1. Introduction

Knowledge of the sea surface displacement field, η(x, y, t), is fundamental to understanding many of the physical processes taking place near the air–sea interface. It is therefore worthwhile to study improved techniques for its measurement. Direct measurement of the surface displacement field is often carried out with individual point measurement devices such as pitch–roll buoys or with arrays of devices such as pressure sensors or wire wave gauges. Such instruments typically provide characterization of the surface frequency–direction spectrum, S_η(f, θ), which can be converted to a wavenumber spectrum, S_η(K, θ), by invoking the dispersion relation for surface waves. Such estimates, however, may be distorted by the presence of near-surface currents, including the orbital velocities of long gravity waves upon which shorter waves ride. Truly spatial measurement of the surface wave field requires dense spatial sampling afforded in practice with imaging or with rapidly scanning sensors.

In this paper we consider the measurement of surface displacement from a stationary platform using interferometric radar techniques developed for terrain mapping over land. These have been extensively investigated in the context of synthetic aperture radar imaging (Li and Goldstein 1990; Rodriguez and Martin 1992; Griffiths 1995; Zebker and Villasenor 1992; Zebker et al. 1994) and may be adapted to the measurement of sea surface topography within constraints dictated by the coherence properties of the sea surface microwave echo. The following section outlines interferometric measurement principles and the major sources of error involved. Section 3 describes a field experiment in which an imaging Doppler radar configured as an interferometer is used to estimate surface displacement. The accuracy of the measurements is confirmed by simultaneous wave-wire measurement.

2. Radar interferometry

An interferometer consists of two antennas separated by a known distance that observe a common scene. The small difference in arrival time of signals incident on the antennas is exploited to estimate the angle of arrival, which, depending upon the arrangement of the antennas, can be related to properties of the scattering surface. Time difference is measured by comparing the phases of the echo recorded by each antenna. In radar interferometry, it is common for one antenna to transmit and receive while the other is receive only.

Following the development of Rodriguez and Martin (1992), we consider an interferometer as depicted in Fig. 1, with antennas separated by a baseline, B_a, tilted by an angle, α, with respect to vertical, and oriented toward incidence angle θ. The height of the surface in Fig. 1 can be reconstructed by estimating the differential range, ΔR = R₂ − R₁. Differential range is related to the differential phase of the echoes received by each antenna,

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where ϕ is the absolute phase difference between antenna 1 and antenna 2, and λ is the electromagnetic wavelength. Because ϕ is only measured modulo 2π, the absolute phase is not actually known, so some form of phase unwrapping (to be discussed later) must usually be employed to obtain the proper value of ΔR. The surface height is calculated using the law of cosines,

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It is assumed in (1)–(3) that R₁, λ, α, B_a, and H are known, while ΔR is obtained from the measured phase, ϕ. The phase is obtained from a short-time ensemble average,

ϕ̂V₁V^*_2N(4)

where V₁ and V₂ are the respective complex echoes (fields) from antennas 1 and 2, and N is the number of independent realizations, or “looks,” that are averaged to reduce the uncertainty due to fading or speckle that is common in coherent radar imagery.

Taking the derivative of (3) with respect to ϕ, the retrieved height uncertainty, σ_h, due to measured phase uncertainty, σ_ϕ, is given by

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Because surface height is retrieved from ϕ, knowledge of phase uncertainties is critical in interferometer design. Sources of phase uncertainty include decorrelation of the backscattered signal and finite signal-to-noise ratio. The Cramer–Rao bound on the phase estimate uncertainty is given by (Miller and Rochwarger 1972; Rodriguez and Martin 1992)

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where ρ is the correlation coefficient between measured signals V₁ and V₂, and SNR is the signal-to-noise ratio. To minimize σ_ϕ and, hence, σ_h, it is desired that V₁ and V₂ be highly correlated and that SNR be large.

Two primary sources of decorrelation are geometric and temporal decorrelation. Geometric decorrelation occurs due to the (usually very small) difference in observation angle of the antennas on the resolution cell under observation. Derived in Li and Goldstein (1990) and Rodriguez and Martin (1992), it may be thought of as a form of the Van Cittert–Zernicke theorem for scattering from a small, pulse-limited area. This theorem describes a Fourier transform relationship between the spatial autocorrelation function of the field observed at an aperture and the far field intensity distribution incident on the aperture. For the radar interferometer, geometric decorrelation is given by

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where the g subscript denotes geometric and Δy is the surface range resolution. While this equation sets an upper bound on the baseline distance (when δ_g → 1), in many practical interferometer designs the maximum baseline is dictated by other factors, such as mechanical constraints. Thus, the actual baseline may be much smaller than the maximum baseline, in which case geometric decorrelation will be small.

Temporal decorrelation may occur if the antennas are not sampled simultaneously, but rather on alternate pulses (as in the system to be described presently). Such a scheme permits a single radar receiver to be time-shared by the antennas. In this case, care must be taken that line-of-sight motions between pulses may either be safely ignored or may be compensated with separate knowledge of Doppler velocity.

If the surface is moving and the two interferometer antennas are sampled sequentially, then the phase difference between V₁ and V₂ will include a component due to the line-of-sight change in range. Also, V₁ and V₂ will decorrelate slightly due to surface scatterer motions within the resolution cell during the short time lag between measurements. Chapman et al. (1994) showed that the probability distribution function (pdf) of the phase differences of two measurements of the ocean surface separated by time lag, τ, is well modeled using a relation for joint Gaussian in-phase and quadrature signals (Middleton 1960):

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where ρ(τ) is the normalized autocovariance of the complex signal, V(t), observed by a single antenna

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For microwave scattering from the ocean surface, the normalized autocovariance may be expressed analytically using

ρτe^−τ2/τ2_ce^−j2kυ_rτ(11)

where τ_c is the coherence time of the surface scatterers, k is the radar wavenumber, 2π/λ, and υ_r is the mean radial velocity of the scatterers. The particular value of τ_c depends upon electromagnetic frequency, illuminated area, and sea state. A value of τ_c ≈ 10 ms is commonly cited for X-band frequencies (Plant et al. 1994). It should be noted that both the radial velocity and finite coherence time are factors limiting the spatial resolving abilities of airborne synthetic aperture radars (Hasselmann et al. 1985; Plant et al. 1994).

Figure 2 shows p_ϕ(ϕ) for several time lags. For short time lags, the pdf is highly peaked about a mean phase consistent with the average motion of scatterers. The spread of scatterer motions contributes to the width of the pdf. For longer time lags, the mean and the variance of the pdf both increase. The standard deviation of the phase difference pdf, σ_ϕ, factors into the height uncertainty through substituting into (5), while the centroid of the pdf, ϕ, contributes a bias in a particular measurement. If the radial velocity can be measured separately, then the primary contributor to the height uncertainty is σ_ϕ. If, however, the radial velocity is not known, then both σ_ϕ and ϕ must be considered.

The measured interferometric phase in the case of surface radial motion has the form

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which for R_1,2 ≫ B_a can be simplified to

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The first term in brackets is the differential range due to the angle of arrival of the echo upon the interferometer antennas (the desired quantity), while the second term is the differential range due to radial scatterer motion. These terms are very nearly independent, since the changes in range due to Doppler velocity have negligible effect on angle of arrival (note that unambiguous measurement of velocity implies radial motions of less than λ/2 between radar pulses).

The angle of arrival, θ, is related to the time-varying surface height, h, through (3), while υ_r is associated with the radial component of the orbital velocities of the surface waves. With the exception of occasional high-velocity excursions in υ_r associated with wave breaking or with scatterers bound to wave crests, both θ and υ_r are slowly time-varying quantities with periods typically measured in seconds. Thus, with a separate measurement of υ_r obtained using either of the interferometer antennas, it is possible to correct each measured ϕ for surface radial motion.

The remaining and usually dominant source of phase uncertainty is finite signal-to-noise ratio. For ρ ≈ 1 and SNR > 10 or so, the dependence of the rms phase uncertainty on signal-to-noise ratio reduces to

View Expanded

This uncertainty, when substituted into (5), yields the rms height uncertainty for a given geometry. Figure 3 shows the rms height uncertainty as a function of signal-to-noise ratio for several practical baselines. These demonstrate the need for high signal-to-noise ratios or large baseline separations for precise height measurements.

A final consideration on the measurement is that the phase sensitivity afforded by large baselines must be traded against the potential error introduced by phase unwrapping needed in the interferometric images. Recall that the interferometer measures the phase difference between the two antennas modulo 2π. Retrieval of surface height, however, requires the absolute phase difference between the antennas, which often exceeds 2π. Several methods of phase unwrapping have been investigated in the literature. All attempt to balance a measure of unwrapping error performance with processing speed. Some of the faster unwrapping methods, notably the residue-cut method (Goldstein et al. 1988), can fail to unwrap an image completely. However, the portions of the image that are successfully unwrapped have very low rms error. A somewhat slower method, the least squares error (LSE) unwrapping algorithm (Ghiglia and Romero 1994) always completely unwraps the entire image, but the rms error is generally higher than that obtained using the residue-cut method. Several other methods approach the residue-cut method in rms error levels, but they are comparativly slow (Zebker and Lu 1998; Ghiglia and Romero 1994). In our application, in which a large number of images must be unwrapped, an efficient unwrapping routine requiring little or no user intervention is required. The LSE routine is therefore chosen despite its suboptimum rms error performance.

3. Experimental setup

This section describes an experiment to estimate surface displacement fields using imaging radar interferometry. We use a second-generation version of the focused phased array imaging radar (FOPAIR) (McIntosh et al. 1995), an X-band radar designed to image the ocean surface with meter-scale spatial resolution. The radar was deployed from a small pier at the University of Massachusetts School of Marine Science and Technology (SMAST) in New Bedford, Massachusetts. During October 1999, a 10-day measurement campaign was undertaken consisting of several 15–20-min-long observations of a fixed area of the sea surface under a variety of environmental conditions. Figure 4 shows the layout of the research pier and a photo of the radar setup. The pier extends into Clark's Cove adjacent to New Bedford Harbor. It lies on an east–west axis with an unobstructed view of Buzzards Bay to the south. The Elizabeth Islands at the mouth of Buzzard's Bay limit the fetch in this direction to about 15 km. Thus, wave heights at this site are generally small.

The radar was placed near the end of the pier and oriented to the south. A vertical baseline interferometer (α = 0) was implemented using two 32-element antenna arrays, as shown in the Fig. 4 photo. The imaged area was a 24° wide sector between 43 and 64 m, resulting in grazing angles between 4° and 2.7°, respectively. Both arrays were measured to be within 0.1° of level along their length.

A resistive wave wire was deployed from the end of a 4-m boom extending off the pier. A low-pass filter band-limited the output of the wave wire to 10 Hz. The wave-wire signal was sampled at 80 Hz (much faster than necessary, but the slowest setting of the A/D used). It was averaged to a 4-Hz time series in postprocessing. Prior to each radar data collection, a 5-min-long calibration was performed in which the wire was shortened by 1 m, allowing a conversion between recorded voltage and height. Wind speed and direction were measured from a small tower erected on the pier. The anemometer height was approximately 10 m above the mean water level.

Complex radar images of the sea surface were obtained with the upper and lower arrays at a 64-Hz frame rate. Image pairs were cross correlated and ensemble averaged over 16 looks, yielding interferometric images at a 4-Hz rate. The phase images, or interferograms, were input into the LSE unwrapping algorithm, and (2) and (3) were applied to the output phase field to estimate the height at each pixel.

One problem encountered in data processing is that this technique produces images with correct height contrast, but not the correct absolute height. Due to tidal variations, the relative height of the interferometer over the water changes. This creates a problem, as ΔR in (2) is obtained from the measured interferometric phase which is known only modulo 2π. And, while phase unwrapping enforces spatial continuity of the resultant height field, it does not determine precisely in which lobe of the interferometer fringe pattern the nearest portion of the imaged surface actually resides. This “fringe uncertainty” is equivalent to uncertainty in the number of integral cycles of 2π required in addition to the measured phase to obtain the correct ΔR in (1).

In our case, an error in the height of the interferometer yields height imagery that, when averaged over time, has a mean slope along the range axis. Luckily, one may safely assume that both the mean height and the mean slope should be zero. A linear extrapolation of the retrieved mean height profile back to the radar location yields the height error, allowing the absolute height to be retrieved.

Thirty-six data collections, all with the radar oriented approximately upwind and upwave, are used in this study. Table 1 summarizes the environmental conditions under which they were acquired. For each radar collection there is a corresponding wave-wire data record.

4. Results

Figure 5 shows a sample image of vertically polarized radar backscatter superimposed on the retrieved surface profile. Here, the vertical scale of the image is greatly exaggerated. It can be observed that radiometrically brighter portions of the image correspond to the front (advancing) faces of the waves, while darker portions correspond to the back faces of the waves. This is consistent with the surface slope-induced modulation of the radar echo from Bragg-resonant capillary waves. A particularly bright feature is evident near the crest of the gravity wave that may indicate a non-Bragg scatterer.

Figure 6 shows range-versus-time images constructed using the central beam of the radar. Surface height (top), Doppler velocity (middle), and backscattered power (bottom) are shown. Surface heights shown are relative to the lower interferometer antenna. Doppler velocities represent radial components of surface wave orbital velocity plus a bias due to Bragg-resonant capillary wave phase velocity and surface wind drift. Backscattered power has been corrected for a cubic dependence on range, so the image, while uncalibrated, is proportional to normalized radar cross section, σ°. Note that both height and velocity measurements (derived from phase differences) are noiselike when the backscattered power is low.

All three images show similar advancing wave features; however, there are also notable differences between them. Wave group structure is evident in all three images; however, a few radiometrically bright, high-velocity features consistent with the wave group velocity (approximately half the phase velocity) are evident in the Doppler and power images but are not discernible in the height image. There is also an increasing prevalence of high spatial and temporal frequency content observed going from the height image to the velocity image to the power image. This is consistent with expectations, as the velocity is proportional to the time derivative of surface displacement, thereby emphasizing higher-frequency waves in proportion to their radian frequency, Ω. The power image is largely modulated by surface slope. Invoking the dispersion relation, the slope, KA, of a wave of amplitude A is proportional to Ω²A. Thus, higher-frequency waves are emphasized in proportion to the square of their radian frequency.

To check consistency between velocity and height estimates, Fig. 7 shows a comparison of predicted rms radial velocities derived from the height measurements to the measured rms Doppler velocities. For this comparison, the center pixel from the displacement image was extracted and the entire time series of 4800 samples was Fourier transformed. Multipication of the result by the radian frequency Ω and inverse Fourier transforming completes the conversion to orbital velocity. Rms values of the resulting time series are then plotted versus the rms Doppler velocity of the same image pixel. The scatter in the plot, with the exception of a few points, is within ±0.05 m s⁻¹ of the unit slope line.

To check validity of retrieved heights, histograms of radar-derived heights and wave-wire measurements are compared in Fig. 8. Twenty-minute-long time series were used to construct the histograms shown. Here the time series were binned into 2.5-cm intervals. It is evident that both the wave wire and the radar are producing nearly identical statistics.

Figure 9 compares wave height spectra from the wave wire and from the radar. Because only a time series is available from the wave wire, one pixel from the center of the displacement image is used in the comparison. The spectra shown are ensemble averages of 18 Hanning-windowed, 256-point variance spectra. The Nyquist frequency of the spectra is 2 Hz, and spectral resolution is 1.56 × 10⁻² Hz. Good agreement between the wave wire (solid line) and the radar (dashed line) is observed for frequencies between 0.2 and 0.8 Hz. Outside of this band, wave-wire spectral densities fall below the measurement noise floor of the radar. The source of the noise floor is intermittent episodes of large variance in interferometric phase when the radar echo signal-to-noise ratio is low. Such episodes occur when the illuminated area (pixel) is, for example, shadowed from the transmitter by an intervening wave crest. Given the impulsive nature of this phase noise, modest improvement on the high-frequency end of the spectrum is obtained using a simple three-point median filter (dotted line). A two-dimensional wavenumber spectrum is shown in Fig. 10 for the same case as the frequency spectrum. This was obtained from radar imagery by integrating a three-dimensional wavenumber–frequency spectrum over positive frequencies. Contours are shown at 0.05, 0.2, 0.4, 0.6, and 0.8 times the spectral peak (linear scale). In this case, the dominant wavelength of 18 m is consistent with the frequency spectral peak of 0.29 Hz.

Figure 11 compares radar-derived and wave-wire-derived significant wave height, H_s, calculated as 4 times the standard deviation of the displacement time series. Again, the center pixel of the radar image was used in each case. The rms uncertainty in a given radar estimate is less than 4.8% of the estimate, and the rms uncertainty in a given wave-wire estimate is less than 4.1% of the estimate. Because significant wave height (SWH) was often quite low, an accounting of excess variance introduced by finite SNR was required, as this biased the radar's estimates. The noise floor of the radar was monitored throughout the experiment, and the median SNR was calculated for each case used in the comparison. The median SNR for the cases shown varies from 16 dB for low SWHs to 31 dB for higher SWHs. Using (5) with median values for SNR, variances due strictly to phase uncertainty can be computed. These are independent of the intrinsic variance of the surface height. These were then subtracted from the variances of the retrieved height time series resulting in Fig. 11.

The correlation coefficient between the radar-derived and wave-wire-derived SWH is 0.92, and a linear fit to the data yields a slope of 0.97 and an intercept of −1.0 × 10⁻³ m. The data has an rms deviation of 4.6 × 10⁻² m from the linear fit. The high correlation for the relatively small SWHs experienced during the experiment demonstrate that the radar is able to precisely measure small surface displacements.

5. Summary

In this paper we have described a method to retreive ocean surface displacements using radar interferometry. Results from a pier-based experiment demonstrate consistency between retrieved heights, Doppler velocities, and echo powers. Retrieved heights also compare favorably with direct measurements by a resistive wave wire. Correlation between in situ and radar-derived significant wave heights exceeds 0.92 over the range of SWHs encountered. A linear fit of the radar measurement with the wave-wire measurements results in a slope of 0.97 and an intercept of −0.001 m⁻² after removing biases due to finite SNR and temporal decorrelation.

The observations in this study were limited to rather low sea states, indicating that the radar technique is amenable to precise height measurements. While the measurements obtained in this study focused on relatively short surface waves imaged within a small area, the technique is scalable to larger illuminated areas. This is provided that the radar can be deployed from a suitable height to avoid significant shadowing of the radar's transmitted signal by wave crests.