a. The importance of precipitation at sea and its measurement

Precipitation rates at sea are an important component of the net freshwater flux (precipitation minus evaporation), which has a major impact on surface water; salinity and temperature are the key parameters affecting the density of sea water and thus the thermohaline circulation. Information concerning rainfall at sea is also clearly important in understanding global atmospheric dynamics and the structure and circulation within major storms. Currently, direct observations of rain at sea have been very limited in nature because of the few vessels crossing the big ocean basins and the difficulties inherent in assessing rain rate when the ship’s superstructure distorts the wind flow around it. Coastal radars provide much useful information but have a range of only a few hundred kilometers at most.

Global precipitation climatologies have been produced before using combinations of data from various satellites (see, e.g., GPCC 1993; Huffman et al. 1997), but the principal measurements over the ocean to date have been by passive microwave instruments [e.g., scanning multichannel microwave radiometer (SMMR)] and also by infrared sensors (e.g., Geostationary Operational Environmental Satellite). While infrared instruments can provide high-resolution imagery, no reliable rain algorithms exist on a pixel-by-pixel basis. Rather, for large regional boxes the proportion of cold cloud-top temperatures (less than a threshold of, say, 235 K) is related to the areal average rain rate. Arkin and Meissner (1987) show a good correlation is achieved in tropical regions when the analysis is performed on boxes 2.5° by 2.5°. This occurs because in deep convective systems there is strong coupling between the high cirrus outflow and the lower-level updrafts that are providing the precipitation. In extratropical regions there is less correlation due to the higher proportion of nonconvective events, for which the temperature of the overlying cirrus provides little guide as to what is happening below. Passive microwave measurements can be more directly related to rain rates, utilizing combinations of brightness temperatures at various frequencies and polarizations. The limiting resolution here is given by the footprint of the instrument, typically of the order of 30 km (Wilheit and Chang 1980), which is much greater than typical length scales within rain events (Goldhirsh 1983). Wilheit and Chang (1980) show that when rain rates are moderate to heavy, the cloud-top temperature varies little with rain rate, causing the estimate to be particularly inaccurate at high rain rates. These two effects (footprints larger than rain cells and nonlinearity in instrument response) bias the retrievals from passive microwave instruments (Nakamura 1991). Active radar (as will be shown later) responds directly to the levels of rain, provides good sensitivity over a wide range of rain rates, and leads to rain information at a much higher spatial resolution.

b. Outline of paper

This paper explores the potential of using altimeter data to determine high-resolution rain rates over the ocean through realistic simulations with known rain events. Sections 2a–c provide a brief summary of how an altimeter operates and the effects of rain on microwave radar data in general and on altimeter data in particular. Having discussed rain’s effect in general terms, the latter part of section 2 deals with the limitations on the spatial resolution attainable and then discusses the specification of the noise parameters, which are an integral part of any realistic simulation.

Section 3 provides the nub of the paper, covering the normalization of the waveform information, the mathematical specification of the problem of unbiased noise-resistant rain-rate retrieval, and its solution. Section 4 then assesses the quality of the retrieved rain information in the presence of realistic noise levels and uncertainties as to the exact form of noncontaminated waveform data. A brief summary and discussion then follow.

This paper uses simulated waveform data (for details of simulation, see the appendix) in order to have a known rain field against which the retrieved parameters may be assessed. In the simulations covered here, parameters were chosen to best represent the waveforms of the TOPEX altimeter (Fu et al. 1994). However, the method is generally applicable and readily adapted to simulate the ERS altimeters (Francis et al. 1991). A subsequent paper will cover the application of the algorithm to real TOPEX waveform data and its validation against independent rain measurements.

a. Operation of an altimeter

Note that the bins in the sampled waveform are limited in coverage (returns beyond a certain time are lost), of finite width, and, when displayed, have often been normalized to fill the “window.” The value of the AGC (automatic gain control) is the sum of the powers in a range of bins spanning a large part of the waveform (bins 17–48 of TOPEX’s 128 bins); this can be used to rescale the power in the bins of the telemetered waveform. The AGC value may be corrected for height of altimeter (spherical spreading loss), instrumental loss, and curvature of the oceanic surface to yield a normalized backscatter strength, σ⁰. Brown (1977) showed that range to the sea surface, local wind speed, and significant wave height H_s could be derived from the position, strength σ⁰, and leading-edge slope, respectively, of the waveform.

b. Mechanisms for interaction of radar with rain

Rain in the region being traversed by the altimeter footprint can affect the altimeter via four different mechanisms.

1. Reflection of the pulse off raindrops produces a signal four orders of magnitude weaker than that from the sea surface (Barrick and Lipa 1985) and, in most cases, such reflections will lie outside the reception window, which on current altimeters starts only 15 m above the surface, except for European Remote-Sensing Satellites (ERS) in ice mode.

2. Delay of the pulse due to radar’s reduced celerity through water compared to air. Even in torrential downpours, the columnar liquid content is only a few centimeters (Goldhirsh and Rowland 1982), and thus the extra path length is negligible compared to other sources of height error.

3. Attenuation of the power of a pulse as it passes twice through the column of rain. Farrow (1975) and Goldhirsh and Rowland (1982) have shown that for a moderate rain rate this effect can be several decibels for a K_u-band radar pulse passing through a 5-km-high rain cell.

4. Changes in sea surface roughness due to damping by impinging raindrops (Tsimplis and Thorpe 1989), consequently increasing the local coefficient of backscatter σ⁰. Also, there may be locally increased winds associated with the downdraft within the rain cell, which will in turn increase the roughness of the sea surface (Atlas 1994).

Analyses of altimeter σ⁰ data for both ERS-1 (Guymer et al. 1995, hereafter GQS) and Topex (Quartly et al. 1996, hereafter QGS) have shown plenty of evidence of sharp attenuation features, many with magnitudes exceeding 2 dB. These two studies also showed evidence of enhanced backscatter possibly associated with damping of the surface roughness, although these effects were often difficult to discern and measure since they occur in regions where attenuation is likely to be important. It is possible that enhanced backscatter (damping of sea surface roughness) is present in the majority of rain events except that its effect is generally obscured by the greater effect of attenuation. In this analysis I examine the change in received power level associated with the ray path through a given patch of atmosphere to a region of sea and back. In terms of the simulations shown in this paper an overall attenuation for a given path is modeled; this may more correctly be thought of as the combined effect of attenuation and enhanced backscatter.

Thus if one can measure the degree of attenuation by rain over a certain region, the rain rate itself can be inferred. If the whole of the region irradiated by the main beam of the altimeter is covered by a 5-km-high rain cell of rain rate 12 mm h⁻¹, then according to Eq. (4) the two-way attenuation of a K_u-band pulse will be 5 dB. A similar change of 5 dB in the observed σ⁰ could be accounted for by a change in wind speed, say, from 2 to 17 m s⁻¹; rain can usually be distinguished from this on account of the different spatial scales (GQS), and, in the case of TOPEX, also in the behavior at its second frequency in the C band (QGS). For the TOPEX altimeter, the backscatter value σ⁰ is calculated from the reflections within a footprint of 8-km diameter at low wave heights (Chelton et al. 1989). However, the dimensions of strong rain cells are often of only a few kilometers (Goldhirsh 1983), and thus it is inappropriate to assume that the whole of the altimeter footprint (the area of the sea surface returning reflections within the reception window) is equally affected. Consequently, a number of differently structured rain cells might account for a drop in σ⁰ of 5 dB. Therefore, to distinguish between these different rain structures, one needs to examine the full waveform data rather than just the σ⁰ values.

c. Effect of small rain events on altimeter waveform data

For a flat surface, it is easily shown (see, e.g., Chelton et al. 1989) that the first returns are from a disc on the sea surface centered at nadir and that subsequent returns come from concentric annuli of equal area. The reflections received at time t relative to those from nadir correspond to the annulus of radius ρ given by

ρ²Hct,(5)

where H is the effective height of the satellite and c is the speed of light. [A derivation of Eq. (5) is given in the appendix.] Thus if a small patch of attenuating rain lies at a distance ρ from altimeter nadir, its effect is observed as a reduction in the total power observed at time delay t. When the altimeter passes close to a small region of attenuation, as depicted in Fig. 2, the reduction in power is observed in different sampled bins, j, of the waveform according to the position of the altimeter along track. As the time of reception relative to the nadir signal may be expressed in multiples of the sampling time between consecutive bins, Δt, rearrangement of Eq. (5) gives

View Expanded

where Y₀ is the distance of the region from the subsatellite track, i is the index number of the observed waveforms corresponding to averaging at intervals of Δx along track, and j₀ is the bin number corresponding to reflection from nadir. Thus the attenuation effect for any small rain cell traces out a parabola in the 2D waveform space with its apex corresponding to the cell’s location along and across track. The magnitude of the effect depends not only on the strength of the rain cell itself but also on the proportion of each annulus that it affects;therefore, for rain cells off nadir, the magnitude of the effect varies as l/ρ, that is, as (j − j₀)^−1/2. More completely, allowance should be made for the smearing effects of finite pulse duration and the SSH PDF, giving a waveform like that displayed in Fig. 1g.

The reverse process, deconvolution, corresponds to the division of the Fourier spectrum of Fig. 1g by the spectra of Figs. 1a,b, which operation is obviously sensitive to noise and uncertainties in the appropriate SSH PDF. Deconvolution has been applied to Seasat data in order to obtain the SSH PDF, assuming the pulse shape and the flat surface response were known accurately (Barrick and Lipa 1985). They were able to use averages of 40 or 160 km along track in order to reduce the effect of noise. However, Rodríguez (1988) has pointed out the sensitivities of such an analysis to uncertainties in the beam pattern or mispointing of the satellite. Such an investigation also assumes that the parameters of interest do not change at all rapidly with distance. This assumption is totally invalid for rain applications as many rain cells are of the order of a few kilometers in size (Goldhirsh 1983) and there is no recourse to averaging of repeat tracks as the rain cells clearly move and develop on much shorter timescales than the repeat period of any altimeter.

Therefore, rather than attempt to perform the deconvolution to recover the effective flat surface response shown in Fig. 1f, I use here the departures from an expected waveform, that is, the discrepancy between the solid and dashed lines in Fig. 1g.

The aim here is to derive a robust inversion to utilize attenuation features observed in waveform data to yield information on the spatial structure and strength of the rain events causing them. In the subsequent discussion it must be noted first that the objective is as much the estimation of the strength of a feature as its location and second that in any realistic situation, small “elements” of rain cells do not exist in isolation, and thus the exercise is one of estimating the magnitude of a given feature in the presence of many others.

d. Spatial resolution

The formulation expressed in (6) and depicted in Fig. 2 hinted at the potential for deriving very high resolution rain information, as each element of attenuation would trace out its own individual parabola in the 2D waveform space. Since for TOPEX averaged waveforms are available at every 0.6 km along track, this might appear an appropriate resolution at which to work. Initial investigations for this paper were performed with a characterization of the rain structure according to a 0.6 km × 0.6 km grid. Square rain cells are of course not physically realistic, but some sort of tessellation is required to represent all parts of the altimeter “swath,” and it is mathematically reasonable to have equal dimensions in the along- and across-track directions.

Although good results are achievable for perfect conditions, there is a high sensitivity to the fading noise described in section 2e. Consequently, the scope of the grid was significantly reduced with simulations and solutions being done with a 2.4 km × 2.4 km grid aligned parallel to the satellite track. This dimension is a multiple of the along-track spacing of the waveforms, enabling a simple formulation in terms of indices, while being smaller than the altimeter swath (the last bin in TOPEX waveforms corresponds to ∼12 km from nadir) and of a similar size to the strong rain events observed by Goldhirsh (1983).

It should also be noted that the trajectory of the attenuation feature given by (6) is independent of the sign of Y₀; this left–right ambiguity cannot be overcome using observations solely from a single-antenna altimeter. Consequently, the grid of “rain pixels” can be specified by two indices: I which denotes the along-track distance is one-quarter of the waveform number i, and J, which denotes the across-track distance of the pixel center, going from 0 (nadir) to 4 (centered at 9.6 km from nadir). Note that this means that rain pixel J = 0 spans from −1.2 km to +1.2 km across track. Higher values represent a symmetrically positioned pair of locations (due to the unavoidable ambiguity referred to earlier); for example, J = 2 refers to +3.6 km to +6.0 km and −6.0 km to −3.6 km, although in most of the simulations, the features are only on the “positive” side of the track.

e. Setting up a reliable simulation

To process waveforms corresponding to the passage of an altimeter over known patterns of rain, a series of simulations were performed. Due to the spatially varying patterns of attenuation, the convolution formulation expressed in (1) could not be adopted; rather, a mean waveform for a particular scenario is determined by the summation of all reflections corresponding to each particular time delay. The details of this are given in the appendix. Also, two important sources of noise must be incorporated in the simulation. These are the additive thermal noise and multiplicative fading noise (cf. section 2a). These are described in further detail below; the parameters were chosen to be consistent with the performance of the TOPEX altimeter. However, similar processing can be performed for the ERS altimeters (Quartly 1997b), and indeed two forthcoming altimetry missions (Envisat and Jason-1) are planned to have dual-frequency altimeters with 128-bin waveforms. Consequently, this paper has not attempted to replicate the particular vagaries of the TOPEX altimeter; TOPEX-specific processing is covered in a subsequent paper.

a. Definition of “attenuation pattern”

The normalization introduced above has two important effects. First, it renders α_i,j independent of σ⁰, the local strength of scattering from the surface. Second, it causes the noise statistics to be nearly uniform across the series of waveform data, which is a condition invoked later. This is because the fading noise on the observed data (see section 2e) has a χ² distribution with its magnitude dependent on ŵ_i,j/(N)^1/2, where ŵ_i,j is the expected (noise free) return and N is the number of independent samples. If there is little or no attenuation, ŵ_i,j ≈ w_ref,j and hence the error in α_i,j is independent of ŵ_i,j and thus has uniform statistics across i and j.

The attenuation patterns for five particular noise-free cases are illustrated in the top half of Fig. 3. These are for the basic rain pixels introduced in section 2d, that is 2.4 km × 2.4 km cells of uniform attenuation located at multiples of 2.4 km from nadir. (Note that the color scale is different for the first two cases to prevent saturation in the display of their much stronger effects.)

b. Simple masking

The attenuation parabolas produced by different isolated rain cells will add linearly. Thus the task of estimating the strength of any possible attenuation feature may be viewed as one of determining the magnitude of the corresponding attenuation pattern. A simple approach would be to use the known ideal pattern as a“mask” or set of weights to be applied to the observed values of α. Clearly, this will give zero if there is no attenuation anywhere nearby and will yield some positive value if the mask coincides with an observed attenuation pattern. However, there is considerable overlap between neighboring parabolas, not so much for different offsets (changes in J) at the same along-track location (I) but between that simulated and the mask for one rain pixel (four waveforms) to the side, or one to the side and one farther away. This is illustrated in Fig. 4, where the attenuation pattern for a single source (Fig. 3b) was processed (convolved) with the masks in Figs. 3a–e. The grid of estimated strength of attenuation features shown in Fig. 4 not only contains a peak at the expected location along and across track but also contains considerable “leakage” into neighboring areas. This leakage is much greater when there are a number of rain cells in the region. Clearly, even for noise-free data, it would be difficult to estimate the strength of weak rain events in the proximity of strong ones.

c. Specification of problem

What is needed is a series of weights that when multiplied by an observed attenuation pattern yield the amount of attenuation associated with each geographical region and whose estimate is unbiased by attenuation occurring in neighboring areas. Here I set up a set of equations governing the required conditions for these weights and then determine a unique, well-defined solution to them. The estimate of source strength S(I, J) can be defined as a weighted sum of the observed attenuation pattern α_i,j by

View Expanded

where λ^J_ij are the appropriate set of weights for estimation of a source at location J from nadir, i is the waveform number along track relative to the position being considered, and j is the bin number. Suitable ranges for these summations are discussed later. The indices I and J span the swath in rain pixels defined earlier. The series of weights will of course be different for each of the five possible values of J.

The estimate S(I, J) must have a definite positive value, Λ, when the weights are applied to an attenuation pattern resulting from a rain cell at the examined location and (in order to be unbiased by nearby features) must be zero when applied to the attenuation pattern resulting from any other rain cell. Expressed mathematically, this is

View Expanded

where α^K_i,j is the attenuation pattern for a known uniform source occupying the Kth rain pixel across track and n, the number of waveforms that the attenuation pattern is shifted, which is constrained to be a multiple of 4. The concept of only prescribing S at intervals of one rain pixel is intuitively sensible; this issue is briefly revisited in section 4b. The actual specified value of Λ is arbitrary; here a value of 1000 was used for a 5-dB rain cell occupying one rain pixel.

To determine the unknown weights λ^I_ij suitable for estimating the strength of a source at each location J, Eq. (9) must be viewed as a host of simultaneous equations—one for each of the possible combinations of K and n. However, the number of equations is less than the number of variables and thus the problem is ill-defined. A very useful constraint that can be applied is to minimize the sensitivity of the retrieved estimate S to any particular realization of the fading noise. The mean-square error in a given estimate of the source strength E² will be given by

View Expanded

where e_ij is the error in each bin due to fading noise. This is taken to be uncorrelated between different bins and waveforms (see Rodríguez and Martin 1994) and, according to section 3a, its statistics are independent of i and j. It may be shown (e.g., Weatherburn 1952) that for a χ² distribution scaled to a mean of unity, σ_e, the standard deviation of e_ij is given by

σ_eN^1/2(11)

Consequently, the expected mean square error in the estimate may be expressed as

View Expanded

Now, the equalities given in (9) combined with the constraint given in (12) uniquely specify a set of weights, which may be solved by use of an appropriate package such as the Numerical Algorithms Group routine f04jdf. For TOPEX, which records data in 128 bins with the leading edge positioned at 32.5, the values selected for j₁ and j₂ are 25 and 124, respectively. The former choice merely ignores data well before the leading edge (and hence containing no appreciable signal at moderate wave heights), whereas the last four bins are discarded because of wrap-around effects caused by the onboard Fourier transformation of the data. Similarly, at moderate wave heights, no attenuation effect can be observed more than 20 waveforms (12 km) away, so Δi was set to 20. Any additional bins or waveforms used would in effect add extra columns or rows of zeros to the solution for λ^J_ij. The resultant series of weights for the five possible source locations are displayed in the lower half of Fig. 3, where again the color scales have been adjusted separately to bring out the details of each.

Each image shows a complicated pattern of positive and negative weights, with the largest positive ones coinciding with the peak amplitude of the relevant attenuation pattern (top half of Fig. 3), and large negative weights being used to force nulls for sources either one rain pixel farther along track (same attenuation pattern shifted by four waveforms) or across track (neighboring attenuation pattern in Fig. 3). As the attenuation pattern for a source far from nadir, for example, J = 4 (Fig. 3e), covers many fewer waveforms and bins than for one at nadir (Fig. 3a), and also the former has a much smaller peak amplitude, the weights for a source far from nadir (Fig. 3j) are nearly an order of magnitude greater than those for a source at nadir (Fig. 3f). Consequently, the estimates of source strengths for the distant sources are much more sensitive to fading noise than for sources at nadir.

d. Demonstration of the efficacy of the algorithm

One realization with fading noise appropriate for 100 independent returns is illustrated by the speckled attenuation pattern in Fig. 5d and the resultant degraded estimates in Fig. 5e. There is considerable error for the rain pixels far from nadir (J = 3 and J = 4), and sometimes β can be undefined for a location, as the random noise contrives to make S(I, J) greater than Λ₀. This is then a problem with estimates of rain rate in intense regions or at least with attempting to convert them back to attenuation losses β, as is necessary for the eventual estimate of rain rate from (3).

Equation (12) gives the expected error in determination of S; multiplying by the derivative of (13) evaluated at S = 0 gives the following expression for σ_β, the rms error in retrievals in the absence of a source:

View Expanded

Note that as both λ^J_ij and Λ₀ scale with the choice of Λ, the overall accuracy of the retrieval is independent of the choice for constant Λ. Since the attenuation pattern is strongest for a source at nadir, its weights are the least and, consequently, so is its sensitivity to the noise. Evaluation of the sum of weights used for each possible source location (J = 0–4) and application of (14) (using N = 100) yields rms errors of 0.20, 0.44, 0.92, 2.05, and 3.44 dB, respectively. Trial runs with fading noise but no attenuation present gave rms values of 0.22, 0.46, 0.94, 2.2, and 4.3 dB for these five locations; the predicted values for J = 3 and 4 are slightly lower than observed because in (14) a linear approximation was used to convert from errors in S to errors in β. This shows that a very reliable recovery of β is possible for nadir and 2.4 km off nadir with a reasonable accuracy achievable for J = 2; however, the estimates for the last two rows can be of use only for the most coarse analyses. Due to the logarithmic inversion in (13), the rms error of β estimates is greater when a signal is present. The values quoted here are for H_s = 1 m, but there is little change with H_s as the rms of the weights increases only slightly with wave height [in response to the increased smearing of any attenuation feature by a larger q(t) in Eq. (1)].

Of course, the rain retrieval exercise can be formulated in terms of smaller rain pixels in order to achieve a higher resolution along a limited section. Solving the inversion for 1.2 km by 1.2 km pixels just along nadir produced a sum of weights similar to J = 3 in the above case, showing that nothing is to be gained by attempting to work at such a high resolution. This result is not surprising as 1.2 km is less than the diameter of the first reflecting region, so the rejection of the effect of neighboring regions across track leads to large opposing weights. (The recovery in such a situation is reliable if no fading noise is applied.)

a. Partial occupation of pixel by rain

In general, of course, the altimeter will not be transiting square rain cells nor will their edges be aligned with the solution space. For rain pixels in the solution space totally within a large uniform cloud there will only be slight errors due to the whole rain cell not matching the grid exactly. However, the greatest effects will occur where the edge of a rain cell lies part way through the pixel of interest. As a simple test of this problem, I illustrate here the effect of squares shifted by one or two waveforms (0.25 or 0.50 rain pixels) with respect to the solution space, such that the apexes of the parabolas no longer coincide exactly with the sets of weights being used.

The results in Fig. 6c show that a shift of two waveforms attributes most of the source strength equally to two boxes, with each getting a source strength, S(I, J), equal to 56% of that of the peak in Fig. 6a. In terms of the derived attenuation factor β the factor is 42% and, if (3) is applied to all locations, the derived rain rate in each of the two main boxes is 48% of that of the one in Fig. 6a. A review of the estimation problems associated with nonlinear algorithms applied to partially filled footprints is given by Nakamura (1991).

This nonlinearity, coupled with the slight leakage observed in many of the other pixels, shows that Fig. 5c portrays a rather optimistic assessment of the algorithm’s performance, since in reality the true conditions will not exactly match the coarse grid of the solution space. However, most of the leakage “errors” are no greater than those due to fading noise (see, e.g., Fig. 5e).

b. Forcing an improvement in performance for partially occupied rain pixels

The errors shown in Figs. 6b,c could be overcome by augmenting the set of equations that the weights must satisfy. All leakage could be removed by requiring (9) to be valid for all |n| ≥ 4 and the appropriate partition of the source strength forced according to area of overlap between simulation and solution space. Noting that the same conditions also apply in the across-track direction leads to the entire set of equations being specified by

View Expanded

where K is the across-track location of the center of the simulated pixel in units of 2.4 km and can increase now in multiples of 0.25 and η_JK, ɛ_n are relatives of the Dirac δ function given by

View Expanded

This has in effect led to a 16-fold increase in the number of equations with the constraint of minimizing E² in (12) still being necessary to define a unique solution. However, the resultant value for E² is many orders of magnitude greater than before, implying a much greater sensitivity to noise.

This is analogous to the fitting of a polynomial to a set of points with the constraint of minimizing the square of the gradient over the given interval: if the number of points is much less than the order of the polynomial, the solution will have much smaller extrema than if the polynomial is tied to a large number of points. Therefore, there is no advantage in adding further constraints—these work well in noise-free scenarios but increase sensitivity to noise in simulations and in real data.

c. Effect of small-scale structure

As rain cells are not of completely uniform rain rates throughout but can contain within themselves small regions of more intense rain, it is important to know how the algorithm copes with small-scale structure. Does it return a meaningful average, or does it tend to be biased?

To test this, simulations were performed of square rain cells with extreme structures within them (see Figs. 7a,b) and the algorithm was applied to the resultant attenuation patterns. In both cases, slight leakage into other rain pixels occurred, but the power was predominantly inferred in the correct location. The estimates for the “check” example were all 2.60 dB, and those for the “tartan” were 3.54–3.65 dB, depending on the across-track location. These results are all close to the true averages (2.60 and 3.63 dB, respectively), indicating that the algorithm effectively averages the attenuation in a region.

d. Errors due to uncertainties in the appropriate processing

In the analyses encountered so far, the unaffected waveform is known exactly a priori. When the algorithm is applied to real altimeter data, the waveform in the absence of rain is not known precisely. These effects are studied here, examining errors in overall strength of signal, thermal noise, and wave height, respectively.

If the power (σ⁰) of the reference waveform is too high (e.g., because there is a genuine increase in wind and hence sea surface roughness in the region of interest), then this will lead to an added positive constant in the attenuation pattern applied to all the bins and all the waveforms and, consequently, increased estimates of rain rate throughout. A 1-dB overestimate of the reference σ⁰ simply translates to an extra 1-dB attenuation loss, β, across the whole swath.

All simulations to date have used the same thermal noise level, which is consequently completely removed by the use of the reference waveforms. If the reference has an incorrect value for this, there will be a constant offset in w_ref,j − w_i,j, which will only lead to a significant proportional change where the reference value is low. This is principally at the very bottom of the leading edge—it does not have any impact before the leading edge, as there all the weights are zero.

An incorrect estimate of the ambient wave height affects the rain retrieval algorithm in two ways. As different wave heights correspond to different degrees of smearing (see Fig. 1b), at large wave heights the attenuation patterns for unit rain cells extend over a greater range of bins than for low wave heights, but their magnitude is diminished slightly. Consequently, the appropriate set of weights for various values of H_s differ. However, a comparison of the weights for H_s = 1 and 1.5 m (not shown) reveals very similar strands of positive and negative weights with comparable magnitude, indicating that failure to use the correct set of weights will have only a minor effect on the retrieval accuracy.

The second and more significant effect is due to the error in the wave height chosen for the reference waveform. The main difference between waveforms with H_s = 1 and 1.5 m occurs at the leading edge, where the waveform for the larger H_s value has more power in the bins preceding the half-power point (bin 32.5) and correspondingly less afterward. It might seem that the errors caused by overestimating and underestimating the reference waveform should be the same, but this is not the case due to the normalization by the reference waveforms in (7). In particular, using a reference H_s value of 1.5 m for the processing of rain-free data with H_s = 1 m leads to values of α ranging from 0.55 in bin 31 to −0.062 in bin 34; whereas, if the reference H_s is 1 m, and the “observed” data have H_s = 1.5 m, then α ranges from −1.23 (bin 31) to 0.056 (bin 34).

Figure 8 shows the result of applying weights to attenuation patterns calculated using erroneous reference waveforms. In both cases the simulation scenario consists of a single square rain cell of strength 5 dB located 2.4 km off nadir. In Fig. 8a the simulation was for H_s = 1.5 m with a reference and weights applied assuming the value to be 1 m; the converse is shown in Fig. 8b. In both cases the rain cell is clearly identified with no spurious along-track structure being produced. The bands running along track correspond to errors in the attenuation patterns. As these errors only occur near the leading edge, estimates in rain pixels far from nadir (J = 2, 3, and 4) are unaffected. The effect on the bands nearest nadir (J = 0 and 1) is clearly more pronounced in the first case than in the second due to the greater errors in α. In Fig. 8a the derived values of β are −1.01 and 1.21 dB for J = 0,1, respectively, with the estimate of the true source being biased by over 5 dB on account of the nonlinear inversion in (13). In Fig. 8b, β is estimated at 0.05 and −0.05 dB with the affect on the true source also being minimal.

The effects detailed here are only pronounced at low wave heights where a change of 0.5 m in H_s makes a large proportional change in the key bins near the tracker point. The algorithm works well at low H_s provided an accurate estimate of H_s is known; in cases of uncertainty an overestimate of the appropriate H_s for processing only degrades the performance slightly. Although altimeter estimates of wave height only agree to within 0.2 m with those from ships or buoys (Callahan et al. 1994), what matters here is the consistency of the estimate. Thus if nearby rain-free waveforms correspond to a wave height of 1.5 m, then that specifies the waveform shape to be used as a reference for the transit of the rain cell. In addition, Quartly (1997a) has shown that for TOPEX-accurate uncontaminated estimates of H_s can be obtained from the waveforms at its second frequency in the C band.

Clearly, very long bands of derived rain features aligned parallel to the altimeter ground track will indicate some sort of error in the reference waveform used.

e. Processing of a complex case

This section provides a further illustration of the ability of the algorithm to handle the various sources of noise and mismatches between simulated and solution space. There are two important aspects to note in this case study. First, a much larger wave height (H_s = 5 m) is used here to demonstrate the efficacy of the algorithm in a very different wind–wave environment; the set of weights applied is not that for H_s = 1 m but a set appropriate to the chosen wave height. Second, apart from a few square rain cells for purposes of demonstration, the simulation is performed for extended round rain cells or long obliquely oriented rainbands, which only map approximately onto the chosen gridded solution space. This is to demonstrate that the processing can automatically handle a wide variety of features.

Figure 9a shows the idealized “real world” representation of the simulation conditions, displaying from the left, four different types of objects:

1. an ellipsoid of uniform (2 dB) attenuation;

2. a set of three square rain pixels of strengths 5, 1, and 3 dB;

3. a rainband of total width 5.6 km, aligned obliquely to the satellite track, with attenuation peaking at 5 dB and decreasing linearly with distance from the center of the band; and

4. a large circular rain cell of radius 9 km with annuli of 3-, 5-, 7-, and 9-dB attenuation.

Figure 9b is the result of mapping Fig. 9a onto the solution space by averaging the transmitted power in square rain pixels of side 2.4 km. Note that as the altimeter cannot distinguish left from right, the oblique rainband maps onto a V shape. Also, in the conversion back into decibels, all the attenuation is assumed to occur on one side; this is a more useful view than assuming all attenuation features are arranged symmetrically with respect to the altimeter track. However, this can lead to computing errors, if, say, the true condition corresponds to an attenuation of 4 dB everywhere since this reduces the signal more than infinite attenuation on one side can. In short, Fig. 9b represents the best possible recovery by an instrument of 2.4-km resolution that cannot distinguish left from right. The actual recovery in the absence of fading noise is shown in Fig. 9c. Very good agreement with Fig. 9b is shown for all the features. (Remember, Fig. 9b is not an intermediate step in the processing. Waveforms are generated for the altimeter traversing the scenario displayed in Fig. 9a, and these waveforms are processed to give Fig. 9c.)

Any one simulation with fading noise may perchance work well for some regions and poorly for others, thus it is better to examine the statistics of the errors due to the noise. One hundred simulations were performed with fading noise according to N = 100. Figure 9d shows the mean recovery for these 100 simulations, while Fig. 9e displays the standard deviation of these estimates. The mean recovery is quite similar to the recovery without noise (Fig. 9c), the main differences being in the fourth and fifth rows where the mean is biased positive on account of the nonlinear inversion in (13). Figure 9e shows the random error in an estimate primarily depends upon its across-track position, thus bearing out the statements made in section 3d, namely, that the rms errors on the first three rows are acceptable and that when the errors are averaged in decibels [which are close to rain rate, see (3)], then larger errors are obtained where there is already significant attenuation.

Walsh (1982) has stated that near the leading edge the decorrelation time between pulses decreases with wave height, and thus for 5-m wave heights a more appropriate value of N would be 200. This translates simply into a reduction by 2^−1/2 of the error levels displayed in Fig. 9e.

To summarize, after an altimeter has traversed the situation displayed in Fig. 9a, suitable processing of the waveform data can produce a pattern of rain pixels such as Fig. 9d, although any particular realization will be subject to errors due to fading noise. The derived attenuation levels in the three rows nearest nadir are shown to be quite robust, while those farther away can reliably discern significant attenuation levels (i.e., greater than 5 dB). There remains the difficulty of overcoming the left–right ambiguity to recover something close to the rain cell distribution shown in Fig. 9a. To some extent this can be achieved by pattern matching, for example, recognizing that the V shape in Fig. 9d probably corresponds to a straight rainband oblique to the satellite path. Guidance as to whether features lie to the left or right of track can also be gained by reference to full 2D imagery from, say, SMMR (see, e.g., Tournadre 1998), although sufficiently contemporaneous images will be rare. The advantage of a multi-instrument mission such as Envisat is that it will incorporate wide swath optical and infrared instrumentation (Medium Resolution Imaging Spectrometer and Advanced Along-Track Scanning Radiometer) along with its dual-frequency altimeter, facilitating much easier interpretation of derived rain-rate patterns.