 21s r22 and the second-order diffraction 0 in the integral of Eq. (26). If these terms term e ik|x|2 r 21 are ignored by considering a plane wave injection from a sufficiently far distance, we can derive the optical specular reflection from the integral part of Eq. (26) without using the stationary phase approximation as follows:

E

e ik⊥ · x ê ik z f [x;O(t)] & dx } e 2k z2sz2 221 d (u 0 ).

(28)

s

In the experimental case, however, the exponential term e 25.2(u 0 /u d) 2 can be practically regarded as a delta function because of | u d | K 1, and Eq. (27) is finite only near the specular/nadir direction of u 0 5 0 in Fig. 2. (coh) The Doppler velocity y dop of the coherent scattering for Fig. 2 is therefore given by the phase term of Eq. (27): (coh) y dop 5 y pl sinu A /cos(u A 1 u B ).

(29)

Near the specular direction of u 0 ø 0, Eq. (29) reduces (coh) to y dop ø 2y pl sinu B , which is the only theoretical observable velocity for a smooth surface. Aside from intensity, the coherent Doppler velocity of Eq. (29) is attributed from the increase in beam path indicated by the line P0(t)P(t) in Fig. 3: →

| P0(t)P(t) | 5 y pl t sinuA /cos(uA 1 u B ).

(30)

For the straight flight of u A 5 0, Eq. (29) gives zero Doppler velocity of the coherent scattering as expected. 5. Incoherent correlation for the coarse surface In this section, the incoherent correlation is derived for the coarse random surface. The general form of incoherent correlation for the nearly nadir direction is found in Eqs. (B5) and (B6) in appendix B: R inc (t 5 t 2 2 t1 ) 5 ^dc s*(r1; t1 )dc s (r2 ; t 2 )& 5 Pt G 2 |R amp | 2 224 p 22 r024 cos22 u 0 3 exp[i2ky pl sinuA t (cosu 0 )21 ]

VOLUME 20

3 exp[ik(|x 2 | 2 2 |x1 | 2 )r021 ]R p (x 2 , t 2 ; x1, t1 ). (31) In the above equation, R p derived in Eq. (B11) represents R p (x 2 , t 2 ; x1, t1 ) 5 e 2k z2R f (0) [e k z2R f (x22x11vct ) e 2(k zssvzt ) 2/2 2 1],

in which the correlation R f of the surface amplitude f [x; O(t)] has been defined as R f (x j 2 x i ) 5 ^ f (x j ; O(t)) f (x i ; O(t))&.

E

(33)

In Eq. (33), the independence from the time parameter t has been induced from the assumption of the isotropic random sea surface. Also notice the relation of R f (0) 5 s 2z .

(34)

Furthermore in the course of deriving Eq. (32), the average sea surface velocity in the vertical direction and its variance have been set as follows: ^y zs & 5 0

(35)

2 ^y zs2 & 5 ssvz [ const.

(36)

Suppose that the following condition of coarse surface is satisfied:

k z2 R f (0) k 1,

(37)

alternately using the radar wavelength of l:

s z k l.

(38)

Then, the coherent scattering represented by Eq. (27) fades out even in the specular direction due to the superposition of waves of different phases. Under this condition, the surface correlation R f in Eq. (32) can be expanded (Beckmann and Spizzichino 1963) in the form 1 R f (x) ø R f (0) 2 x t R df x, 2

(39)

in which the correlation matrix of the first derivatives of f [x; O(t)] has been introduced: R df 5 2

[

^ fx fx& ^ fy fx&

]

^ fx fy& . ^ fy fy&

(40)

For weather radar, especially for millimeter wave radar, the condition of Eq. (38) is easily satisfied for the sea surface scattering. This point will be revisited in section 6. Since the nondiagonal elements of Rdf , which are odd functions with respect to x and y on the isotropic random surface, will vanish for x 5 0, we calculate Eq. (39) into the form of R f (x) ø R f (0) 2

dx1 dx 2 e ik⊥ · (x22x1 )

(32)

s12 2 (x 1 y 2 ) 2

(41)

along with the variance s 21 of random surface slopes (i.e., first derivative of f )

MARCH 2003


s 21 [ ^ f x f x &x50 5 ^ f y f y &x50 .

377

(42)

Substitution of Eq. (41) into Eq. (32) yields R p (x 2 , t 2 ; x1, t1 ) 5 e 2k z2R f (0) [e k z2R f (0) e 2s12j 2221 e 2(k zssvzt ) 2 221 2 1] ø e 2k z2s12j 2 221 e 2(k zssvzt ) 2 221,

(43)

in which the vector j designates

j [ x 2 2 x1 1 v c t.

(44)

Further calculations for Eq. (31) along with Eq. (43) yield the final form of the incoherent correlation, as described in appendix C: R inc (t 5 t 2 2 t1 ) 5

[ ]

Pt G 2 u d2 l2 |R amp | 2 tanu 02 exp 2 10 2 2 2 4 2 p (ln2)r0 s1 cos u 0 2s12 3e

2 1s 2 )t 2 22k 2 (sdop svz

e 2ik⊥ · vct


(45)

in which u d is the beamwidth in radians; s 2svz has been defined as in Eq. (36), and the spectral broadening due to Doppler fading s 2dop is represented in the form of Eq. (C4) along with Eq. (11): 2 sdop 5

FIG. 4. Geometrical interpretation of the terms of Eq. (47) for pulse pair emitted with the time interval of t. The first term of Eq. (47) can be obtained by projecting the horizontal component (y pl t cos u A ) of the platform velocity onto the beam line. The additional displacement (y pl t sinu A tanu 0 ) of the footprint center is caused by the vertical movement (y pl t sinu A ) of the platform. The third term arises from the increase in beam path in the same manner as coherent scattering.

2.61 2 2 u y . 4p 2 ln2 d c

(46)

The Doppler velocity can be derived from the last two phase terms in Eq. (45). It is also noted that for the condition t 5 0, Eq. (45) reduces to the surface radar equation for the nearly vertical incident onto a sea surface (Kozu 1995). As a conclusion of this section, the Doppler velocity from the coarse surface is represented for the configuration of Fig. 2: (inc) y dop 5 2y pl cosuA sinu 0 cosw 0

2 y pl sinuA sinu 0 tanu 0 1

y pl sinuA . cosu 0

(47)

The first and second terms reflect the effect of randomness. The third term, equal to the Doppler velocity of the coherent scattering [Eq. (29)], is due to the increase in beam path due to a platform displacement, which is characteristic of incoherent surface scattering. A geometric interpretation of the first and second terms in Eq. (47) is schematically depicted in Fig. 4. The first term depending on w 0 can be obtained by projecting the horizontal component (y pl cosu A ) of the platform velocity onto the beam line. As shown in the figure, the second term arises from the additional displacement y pl t sinu A tanu 0 of the footprint center caused by the vertical movement (y pl t sinu A ) of the platform. Since this displacement is always included in the same plane as the beam line, the projection of the corresponding velocity onto the beam line (y pl sinu A sinu 0 tanu 0 ) becomes indepen-

dent of w 0 . Further, notice in Eq. (45) that the timedependent decorrelation of the signal is given by the sum of the Doppler fading s 2dop and the variance of surface vertical velocity s 2svz , but not by the surfaceslope roughness s 21. In other words, the surface-slope roughness only decreases the total intensity of the incoherent scattering proportional to s 22 and deteriorates 1 the signal-to-noise ratio to give a worse measurement accuracy, but has no effect on the time-dependent coherence itself. 6. Conclusions and discussion The correlation signal from the sea surface has been formulated in order to remove a contaminated-platform velocity from a water droplet Doppler velocity on pulsepair operation with the slant/incident angle of | u 0 | K 1. The Doppler velocity of the coherent scattering is given by Eq. (29) for the configuration shown in Fig. 2. Nonetheless in experiments, the coherent correlation cannot be observed even in the specular direction, because the coarse surface condition of Eq. (38) is usually satisfied for centimeter to millimeter wavelengths. For incoherent scattering, the Doppler velocity is represented by Eq. (47) constituted of three terms. The first and second terms are caused by randomness of the surface. The third term, identical to Eq. (29), is due to the change in beam path due to the movement of a platform, which is a characteristic feature of incoherent surface scattering, because the scattering points on a random surface are constrained by the averaged random surface, namely, by the macroscopic reflecting surface mentioned earlier in section 3. Previous experiments reviewed in Beckmann and Spizzichino (1963), Valenzuela (1978), and Furuhama et al. (1986) showed that backscattering intensities/cross sections were well represented by Eq. (45) along with t 5 0, as far as the nearly nadir operation was concerned

378


in the band of centimeter wavelength. In addition, a recent experiment of an airborne cloud profiling radar (called SPIDER) with the millimeter wavelength of 95 GHz (Horie et al. 2002) showed that the backscattering intensity from the sea surface has the dependence of the incident angle u 0 in the form of 1 exp(2a tan 2 u 0 ), cos 4 u 0

(48)

with a constant a. These facts indicate that the sea surface echo in the centimeter to millimeter wavelength can be represented by the incoherent scattering of the Kirchhoff theory for the nearly nadir operation (i.e., quasi-mirror scattering regime). In turn, the derivation in section 5 as well as the general formalism throughout this paper, involving the Kirchhoff theory with the isotropic Gaussian random sea surface, can be legitimized. Hence for Doppler experiments, the measured correlation and Doppler velocity are expected to be represented by Eqs. (45) and (47) for | u 0 | K 1, respectively. A procedure to remove a contaminated platform velocity is summarized for the configuration of Fig. 2. Assuming a flight angle of | u A | K 1, then the Doppler velocity from sea surface given by Eq. (47), can be approximated to (inc) y dop ø 2y pl sin(u A 1 u B ) 1 y pl sinu A ø 2y pl sinu B , (49)

while that from clouds can be written from Eq. (1):

y dop ø 2y pl sinu B 1 y cld .

(50)

This means that the direct subtraction of Eq. (49) from Eq. (50) retrieves the line-of-sight velocity of water droplets (y cld ) for the condition of | u A | K 1. The results of this paper can be applied to the reflection from a ground surface, as far as the condition of large-scale roughness, equivalently the Kirchhoff theory, is satisfied. For this application, the flight angle u A must be replaced with the sum of the averaged slope angle of a ground surface (including a curvature of the earth) and a flight angle against the geoid, which will be referred to as the effective flight angle of uAeff . For | uAeff | K 1, the removal method in the last paragraph is still in effect. However, it seems difficult to determine the effective flight angle uAeff , because the average slope of the ground surface changes from one place to another. The Communication Research Laboratory recently observed the third term effect in Eq. (47) due to the finite uAeff above mountain slopes in a preliminary airborne radar experiment. In the case of | uAeff | t 1, the formulation of this paper may be incompletely satisfied. Nonetheless, considering the plane incident of unit amplitude,—that is, exp[ikin · r] instead of the spherical wave of Eq. (A1)— and also omitting the second-order terms | x | 2 in the Green function of Eq. (A7) on the scattered wave cal-

VOLUME 20

culation, the same procedures as in sections 2 and 5 derive the incoherent correlation: R inc (t 5 t 2 2 t1 ) 5

[ ]

|R amp | 2 S tanu 02 22k 2 s 2 t 2 2ik · v t svz exp 2 e e ⊥ c 2 3 pr02 s12 cos 4 u 0 2s12 3 exp[i2kiy pl sinuA t (cosu 0 )21 ],

(51)

in which S designates the illuminated area on the surface. Since we have ignored the beam spread from an antenna for Eq. (51), the Doppler fading s 2dop does not appear, and the expression of the amplitude for t 5 0 will never correspond to the experimental case. However, the Doppler phase term is identical to Eq. (45). It means that the phase part of the incoherent correlational signal is still correct for the general angles of uAeff(u A ) and u B . This result can also be supported by the geometrical interpretation in Fig. 4, which is satisfied regardless of the values of uAeff and u B . Thus as far as the Doppler velocity is concerned, the expression of Eq. (47) is always valid for the incoherent scattering. Furthermore, in practice, the informational defect of the correlation amplitude for large uAeff(u A ) and u B will not matter so much, because the intensity is only considered implicitly for Doppler measurements via the signal to noise ratio in order to evaluate measurement accuracy (Doviak and Zrnic 1993). In airborne experiments, the characteristic third term in Eq. (47) will play an important role in case of uAeff(u A ) around 0.38, while in spaceborne experiments the term will always be important. In such situations, once we somehow know the flight velocity y pl and the effective flight angle uAeff , we can solve for a mispointing angle u B from Eq. (47). Acknowledgments. The authors wish to thank Prof. T. Sato at Kyoto University, Japan, for a suggestion on discrete Fourier transform operation. A special thanks is to be dedicated to Dr. S. Iwasaki at Frontier Observational Research System for Global Change, Japan, for his unpublished but careful analysis on sea surface Doppler velocities by an airborne 95-GHz radar (SPIDER), which is the incentive to this paper. The authors also wish to thank our colleague H. Horie for showing preliminary data of sea surface scattering data with the SPIDER. APPENDIX A Derivation of the Scattered Wave On the incident wave of Eq. (15),

c in (r, r9; t) 5

e ik|r2r9| 2|x9| 2 421s 22 r , e |r 2 r9|

(A1)

we can expand the phase term up to the second order as

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k|r 2 r9| ø kr 1 k in · r9 1

k |r9| 2 . 2r

(A2)

Equation (A2) rewrites the incident wave and its normal derivative at r9 in the forms of

c in (r, r9; t)

51

2

e ikr ik · r9 k 1 k ø e in exp i 2 2 |x9| 2 1 i f 2 [x9; O(t)] r 2r 4sr 2r

[1

2

]

e ikr ik · r9 k 1 ø e in exp i 2 |x9| 2 r 2r 4sr2

6

5

1

in which the negative of the scattering wave vector has been introduced as

k 5 k⊥ 1 k z zˆ [ kin 2 k 5 22k.

c s (r; t) 5 i

3

2

c · k in )c in (r, r9; t). ø i(n9

(A4)

In Eqs. (A3) and (A4), nˆ9 denotes the normal vector at r9 on the random surface. Under the general Kirchhoff approximation, the resultant wave c (r, r9; t) in the random surface can be approximated as the sum of the incident and scattered waves that are mutually related through Snell’s law. The scattered wave c s (r, r9; t) at r9 can be represented through the amplitude reflectivity Ramp (r9), as

c s (r, r9; t) 5 Ramp (r9)cin (r, r9; t).

s

(A5)

1/r 2 ø 1/r02 , r 0 5 z 0 /cosu 0 ,

(A13)

we can rewrite Eq. (A10) into the form of

c s (r; t) 5 i

!

A 0 e i2kr0 pG 2 f (k ; t). 4pkr02 k2

(A14)

Dropping the primes of the coordinates, f (k ; t) can be represented as

f (k ; t) [ R amp k exp[i2ky pl sinuA t(cosu 0 )21 ] 3

E

e ik|x| 2r021 e 2|x| 2 221 sr22 e ik z f [x;O(t)]

s

3 k · {=⊥ f [x; O(t)] 2 zˆ} dx. (A6)

Furthermore, the Green function can be approximated: e ikr 2ik · r9 ik|x9| 2 221 r 21 e e r

]g(r, r9) c · k)g(r, r9). ø 2i(n9 ]n9

(A12)

along with the definition of

c (r, r9; t) 5 [1 1 R amp (r9)]c in (r, r9; t) ]c (r, r9; t) c · k in )[1 2 R amp (r9)]c in (r, r9; t). ø i(n9 ]n9

(A11)

kr 5 kz(t)/cosu 0 5 kr0 1 ky pl sinuA t/cosu 0

The resultant wave c (r, r9; t) and its normal derivative can therefore be calculated:

(A15)

For the small slant angle u 0 and the small beam width u d , Eq. (A15) can be further approximated through the 2D Gauss–Green theorem in a similar manner to Ogura (1990, chapter 5):

f (k ; t) (A7)

Substitutions of Eqs. (A6) and (A7) into the Green’s theorem of Eq. (14) yields the scattered wave in the form of

E

e ik · r9 e ik|x9| 2 r 21 e 2|x9| 2 221s r22 (k · c n9) dS,

c n9dS 5 {=⊥ f [x9; O(t)] 2 zˆ}dx9dy9

6

e i2kr 4pr 2

E

where the amplitude reflectivity Ramp (r9) has been replaced with the averaged value of Ramp over the integral surface. Nothing the following relations of

k 1 i f [x9; O(t)]=⊥ f [x9; O(t)] c in (r9; t) r

5i

!

A 0 e i2kr pG 2 R 4pr 2 k 2 amp

(A10)

k 1 øc n9 · ik in 1 i 2 x9 r 2sr2

g(r, r9) ø

(A9)

Taking the maximum gain G of the antenna into account, the received signal can be represented in the form of

(A3)

]c in (r, r9; t) ]n9

c s (r; t)

379


ø R amp k exp[i2ky pl sinuA t(cosu 0 )21 ] 3

[

2|k | 2 kz 2

e ik · r9 e ik|x9| 2 r 21 e 2|x9| 2 421s r22 R amp (r9)(k · c n9) dS,

s

(A8)

i kz

E R

e ik|x| 2r021 e 2|x| 2 221 sr22 e ik · r dx

s

]

e ik|x| 2 r021 e 2|x 2 221 sr22 e ik · r (k⊥ · c n c ) dc . (A16)

Since the last line integral may be ignored for an integral

380


dimension much larger than the correlation length of the surface function f , the scattered wave received by the antenna can finally be calculated from Eq. (A14) in the form of

!

A e i2kr0 pG 2 |k | 2 k c s (r; t) 5 2i 0 2 R 4pkr0 k 2 k z amp

E

e ik|x| 2r021 e 2|x| 2 221sr22 e ik · r dx.

R p (x 2 , t 2 ; x1, t1 ) [ ^ p(x 2 ; t)p(x1; t)& 5 ê ik z{ f [x 2;O(t 2 )]2 f [x1;O(t1 )]} & 2 |ê ik z f [x;O(t)] &| 2 .

ê ik z{ f [x 2;O(t 2 )]2 f [x1;O(t1 )]} & ø ê ik z{ f [x 2;O(t 1 )]2 f [x1;O(t1 )]} e ik zy zs[t1,x 21vct ;O(t)]t &.

(A17)

s

APPENDIX B General Formalism of Incoherent Correlation

!

(B1)

2|k | 2 R . 2kk z amp

(B2)

A 0 e i2kr0 pG 2 2F(k)k 2 I(t; k), 4pkr02 k2

in which F(k) has been set as F(k) 5

^y zs & 5 0 2 ^y zs2 & 5 ssvz [ const.

(B10)

Taking the ensemble average of Eq. (B8) with the relations of Eqs. (B9) and (B10), Eq. (B7) can be calculated

5 e 2k z2R f (0) [e k z2R f (x22x11vct ) e 2(k zssvzt ) 2221 2 1].

(B11)

In the above equation, the correlation R f of the surfaceamplitude f [x; O(t)] at an arbitrary time t has been defined as

I(t; k) 5 exp[i2ky pl sinuA t (cosu 0 )21 ]

E

(B9)

R p (x 2 , t 2 ; x1, t1 )

The integral part I(t; k) has the form of

3

(B8)

Suppose that the distribution of y zs [t, x; O(t i )] is independent of x and its referenced origin O(t i ), which can be satisfied for the isotropic random sea surface, then the average and variance can be set to

The deviation dcs (r; t) of the scattered-wave amplitude from the average c 0 (r; t) is written from Eq. (A17),

dc s (r; t) 5 i

(B7)

Substitution of Eq. (10) into the first term of Eq. (B7) yields for a short time interval t :

3 exp[i2ky pl sinuA t(cosu 0 )21 ] 3

VOLUME 20

e ik|x| 2 r021 e 2|x| 2 221s r22 e ik⊥ · x p(x; t) dx,

R f (x j 2 x i ) 5 ^ f [x j ; O(t)] f [x i ; O(t)]&,

(B3)

(B12)

s

along with the deviation function p(x; t) defined by p(x; t) 5 e ikz f [x;O(t)] 2 ê ikz f [x;O(t)] &.

(B4)

where the independence from the time parameter t can be induced from the assumption of the isotropic random sea surface.

Hence we obtain the incoherent correlation along with the transmitted power P t 5 4p | A 0 | 2 : R inc (t 5 t 2 2 t1 )

APPENDIX C Final Form of Incoherent Correlation for the Coarse Surface

5 ^dc s*(r1; t1 )dc s (r2 ; t 2 )&

Using Eq. (43), Eq. (B6) can be calculated:

5 Pt G 2 |R amp | 2 224 p 22 r024 cos22 u 0 Î*(t1; k)I(t 2 ; k)&. (B5) In Eq. (B5) the correlation of I(t; k) can be written in the form of Î*(t1; k)I(t 2 ; k)& 5 exp[i2ky pl sinuA t (cosu 0 ) ] 21

E

dx1 dx 2 e

ik⊥ · (x 22x1 )

3 exp[2(|x1 | 2 1 |x 2 | 2 )221 sr22 ]

5 exp[i2ky pl sinuA t (cosu 0 )21]e 2(k zssvzt ) 2 221 3

E

dx1 dx 2 e ik⊥ · (x 22x1 ) exp[2(|x1 | 2 1 |x 2 | 2 )221sr22 ]

3 exp[ik(|x 2 | 2 2 |x1 | 2 )r021 ] exp[2s12 k z2 |j | 2 221 ]. (C1) To separate the integral variables in Eq. (C1), the following variables are introduced:

3 exp[ik(|x 2 | 2 2 |x1 | 2 )r021 ] 3 R p (x 2 , t 2 ; x1, t1 ), (B6) in which the correlation R p designates

Î*(t1; k)I(t 2 ; k)&

R g 5 (x1 1 x 2 )/2

r g 5 x 2 2 x1.

(C2)

Substitution Eq. (C2) into (C1) followed by order estimations yields

MARCH 2003


Î*(t1; k)I(t 2 ; k)& ø 2.61l2 r02 u d2 224 p 22 s122 cos22 u 0 (ln2)21 2 2 3 exp[22k 2 (sdop 1 ssvz )t 2 ]e 2| k⊥ | 2221s122k z22 e 2ik⊥ · vct


(C3)

in which the spectral broadening s due to Doppler fading has been calculated in the same form as Sloss and Atlas (1968) and Kobayashi (2002): 2 dop

2 sdop 5

2.61 2 2 u y . 4p 2 ln2 d c

(C4)

Finally, Eqs. (B5) and (C3) lead the correlation signal of the incoherent scattering to the final form of R inc (t 5 t 2 2 t1 ) 5

[ ]

Pt G 2 u d2 l2 |R amp | 2 tanu 02 22k 2 (s 2 1s 2 )t 2 svz dop exp 2 e 2 2 2 4 2 p (ln2)r0 s1 cos u 0 2s12 10

3 e 2ik⊥ · vct exp[i2ky pl sinuA t (cosu 0 )21 ].

(C5)

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