## Background and introduction

Weather radars that operate at frequencies higher than about 5 GHz can be affected adversely by rain attenuation. However, for spaceborne applications, where the size and mass of the antenna are limited, adequate spatial resolution can be obtained only by increasing the frequency. For the Tropical Rainfall Measuring Mission (TRMM) precipitation radar (PR), the use of 13.8-GHz radar frequency (2.17-cm wavelength) with a 2-m antenna size represents a compromise between the desire to minimize the antenna size and rain attenuation and maximize the spatial resolution. As demonstrated by Hitschfeld and Bordan (1954) in their classic study, estimates of rain rate from a single–attenuating wavelength radar are unstable when the path attenuation is large unless the radar constant and the drop size distribution are known to a high degree of accuracy. Because these conditions are seldom met, an alternative strategy is needed to complement the Hitschfeld–Bordan method for moderate and high rain rates. One of the proposed ways of estimating attenuation is the surface reference technique (SRT), developed over a number of years and extensively tested by airborne weather radar data (Durden and Haddad 1998; Fujita 1983; Iguchi and Meneghini 1994; Iguchi et al. 2000; Kozu and Nakamura 1991; Marecal et al. 1997; Marzoug and Amayenc 1994;Meneghini et al. 1983, 1992).

The primary objective of the paper is to describe and to illustrate the operational SRT algorithm for the TRMM PR. After a brief description of the TRMM PR in section 2, the basic equations of SRT are given in section 3. Because the stability of the reference target is critical to the accuracy of the method, several reference estimates and the behavior of the surface cross sections are described in sections 4 and 5. Examples of the SRT-derived attenuation are shown in sections 6 and 7, followed by a summary.

## TRMM PR specifications

The TRMM PR, built by the National Space and Development Agency of Japan, operates at a frequency of 13.8 GHz and uses a 2 m × 2 m phased-array antenna consisting of 128 slotted waveguides. A peak power greater than 500 W is achieved using 128 solid-state power amplifiers. The horizontally polarized beam is electronically scanned cross track ±17° with respect to nadir. Using frequency agility (at 13.796 and 13.802 GHz) and a pulse repetition frequency of 2776 Hz, 64 independent samples are collected at each of the 49 fields of view that compose the scan. The 3-dB beamwidth at nadir is approximately 0.71° (antenna gain of 47.4 dB), which gives a horizontal resolution near the surface of about 4.3 km and a swath of 215 km. The transmit pulse of 1.67 (×2) *μ*s is sampled every 0.833 *μ*s, giving a range resolution of 125 m. However, because the high-resolution data are retained only for the near-nadir and near-surface samples, the nominal range resolution is 250 m. Because SRT uses a rain/no-rain categorization it should be noted that the “rain-certain” flag is set when the signal level exceeds the sum of the mean noise level and three times its standard deviation over at least three consecutive range gates. For purposes of the algorithm, all other cases are considered to be rain free. Details of the system design, data system, and internal and external calibration procedures can be found in Kawanishi et al. (1998).

Although the focus of this paper is on one of the attenuation techniques used in the TRMM PR, the TRMM set of sensors also includes the TRMM Microwave Imager (TMI), the visible and infrared scanner (VIRS), as well as two related Earth Observing System instruments: the Clouds and the Earth’s Radiant Energy System and the Lightning Imaging Sensor (Kummerow et al. 1998). The complementarity of the sensors is a critical aspect of the TRMM concept. Because the PR provides high-resolution vertical information on rain rate, hydrometeor phase state, and stratiform/convective classification over both land and ocean, it can be used to improve the TMI rain retrievals over the broader swath of that instrument. The TMI PR results can be used to test, and potentially to improve, the accuracy of the VIRS retrievals not only on the TRMM satellite but on the instruments on geostationary platforms.

## Path-attenuation estimates and reliability

### Rain-rate algorithms

*Z*

_{m}and its relationship to the actual value

*Z*and the radar return power

*P*at range

*r*by

*k*is the specific attenuation (dB km

^{−1}),

*C*is the radar constant, and |

*K*|

^{2}, the dielectric factor, is by convention taken to be equal to 0.93. Using

*k*=

*αZ*

^{β}, the solution to

*Z*can be written as

*Z*

*r*

*Z*

_{m}

*r*

*c*

*qS*

*r*

^{−1/β}

*q*= 0.2 ln10

*β, c*is a constant, and

*S*(

*r*) =

^{r}

_{0}

*α*(

*s*)

*Z*

^{β}

_{m}

*s*)

*ds.*If the condition

*Z*(

*r*= 0) =

*Z*

_{m}(

*r*= 0) is imposed, then

*c*= 1 and the Hitschfeld–Bordan solution is recovered. If

*Z*(

*r*) at the range

*r*

_{s}just above the surface is set equal to

*Z*

_{m}(

*r*

_{s}) exp(0.1 ln10

*A*

*A**c*=

*qS*(

*r*

_{s}) + exp(−0.1

*β*ln10

*A**Z*(

*r*= 0) =

*Z*

_{m}(

*r*= 0) but modify the path attenuation to be consistent with the SRT estimate, either

*α*or

*C*(calibration constant) can be adjusted (Meneghini et al. 1983; Meneghini and Nakamura 1990). In the first case, the

*α*-adjustment estimate is given by

*Z*

_{α}

*r*

*Z*

_{m}

*r*

_{o}

*qS*

*r*

^{−1/β}

_{o}= [1 − exp(−0.1

*β*ln10

*A*)]/

*qS*(

*r*

_{s}).

*p*

*θ*

_{1}

*θ*

_{2}

*θ*

_{1}and

*θ*

_{2}are related to the PIA estimates derived from the SRT and HB approaches, respectively. Using Bayes’s theorem and approximating the a priori densities,

*p*(

*θ*

_{1}|PIA) and

*p*(

*θ*

_{2}|PIA), the maximum value of (6) is determined as a function of the individual PIA estimates and their variances. From PIA, a corrected

*Z*profile can be constructed. After accounting for effects of nonuniform beam-filling, the rain rate

*R*follows from a power-law relationship of the form

*R*=

*aZ*

^{b}(Iguchi et al. 2000). The remainder of the paper will focus on estimating the attenuation

*A*and its reliability by means of SRT. The data are taken from version 4 of TRMM algorithm 2a21.

### Path-attenuation estimation

*P*

_{s}from the surface at an incidence angle

*θ*with respect to nadir at a height

*H*above the surface is related to the normalized radar cross section (NRCS)

*σ*

^{0}

_{L}

*P*

_{s}

*θ*

*P*

_{0}

*λ*

^{2}

*G*

^{2}

*Lσ*

^{0}

_{L}

*θ*

*a*

*θϕ*

_{B}

*θ*

_{BP}

*π*

^{2}

*H*

^{2}

*θ*

_{C}

*θ*

^{2}

*θ*

^{2}

_{B}

*θ*

^{2}

_{P}

*ϕ*

_{B}and

*θ*

_{B}are the 3-dB one-way along- and cross-track beamwidths,

*θ*

_{C}is the angle measured from nadir to the vector along the antenna main lobe,

*P*

_{0}is the transmit power,

*G*is the antenna gain,

*L*is the system loss factor,

*λ*is the radar wavelength, and

*a*is the atmospheric attenuation along the main lobe. Also,

*τ*is the 6-dB width of the receiver bandpass-filter output waveform, and

*c*is the speed of light. In processing the data, we assume that the local maximum in the return power profile corresponds to

*θ*=

*θ*

_{C}so that the exponential term in (7) is equal to 1. We also assume that at this range the radar return power from the rain can be neglected so that the radar return power and

*σ*

^{0}

_{L}

*θ*) are directly proportional. In writing the equations for SRT, we use the normalized surface cross sections;an equivalent formulation can be written in terms of the surface return powers. It is convenient, moreover, to use decibel units: for the remainder of the paper, the quantities

*σ*

^{0}and

*A*will be used, where

*σ*

^{0}= 10 log

*σ*

^{0}

_{L}

*A*= 10 log

*a.*

*σ*

^{0}(

*R, θ*), is related to the NRCS that would exist in the absence of rain,

*σ*

^{0}(NR,

*θ*), by

*σ*

^{0}

*R, θ*

*σ*

^{0}

*θ*

*A*

*A*is related to the specific attenuation

*k*(dB km

^{−1}) and the extinction cross sections

*σ*

_{ext}of the scatterers by

*s*” runs from the radar to the surface;

*N*(

*D*) is the drop size distribution, and

*D*is the equivolume diameter.

*σ*

^{0}(NR,

*θ*) in (10) is not measurable, it is replaced by a reference value, 〈

*σ*

^{0}(NR,

*θ*)〉, which leads to the attenuation estimate,

*A*

*A**σ*

^{0}

*θ*

*σ*

^{0}

*R, θ*

*σ*

^{0}(NR,

*θ*)〉 is that it be obtained from an average over nonraining areas. Candidates for the reference estimate are discussed in the following section.

*A*

*σ*

^{0}

*θ*

*σ*

^{0}

*R, θ*

*N,*the number of samples from which the reference value is computed, is large, then var(

*A**σ*

^{0}(

*R, θ*)] = var[

*σ*

^{0}(NR,

*θ*) −

*A*] = var[

*σ*

^{0}(NR,

*θ*)], where the second equality follows from the fact that we are interested in the variability of the PIA estimate for a fixed value of the true path attenuation

*A.*Therefore,

*A**σ*

^{0}

*θ*

*A**E*

*A*

*A**A*

*σ*

^{0}

*θ*

*A*is the true two-way path attenuation. The variance of

*σ*

^{0}(NR,

*θ*) consists of several independent components. The first results from the inherent variability of the surface scattering such as variable wind velocities over ocean and changes in vegetation, surface roughness, and soil moisture over land. A second component arises from finite sampling: if the surface is modeled as a Rayleigh target composed of a large number of independent and roughly equal scattering contributions, then a group of 64 independent samples yields a standard deviation of 5.57/(64)

^{0.5}, or 0.7 dB (Doviak and Zrnić 1993). Conversion of the analog voltage into an 8-bit word gives quantization steps of 0.38 dB, so that the maximum error is 0.19 dB with an associated standard deviation of 0.38/12

^{0.5}, or 0.11 dB. As noted above, one other source of error is the effect of rain on the surface cross section, an effect that varies as a function of wind speed, rain rate, and incidence angle (Bliven and Giovanangeli 1993; Tsimplis and Thorpe 1989; Yang et al. 1997). Evaluation of this error, however, requires independent estimates of path attenuation and is beyond the scope of the paper. Attenuation effects from cloud water and, to a lesser extent, atmospheric gases also can be considered as error sources in that the estimated path attenuation is usually attributed to the precipitation when, in fact, a fraction of it may be caused by changes in cloud water outside and within the rain. Although this can be a significant source of error at 35 GHz, at 13.8 GHz the contribution is usually negligible.

*σ*

^{0}(NR,

*θ*)] by the sample standard deviation of the data used to derive the reference measurement. The output product from the operational system includes the path attenuation estimate

*A,*the reliability factor

*A*/std[

*σ*

^{0}(NR,

*θ*)], and a reliability flag. The purpose of the reliability flag is to provide the user with information about the attenuation estimate, including the surface background (land, ocean, or coast), the type of surface reference data used to estimate attenuation (see section 4), and a qualitative indication of its accuracy. The quality flag can take on one of three values:

## Reference estimates

The reference estimate 〈*σ*^{0}(NR, *θ*)〉, which is used to approximate the quantity *σ*^{0}(NR, *θ*) in (10), can be obtained by several different averaging schemes. The four reference estimates that have been implemented in the operational code are outlined below.

### Temporal reference

A temporal reference dataset can be derived by fixing the spatial location and calculating the statistics over the time history of the rain-free measurements. This reference dataset comes closest to the usual idea of a fixed target reference in that the same set of target cross sections is measured in the presence and absence of rain. The temporal reference dataset consists of the monthly rain-free NRCS statistics compiled over a latitude–longitude grid of 1° × 1°. Because the PR coverage extends from 37°S to 37°N, 26 640 such grid elements are needed. Each element is further subdivided into 26 “angle bins” so that statistics are categorized according to incidence angle. During the processing of each orbit, the running sample mean, mean square, and number of data points are continuously updated. At the end of the month the sample mean and mean square and number of points are computed and stored. To simplify the processing, separate read-only and write-only files are used; for a particular month, the write-only file is updated while the read-only file, containing the reference data from the previous month, is used to compute PIA when rain is encountered. At the end of the month, the write-only file becomes the read-only file for the next month and the read-only file is initialized to zero. Note that, although the cross-track scan consists of 49 angle bins or beam positions, there are only 25 distinct off-nadir incidence angles. In the terminology used here, the incidence angle takes on one of 25 values whereas the scan angle takes on all 49 values.

### Alongtrack spatial reference

For the spatial reference, the sample mean and standard deviation at each scan angle are computed over *N*_{s} fields of view prior to the detection of rain. This computation is done by calculating the statistics over a moving window in the alongtrack direction at each of the 49 scan angles. Separate arrays are defined for land, ocean, and coastal backgrounds so that, for instance, when rain is encountered over land at an incidence angle *θ,* the spatial reference data are derived from the last *N*_{s} rain-free fields of view measured over land at angle *θ.* Because *N*_{s} has been set to 8 and the time for a full scan is about 0.6 s, the statistics can be interpreted as characterizing the spatial variations of the cross section at a fixed time and incidence angle.

### Global reference

*σ*

^{0}(NR,

*θ*)〉

_{GL,O}and {std dev[

*σ*

^{0}(NR,

*θ*)]}

_{GL,O}this condition can be written:

*σ*

^{0}(NR,

*θ*)〉 by 〈

*σ*

^{0}(NR,

*θ*)〉

_{GL,O}; that is, if the reference value obtained from either the spatial or temporal reference set differs significantly from the global mean, use the global mean and standard deviation as the reference data. Examination of the results of the algorithm shows that, although the global reference is seldom used, it is usually invoked when rain is present over a coastline. This result occurs because spatial and temporal reference data at a land–ocean boundary are often inaccurate because of the significant differences in the surface scattering properties of the two surfaces. Thus, in computing the statistics over 1° × 1° boxes that include land and ocean, the variance is high and the mean is usually representative of neither surface type. Because the spatial reference is defined by prior rain-free measurements, the situation can arise that the reference data are gathered from a region far from the rain area. For example, when rain is present over the Atlantic coastline of North or South America, the closest rain-free oceanic data are taken from measurements in the Pacific. Clearly, a better solution in these cases would be to use the rain-free data “beyond” the storm. However, because the operational requirement is to process the data scan by scan, only prior measurements can be used.

### Cross-track reference

*σ*

^{0}(NR,

*θ*) can be modeled by a quadratic function of the form

*γθ*

^{2}+

*η.*In this equation, the parameters

*γ*and

*η*are determined by minimizing the mean-square error between this function and the rain-free measurements within the scan. The best-fit quadratic provides a reference curve against which the two-way path attenuation can be estimated. In particular, if rain is present at an incidence angle

*θ,*the path attenuation derived from the cross-track reference is

*A*_{XTO}

*θ*

*γθ*

^{2}

*η*

*σ*

^{0}

*R, θ*

*θ*greater than and less than 0° provide better reference curves. Denoting the polynomial for

*θ*≥ 0 by

*p*

_{1}(

*θ*) and for

*θ*< 0 by

*p*

_{2}(

*θ*), where the unknown parameters are determined by minimizing the mean-square error between the function and the NRCS data at the rain-free incidence angles, then the estimate of path attenuation becomes

*θ*

_{j};

*j*= 1, . . . ,

*n*} the set of

*n*angles within the swath over which rain is absent, the reliability can be defined in a way similar to that of (15):

*A*_{XT}

*θ*

*A*_{XT}

*θ*

*A*_{XTO}(

*θ*)] is approximated by

*γθ*

^{2}

_{j}

*η*) in (21) is replaced by the polynomials

*p*

_{1}(

*θ*) and

*p*

_{2}(

*θ*) for the right- and left-hand sides of the scan. Illustrations of the cross-track and alongtrack reference data and the associated path attenuation estimates are given in sections 6 and 7. Note that, in the current operational algorithm, the attenuation estimates from the cross-track reference are written to a diagnostic file and are not considered candidates for the attenuation estimate written to the standard output file.

## Characteristics of the surface cross sections

The SRT is expected to work well (i.e., small relative error) when the rain attenuation is much larger than the inherent variability of the NRCS of the surface. However, because NRCS varies with space, time, surface type, and incidence angle, the statistical characterization of this quantity is not simple. Nevertheless, some insight into the problem can be gained by examining the “global” statistics over monthly periods. Shown in Table 1 are the mean and standard deviation of the rain-free NRCS over ocean as a function of the off-nadir incidence angle. The results are shown for October 1998 and January 1999 where the data for the latter month are shown in parentheses. The corresponding statistics for land backgrounds are shown in Table 2.

An important objective for the TRMM radar is to compare mean NRCS with those derived from scatterometer and altimeter datasets (e.g., Brown 1979; Guymer et al. 1981; Ulaby et al. 1982; Witter and Chelton 1991; Stoffelen and Anderson 1997). This comparison is useful not only in instrument calibration but in developing algorithms for wind speed estimation. However, because the focus of the paper is SRT, the critical variable in the method is not the mean but the standard deviation of the reference. Over ocean, the smallest standard deviations occur at incidence angles from 3° to 7° with a minimum of about 1.2 dB at angles of 4° to 5°. To put these numbers in the context of the attenuation estimates, we use a *k–R* relationship for 13.8 GHz at *T* = 20°C of *k* = 0.024*R*^{1.15} derived from Swiss disdrometer data (courtesy of Dr. Matthias Steiner). Here, *R* is the rain rate (mm h^{−1}), and *k* is the specific attenuation (dB km^{−1}). For a rain rate of 5 mm h^{−1} over a 5-km path at nadir, the two-way path attenuation is 2 × *k* × 5 = 1.53 dB. According to the convention of (16), this value would be considered to be a marginally reliable estimate for angles between about 2.8° and 7.8° and to be unreliable at other angles. At 10 mm h^{−1}, the two-way path attenuation increases to 3.4 dB, a value that would be considered to be marginally reliable for incidence angles out to about 15°.

Over land, the standard deviation at nadir is in excess of 8 dB. Contributing to the large standard deviation are the frequent occurrences of extremely high cross sections. For example, under rain-free conditions, about 10% of the *σ*^{0} data exceed 20 dB, and about 2% exceed 25 dB. In contrast, at nadir incidence over ocean, only about 1.4% of the data exceed 20 dB and 0.2% exceed 25 dB. In moving to off-nadir angles, the standard deviation decreases rapidly. For angles greater than 10°, the standard deviation is approximately constant, with values between 2 and 2.3 dB. An incidence angle of 10° also represents the angle beyond which the average standard deviation over land surfaces is smaller than the average standard deviation over ocean (Tables 1 and 2).

It should be mentioned that the standard deviations obtained from the spatial and temporal references datasets are often significantly smaller those in Tables 1 and 2, because the former are restricted either in space (temporal) or in space and time (spatial) where the data tend to be more homogeneous. Shown in Figs. 1–3 are plots of the mean and standard deviation of NRCS (horizontal polarization) at incidence angles of 0°, 5.68°, and 13.49°, respectively, where the statistics have been computed for the rain-free data in each 1° × 1° latitude–longitude box for the month of November 1998.

An examination of the results shows several properties relevant to the reliability of SRT. Over ocean, the standard deviations (SD) of NRCS at angles between about 3° and 8° are usually small and spatially uniform (Fig. 2). Above 10°, SD becomes more nonuniform in space (Fig. 3). Although SD remains small over much of the ocean, it can become quite large in regions of light wind, with values sometimes exceeding 4 dB. (Light-wind regions can be identified by occurrences of high near-nadir NRCS and low off-nadir NRCS.) At nadir and near-nadir angles, a similar, but less pronounced spatial inhomogeneity is apparent over the same areas of light wind (Fig. 1). Over land, the situation is more complex spatially; however, as noted above, the general trend is for SD to decrease with incidence angle out to angles of about 10°, beyond which it remains relatively constant.

That the minimum variance in the global data of ocean cross sections occurs near 4° rather than 10° is somewhat surprising. The results, moreover, are consistent in that monthly averages over 2 yr of data show that the minimum variance occurs between 4° and 5°. This characteristic of the global data does not hold at smaller scales, however. Comparisons of the maps in Figs. 3 and 4 show that over large regions of the oceans the standard deviation at 13.49° is comparable to that at 5.68°. On the other hand, the variance at 13.5° in the light wind regions is significantly larger than that at 5.68°. A similar pattern is seen in comparisons of the statistics at 4° and 10°, in which the global variance at 4° is smaller than that at 10° despite large areas in which the opposite behavior is observed.

In the terminology introduced in section 4, Figs. 1–3 represent the temporal reference data at three incidence angles, and the results in Tables 1 and 2 represent the global statistics over ocean and land. The variances associated with these reference measurements are normally larger than those associated with the spatial reference data. Because the dataset associated with the smallest variance is selected, it follows that the spatial reference set is most often used to compute path attenuation and reliability. The alongtrack and cross-track spatial reference data are discussed in the following sections.

## Sample results of the method

The top plot of Fig. 4 shows the profile of the radar return power (dB*m*) at an off-nadir incidence angle of 7.1°. The PR measurements were taken on 25 August 1998 over Hurricane Bonnie located off the southeast coast of the United States. The overpass occurred at 1114 UTC where the location that corresponds to the midpoint of Fig. 4 is (26.4°N, 71.1°W). The vertical scale is given in terms of range bin numbers, where the length of a range bin is 250 m. In this example, the surface return occurs at range bin 80 and a bright band is evident at about 5 km above the surface, near bin number 60. The scan numbers are given along the abscissa. Because a single scan represents an alongtrack distance of 4.3 km, the horizontal scale corresponds to an 860-km segment along the satellite track. From the data in Fig. 4, an approximate rain-free spatial *σ*^{0} reference is about 8 dB so that the maximum value of path attenuation over this segment, occurring at scan number 3828, is (8 dB − −7 dB) = 15 dB. Two sets of broken horizontal lines are shown at the bottom of the figure, where a black line indicates the presence of rain (as detected by the PR) and the gray line shows where the SRT-derived path attenuation is judged to be reliable or marginally reliable according to (16). For example, rain is detected over the interval from scan number 3890 to 3916, but the estimates of *A* are judged to be unreliable. Two other examples over the same time period are given in Figs. 5 and 6. In Fig. 5, where the incidence angle is 3.55°, the strong convective cell at scan number 3882 is associated with a two-way path attenuation of about 24 dB. Note that regions of strong attenuation at scan 3882 and from scans 3804 to 3830 correspond to a loss of the mirror-image signal. This result is not surprising, because the mirror-image return is proportional to the four-way rather than the two-way attenuation and is normally not detected over heavy rain (Liao et al. 1999). As the scan angle increases, the surface return is broadened in range and the rain-free NRCS decreases. These features are evident in Fig. 6, in which the rain and surface returns are shown for an incidence angle of 10.65°. In this case the mean background level in the absence of rain is about 6 dB, with a sample standard deviation of approximately 1 dB.

It is apparent from the examples shown in Figs. 4–6 that the spatial reference value depends on whether the reference data are taken prior to or subsequent to the detection of rain. As already noted, the rain-free data acquired after the rain measurements cannot be used in the reference estimate. In cases such as this one of strong and highly variable winds, the basic assumption of SRT—that the surface cross section within and outside the rain remains unchanged—is questionable. As noted in the next section, the primary motive for examining several reference datasets, obtained from different spatial and temporal averages, is to assess the magnitude of this kind of error.

The path attenuation estimates also depend on the number of rain-free measurements used to estimate the reference value. The change in the path-attenuation estimates with the length of the averaging window has been examined for averaging windows of 4, 8, 16, and 20 fields of view. Histograms of (*A*_{8} − *A*_{16}), where *A*_{j} denotes the path attenuation estimate using a window consisting of *j* observations, and other combinations, have a sharp maximum at 0, indicating that over ocean the length of the averaging window is not a critical factor.

Figures 4–6 are similar to the standard range–height indicator plots that are useful in examining the vertical structure of the storm. To view the horizontal structure, a constant-altitude plan position indicator is appropriate. Shown in the top of Fig. 7 is the radar return power (dB*m*) from a fixed height of 2 km above the surface. The horizontal scale is identical to that of Figs. 4–6 and represents an 860-km segment along the satellite track. The vertical scale, equal to the swath of the PR, is 215 km. Note that, because the angle bins are numbered from 0 to 48, the angle bins corresponding to nadir and to ±17° are 24, 0, and 48, respectively. An image of the apparent normalized radar cross section of the surface is shown in the bottom of Fig. 7 over the same region. The dark bands in the image correspond to the heavier rain regions of the top figure and clearly show the effects of rain attenuation. The white regions indicate a folding in the plotting scale for which the actual levels of *σ*^{0} extend below the minimum of −10 dB. These areas are associated with the strongest attenuations and highest rain rates.

## Alongtrack and cross-track PIA estimates

The alongtrack spatial reference estimates, illustrated in Figs. 4–6, are calculated from the sample mean of NRCS over eight fields of view prior to the detection of rain. Data at each scan angle are processed independently. For the cross-track reference, we make use of the functional behavior of *σ*^{0} with angle. Two examples of cross-track reference estimates are shown in Fig. 8 in which (×) and (∗) are used to represent the NRCS measurements at angles along which rain is absent and present, respectively. In both examples, a region of moderate rain occurs at angles between −14° and −8°; at the top, a more extensive region, consisting primarily of light rain, is evident over most of the right side of the swath from 2° to 14°. As noted in section 4d, the path attenuation *A* at the angles at which rain is detected can be estimated by the difference between a quadratic function and measured NRCS. The reliability is taken to be the ratio of *A* to the rms difference between the quadratic function and the rain-free data. The solid lines in the figures show the best-fit quadratic curves. According to the definition of reliability given by (20) and (21), the estimates over regions of moderate rain would be judged reliable or marginally reliable, and those over the light rain area would be judged unreliable.

The shape of the curves in these examples is characteristic of light-wind cases for which high values of NCRS at nadir decrease rapidly with angle. Two scans taken over a high-wind region in Hurricane Bonnie (24 August 1998) are shown in Fig. 9. As compared with the previous cases in Fig. 8, the nadir values are lower (7 dB), with a smaller variation of NCRS with angle. The results are typical of other cross-track and alongtrack estimates in the sense that the relative error in the path attenuation estimate decreases in going from lighter to heavier rains. Images of radar return power at 4 km (top) and NRCS (bottom) during this time period are shown in Fig. 10.

Insight into the reliability of the estimates can be gained by examining plots of the path attenuation derived from the along- and cross-track reference data. A scatterplot of the estimates is shown in Fig. 11, for which the data are taken from orbit 7056 of 18 February 1999. Only cases of rain over ocean are displayed. The number of data points (14 753) represents the number of rain observations over ocean for which both estimates yield positive values of *A.* If we require that both estimates be at least “marginally reliable,” as defined by (16), and that the absolute value of the difference be less than 0.5 dB, the number of data points reduces to 5851 or about 40% of the total shown in Fig. 11.

*A*

_{i}(

*R*

_{j}) denotes the path attenuation for the

*i*th rain observation using the reference dataset

*R*

_{j}. Values of

*d*and

*D*are shown in Table 3. For data judged to be marginally reliable or reliable, the mean absolute difference

*D*between the attenuations derived from the along- and cross-track references is 0.44 dB. For the alongtrack and temporal reference datasets, the mean absolute difference increases to 0.74 dB and increases further to 0.85 dB in comparing the attenuations obtained from cross-track and temporal reference data. As shown by the results in Table 3, for a given pair of reference datasets the value of

*D*is nearly independent of the reliability category. By contrast, the normalized error

*d*clearly decreases as the category changes from all data to marginally reliable data to reliable data. For example, for the along- and cross-track reference data, the normalized difference in attenuation decreases from 0.42 to 0.21 to 0.10 as less reliable data are eliminated. The improvement, however, comes at the expense of removing a large fraction of the data: of the 23 783 rain observations over ocean in this orbit, the attenuations derived from the three reference datasets are positive in 12 395 or 52% of the cases. The attenuation estimates are judged to be marginally reliable or reliable in 6062 (25%) cases and reliable in only 1549 (6.5%) cases. It should be recalled, however, that for the operational algorithm the reference dataset having the smallest variance is used to estimate path attenuation, and the reliability is assessed only from that reference dataset. Because only a single “best” reference dataset is used, the fractions of data judged to be marginally reliable or reliable data are significantly higher. Moreover, in most instances in which the path attenuation is judged to be unreliable for all available reference datasets, the rain rate is light and the Hitschfeld–Bordan method can be expected to provide reasonably accurate attenuation correction. Nevertheless, the issues of defining the reliability of the SRT estimate and determining a weighting factor between this estimate and the Hitschfeld–Bordan estimate are under study and will require extensive comparisons with independent measurements of the reflectivity factors and rain rates (Iguchi et al. 2000).

## Summary

Accurate estimates of rain rate from the TRMM precipitation radar require correction for attenuation. Because the Hitschfeld–Bordan method tends to become unstable at higher rain rates, the surface reference technique is used in the TRMM PR algorithms to complement it. Although the basic concept of the method is simple, the implementation is complicated by the fact that a number of different rain-free reference datasets can be defined. The decision as to which of these sets to use is based upon the ratio of the path attenuation estimate to the standard deviation of the rain-free reference data: in essence, the reference dataset used to estimate path attenuation is that which gives the largest ratio. This ratio is also used to define the reliability of the estimate. Application of the method to rain over ocean shows that SRT typically performs well in moderate to heavy rain rates but is unreliable in light rains. Comparisons among the attenuations derived from three reference datasets (temporal, along-, and cross-track) were shown for a single orbit of data. In general, as the reliability of individual estimates increases, the fractional error between them decreases. Whether a weighted combination of the path attenuations can be used to increase the accuracy of the estimate or to improve the determination of reliability is an open question.

Comparisons of this kind are useful in understanding how the attenuation estimates change with different reference datasets; however, it remains to establish the validity of the results by more independent means. Preliminary comparisons of the SRT-derived attenuation with those derived from the Hitschfeld–Bordan and mirror image methods show fair to good agreement in moderate stratiform rain, a situation for which the three methods are expected to be most accurate. Comparisons of dB*Z* maps derived from the PR with those from several Weather Surveillance Radar–1988 Doppler ground-based weather radars are encouraging. These comparisons, and an assessment of the technique over land, will be considered in future work, with the goals of assessing the validity of the SRT specifically and of attenuation correction methods generally for spaceborne weather radars.

## Acknowledgments

The work is supported in part by Dr. Ramesh Kakar of NASA Headquarters under the TRMM Science Program.

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Same as Fig. 1 but for an off-nadir incidence angle of 5.68°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 1 but for an off-nadir incidence angle of 5.68°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 1 but for an off-nadir incidence angle of 5.68°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 1 but for an off-nadir incidence angle of 13.49°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 1 but for an off-nadir incidence angle of 13.49°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 1 but for an off-nadir incidence angle of 13.49°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

(top) Range profile of the radar return power (dB*m*) at an incidence angle of 7.1° for scan numbers 3760 to 3960 of orbit 4267 over Hurricane Bonnie taken on 25 Aug 1998. (bottom) Corresponding plot of the apparent NRCS. Dark bar indicates PR rain detection; light bar indicates regions where the SRT-derived path attenuation is judged to be reliable or marginally reliable

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

(top) Range profile of the radar return power (dB*m*) at an incidence angle of 7.1° for scan numbers 3760 to 3960 of orbit 4267 over Hurricane Bonnie taken on 25 Aug 1998. (bottom) Corresponding plot of the apparent NRCS. Dark bar indicates PR rain detection; light bar indicates regions where the SRT-derived path attenuation is judged to be reliable or marginally reliable

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

(top) Range profile of the radar return power (dB*m*) at an incidence angle of 7.1° for scan numbers 3760 to 3960 of orbit 4267 over Hurricane Bonnie taken on 25 Aug 1998. (bottom) Corresponding plot of the apparent NRCS. Dark bar indicates PR rain detection; light bar indicates regions where the SRT-derived path attenuation is judged to be reliable or marginally reliable

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 4 but for an incidence angle of 3.55°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 4 but for an incidence angle of 3.55°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 4 but for an incidence angle of 3.55°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 4 but for an incidence angle of 10.65°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 4 but for an incidence angle of 10.65°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 4 but for an incidence angle of 10.65°

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

(top) Radar return powers (dB*m*) at a height of 4 km above the surface over Hurricane Bonnie measured on 25 Aug 1998. Vertical and horizontal scales represent distances of 215 and 860 km, respectively. (bottom) The apparent NRCS of the surface *σ*^{0} over the same region

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

(top) Radar return powers (dB*m*) at a height of 4 km above the surface over Hurricane Bonnie measured on 25 Aug 1998. Vertical and horizontal scales represent distances of 215 and 860 km, respectively. (bottom) The apparent NRCS of the surface *σ*^{0} over the same region

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

(top) Radar return powers (dB*m*) at a height of 4 km above the surface over Hurricane Bonnie measured on 25 Aug 1998. Vertical and horizontal scales represent distances of 215 and 860 km, respectively. (bottom) The apparent NRCS of the surface *σ*^{0} over the same region

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

NRCS of the surface versus incidence angle for two scans from measurements taken on 2 Feb 1999. Data from over rain and rain-free areas are represented by the symbols (∗) and (×), respectively. Best-fit quadratic curves are shown as solid lines

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

NRCS of the surface versus incidence angle for two scans from measurements taken on 2 Feb 1999. Data from over rain and rain-free areas are represented by the symbols (∗) and (×), respectively. Best-fit quadratic curves are shown as solid lines

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

NRCS of the surface versus incidence angle for two scans from measurements taken on 2 Feb 1999. Data from over rain and rain-free areas are represented by the symbols (∗) and (×), respectively. Best-fit quadratic curves are shown as solid lines

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 8 but for two scans taken over Hurricane Bonnie on 24 Aug 1998

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 8 but for two scans taken over Hurricane Bonnie on 24 Aug 1998

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Same as Fig. 8 but for two scans taken over Hurricane Bonnie on 24 Aug 1998

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

(top) Radar return power (dB*m*) at a 4-km height taken over Hurricane Bonnie on 24 Aug 1998. (bottom) The apparent NRCS of the surface over the same region

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

(top) Radar return power (dB*m*) at a 4-km height taken over Hurricane Bonnie on 24 Aug 1998. (bottom) The apparent NRCS of the surface over the same region

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

(top) Radar return power (dB*m*) at a 4-km height taken over Hurricane Bonnie on 24 Aug 1998. (bottom) The apparent NRCS of the surface over the same region

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Scatterplot of the path attenuations as estimated by the cross-track reference data (vertical scale) and the alongtrack reference data (horizontal scale) for all data in which both attenuations are nonzero

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Scatterplot of the path attenuations as estimated by the cross-track reference data (vertical scale) and the alongtrack reference data (horizontal scale) for all data in which both attenuations are nonzero

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Scatterplot of the path attenuations as estimated by the cross-track reference data (vertical scale) and the alongtrack reference data (horizontal scale) for all data in which both attenuations are nonzero

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Scatterplots of the path attenuations as estimated by the alongtrack, cross-track, and temporal reference data for cases where all three attenuations are judged to be reliable or marginally reliable. (top) Path attenuations derived from the cross-track and alongtrack reference data. (bottom) Path attenuations derived from the temporal and alongtrack reference data

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Scatterplots of the path attenuations as estimated by the alongtrack, cross-track, and temporal reference data for cases where all three attenuations are judged to be reliable or marginally reliable. (top) Path attenuations derived from the cross-track and alongtrack reference data. (bottom) Path attenuations derived from the temporal and alongtrack reference data

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Scatterplots of the path attenuations as estimated by the alongtrack, cross-track, and temporal reference data for cases where all three attenuations are judged to be reliable or marginally reliable. (top) Path attenuations derived from the cross-track and alongtrack reference data. (bottom) Path attenuations derived from the temporal and alongtrack reference data

Citation: Journal of Applied Meteorology 39, 12; 10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2

Mean and standard deviation of NRCS over ocean for Oct 1998 (Jan 1999)

Mean and standard deviation of NRCS over land for Oct 1998 (Jan 1999)

Mean absolute difference *D* and normalized mean absolute difference *d* for path attenuations derived from the alongtrack (AT), cross-track (XT), and temporal (Temp) reference datasets