Remote Sensing of Sea State by Analysis of Backscattered Microwave Phase Fluctuations

The relative phase of a normally incident microwave signal scattered off the sea surface is a random function of space and time. The statistics of these random phase fluctuations contain information about the wave-height statistics. This study demonstrates that the wave-height spectra can be deduced directly in terms of the spectra of the phase fluctuations without recourse to inversion techniques even when surface roughness exceeds many wavelengths of the incident signal. In the particular case of a nadir-directed satellite-mounted microwave source operating in the backscatter mode, the wave-height and phase spatial spectra are proportional with a constant proportionality of (2k_{0})^{2} , where k_{0} is the wavenumber of the scattered signal.


I. INTRODUCTION
M ANY techniques have been suggested for deriving knowl- edge of the sea state from both passive and active electromagnetic observations.These include various high-frequency (HF) scatter techniques [ I ] , [2], the laser profilometer [3], the short pulse microwave altimeter [4], and the microwave scatterometer [5].We propose a new technique that obtains direct information about sea state parameters from analysis of the phase fluctuation spectrum of a microwave signal reflected off the sea surface.The technique requires an airplane-borne or satellite-mounted microwave source that illuminates the sea surface at normal incidence and detects the backscattered return.By applying the theoretical model of scattering that found such useful application to the problem of optical and microwave propagation in random media, we are able to derive U.S. Government work not protected by U.S. copyright.the wave-height spectrum directly in terms of the spectrum of backscattered phase fluctuations.

ANALYSIS
From existing theoretical formulations based on physical optics [ 6 ] , [7] we can derive the following equation for the magnetic field backscattered from a perfectly conducting ocean surface for a normally incident spherical wave e i 2 k 0 r r2 sec 0 2 where ko is the wavenumber of the incident radiation, Co is the incident polarization vector of the magnetic field, and r is the distance from the scattering point P to the point of observation P' (see Fig. 1).The origin of coordinates is at (p', 0 ) = (x', y ' , 0 ) ; r = d p 1 2 + [z' -t(pl + p ' ) ] 2 ; $ is the wave height at P, 6 is the angle from the nadir to the scattering point P ; A is the illuminated area, and z' is the height of the transmitterreceiver above the mean sea surface.To a close approximation z', the height of the transmitter, can replace Z' -in sec 6 / r 2 ; however, in the phase we must retain because we assume that the wave-height variations are much greater than a wavelength of the microwave signal.These approximations give where ro = d m .
We now introduce the angle el , which is the angle from the nadir to the mean sea surface plane beneath the point P, and expand the argument of the exponential in terms of 61, where we have approximated (1 -cos 0,) -1 -2/2 which is valid if O1 < 1.Since 0 < Bo where Bo is the half beamwidth, assumed to be less than a few degrees, this approxima- If we ignore the normalized amplitude fluctuations with respect to the enormously larger phase fluctuations, which is a reasonable approximation in the case of scattering from the sea surface (see Appendix), the flcctuations in y can be entirely attributed to phase fluctuations, i.e.,

? ( P I )
The problem remaining is to relate the statistics of y to the wave-height statistics by analysis of (5).
Consider the equation for y , < ,  , i 2 k O I r O -t ( P l + P '
(19) Equation (1 1) demonstrates that dy, which can be related to the phase fluctuations by means of (7), is simply the random spectral amplitude dB ( K ) , which depends upon the wave-On the left side of (19) we have a function of (plp 2 ) alone height variations times a weighting function I ( K ) , that is, (a consequence of the statistical homogeneity assumption).In order to have a similar dependence on the right side, the quan- tity in the angle brackets must have the form where (dB Substituting this result into (19), we find that F satisfies the ikoCo . ( 13) relation For a circular illumination pattern of diameter D, substitute From (1 2) and ( 14) we can calculate the power spectral density of dy in terms of the power spectral density of dB and ultimately relate the phase statistics to wave-height statistics.First, we multiply (1 2) by its complex conjugate average received power that appears in nearly all treatments of plane-wave scatter from rough Gaussian surfaces since Isakovich's [ 101 pioneering work in 1952.The principal contribution of the subsequent analysis is (1) to show how this result can be extended to spherical-wave scatter via simple transfer-function concepts, and (2) to employ these concepts to derive the statistics of the received signal phase, rather than its average intensity.the left side of (1 5).From ( 9 ) and (7), (dy(K1)dy*(K2)) = I(Kl)l*(Kz)(dB ( K 1 ) dB* (15) By an analogous procedure we can determine the form of and from the definition (1 0), calculate the expression in the angle brackets on the right side of (1 5), that is

(rcol)TY*@z)) = (e ~C O ~( P ~) -O ~( P ~) I )
Again, if it is assumed that the phase fluctuations 9 1 are nor-Gaussian random variables as in the steps leading to (1 7); then, If $ is assumed to be a zero-mean homogeneous mally distributed, we can take recOuISe to the theorem for Gaussian random variable, which is a reasonable assumption for the wave-height fluctuations [ 2 ] , it is not difficult to show

that [91 ( e i [ 0 1 ( P 1 ) -0 1 ( P 2 ) 1 ) = ~-G D O ( P ~-P ~) = \ e j ~1 -0 1 ( , -~Z ~O C E C P ~) -~C P Z
where Dg is the so-called structure function of the wave-height variations defined by where Do is the structure function of the phase fluctuations.

Dg@1 -
The result of the steps leading to (28) is to transform the scattering problem to the spatial frequency domain.The conventional approach to sea scatter calculations is to take a form of the physical optics integral (5) and to determine the desired statistics of H by approximating the integral, e.g., assuming plane wave incidence, applying stationary phase, etc., using information about typical surface roughness parameters, and averaging.Equation (28) demonstrates that the statistics of H may be determined exactly by transforming the scattering problem into the spatial frequency domain.Here the linear system aspect of the problem is emphasized, and the spectral statistics of the received field G(K) factor conveniently into two terms, separating the spectral quantity that depends on the wave-height statistics for plane-wave incidence F(K) from a quantity I I i2 that depends solely on the propagation geometry.Equation ( 28) is the equation of a linear system in the spatial frequency domain that describes how the input spectrum (the angular spectrum of scattered plane waves leaving the sea surface F ) , is multiplied by a transfer function1 I l2 so that the resultant G = I I I2F is the angular spectrum of scattered waves at the receiver.Examination of (13), which is the definition of I , reveals that it is also the diffraction integral for the field backscattered from a sinusoidal surface irregularity of wavenumber K whose amplitude is small in terms of the radio wavelength, observed at a distance z' from the surface.Therefore, the physical interpretation of I I l2 is that it describes the mgular distribution of power scattered by a single Fourier surface component in the "slightly rough" or Rayleigh limit [2], [ l l ] , [ 121.This result could have been anticipated, since from linear system theory, I I l2 is the transfer function or the frequency response of the medium.Equation ( 28) can be solved formally to give the waveheight statistics by using the Fourier transform.By inserting the definition of F from ( 22) into (28), we obtain or, by substituting the definition of G, Without any further approximation we can find the waveheight structure function, a quantity directly related to the power spectral density, by measuring the structure function of the phase fluctuations of the backscattered signal, computing the Fourier transform of exp [--+Do], multiplying by 1 1 [found from (14)], Fourier transfonning the result and, finally, computing the logarithm.In a practical case, we would use the satellite or airplane velocity v times the time lag T as the vector p l .In this instance, the Do would be computed from the recorded time series of the phase fluctuations, where a&) = q 91 ( 0 and the transforms in ( 3 0 ) would reduce to their one-dimensional analogs.

APPROXIMATIONS
In t h i s section, we w i 4 sin2 [k,z'(lsec e,)], the scattering cross section of a circular plate of diameter 22' tan 80. Second, the curve asymptotically approaches its final value of unity beginning at K = 2(2' tan Bo)-1, indicating the irregularities in the surface roughness that are larger than, or of the same order as, the beam diameter are severely affected by the truncated illumination of the beam pattern; whereas, higher spatial frequencies are not.Third, the cutoff of j I l2 occurs at a spatial wavenumber K = 2ko tan do.These three characteristics have simple physical interpretations.The quantity I I l2 describes the first-order Bragg scattering properties of different Fourier components of the surface roughness that are truncated by the finite flumination pattern of the transmitter.For very low spatial frequencies ( K -+ 0) the irregular surface resembles a flat, circular plate so that the power at K = 0 should be proportional to the scattering cross section of that particular geometry.As observed, the scattering properties of shortwavelength Fourier components of surface roughness (much less than the beam diameter 22' tan d o ) are essentially unaffected by the truncated illumination pattern, whereas larger spatial wavelengths w i l l have their scattering lobes considerably broadened.Finally, from the Bragg theory, surface irregularities of spatial wavenumber K will backscatter energy at an angle 19 = sirp1 [ K / 2 k o ] .Clearly, when K/2ko > tan 8, most of the energy will be scattered out of the receiving cone.This accounts for the precipitous decrease of I I l2 beyond the point K.= 2ko tan 8,.
A useful engineering approximation for I I l2 can be obtained from (34): Because there is negligible area under I I l2 in the region 0 < K < 1O(z' tan compared with the region 1O(z' tan < K < 2ko tan 8 0 for a typical satellite geometry, we make a negligible error by assuming that 11 l2 is roughly constant throughout the entire region 0 < K < 2ko tan 80. From ( 2 8 ) we see that G(K) = F(K) I I 12.If F ( K ) has no significant power beyond K = 2ko tan Bo, then G(K) = F(K), and from their respective definitions, [ ( 2 2 ) and (27)J D@@) = 4 k 0 2 D ~@ ) .Hence, the wave-height statistics are proportional to the phase statistics.
To estimate the bandwidth of F(K), note that F ( K ) is the power spectral density of exp [i2k0[] with a random variable.A reasonable estimate of the bandwidth of a phase modulated signal is the maximum instantaneous spatial frequency.For a phase modulated wave exp [$(x) J the bandwidth 6 < In the case of phase modulation due to ocean waves 6 -2kos where s is the rms wave slope.Again, to minimize spectral distortion, the bandwidth of F(K) must be less than the spectral width of I I 12, i.e., For s 0.04 a half-beamwidth of approximately three degrees will satisfy (36).Under this approximation, the phase statistics and wave height statistics are proportional, that is

IV. CONCLUSIONS
We have shown that, for reasonable microwave propagation parameters (fo -10 GHz, ko -200 m-l, 8, -0.05 rad antenna half-beamwidth), the phase structure function2 can be written directly in terms of the wave-height structure function giving DO@) = 4ko2DtCo>. (39) This also implies the following relationship between the signal phase spatial spectrum S@(K) and the two-dimensional waveheight spatial spectrum S(K): A satellite traversing a line can measure the above quantities only along a single direction, i.e., K = K cos OLi--t K sin 6, where a is the nadir-pointing track angle with respect to an arbitrary x-y coordinate geometry at the earth's surface.This unidirectional wave spectral measurement is identical to that obtained from an airborne laser proflometer, where the received signal phase (or time of flight) varies in direct proportion to the height of the sea as the instrument "profiles" the ocean waves.The important conclusion to be drawn here is that the final spectral results (40) are identical for the laser profdometer and the microwave nadir-pointing phase-measuring radar, even though the scattering mechanisms are entirely different.In the former case, the reflection occurs from an

r0
Both y and exp [-i2koEl may be written in terms of their two-dimensional Fourier-Stieltjes amplitudes (a generalization of the Fourier transform for random functions; see Yaglom 181) as, i 2 k o [ r g -E ( ~1 + ~' ) 1 (5 ) r0Substituting ( 9 ) and (10) into (8), we obtain the relationship between the random amplitudes dy and dB in the form result into (16), we obtain 0) and integrate over angles and of course, by the Fourier inversion theorem (I4) Equation (22) has the same form as the classic integral for the where I ( K ) is now a function of only the magnitude of the two-dimensional wavenumber K .
18) In order to have the p1 -p2 dependence on the right side of (241, and at2 and Rg are, respectively, the variance and the spatial correlation function of the wave height.Substituting this

ll 3 )Fig. 2 .
Fig. 2. Plot of transfer function I Z 12 versus K .For K / 2 k o = sin 0, I Z l2 is also angular spectrum of plane waves scattered from truncated Fourier component of surface roughness of wavenumber K.

From
Fig. ( 2 ) , we see that I I l2 % 1 in the region (2' tan Bo)-1 4 K < 2ko tan Bo.For typical satellite parameters 8, -0.05, ko S 200 m-l, and z' -IO6 m, we have 2 X 1 0-5 rn-l < K 2 20 m-l.Over this broad spectral range the angular spectrum of plane waves arriving at the receiver G ( K ) is identical to the an-glar spectrum of scattered waves just above the ocean surface F(K).

3) we obtain the final working equation
tion i s reasonable.The second term i n (3) may be ignored if For a typical microwave signal (A = 3 cm) ko -200 and for a rough sea surface g,,, -I m, the approximation is valid if Bo < 4'.With this approximation applied to (

shown by writing H and Ho in phasor notation, H = Aei@ and Ho = AOe@0. The
total field H is a random variable whose amplitude and phase have both mean A