A Ku-Band Laboratory Experiment on the Electromagnetic Bias

Sea-surface electromagnetic bias (EM bias), the difference between the mean reflecting surface and the geometric mean sea level, must be accurately determined to realize the full potential of satellite altimeters. A uniformly valid algorithm relating the normalized (or nondimensional) EM bias, Le., “bias/significant wave height,” to physical variables has not yet been established, so we conducted laboratory experiments to guide model development. Simultaneous and collocated measurements of surface topography and altimeter backscattered power were made in the large IMST wind-wave facility for a wide range of wind and mechanically generated wave conditions. A small microwave footprint on the water surface was produced by a focused-beam 13.5 GHz radar system that has a high signal-to-noise ratio. Consequently specular facets are easily identifiable and the data show that troughs are on average better reflectors than crests. Dimensional relations seldom yield robust algorithms and in fact, although rather high correlation is found between normalized EM bias and either wind speed or wave height, the laboratory coefficients are considerably greater than those of in situ algorithms. Nondimensional parameterization is more useful for deriving scaling laws, and when the normalized EM bias is displayed as a function of wave height skewness or wave age, laboratory and field data converge into consistent trends. In particular, normalized bias decreases with wave age, but unfortunately, even the wave age model does not account for the effects of mechanically generated waves, which produce appreciable scatter relative to the pure wind cases. Thus, we propose a two-parameter model using 1) a nondimensional wave height, which is computed for local winds, and 2) a significant slope, which is computed for nonlocally generated waves. Analysis of the laboratory data shows that the normalized EM bias for mixed conditions is well modeled as a product of these two parameters.


I. INTRODUCTION
S PACEBORNE radar altimeters are used to map mean sea level (msl), which is of crucial interest in geodetic, topographic, and large-scale ocean circulation studies.A radar altimeter is an active microwave instrument from which short pulses are transmitted vertically towards the sea-surface.When a pulse hits the ocean, the echo is par tially reflected by specular facets towards a receiver on-board the spacecraft.The round-trip travel time of the pulse is related to the altitude of the satellite above the sea.If the satellite orbit is known, then the altimeter range measurements can be used to determine the sea surface topography.Horizontal gradients of dynamic sea-surface topography provide a measure of oceanic geostrophic cur rents.The dynamic topography of a western boundary current like the Gulf Stream can be about 2 m, but many other gradients are so weak that a precision of few centi meters is required for msl measurements to be useful.
In fact, altimeter range measurements yield the mean reflecting surface level, which generally differs from the true msl-the deviation is known as electromagnetic bias (EM bias).The bias occurs because sea-surface roughness is spatially inhomogeneous.For microwave frequencies, troughs typically are better reflectors than crests so the level of the mean reflecting surface is usually lower than the true msl.EM bias can be large enough that it needs to be accounted for to get optimum use of altimeter data.
In spite of intense work since the first observation by Yaplee et al. [I], there is still no robust EM bias empirical algorithm or analytical model.EM bias is a function of the joint wave height and wave slope probability density function (pdf), which is not measured.There is as yet no unique inversion algorithm for deriving EM bias from re turn waveforms [2], [3].EM bias is primarily dependent upon sea-surface topography, but radar pointing angle [ 4 ] [ 6] and radar frequency [7], [8] are also important.Skew ness of the wave height distribution has been theoretically related to EM bias [9], and for suitable model assump tions, it can be deduced from the return waveform.But skewness estimates from waveforms are frequently not accurate enough for this method to provide an operational algorithm [10] so its operational use is questionable.As a result of these factors, potential operational algorithms are semi-empirical relationships that use l) the significant wave height SWH, which is related to the rise time of the return waveform leading edge, and 2) the wind speed U, which is related to the backscattered radar cross section U 0 [11].
Considerable effort has been devoted to modeling EM bias, and comprehensive reviews of theoretical and ex perimental studies can be found in [3] and [ 12).Table I summarizes assorted theoretical and observational expres sions for computing the so-called normalized or nondi mensional EM bias, (3 = EM bias/SWH.Relationship based on the cross-skewness coefficient -y (formula (3) in   I) seems to be the most defendable considering mi crowave scattering theorr, but r© is a complicated com bination of second-and third-order statistical moments of re joint wave height and wave slope pdf [13)-[l 6], which are usually obr}ined from sea-surface wavenumber spec tral models.Since wavenumber spectra are not measured by altimeters, this approach will require input from other sensors and so it is not likely to be used as primary dar} reduction algorithm.A wave-age dependence has been proposed [5] (formula (5) in Table I), and for a power law form of re wavenumber spectrum [17], [18] I).Younger seas are expected to yield higher bias than older seas.The nondimensional significant wave height model could be used as a primarr data reduction algorithm since altime ters provide measures of U and SWH; however, the coef ficients need to be checked to establish their range of va lidity.Finally, the TOPEX team selected the relation given by expression (13) in Table I [22].Equation ( 13) contains variables which are known to be highly corre lated and thus not independent.Furthermore, they do not use nondimensional parameters which have been found to be useful and robust for characterizing the sea surface.

TABLE I THEORETICAL AND OBSERVATIONAL (Ku-BAND) PARAMETERIZATIONS OF THE EM BIAS
The reader should be aware that EM bias is a function of 1) microwave wavelength, and 2) the height of the ra dar above the water surface [7], [8], [23].While these effects are important, they are beyond the scope of this investigation since as in many other studies, we consider only a single radar frequency and one experimental con figuration.Indeed, field measurements from towers differ from aircraft and satellite measurements so a more gen eral representation will be required to provide an unified model.The data from laboratory experiments may be use ful to extend the range of values for validation of models that include range effects.
The feasibility of bias measurements in a laboratory wind-wave tank was demonstrated by Lifermann et al. [24] .Parsons and Miller [25] measured the bias from var ious combinations of wind and mechanically generated waves in the NASA Wallops Flight Facility wind-wave tank using a focused beam radar.Their data from wind cases showed agreement with models by Jackson [91 and Huang et al. [26] , but the influence of background waves was unexplained.
In this paper, we examine some empirical models for EM bias using new data from experiments that we con ducted with a 13.5 GHz Ku-band radar in a large wind wave tank.This microwave frequency is close to one of the TOPEX/POSEIDON altimeter frequencies .These new data and existing data are used in the continuing search for a robust model .The experimental setup and data processing procedures are described in Section II.Section III documents the observational conditions .We present the radar measured wave height distributions in Section IV and modeling results in Section V.In the final section, we discuss the implications of these laboratory experiments for ocean altimetry .

II. EXPERIMENTAL ARRANGEMENT AND DATA PROCESSING PROCEDURES
The experiments were conducted in the large air-sea interaction simulation facility of the IMST Laboratory (Fig .1), which is described in detail in [27].The water tank is 40 m long , 3 m wide, and approximately 1 m deep, and the height of the aerodynamical flow above the water surface is about 1.5 m.Waves in the frequency range 1-2 Hz with various steepness up to wavebreaking can also be generated by a completely immersed electrohy draulic wavemaker.
The wind velocity can be adjusted from 0.5 to 14 m • st.We measured the mean wind speed at l m above the water surface with a Pitot tube with a precision of a few cm • s _,.For the range of wind speed considered in this work, the air turbulent boundary layer thickness re mains smaller than about 50 cm [281 , so the reference velocity U measured at l m height represents U 00 • As usual, when relating laboratory and field observations, the U value is taken as equivalent to U 10 from field measure ments [17] , [29] .While the drag coefficient over wind waves is still an open question, the above

A. Exp erimental Setup
We operated a focused beam, bistatic, 13.5 GHz radar system in the IMST wave tank at a fetch of 28 m, as shown in Fig. 2. The bistatic configuration is similar to a nadir looking monostatic configuration because in both cases the specular points are the zero slope facets of the water-  waves.This radar system evolved from a scatterometer system, which was developed by Bliven and Norcross [30] and which has been used in wave-tanks by Giovanangeli et al. (3 1] and Bliven and Giovanangeli [32] .The 1 kHz square-wave modulated source is from a Gunn oscillator, which is attached to a 10 ° standard gain horn that is aimed at a large focusing lens.The modulated signal is scattered from the water surface and detected by a square-law crys tal diode that is attached to another 10 ° standard gain horn.The signal is amplified and demodulated by two narrow-band HP-4 15E SWR meters, which were cali brated with a signal generator of known amplitude.The two amplifiers are used in a parallel configuration with gains of IO and 20 dB, which provides good precision for both low and high signals.

>----t 3m
An important consideration for laboratory , tower, and aircraft EM bias studies is to have the radar illuminated spot size much smaller than the length of the dominant waves because we need to know the scattered power level as a function of long wave elevation.So to focus the radio waves , we used a 42 cm diameter dielectric lens, which produces a spot size of 3 cm diameter at the l m focal length.This lens was designed and constructed by Par sons and Miller [25].
At first glance, this bistatic configuration is equivalent to a nadir-looking monostatic configuration because in both cases specular points are the zero slope facets.The situation, however, is compounded by the proximity to the surface.The distance between the transmit horn and the lens is l m.The distance between the water level at rest and the receive horn is 0. 7 m.The energy coming from the standard gain horn is focused by the 42 cm di ameter lens to essentially a point at a distance of 1 m from the lens.Therefore, energy is converging on that point from directions spanning ± 12 °; which suggests that specular facets contributing to the fĕceived energy will have orientations between ± 12 ° of horizonfĦl.But the actual distribution includes the weighting function.from the source 10 ° standaf gain horn, which has a relatively narrow beamwidth, i.e., a -3 dB attenuation at +5 ° and -5 ° and a -10 dB attenuation at -8 ° and + 8 °.The two sidelobes are centered around -14 °•and + 14 ° and their relative level is -15 dB .So they do not contribute to il luminate the spot.Consequently, the wave tank sfĉcular facets are predominantly from within ±5 ° of horizontal.The reader should be aware that fħis differs somewhat half wavelength intervals.The interference effect is sig nificantly reduced by the bistatic configuftion and by ab sorbing material placed at appropriate locations (25), (35) .To obfĦin the refaeectivity signal, a range scaling fâctor must be determined as a function of water level.The eas iest way to calibrate was to recof the backscattered power while slowly draining or fãlling the tank with water, which was the procedure used by Parsons and Miller [25).Thus, for a given tank water level h, the acquired faeat surface fflected power Peal (h) was considered as the basic ref erence power fêr calibration, and the reflectivity signal is r(t) = P(t)/P cal (h (t)) (1) from the altimeter case where essentially horizonfl facets with p (t) and h (t) being, respectively, the backscattered contribute.Defils on the lens design and on properties power and water elevation at instant t.Kirchner et al. [35] of the focused apertufĕs can be found in (33].Installation féund theoretically a radar power return proportional to and performance of a radar focused by the lens used in v-2 due to range effects and spot size variations with the this study is described in [34] and l 35 1lens configuration (D being the distance between the lens The radar functioned as a scatterometer because it and the water level).From the drain tests, we found the _measured re _ flected power but � ot range.!herefire, the backscattered power p to be a decreasing function of D, water elevat10n was measured with a capacitance probe of but the main variations in p were due to interference ef-0.3 mm outer diameter located within the radar spot.Con-� fects (radio waves going back and forth).The correction s � quently backsca : tered _ power and water surface level fâctor oscillates evef half radar wavelength with approx signals were acquired simultaneously at the same locaimately a 0_ 7 dB amplitude.Interferences effects are dis tion, thus allowing EM bias evaluations.The accuracy of cussed in detail in [33] and (34] .Measured variations of the water level me � suremen � s by the probe was ±0.2 mm. the scaling factor Peal (h) were consistent with our as Test runs made with and without the wave gauge showed sumptions and with the above cited pfĖvious studies.that the reflected power was decfased by less fħan 1 % Another concern for simulating satellite conditions by when fĨe probe was present within the radar spot, but this ground based exfĊriments is the so-calledfocusing effect.d � crease was not wa _ ter level d � pe � dent, and consequently When operating a radar from space, the facet-like reflec did not affect E � bias de !ermtna � on.
. tors on the surface that cause the fturned signal are nearly Ra _ dar absorbmg matenal was mstalled at appropn � te horizontal.It is desirable to simulate these conditions from locations around and above the radar setup, on the ceiling, and on the side walls of the tank in order to minimize multipath reflection.Numerous tests made with the same environmental conditions but with different configura tions of absorbing material showed that EM bias deter mination was insensitive to the placement of absorbing material.This is probably because the received power from multipath reflections was much lower than from specular facets.Overall, the signal-to-noise ratio was féund to be excellent: a small disk of absorbing material just above the spot decreased the reflected power by more than 30 dB .

B. Radar Calibration
To compute EM bias, only reflectivity measurements are needed-so absolute calibration of the radar system is unnecessary.We defined a reflectivity signal as the ratio between the power reflected from the roughen surface to the corresponding faeat surface.A range scaling factor is needed since measufĕments showed that the backscattefėd power from a calm water surface is a function of water level elevation (range effects and spot size variations).There is also an interference effect: radio waves going back and forth along the path between the transmitter, the lens, the water surface, and the receiver produce station ary electromagnetic waves with nodes and antinodes at onboard an aircraft, from a tower, or in a wave tank.One potential problem that can arise is due to focusing facets that are attributable to the possibility of concave (upward) facets focusing the incident signal back toward the re ceiver.Reinforcement, which may be viewed as a gener alized representation of focusing, occurs when there is coherent combination of reflection elements distributed across all, or a portion of the féotprint.At the surface, such reinforcement is a consequence of favorably oriented plane facets concentric to the illumination wavefront, and having a range distribution that is relatively concentrated with respect to the radar wavelength.These facets will give a high power level at the radar and since these fâcets will be more predominant in the troughs due their concave shape, there could be a built-in bias.While this effect may be important, a complete analysis has not yet appeared in the literature and it is beyond the scope of this study .
We have, however, investigated scattering using a sim ple geometric ray-tracing model.We simulated the traj ec tories of optic rays reflected by the water surface using geometric optic approximation.Given an illuminated area, we computed the density of rays that come back to .the receiver.The built-in bias due to focusing effect was de fhned by the mean of the wave profhle weighted by this density function.We quantified this bias for cases of sinus waves fir which no EM bias should occur due to the crest-to-trough symmetry .Waves were characterized by their wavenumber k and their steepness E = ak, a being the amplitude.The computations showed that the built-in bias due to focusing was maximum when the radius of curva ture above the trough, r e = (Ek) -1 , was equal to radar elevation R. The results are summarized in Table II.The parameters of interest are the ratios R / r, and d / L (d is the diameter footprint, and L is the wavelength : L = 21r / k) .When R / r e >> 1, there is no focusing effect due to geometric consideration_ This is the case for satellite data and some aircraft data.When R / r, -1 the built-in bias is significant when the spot size is not small relatively to the dominant wavelength.This is typical of patterns seen on the bottom of illuminated swimming pools, and this could occur in wave-tanks for laboratory radar sys tems without a lens to focus the electromagnetic waves .When R / r, -I and d / L << I. which is the normal situation for wave-tank with focused radar systems, tower, and some aircraft data, the focusing effect is relatively small: the built-in bias is of order 0%-2 % , depending upon R. L and the wave steepness E.
This simple model does not solve the full focusing problem, but it gives some clues .An analytical approach of the focusing question is not feasible because the sur face is unknown_ If the surface characteristics were known, the EM bias issue would be completely solved.Generalized focusing is determined by spatial coherence of the collective facet reflections .The fundamental spatial scale parameter is not the footprint diameter, but the ef fective diameter of the coherent footprint.The relative coherent scattering component may differ depending upon radar range and radar wavelength.A broad set of data from wave-tanks , towers , and aircraft will be useful to validate more precise physical models of focusing effects and the data presented herein can contribute to those ef forts_

C. Data Processing Procedures
For each run, the two analog 10 and 20 dB gain radar outputs and the wave gauge analog signal were acquired at 300 Hz.The analog signals were low-pass filtered with a cut off frequency of 100 Hz in order to avoid aliasing.180 000 samples per line and per run were stored in mem ory, so each run lasted 10 minutes.Data were acquired both on a PC 386 for real time analysis and on a HP 1000 computer for post processing.The 10 and 20 dB gain ra dar signals were combined to yield double-decade high precision backscattered power signals P(t) .and the re flectivity signals r(t) defined in ( 1 ) were computed.
Probability density functions (pdf) of wave height el evation were estimated in a standard manner from the wave gauge signal.Elevation data were normalized by the standard deviation of the wave height distribution.Then the normalized values were sorted into elevation bins of width 1 / 15 standard deviation.The pdf value of bin num ber "i" was equal to the ratio of the number of samples that belonged to this bin, namely N, , to the total number (2) In order to compute the elevation pdf of the facets that reflect the radio waves, each sample was weighed by its reflectivity value.So for each elevation bin, the pdf of radar elevation value was equal to the amount of reflec tivity produced by the samples that belong to that bin, normalized by the total reflectivity.If r u is the reflectivity of sample j of bin i, then pdf Radar Heighi ( i ) = I; f;; The EM bias is just the difference between the means of distributions ( 2) and (3): EM bias = L h; (pdf Radar Hei g h,(i) -pdf Hei g ht ( i)) . (4) ' For each test run, we computed distributions (2) and (3) and compared their first four moments.The significant wave height, SWH, was taken as four times the standard deviation.FFT analysis gave us the dominant frequency Measurements were made at a fetch of 28 m.At this distance, the wave field was free from refçection from the wave absorber located at the downwind end of the tank.In the next section, we describe some illustrative results concerning the radar and wave height signatures related with the EM bias determination.

IV . RADAR SIGNALS AND WAVE HEIGHT DISTRIBUTIONS
A. Time Series  [24] under similar environmental conditions.The peaks are probably due to the flat surface that exists ahead of a near breaking crest, as identified by Bonmarin [37] using video techniques.Note that there is no backscatter ing at nadir of the breaking crests.For moderate winds (wind speed ranging from 3 to 8 m • s-1 ), the radar is also sensitive to the water wave nat ufl modulations: on the average, the reflected power is smaller over groups of steep waves than between wave groups.The obsefěed low-frequency variation in the backscattered power is due to local changes in the slope variance produced by subharmonic wave instabilities.
For cases of mechanically generated waves with no wind forcing, the water surface elevation h (t) and the as sociated radar signal r(t) are shown on Figs. 4 and 5.For small steepness ( § = 0.f®7-Fig.4), the radar signal is periodic with a high peak over each wave trough and a U= 0 m s-1 -------------------- Wave1--------------'----------1 ---------..,..,,-�-----i  of order -,r * § {26], i.e., -2.2%.For large steepness ( § = 0.012-Fig.5), time series of wave elevation show clearly the presence of wave groups due to natural sub harmonic instabilities.The amount of reflected power in creases significantly between two wave groups thus lead ing to low frequency modulations of the radar signal, and there is a radar enhancement above each wave trough thus giving a significant negative EM bias (/3 = -5.5%).An example of a combination of wind waves and pad dle waves is shown on Fig. 6.In this case, a light wind ( U = 2. 7 m • s -t) is acting on the gentle paddle waves described in Fig. 4. If compared with this previous case, the radar response changes drastically with no evident pe riodicity due to the presence of small ripples.The effect of the light wind is to decrease the bias (/3 = -1.6%)probably because the wind produces more uniform rough ness along the paddle waves.Then, when we increase the wind speed above such paddle waves.we find the EM bias to be an increasing function of wind speed as it is the case for pure wind waves.Time series of mixed waves conditions are quite difficult to interpret because there is no obvious way to separate the backscattering due to wind waves from the backscattering due to paddle waves.Non linear interactions between waves at different scales are likely to occur and how they affect the probability density fvnctions is of crucial interest here.

B. Probability Density Functions
Fig. 7 displays the pdf of the actual and the radar ob served wave-height distributions as defined in Section 11-C, for various observational conditions of pure wind waves.Some features are of interest.For example, both wave and radar distributions depart from Gaussian distri butions, and as the wind speed increases, the asymmetry increases.Also the two pdf depart more and more from each other, the radar pdf being more asymmetrical than the wave-height pdf.Lastly , at U = 14 m • s _,, the right side of the wave-height pdf exhibits a small bump, known to be due to wavebreaking [38].Such a bump does not appear on the radar pdf, which suggests that the crests of breaking waves are not significant scattering sources for altimetry.
Moments up to order four were computed from these distributions.EM bias values, which are related to the mean values of the distributions, will be presented in the next section.Results on SWH, computed from second order moments of both the wave height and the radar pdf are compared on Fig. 8(a).There is a systematic bias be tween the SWH observed by the radar and the actual SWH.Radar computed SWH is approximately 90% of the actual SWH.This is in accordance with recent results, namely, the GEOSA T measured SWH appeared generally smaller than the SWH values delivered by buoys [11], [39] and wave models [40] .
The skewnesses of the wave height and the radar dis tributions displayed in Fig. 7, are compared on Fig. 8(b) .For values smaller than 0.3, the skewnesses are identical.But at higher values, the radar skewness is almost three times greater than the actual wave height skewness.This is important with regard to the oceanic remote sensing of the wave height skewness [2].[ 10].Indeed, no difference is usually made between the skewness of the specular fac ets distribution and the actual wave height skewness.For wind generated waves.at short fetches the two skew nesses clearly depart significantly from each other.No specific relationship was found between the kurtosis of the two distributions .
Examples of pdf corresponding to mixed wind waves and paddle waves are shown on Fig. 9. Pure mechanical For each test, the reflectivity is on average much higher near the troughs than near the crests of the waves, thus yielding negative values of the bias.

A. Statistics
For each run, we computed the EM bias as indicated in Section II-C.
Table III      waves or mixed waves.The scatter of the data is quite large.The standard deviations are as high as the mean values.
From Table Ill, the bias as a percentage of SWH pos sesses a mean value of order -7 .0%.This is significantly higher than values usually found in open oceans in Ku band domain ( -2 % to -5% of SWH) [7], [12].Reasons for such difference will follow.It is seen that pure wind waves yield, on the average, values about 52 % higher than the values corresponding to mixed waves and about 110% higher than the values corresponding to pure paddle waves.This is qualitatively consistent with the numerical work of Ioualalen et al. [ 41]. who found higher biases for three dimensional wavefield than for two dimensional Clearly, a constant value is not appropriate for character izing the bias.

B. Parameterization with the Wind Sp eed and the Significant Wave Height
In order to understand the bias sea-state dependence, we have made regression analyses between absolute EM bias values and some other parameters such as SWH, U, §, skewness, / 0 , 11 0 , r 0 , as defined in Section II-C.We found quite good correlations with wind speed (r 2 0.78), significant slope (r 2 = 0.71) and skewness (r 2 = 0.52), but not with dominant wavelength or dominant fre quency (r 2 lower than 0.2).However, the best correlation was between EM bias and significant wave height.The relationship between these two parametef is not linear but quadratic (Fig. 10

Quadratic relationships between absolute EM bias and signifãcant wave height as well linear relationships be tween normalized EM bias and wind speed have also been obsefěed in open seas by various authof (see Table I, and [7], [12] fêr a review).
Note that if we consider only the pure wind wave runs, a mean square regression analysis conducted with a par abolic representation produces a high correlation coeffi cient (Fig. 11 This parabolic expression suggests a saturation effect at high wind speed with a maximum EM bias occurring at U = 14 m•s-1 • From the wind run data of Parsons and Miller [25] , we found a saturation effect occurring at U = 18 m • sl.This saturation effect has been mentioned in [23] (expression (9) in Table I), from the Melville et al. [12] open field tower measurements, the maximum bias occurring at U = 25 m • s -I.Also reported is the Ku-band parabolic expfssion proposed by Walsh [42] as an in terim algorithm for the TOPEX mission, the maximum bias occurring at U = 16 m • s-1 (expression (11) in Table I).Recently, Rodriguez et al. [8] have numerically pre dicted that the bias would increase with wind sfĉed up to about U = 10 m • s -1 then it would decrease with increas ing wind speed.Analyzing GEOSAT altimeter data, for wind speeds above 10 m • sl and SWH greater than 4 m, Witter and Chelton [43] found that biases decreased with increasing wind speed, which they attributed to effects of satellite pointing attitude angle and sea state.
The various parameterizations relating fĲ to the wind velocity U, listed on Table I, need clearly to be compared and discussed.The coefficients from our laboratory data are signifãcantly higher than those of the expressions es- Obsefěations from a large variety of conditions led us to think that the required coefficients a, b and c would be themselves sea-state dependent.The expressions give only explicit defĊndence of /3 on wind velocity U. We note, however, that at the same wind speed U, the wavefield from different exfĊriments may greatly differ from each other depending upon the fetch, the water depth, the pres-sure of currents or swell, etc .As a consequence, the biases would differ from each other.Note that the results of [44] (expression (12) in Table I) addressed mainly to deep ocean while those of [12] (expressions ( 8) and ( 9) in Ta ble I) were obtained in shallow water area, and that IMST wave-tank results came from young waves on deep water.Clearly , additional parameters related to the sea-state need to be introduced.
In that respect Melville et al. [12] chose SWH, a pa rameter directly available from altimeter measurements .They proposed the following expression, (3 = -0.0146-0.002 1 * U -0.0039 * SWH r 2 = 0.74 with the same correlation coefficient, IMST data fit with the expression: In these formulas U is in m • s-1 and SWH in m.
Again the coefficients greatly differ from one expres sion to the other.More particularly the SWH coefficients are not the same order of magnitude because tank-gener ated wave heights and open-sea wave heights are not the same order of magnitude.These later formulas involving SWH would have the same disadvantage as the previous formulas relating (3 to U, that is they apply only for par ticular observational conditions.

C. Representation with Dimensionless Parameters
There exists at this time no fu lly satisfactory theoretical way to relate the EM bias to the sea-state because it is necessary to predict the joint probability density function of wave-height and wave slope starting from the basic hy drodynamic equations.This is still actually out of reach of the theoretical models of wind waves.However the characterization of the sea-state has progressed quite sig nificantly on the basis of semi-empirical investigations.The sea-state characteristics are known to be better char acterized by dimensionles� parameters such as g * SWH /U 2 , n 0 * U/g, g *X/U 2 , g * T/ U • • • where X is the fetch and T the wind duration.An homogeneous relationship relating (3, which is a dimensionless param eter, to the sea-state would involve such parameters .Note that in existing analytical work, the EM bias is related with dimensionless variables such as the wave-height skewness [9] , significant slope [26] , the height slope cross skewness [ 13], the wave age and the nondimensional wave height [5], I 181, [ I 9] .Fig. 12 displays (3 versus the wave-height skewness for both airborne 10 GHz data [45] and IMST wind-wave tank 13.5 GHz data.Clearly, there is no gap between the two data sets.In addition, the evolution over the large range of the wave-height skewness is quite well predicted by the Jackson [9] theoretical expression.Unfortunately, from a practical point of view, the skewness cannot be used to predict the bias because the altimeter itself is not able to measure this parameter with enough precision or confidence [10].so other dimensionless parameters are required for the EM bias determination. The dimensionless mean reflectivity coefficient r de fined in Section II-C, is a candidate for the EM bi;� pa rameterization because it can be obtained from altimeter data.Theoretically, r0 "" 1 / a; where a; is the filtered slope variance, the low-pass cutoff wavelength being the 2.2 cm radar wavelength [36] .Following Cox and Munk [46] , a; "" U. We found previously (3 "" U, so we ex pected {3 "" 1 / r 0 • Considering all the nonzero wind ve locity runs, we found a good correlation, r 2 = 0.86, be tween (3 and I/ r 0 • The correlation decreased to r 2 = 0. 73 if 1 /r�, as proposed by Melville et al. [ 12], is considered instead of 1 / r 0 • When the overall set of 83 runs is taken into account, a completely empirical approach yields a highly correlated linear relationship between (3 and r expressed in decibels (Fig. 13): 0 (3 = -0.00852+ 0.0123 * r odB r 2 = 0.89.(9) This is the highest correlation coefficient fo und between fĽ and another parameter when all the 83 runs are con.sidered all together.The reflectivity coefficient is a good parameter to predict the bias probably because it contains intrinsic information on sea state.Nevertheless, we did not find the inverse square law dependence between the normalized bias and the backscattered power [see Table I, formula (8)] to be the optimal representation.
It is quite well established that the wave age C 0 / U, where C 0 is the phase speed of the dominant wave, and the nondimensional significant wave-height g * SWH / U 2 are two basic quantities which characterize the sea state on statistical grounds.These quantities have also been used to parameterize EM bias.Note that: 1) the formula agrees with [5] (formula (5) in .� Table I) .Young waves yield higher bias than waves; � 2) daf from wind waves fit closely to the regression cufĚe; and 3) the correlation coefficient is relatively high, but large scatter is obsefĚed fër mixed-waves runs.This is discussed elsewhere.
The average of the exponents of ( 11) and ( 12) is about -0.5.Such an exponent would make the resulting equa tion lineafęy proportional to wind speed.It may really be that this parameterization works because it is a surrogate for wind speed, with a randomizing efgect from wave height.
A point of interest is that formulas (10) and ( 11 one would expect significant reduction of the scatter no ticed for mixed-waves runs.
In the IMST large wind wave facility, expressions sim ilar to those of Hasselmann et al. [ 17] apply to the evo lution of the dimensionless dominant frequency n 0 * U / g and the dimensionless mean energy g 2 * SWH 2 / U 4 with respect to the dimensionless fetch g * X/ U 2 [47] .W hen used to display the EM bias as a function of a pseudowave age, they do not yield any reduction of the scatter, which remains similar to what is shown on Fig. 15 for mixed wave runs.This is not surprising as the pseudowave age is determined from expressions strictly valid for pure wind-wave fields in local equilibrium state.In this case there exists a one to one correspondence between the wave-age and the dimensionless wave-height, whereas with a swell the correspondence breaks down.Given a dimensionless significant wave-height.it appears not ten able to define an "equivalent " equilibrium state by sim ply introducing a pseudo wave-age.This leads us to con clude that in the presence of swell a simple quantity i; or g * SWH/ U 2 would be insufficient to predict EM bias.
The data scatter using formula ( 11) is shown on Fig. 16 and clearly the scatter is large.Recently Rodriguez et al. [8] pointed out two mechanisms underlying the EM bias: the modulation of small waves along the long waves and the long wave tilt modulation.So we proposed to ex plain the origin of the scatter by a two-parameter model, using dimensionless wave-height and wave-slope.Both pure wind waves and mixed seas can be represented by this model .A two parameter representation should in volve information on both wave-height and wave-slope because EM bias is the mean height of the zero slope fac ets.So we introduce the significant slope §, and the two parameter representation of the normalized EM bias is in terms of sign(ficant slope and nondimensional significant This model provides close agreement with all of the laboratory data for both pure wind and combined condi tions as shown in Fig. 17.This result helps us to under stand EM bias on physical basis but its application awaits a viable technique for measuring § from altimetric data.

VI. DISCUSSION AND CONCLUSION
During this work we were primarily concerned with the physical mechanism involved in the interactions between microwave and water-surface waves.Of main interest was the dependence of the microwave backscattered signal on the sea state or on specific event occurring in the wave field.In that respect, it is known from previous investi gations of Kwoh and Lake [48], Banner and Fooks [49] , Lifermann et al. (24] that experiments in a wave tank of fer unique advantage of allowing accurate control of the observational conditions.As a consequence, basic facts such as the sensitivity of the backscattered signal to the modulational instability or breaking of the surface waves were identified.
The illuminated spot size was small, so time series data were used in an analysis of pdf distributions to quantify the backscattered power as a function of wave phases.Generally the long-wave troughs were better reflectors than the long-wave crests.But this crest-to-trough asym metry, which was measured in terms of EM bias, de pended on local conditions.The goal of this paper was not to produce results for direct application to satellite altimetry.but rather to help to evaluate how robustly some empirical EM bias models characterize diverse data sets.
The analysis of open field observations and laboratory ob servations shows the inadequacy of EM bias parameter izations in terms of dimensional quantities.On the other hand, the data sets provide evidence that some recent pa rameterizations using nondimensional variables are quite robust.The laboratory data, however indicate that a new combination of nondimensional variables, which uses sig nifhcant wave-height and significant wave-slope, is needed to effectively model diverse wave conditions.
Although some of the EM bias models we presented include field data, fjll application to field conditions needs further consideration of specific problems.Among others is the focusing effect due to range limitations.This may lead to an additional bias towards the wave troughs.Un der the specifhc geometry of our observations; this bias would be small with respect to the first order bias asso ciated with the wave-field nonlinearity.The high EM bias values measured in the tank were not due to an additional built-in bias: they were mainly due the nonlinear wave field characterized by high skewness values of the wave height distributions .This is the physical law demon strated by Jackson [9] and illustrated in Fig. 12.
While comparing laboratory and field situations, of more importance would be the fĖinforcement effect which occurs when there is a coherent combination of reflection elements distributed across all, or a portion, of the foot print.The associated spatial scale parameter may signifi cantly differ from laboratory to field conditions.This lab oratory data set can contribute to a better understanding of what is going on.Clearly, further progress will require both improved instrumentation and analytical studies.For example, significant slope seems to be unobtainable with present single-frequency altimeter design but perhaps fj ture satellite-borne systems will use an enhanced design (building upon ROWS concepts [50]) or include comple mentary instruments such as an altimeter and a SAR.On the other hand, only a few investigations have been de voted to understanding the physical mechanisms.A nu merical study by Poitevin et al. [51) shows that a modu lated waveform produced by the instability of an initially uniform Stokes wave at moderate steepness yields abso lute bias oscillating between one to four .times the bias value for the Stokes wave.Ioualalen et al. [4 1] found that a two-dimensional wavefield with shof cfĖsted waves yields an absolute bias higher than that of a one�dimen sional wavefield for a same value of the significant slope.Rodriguez et al. [8] reported that both the modulation of small-scale waves and the modulation of large-scale sur face tilt contribute roughly the same amount to the bias.Additional research is needed to quantify local variability of wave fields in order to develop more realistic sftistical models for EM bias.
11 0 of the energy containing the waves.The dominant wavelength / 0 of the wave field was obtained from n 0 us ing the linear dispersion relation for deep water gravity waves.We calculated also the significant slope §, § = SWH / 4 * / 0 , the phase speed of the dominant waves C 0 , C 0 = I ,, * 11 0 , and the mean r 0 of the re flectivity coefficient r,J • III.OBSERVATIONAL CONDITIONS Our goal was to cover the whole set of experimental conditions allowed by the IMST facility.Consequently, 83 different experiments have been conducted and were scheduled as follows: 1) 22 wind runs with wind speed ranging from Ot o 14 m•s-1 • 2) 19 runs with paddle generated waves: the first I 0 with a paddle frequency of 1. 75 Hz but with different pad dle amplitudes and the last 9 with the same wave ampli-tude but with paddle frequencies ranging from 1. 1 to 2.1 Hz.The significant slope § ranged from 0.002 to 011 m • s -l (11 runs) • paddle fìequency 1.75 Hz, § = 0.014, U ranging from Oto 11 m • s-1 (10 runs)

Fig. 3
displays time series of the water surface eleva tion h (t) and the radar reflectivity r (t) for different wind speeds.With the water surface at rest, both signals are fçat.At U = 1.7 m • sl, the radar signal is still quite fçat as the wavelength of the dominant capillarity-gravity waves (1 0 = 5 cm) is shorter than three times the 2.2 cm radar wavelength [36].At U = 2. 7 m • s-1 , the wave- .length of the dominant waves is 14 cm and the radar sig nal changes drastically.Significant oscillations appear at the frequency of the dominant waves.The reflected power is much higher over the wave troughs than over the wave crests.This asymmetry, which characterizes the bias, is even more noticeable at U = 4. 0 m • s -1 • As the wind speed increases, the average value of the backscattered signal decreases and the crest-to-trough asymmetf in creases.At U = 1 1. 0 m • s -I the radar signal appears to be dominated by peaks occurring ahead of near breaking crests .This has been already obsefĚed by Liferrnann et al.

Fig. 3 .Fig. 4 .Fig. 5 .
Fig. 3. Samples of time series of the war~r surface deflection level h (t) and the radar reflectivity signal r(t) for pure wind waves.low peak above each wave crest, thus leading to a nega tive EM bias (� = -2.7%).This differs from the Pafons and Miller fėsults [25] for which positive bias was re ported for paddle waves.Theoretically, the bias should be zero for a sinus wave because the mean level of the horizontal facets is zero, but for Stokes waves fĨe bias is

Fig. 6 .
Fig. 6.Samples of time series of the water surface deflection level h (t) and the radar reflectivity signal , (r) for case of mechanical wave at 1.3 Hz frequency and small steepness ( § = 0.007) with wind effect (U = 2.7 m•s-1 ).

6 )
bias and SWH in cm) Because the absolute EM bias is so cleafęy fėlated with SWH, we have focused our attention on the so-called nor malized or nondimensional EM bias ratio {3 = EM bias/SWH.Considering all the 83 runs, we found a better correlation between fĲ and U (r 2 = 0. 74) than between fĲ and SWH (r 2 = 0.53).Fig. 11 shows clearly the wind speed dependence of the normalized EM bias.Taking into account runs with nonzero wind speed, we found the fêl lowing flationship: {3 = -0.0190-0.0100 * U r 2 = 0.77 ((with U in m•s-1 )