## 1. Introduction

The accurate retrieval of oceanic rain rates from satellite microwave measurements has been an elusive goal since the concept was first proposed by Buettner (1963) and then demonstrated with satellite data by Wilheit et al. (1977). While the physical basis for these retrievals is sound, we believe that there are three significant assumptions inherent to these methods that make the measurement of tropical average rain rates to better than 10% problematic. These assumptions concern the specification of the following characteristics of a rain system: 1) the rain-layer thickness (often assumed to extend from the surface to the freezing level), 2) the relative amount of cloud water versus rain water, and 3) the varying rain intensities across the radiometer footprint (which is commonly called the “beamfilling effect”). The observed brightness temperature (*T*_{B}) is strongly influenced by these three characteristics. Significant newinformation on these three issues will have to await the Tropical Rain Measurement Mission (TRMM; Simpson et al. 1988), scheduled to be launched in 1998. The combination of TRMM’s microwave radiometer and rain radar will help to quantify the above three processes. In the meantime, while the new rain retrieval method described herein does not solve these problems, it does attempt to explicitly address them in a physically realistic way.

We present a unified, all-weather ocean algorithm that simultaneously finds the near-surface wind speed *W* (m s^{−1}), columnar water vapor *V* (mm), columnar cloud liquid water *L* (mm), rain rate *R* (mm h^{−1}), and effective radiating temperature *T*_{U} (K) for the upwelling radiation. This algorithm is a seamless integration of the Wentz (1997) no-rain algorithm and a newly developed rain algorithm. The algorithm is based on the fundamental principles of radiative transfer and explicitly shows the physical relationship between the inputs (*T*_{B}) and outputs (*W, V, L, R,* and *T*_{U}). The wind speed retrieval must be constrained to an a priori value for moderate to heavy rain, and *T*_{U} must be constrained by a statistical correlation for clear skies and light rain. The other retrievedparameters are unconstrained over the full range of weather conditions. Wentz (1997) discusses the algorithm’s performance in the absence of rain, and herein we focus on the rain component of the algorithm.

A particular strength of the new method is its ability to “orthogonalize” the retrievals so that there is minimum cross talk between the retrieved parameters. With respect to estimating rainfall, it is important to remove the water vapor contribution to the observed brightness temperature. We will present results showing that the error in retrieved water vapor (as determined from radiosonde comparisons) is uncorrelated to the retrieved rain rate. Likewise, the influence of the radiating temperature *T*_{U} is separated from the liquid water signal by using the polarization information contained in the observations. Because the rain rates are retrieved only after all other significant influences on *T*_{B} are quantified, the various retrievals can be analyzed for climate relationships between them, with high confidence that there is a minimum of algorithm cross talk.

Conceptually, the rain retrieval involves the following steps. The physics of radiative transfer shows that there is a direct and unique relationship between brightness temperature and the atmospheric transmittance *τ*_{L} of liquid water. In view of this, the first step is to directly retrieve *τ*_{L} along with the other directly observable parameters *W, V,* and *T*_{U}. In the context of rainfall, *τ*_{L} is related to the columnar water in the rain cloud, and *T*_{U} provides information on the height from which the radiation is emanating and whether radiative backscattering by large ice particles is occurring (Spencer 1986). The retrieval of *τ*_{L} is done by solving a set of simultaneous brightness temperature equations. A basic premise in this retrieval is that the polarization signature of the *T*_{B} allows for the separation of the *τ*_{L} signal from the *T*_{U} signal. The *T*_{B} model is formulated such that the *T*_{U} parameter includes both radiative scattering effects and air temperature variability. In the next step, the spectral signature of the retrieved *τ*_{L} at 19 and at 37 GHz is used to estimate the beamfilling effect. A beamfilling correction is applied, and the mean atmospheric attenuation *A*_{L} for liquid water over the footprint is found. Mie scattering theory and an assumed relationship between cloud water and rain water are then used to convert the *A*_{L} to a columnar rain rate, which is defined as the vertically averaged rain rate times column height. Finally, the columnar rain rate is converted to a vertically averaged rain rate by dividing by an assumed rain column height that is a function of a sea surface temperature climatology. The final assumption is that the surface rain rate equals the vertically averaged rain rate. In this way, we explicitly handle the three rain cloud characteristics listed above.

The algorithm is developed and tested using the observations taken by the Special Sensor Microwave/Imager (SSM/I; Hollinger et al. 1987). The SSM/I is a scanning radiometer that operates at four frequencies: 19.35, 22.235, 37, and 85.5 GHz. It is flown by theDefense Meteorological Satellite Program (DMSP) on operational polar orbiting platforms. The results herein are based on SSM/I observations for the 4-yr period from 1991 to 1994. Observations from two satellites, *F10* and *F11,* are used. The *F10* observations cover the entire 4-yr period, while the *F11* observations begin in January 1992.

The algorithm described herein is being used to produce the NASA Pathfinder Data Set for Scanning Multichannel Microwave Radiometer and SSM/I. This dataset will be a 20-yr time series of geophysical parameters, which will be broadly distributed to the research community.

## 2. The no-rain algorithm

We begin by reviewing the no-rain algorithm described by Wentz (1997). Then section 3 shows how this algorithm is extended to include rain observations. In the absence of rain, there is a relatively simple and unique relationship between the ocean brightness temperature (*T*_{B}) measured by SSM/I and *W, V,* and *L.* As a consequence of this simple relationship, these parameters can be retrieved to a high degree of accuracy. The retrieval of (*W, V, L*) is accomplished by varying their values until the *T*_{B} model function matches the SSM/I observations. After a precision calibration to in situ observations, the rms retrieval accuracies for *W, V,* and *L* are 0.9 m s^{−1}, 1.2 mm, and 0.025 mm, respectively (Wentz 1997). We now give some details on the no-rain algorithm so that one can then see how it is extended to include rain.

*W, V,*and

*L*are done at this resolution, while the rain rate retrievals are done at the spatial resolution of the 37-GHz footprint, which is 32 km (see section 5). The no-rain algorithm then retrieves

*W, V,*and

*L*by solving the three model function equationswhere

*F*is the model function. We have not included the wind direction term in the above equations because it is a small effect that is unimportant in the context of the rain algorithm. The SSM/I brightness temperatures at 37 GHz are denoted by

*T*

_{B37V}and

*T*

_{B37H}for vertical and horizontal polarization, respectively. The conversion of

*T*

_{A}to

*T*

_{B}requires dual-polarization observations, but only

*υ*-pol observations are taken at 22 GHz. Thus at 22 GHz, we work in terms of the antenna temperature

*T*

_{A22V}rather than

*T*

_{B}. The term

*T*

_{BC}is the cold spacebrightness temperature equaling 2.7 K. The

*G*factors are the antenna pattern coefficients that account for antenna spillover and cross-polarization leakage (Wentz 1991). Equations (1) represent three quasilinear equations in three unknowns:

*W, V,*and

*L.*This system of equations is solved by Newton’s method, as is explained in Wentz (1997).

## 3. Extending the algorithm to include rain

To create an all-weather algorithm, the no-rain algorithm is extended in the following ways.

- The cloud water parameter
*L*is replaced by the total transmittance of cloud and rain water at 37 GHz,*τ*_{L37}. - An additional parameter is added to the retrieval: total transmittance of cloud and rain water at 19 GHz,
*τ*_{L19}. - When rain occurs, the wind speed retrieval is constrained to an a priori value.
- When rain occurs, the effective air temperature
*T*_{U}becomes a retrieved parameter.

^{−1}the algorithm no longer retrieves wind speed. The first three modifications are discussed in this section, and the fourth modification involving the effective air temperature is discussed in section 5.

*L*by the 37-GHz liquid water transmittance

*τ*

_{L37}for both cloud and rain water. In the no-rain algorithm, the cloud liquid water

*L*enters the equation only through the transmittance. This dependence can be expressed aswhere

*θ*is the incidence angle,

*A*

_{O}and

*A*

_{V}are the atmospheric oxygen and water vapor absorptions, and

*A*

_{L}is the cloud liquid water absorption. Wentz (1997) gives expressions for

*A*

_{O}and

*A*

_{V}as functions of the effective air temperature and columnar water vapor. In the absence of rain, the radiative transfer through the cloud droplets, which are much smaller than the radiation wavelength, is governed by Rayleigh scattering, and the absorption is proportional to the columnar liquid water content

*L*(mm) of the cloud (Goldstein 1951), as indicated by (3). The absorption has a small dependence on the temperature

*T*

_{L}of the cloud water.

*τ*

_{LF},where

*τ*

_{LF}now replaces

*L*as a retrievable parameter. The subscript

*F*is introduced to denote the dependence on frequency. Often it is more convenient to work in terms of the absorption than the transmittance. Given the retrieved value for

*τ*

_{LF}, the absorption (before doing the beamfilling correction) is given by

*A*

_{LF}

*θ*

*τ*

_{LF}

When rain is present, the relationship between *τ*_{LF} and liquid water content is more complex, as discussed in section 7, and the simple Rayleigh expression is not valid. However, by parameterizing the *T*_{B} model in terms of *τ*_{LF} rather than *L,* we defer the problem of relating *τ*_{LF} to the liquid water content. In other words, we are dividing the rain retrieval problem into two steps. The first step involves separating the liquid water signal, expressed in terms of *τ*_{LF}, from the signal of the other parameters. Since *T*_{B} is nearly proportional to *τ*^{2}_{LF}*τ*_{LF}. In the second step, a rain rate is inferred from *τ*_{LF}. It is only in the second step that one is required to make assumptions regarding the beam filling, the cloud/rain partitioning, and the rain column height.

The second modification is to introduce *τ*_{L19} as an additional parameter to be retrieved. For the no-rain algorithm, Rayleigh scattering gave a fixed relationship between the transmittances at 19, 22, and 37 GHz, and hence it was not necessary to separately retrieve *τ*_{L19}. However, when rain is present there is no fixed spectral relationship between the transmittances. Accordingly, we directly retrieve *τ*_{L19} by introducing a fourth equation into the retrieval process.

The third modification is to eliminate wind speed as a retrieved product when there is significant rain. The decrease in the atmospheric transmittance obscures the surface and degrades the ability to retrieve the wind speed. Furthermore, the *T*_{B} modeling error is larger for raining observations due to errors in specifying the effective air temperature, as is discussed in section 5. For moderate to heavy rain it is best to constrain the wind parameter to some specified a priori value. To do this, we use the SSM/I wind retrievals in adjacent, no-rain areas to specify *W.* If no such wind retrievals are available, we use a monthly, 1° latitude by 1° longitude wind climatology to specify *W.* This climatology is produced from 7 years of SSM/I observations.

*τ*

_{L22}is specified by an interpolation between

*τ*

_{L19}and

*τ*

_{L37}. The constraint on wind is accomplished by the Λ(

*x*) term, which is the weighting function given by

*x*) goes smoothly from 0 to 1 as its argument

*A*

_{L37}goes from 0.04 to 0.15. The lower bound of 0.04 represents the onset of rain and was found from an investigation of 38 northeast Pacific extratropical cyclones (Wentz 1990). The upper limit represents a rain rate of about 2 mm/h, depending on the rain column height and other factors. When Λ(

*x*) = 0, the above equations have the same form as the no-rain equations (1). When Λ(

*x*) = 1, Eq. (7d) simply becomeswhere

*W*

_{0}is the a priori wind speed. Equation (10) forces the wind retrieval to equal

*W*

_{0}. Thus

*A*

_{L37}plays the role of a blending parameter for joining the no-rain algorithm with the raining algorithm. As

*A*

_{L37}goes from 0.04 to 0.15, the algorithm smoothly transforms from the no rain algorithm to a rain algorithm. The set of equations (7) is solved in the same way as was discussed above for the no-rain algorithm. The equations are assumed to be stepwise linear in terms of the unknowns (

*W, V, τ*

_{L19},

*τ*

_{L37}), and the equations are solved in an iterative manner. Typically about five iterations are needed to reach the 0.1-K convergence level.

## 4. Retrieval of water vapor in rain

Figure 1 shows the difference between the SSM/I retrieved water vapor and the value obtained from collocated radiosonde observations (RAOB). The difference is plotted versus rain rate. The quality control of the radiosonde data and the collocation with the SSM/I are discussed in Wentz (1997). There are a total of 35108 SSM/I overpasses of radiosonde sites. For these overpasses, a total of 81922 rain observations are found within a 112-km radius of the site. The solid curve shows the mean difference and the dashed curves show the ±1 standard deviation of the difference. The rain rate is computed from the SSM/I observations, as described in this paper. The statistics are computed by first binning the observations into 0.5 mm/h rain-rate bins. For rain rates between 1 and 15 mm/h, the typical rms difference between the SSM/I and radiosonde vapor is 5 mm. In comparison, the rms difference for the no-rainobservations is 3.8 mm. The error analysis in Wentz (1997) indicates that the spatial and temporal sampling mismatch between the SSM/I 56-km footprint and the radiosonde point observation contributes about 3.7 mm to the total rms difference. Thus nearly all of the rms difference for the no-rain observations is due to the spatial–temporal mismatch. For the rain observations, about half of the rms difference is due to the spatial–temporal mismatch.

*A*

_{L19}, the SSM/I retrievals were biased low (high) relative to the radiosonde values for high (low) values of sea surface temperature

*T*

_{S}. In view of this, we apply the following ad hoc correction to the retrieved values of

*V*in order to correct the observed systematic error:This correction has been applied to Fig. 1. We believe that this systematic error is due to radiative scattering and the degradation in the assumed relationship between the effective air temperature and water vapor, as is discussed in the next section.

We find that when *A*_{L19} exceeds about 0.3 (which corresponds to *R* ∼ 15 mm/h, depending on rain column height), the atmosphere is too opaque and/or scattering is too strong to obtain a useful estimate of *V.* The procedure discussed in section 5 for obtaining *τ*_{L} when radiative scattering is significant requires that *V* be specified. Thus, for *A*_{L19} > 0.3, we use an a priori value for *V* based on the SSM/I vapor retrievals in adjacent, norain, and light-rain areas. If no adjacent retrievals areavailable, we use a monthly, 1° latitude by 1° longitude vapor climatology to specify *V.* This climatology is produced from 7 years of SSM/I observations.

## 5. Effective air temperature and radiative scattering

*T*

_{B}model function

*F*(

*W, V, τ*

_{L}) is an assumed relationship for the effective air temperature

*T*

_{U}versus the retrieved columnar water vapor

*V*and a climatological sea surface temperature

*T*

_{S}. The effective air temperatures, which we also call the effective radiating temperatures, for the upwelling and downwelling atmospheric radiation are defined aswhere

*T*

_{BU}and

*T*

_{BD}are the upwelling and downwelling atmospheric brightness temperatures and

*τ*is the atmospheric transmittance. An analysis of 42195 radiosonde observations shows that in the absence of rain

*T*

_{U}and

*T*

_{D}are well correlated with the

*V*and

*T*

_{S}:where the function Ψ (

*V, T*

_{S}) and the coefficients

*c*

_{6}and

*c*

_{7}are given by Wentz (1997). At 19 and 37 GHz,

*T*

_{U}and

*T*

_{D}are very similar, with

*T*

_{U}being about 2 K colder.

*V, T*

_{S}) degrades because 1) precipitation and associated convection alters the air temperature and 2) radiative scattering by large raindrops and ice particles reduces

*T*

_{BU}and

*T*

_{BD}. Since by definition

*T*

_{U}and

*T*

_{D}are proportional to

*T*

_{BU}and

*T*

_{BD}, they also decrease when scattering occurs. Fortunately, the radiosonde comparisons in the previous section show that the vapor retrieval error due to the degradation of Ψ(

*V, T*

_{S}) is not large and can be partially corrected by the ad hoc adjustment (11). However, the error in

*τ*

_{L19}and

*τ*

_{L37}can be large if scattering effects are ignored. The rain algorithm accounts for scattering (and rain-induced variations in air temperature) by letting

*T*

_{U}become a retrieved parameter rather than being specified by Ψ(

*V, T*

_{S}). Since

*V*and

*W*have already been found by the procedure described in section 3, the retrieval problem is reduced to two equations in two unknowns (i.e.,

*T*

_{U}and

*τ*

_{L}):There is a separate pair of equations for 19 and 37 GHz, and we have temporarily dropped the subscript denoting frequency. Two assumptions are required to solve these equations. First,

*T*

_{U}and

*T*

_{D}are assumed to be closely correlated so that

*T*

_{D}can be specified as a function of

*T*

_{U}according to (14). The second assumption is that

*T*

_{U}has the same value for vertical and horizontal polarization. In the absence of scattering,

*T*

_{U}is completely independent of polarization. For moderate to heavy rain, for which scattering is significant, SSM/I observations show that the saturation values for the

*v*-pol and

*h*-pol

*T*

_{B}observations are nearly the same (Spencer et al. 1989). Thus, the assumption of polarization independence seems reasonable.

*τ*

_{L}and

*T*

_{U}assuming a simplified

*T*

_{B}model. We denote the solution for

*T*

_{U}for the complete

*T*

_{B}model by the function

*T*

_{U}

*ξ*

*T*

_{BV}

*T*

_{BH}

*T*

_{U}with the no-rain estimate given by Ψ(

*V, T*

_{S}), we use the expression:and

*τ*is the total transmittance

*τ*of liquid water, water vapor, and oxygen. Equations (15a) and (17) are then combined to retrieve

*τ*

_{L}. When

*τ*≥ 0.7, then

*T*

_{U}= Ψ(

*V, T*

_{S}), and the retrieved

*τ*

_{L}is identical to that found by the no-rain algorithm described in section 2. When the

*τ*⩽ 0.5, the retrieved

*τ*

_{L}and

*T*

_{U}are based solely on the magnitude and polarization signature of the dual-polarization observations (either 19 or 37 GHz).

*T*in the effective temperature aswhich is a measure of the decrease in brightness temperature due to radiative scattering. In addition to radiative scattering, Δ

_{U}*T*

_{U}is also a measure of the decrease in the air temperature due to most of the radiation coming from the cold cloud tops. Figure 2 shows Δ

*T*

_{U}plotted versus

*τ*for the time period from July to September 1992, as derived from the

*F10*observations. For this time period, the SSM/I retrieval algorithm finds 7859295 rain-influenced footprints over the world’s oceans. The curves are generated by binning these observations into Δ

*τ*= 0.01 bins. The solid curves show the mean value for each bin, and the dashed curves show the ±1 standard deviation for each bin. The depression in the effective temperature for the 37-GHz observations, which are most affected by scattering, is about twice that of the 19-GHz observations. For low values of

*τ,*Δ

*T*

_{U}is about −10 K and −20 K for 19 and 37 GHz, respectively. For

*τ*> 0.7, the retrieved value of

*T*

_{U}given by

*ξ*(

*T*

_{BV},

*T*

_{BH}) for 19 GHz becomes noisy and unreliable.

The retrievals *W, V, τ*_{L}, and *T*_{U} are all done at the common spatial resolution of the 19-GHz channels, which is about 56 km. For the rain rate retrievals, we want as much spatial resolution as possible. In order to obtain a rain rate at the resolution of the 37-GHz footprint, we make the assumption that *W* and *V* are uniform over the 19-GHz footprint. The above equations are then used to find *τ*_{L37} and *T*_{U37} given the 37-GHz *T*_{B} at their original resolution of 32 km. In Fig. 2, the spatial resolution for the Δ*T*_{U} values is 56 km for 19 GHz and 32 km for 37 GHz.

*τ*

_{L19}and

*τ*

_{L37}. Thus to compute a rain rate at the 32-km resolution, a value of

*τ*

_{L19}at the spatial resolution of 32 km is required. We use the following expression to specify a high-resolution

*τ*

_{L19,HI}:where the subscript HI denotes the higher spatial resolution. Note that (20) is in terms of the absorption, that is, ln(

*τ*

_{L}). The assumption behind Eq. (20) is that, even though the cloud and rain water may significantly vary over the footprint, the ratio of 19 to 37 GHz liquid water absorption is relatively constant. If this is true, then the observed spatial variation in

*τ*

_{L37}can be used as a scaling factor to compute

*τ*

_{L19}at the higher spatial resolution. In reality, this spectral ratio will have some interfootprint variability, and (20) will introduce some error into the rain retrieval. However, this error will tend to be unbiased and will tend to zero when doing spatial and temporal averages of rain rate.

## 6. The beamfilling effect

*R*and brightness temperature

*T*

_{B}that occurs when averaging over the radiometer footprint. To illustrate this effect, we use a relatively simple model for the brightness temperature:

*T*

_{B}

*T*

_{E}

*τ*

*τ*

^{2}

*ρ*

*T*

_{E}(

*τ*) is the effective temperature of the combined ocean and atmosphere system,

*τ*is the total transmittance through the atmosphere, and

*ρ*is the reflectivity of the sea surface. The effective temperature is a function of

*τ.*For

*τ*= 1,

*T*

_{E}equals the sea surface temperature

*T*

_{S}and, for

*τ*= 0,

*T*

_{E}equals the effective temperature

*T*

_{U}of the upwelling atmospheric radiation;

*T*

_{E}smoothly varies from

*T*

_{S}to

*T*

_{U}as

*τ*goes from 1 to 0.

*T*

_{B}model function described in section 2 and provides considerable insight into the retrieval algorithm discussed in the previous sections. As discussed in section 5, to separate

*τ*from the

*T*

_{E}signal requires dual-polarization observations, either at 19 or 37 GHz. Looking at the simple

*T*

_{B}model above, we see that the

*T*

_{E}term is easily eliminated, and the transmittance is given byValues for the wind speed and water vapor come from the procedure discussed in section 3. The wind speed and climatology sea surface temperature are used to specify

*ρ*

_{V}and

*ρ*

_{H}. The oxygen and water vapor components of

*τ*

^{2}are factored out using Eq. (4), thereby obtaining just the liquid water transmittance

*τ*

^{2}

_{L}

*τ*signal from the

*T*

_{E}signal is similar to that proposed by Petty (1994).

*τ*

^{2}

_{L}

*τ*

^{2}

_{L}

*P*(

*A*′) for the cloud and rain water absorption

*A*′ within the footprint. The desired retrieval is the mean absorption over the footprint, which is given byWithout any correction for the spatial averaging, the estimate of the absorption would beAny variation of the absorption within the footprint will result in the estimate

*Â*

_{L}being lower than the true mean value

*A*

_{L}. This systematic underestimation of the absorption is called the beamfilling effect.

*P*(

*A*′) is an exponential distribution of the formthen the integrals in (23) and (24) are easily evaluated, and one obtains the following relationship between

*A*

_{L}and

*Â*

_{L}:Here Δ

*A*

_{L}is the rms variation of

*A*′ within the footprint and

*β*is the normalized rms variation of

*A*′. For an exponential distribution

*β*equals unity. An expansion of statistical moments to the second order in

*Â*

_{L}

*β*

^{2}shows that (27) is correct (to second order) for any distribution

*P*(

*A*′) having a mean

*A*

_{L}and an rms variation of Δ

*A*

_{L}. Thus if

*β*is known, (27) can be used to correct the beamfilling effect. Here

*β*is a normalized quantity that is related to the variability of liquid water in the footprint and hence is essentially independent of frequency.

*β,*we note that the ratio

*Â*

_{L37}/

*Â*

_{L19}will be less than that predicted by Mie scattering theory when the beamfilling effect is significant. Thus comparing

*Â*

_{L37}/

*Â*

_{L19}to the Mie absorption ratio provides the means to determine the beamfilling effect. The relationship between the Mie ratio and

*Â*

_{L37}/

*Â*

_{L19}is given by the 37 to 19 GHz frequency ratio of Eq. (27):where the left-hand side is the Mie ratio given by Eq. (32) below. This ratio varies from 3.5 for light absorption values to 2.8 for heavy absorption. If the beamfilling effect is not significant, then

*Â*

_{L37}/

*Â*

_{L19}will equal

*A*

_{L37}/

*A*

_{L19}. Thus when

*Â*

_{L37}/

*Â*

_{L19}≥

*A*

_{L37}/

*A*

_{L19}, no beamfilling correction is done (i.e.,

*β*is set to 0). Otherwise (29) is inverted to find

*β*as a function of the two ratios:

*Â*

_{L37}/

*Â*

_{L19}and

*A*

_{L37}/

*A*

_{L19}. The inversion is readily done because the expression on the right-hand side of (29) increases monotonically with

*β.*It equals

*Â*

_{L37}/

*Â*

_{L19}when

*β*= 0. Thus, the beamfilling correction consists of finding a value for

*β*that when substituted into (29) yields the absorption ratio given by the Mie theory. Given

*β,*the mean absorption for the footprint is then found (27).

The magnitude of the beamfilling correction is characterized in terms of the ratios *A*_{L19}/*Â*_{L19} and *A*_{L37}/*Â*_{L37}, which are called the beamfilling correction factors (BCF). When *Â*_{L37}/*Â*_{L19} is significantly less than *A*_{L37}/*A*_{L19}, large values for *β* and BCF are found. For example, when *Â*_{L37}/*Â*_{L19} = 2, the BCF is 1.4 and 2.0 for 19 and37 GHz respectively. For even smaller values of *Â*_{L37}/*Â*_{L19} the BCF increases exponentially, and we must impose the following limits. The maximum values of 3.4 and 6.4 are used for the 19-GHz and 37-GHz BCF, respectively, which corresponds to the exponent 2*Â*_{L37}*β*^{2} sec*θ* in (29) reaching a value of 3.0. If the BCF exceeds the maximum, it is reset to the maximum. Another overall limit is placed on *A*_{L19} and *A*_{L37}. Neither value is allowed to exceed 1.2. These limits correspond to observations of heavy rain for which the 37 GHz and, to a lesser degree, the 19-GHz brightness temperatures have reached saturated levels. The effect of these limits is to place an upper hound on the retrieved rain rate. For the extreme case of *A*_{L19} reaching a value of 1.2, the retrieved rain rate will be about 25 mm/h (75 mm/h) for a rain column height of 3 km (1 km). We consider the 25 mm/h limit as an extreme upper bound on the algorithm’s ability to retrieve rain. For such high rain rates, both the 19-GHz and 37-GHz observations have saturated, and the retrieval error can be very large.

Figure 3 shows the 37-GHz absorption plotted versus the 19-GHz absorption for the July–September 1992 period discussed in section 5. The bottom curve in Fig. 3 shows the retrieved absorptions *Â*_{L37} versus *Â*_{L19} before the beam-filling correction. The middle curve shows the absorptions *A*_{L37} versus *A*_{L19} after the beam filling correction, and the top curve shows the theoretical curve computed from Mie scattering computations. The curves are generated by binning the 7859295 observations into absorption bins having a width of 0.005. The solid curves show the mean value for each bin, and the dashed curves show the ±1 standard deviation for each bin. The *A*_{L37} versus *A*_{L19} curve closely follows the theoretical curve up to values of *A*_{L19} ≈ 0.4. Above this value, therestriction that *A*_{L37} ⩽ 1.2 becomes important, and the curve asymptotically tends to the 1.2 value. For the high absorption bins, *A*_{L37} is a constant 1.2, and hence standard deviation envelope goes to zero.

Figure 4 shows the effect of the normalized rms variation *β* on the computation of *A*_{L37} and *A*_{L19}. For this figure, Eq. (27) is used to compute *A*_{L37} and *A*_{L19} from the retrieved values *Â*_{L37} and *Â*_{L19} using four different *β* values: 0.7, 0.8, 0.9, and 1.0. That is to say, rather than computing *β* for each observation, we use an average value. The theoretical Mie curve lies between the *β* = 0.8 and *β* = 0.9 curves. This indicates that, on the average, the beamfilling effect is characterized by a normalized rms variation *β* ≈ 0.85, which is somewhat less than the *β* = 1 value given by an exponential probability density function for *A*_{L}.

## 7. Inferring rain rate from liquid water attenuation

*A*

_{L19}and

*A*

_{L37}once the beamfilling correction has been applied. The retrieval method discussed above results in

*A*

_{L}being directly related to the transmittance

*τ*of the radiation from the sea surface upward through the atmosphere. Thus, more generally speaking, the retrieved

*A*

_{L}is an attenuation factor for the polarized sea surface signal traveling from the sea surface through the atmosphere. In the absence of scattering, the attenuation and absorption are equivalent and are given bywhere the integral is over altitude

*h*(km),

*κ*(

*h*) (km

^{−1}) is the Mie absorption coefficient, and

*H*(km) is the height of the liquid water column. Radiative scattering from rain drops and ice modify the attenuation. For example, the attenuation for point-to-point microwave communication links is still given by (30), but

*κ*(

*h*) is the Mie extinction coefficient, rather than absorption coefficient. However, the sea surface is an extended source, as opposed to a point source. For an extended source, the polarized surface signal traveling along the SSM/I viewing direction is scattered in other directions, while the surface signals traveling in other directions are scattered into the SSM/I viewing direction. These two effects tend to compensate, and using the extinction coefficient in (30) would overestimate the attenuation of the surface signal.

For moderate to heavy rain (R ≥ 10 mm/h), the 19-GHz (37-GHz) extinction coefficient is about 20% (60%) higher than the absorption coefficient. One distinguishing characteristic between the extinction and absorption coefficients is their spectral signature. For light to moderate rain (5 mm/h) the 37 to 19 GHz ratio for the extinction coefficient is 3.8 as compared to 3.0 for the absorption coefficient. Figure 3 shows that the spectral signature of the SSM/I retrieved *Â*_{L37}/*Â*_{L19} is about 2 for light to moderate rain. Thus, a significantly larger beamfilling correction would be needed for the extinction coefficients as compared to the absorption coefficients. We decided to use the absorption coefficient to evaluate (30) because 1) its spectral signature is closer to the observed *Â*_{L37}/*Â*_{L19} and 2) we expect that the attenuation of the polarized surface signal due to scattering will be small (i.e., the scattering into and out of the viewing direction will tend to cancel).

Fortunately, the choice of the attenuation coefficient does not have a large effect on the retrieved rain rate. The larger extinction coefficients would give a lower rain rate except for the fact that the beamfilling correction is larger for the extinction coefficients. These two factors tend to cancel each other, and, in general, the rain retrievals using the absorption coefficients are only about 10% higher than using the extinction coefficients. For example, if the best choice for *κ* in (30) is halfway between the absorption and the extinction coefficient, then our rain retrievals would be biased about 5% high.

*r*is the drop radius,

*N*

_{C}(

*r*) and

*N*

_{R}(

*r*) are the drop size distributions for cloud and rain water respectively,and

*σ*(

*r*) is the Mie absorption cross section. When

*r*is much smaller than the radiation wavelength, which is the case for the cloud-water integral,

*σ*(

*r*) is simply proportional to

*r*

^{3}, and hence the cloud integral is proportional to the total volume of cloud water per unit volume of atmosphere. Thus the vertically integrated cloud absorption given by (30) is proportional to the columnar cloud water

*L.*

For the rain integral, the simple *σ*(*r*) ∝ *r*^{3} does not hold, and the absorption depends on the details of the drop size distribution. We use the Marshall and Palmer (1948) drop size distribution to specify *N*_{R}(*r*). The Marshall–Palmer distribution is parameterized in terms of a nominal rain rate. Following the method described by Wilheit et al. (1977), we vary this nominal rain rate from 0.01 to 50 mm/h and compute the above rain absorption integral, denoted by *κ*_{R}, and the actual rain rate assuming the fall velocity given by Waldteufel (1973). We find that the *κ*_{R} versus rain rate relationship in the 19–37-GHz band is well approximated by a power law relationship, which is close to linear.

*H*is the height of the rain column,

*L*is the columnar cloud water (mm), and

*T*

_{L}is the rain cloud temperature. The rain rate

*R*(mm/h) is the rain rate averaged over the rain column given bywhere

*R*(

*h*) is the rain profile. The difference between

*R*and

*R*(0) is an additional source of error when comparing to in situ surface rain measurements. Evaporation under the rain cloud will tend to make

*R*greater than

*R*(0), while a decrease in

*R*(

*h*) at the top of the rain cloud will tend to make

*R*less than

*R*(0). The rain cloud temperature is assumed to be the mean temperature between the surface and the freezing level,

*T*

_{L}

*T*

_{S}

*T*

_{S}is the climatology sea surface temperature.

Equation (32) reveals a fundamental problem in retrieving rain rate. Given just *A*_{L19} and *A*_{L37}, it is not possible to uniquely separate and retrieve *L, R,* and *H.* The spectral dependencies of the cloud water term and the rain rate term are nearly the same, as can be seen by the spectral ratio of the coefficients (0.208/0.059 = 3.5; 0.0436/0.0122 = 3.6). By doubling the rain rate *R* and halving the height *H,* one obtains about the same *A*_{L}. Thus to obtain an estimate for *R,* one must make apriori assumptions regarding *L* and *H.* Potentially, these assumptions can produce significant errors in the rain retrievals.

*L*on the order of 1 to 2 mm. It seems reasonable to assume that, in general,

*L*increases with

*R*and then tend to level off at very high rain rates. Furthermore, an investigation of 38 northeast Pacific extratropical cyclones (Wentz 1990) indicates that when the SSM/I retrieval of

*L*reaches a value of 0.18 mm, a drizzle or light rain is likely. The following expression incorporates the features of 1) rain beginning at L = 0.18 mm, 2)

*L*increasing with

*R,*and 3) the

*L*versus

*R*relationship leveling off at high

*R,*with

*L*reaching a maximum value between 1 and 2 mm,

*L*

*HR*

The specification of the rain column height *H* is based, in part, on the altitude *H*_{F} of the freezing level as derived from the radiosonde observations. The global radiosonde observations for the 1987–90 period collected by Wentz (1997) are used to find *H*_{F} as a function of the sea surface temperature *T*_{S}. Out of the total 42195 radiosonde soundings, we only use the 9120 soundings for which the surface relative humidity is ≥90%. By restricting the dataset to high humidity cases, the results should be more indicative of rain observations. Figure 5 shows the height of the freezing level measured by the radiosondes versus the climatological sea surface temperature at the radiosonde site. For the stations at very high latitudes, the typical value of *H*_{F} is about 1 km. The midlatitude value of *H*_{F} ranges from 2 to 4 km, and in the Tropics *H*_{F} reaches a value of 5 km.

Equation (32) shows that the retrieved rain rate isvery nearly proportional to *H*^{−1}. Thus the proper specification of *H* is critical to obtaining good rain rate retrievals. In a preliminary analysis, we used the *H*_{F} values shown in Fig. 5 to specify *H* and found that the rain rates in the Tropics were about 40% lower in comparison to other climatologies (see section 8). We find that reducing to 3 km in the Tropics corrects the underestimation of rain relative to the climatologies. It is not unreasonable to expect that *H* is somewhat less than the freezing level because warm tropical rain does not extend up to the freezing level (Fletcher 1969). However, a reduction from 5 to 3 km seems extreme since warm rain is not that prevalent. Probably, this adjustment is compensating for some other deficiency in the algorithm, such as the algorithm’s inability to accurately measure high rain rates. In any event, we let *H* be the one tuning parameter in the algorithm.

*H*

_{F}to specify

*H.*The following simple regression, which is shown in Fig. 5, is derived so as to match

*H*

_{F}for low values of

*T*

_{S}and to reach a value of 3 km for high values of

*T*

_{S}:

Having specified *H* and the relationship between *R* and *L,* one can invert Eq. (32) and find a value for *R* given *A*_{L}. Note that, as a result of the beamfilling correction discussed in section 6, the retrieved values of *A*_{L19} and *A*_{L37} are not independent. Rather, they are computed such that their ratio is consistent with Eq. (32) above. For this reason, the same value for *R* is found from either *A*_{L19} or *A*_{L37}. The one exception is when *A*_{L37} exceeds the maximum value of 1.2. In this case, *A*_{L19} is used to compute the rain rate.

## 8. Rain-retrieval results

### a. Probability density function of SSM/I rain rates

All results in this section are based on SSM/I observations for the 4-yr period from 1991 through 1994. Observations from two satellites, *F10* and *F11,* are used. The *F10* observations cover the entire 4-yr period, while the *F11* observations begin in January 1992. The top frame of Fig. 6 shows the probability density function (pdf) for the rain rates inferred from the two SSM/Is. The thick curve shows global results, and the thin curve shows tropical results (20°S–20°N). The computation of any rain pdf is very dependent on the temporal and spatial averaging. For the SSM/I, the temporal averaging is essentially instantaneous, and the spatial averaging has a resolution of about 32 km. A rain pdf computed from rain gauges looks very different than that shown in Fig. 6 because the spatial averaging is very different. The leftmost point on the pdf curves corresponds to the number of no-rain observations. A total of 85.9% of theSSM/I observations indicated no rain. An additional 8.3% of the observations indicated very light rain not exceeding 0.2 mm/h, and the remaining 5.8% of the observations indicate rain exceeding 0.2 mm/h. We consider the accuracy of the “very light rain” retrievals as questionable. Some or many of these observations may actually be heavy nonraining clouds. Note that the contribution of the very light rain observations to the total rainfall is very small (see below).

To determine the contribution of the various footprint-averaged rain rates to the overall rainfall amount, we multiply the rain pdf by the rain rate, as shown in the bottom frame in Fig. 6. In this case, the area under the curve equals the average oceanic rainfall, which is 0.12 mm/h (2.9 mm/day) globally and 0.16 mm/day (3.9 mm/day) in the Tropics. The questionable very light rain observations (R < 0.2 mm/h) only contribute 0.007 mm/h (0.17 mm/day) to this total. One-half of the total global oceanic rainfall occurs at footprint-averaged rates above (and below) about 3.5 mm/h. For rainfall in the Tropics, this midpoint value increases to 5.5 mm/h. Due to the large size of the footprint (32 km) over which the enveloped rainfall is averaged, this midpoint value is much lower than that obtained from rain gauges. Four-minute rain gauge statistics (Jones and Sims 1978) suggest that about half of tropical rainfall occurs at rates above about 20 mm/h. One possible interpretation ofthis result is that, on the average when significant rain is being observed, only about one-quarter of the SSM/I footprint is actually covered by rain.

### b. Global distribution of SSM/I rain rates

Figure 7 shows the seasonal and annual zonally averaged rainfall computed from the SSM/I observations for 1991–94. The meridional structures revealed by the SSM/I are similar to previously published climatologies. The maximum oceanic rainfall occurs at the equatorial latitudes associated with the strong convection in the intertropical convergence zone (ITCZ) for all seasons. This peak is quite narrow in meridional extent and varies from about 7 mm/day in the winter to a maximum 11 mm/day in the summer. The seasonal north–south migration of the ITCZ, which is in phase with the solar insolation, is also apparent in the figure. The extratropical rainfall is greater in the Northern Hemisphere than in the Southern Hemisphere for all seasons. Low precipitation rates (∼1 mm/day) are observed in those zones of subsidence influenced by the large semipermanent anticylones.

Figure 8 shows color-coded global maps of the SSM/I annual and seasonal rainfall average over the four yearsfrom 1991 through 1994. The major features of the spatial distribution of the average annual rainfall are quite similar to those revealed in other satellite climatologies (see below). The largest annual rainfall amounts are seen to occur in the tropical Pacific, extending from South America to Papua New Guinea. Peaks of 15 mm/day occur throughout this band. Additional heavy rain associated with the Indian summer monsoons is apparent in the Bay of Bengal. The other major feature of the global rainfall maps is the extremely dry areas associated with the large semipermanent anticylones in the southeast Pacific and southeast Atlantic. These areas are essentially void of rain (*R* < 0.3 mm/day).

### c. Comparison to other satellite climatologies

We now compare our rainfall estimates (hereafter WS) to two other emission-based rain climatologies: Spencer (1993, hereafter MSU), and Wilheit et al. (1991, hereafter WCC). The MSU rain rates are inferred from the 50.3-GHz *T*_{B} observations taken by the Microwave Sounding Unit (MSU). The WCC rain rates are inferred from the SSM/I *T*_{B} observations. The same period of record (1991–94) is used from these datasets. Figure 9 compares the three estimates of the annual zonally averaged rainfall. In general, the three rainfall estimates are similar, but there are some notable differences. We first note that above 50°N and below 55°S, the MSU rain data are contaminated by sea ice (see below). This explains the upturn at the two ends of the MSU curvein Fig. 9. In the ITCZ, the WS, MSU, and WCC reach maximum values of 8.1, 7.4, and 6.9 mm/day, respectively. This represents about a 15% difference between the highest estimate (WS) and the lowest estimate (WCC). In the extratropics storm track regions, the situation changes. Here the WS rainfall is the lowest and MSU is the highest. Very close agreement is seen in the very dry areas associated with the semipermanent anticylones.

Figure 10 shows color-coded global maps of the MSU minus WS rainfall and the WCC minus WS rainfall. To compute these differences, the rainfall is averaged over the four years (1991–94) and then smoothed to a spatial resolution of about 300 km. The largest differences are seen between the MSU and WS. The MSU produces more rainfall in the downstream portions of the extratropical storm tracks and less rainfall over most portions of the Tropics, particularly in the tropical west Pacific. Comparisons of Fig. 10 to SSM/I retrievals of cloud water (not shown) suggest that the MSU–WS differences might be related to cloud water. Areas where the MSU–WS difference is significantly positive (negative) are moderately correlated with areas having a relatively high (low) cloud amount as compared to the rainfall. One example is the downstream portions of the extratropical storm tracks where there is significant cloud coverage but relatively little rain. In these regions the MSU rainfall is about 2 mm/day higher than WS. In contrast, along most of the ITCZ, the cloud content is relatively small compared to the heavy rain, the MSU rainfall is about 2–3 mm/day lower than WS. An interesting ocean area is seen just west of Central America and Columbia. The north (south) part of this area shows large negative (positive) MSU–WS differences. An analysis of SSM/I retrievals shows moderately heavy rain and relatively small cloud contents in the north andthe reverse situation in the south, which is the same correlation as seen in the storm tracks and the ITCZ. The correct partitioning of cloud and rain water is a problem for both MSU and SSM/I. As pointed out by Spencer (1993), the hypersensitivity of the MSU 50.3-GHz channel to both cloud water and rainwater makes the MSU unable to distinguish between the two. We have somewhat more confidence in the SSM/I rainfall because the frequencies of 19.3 and 37 GHz are less sensitive to cloud water, and we have attempted to do a cloud versus rain partitioning. This confidence is bolstered by the fact that the cloud to rain ratio derived from SSM/I seems realistic. It is a minimum just off the east coasts of the continents where baroclinic wave activity is the strongest. Then this ratio increases eastward across the ocean basins, consistent with weaker wave activity.

The difference map between WS and WCC shows better agreement. The major difference is in tropical areas of heavy rain, where the WS is about 2 mm/day higher. In the extratropical storm tracks, the WCC is typically about 1 mm/day higher. In the dry areas, all three rain estimates (WS, MSU, and WCC) agree well. We find no obvious correlation between the WS–WCC difference and other parameters, except for the rainfall itself. When the rain is very heavy, WS tends to be higher than WCC.

Note that in the MSU–WS figure, the red areas in the Sea of Okhotsk, the Bering Sea, Hudson Bay, Labrador Sea, and off Antarctica are sea ice contamination in the MSU rain product. A very small amount of ice contamination is also seen in the WCC product just north of Japan.

## 9. Conclusions

A new method for the physical retrieval of rain rates and the effective radiating temperature *T*_{U} from the SSM/I has been presented. The method is part of a unified ocean parameter retrieval algorithm that also diagnoses total integrated water vapor, cloud water, and wind speed. We find that the water vapor retrievals maintain reasonably good accuracy when there is rain in the field of view. The rms difference between the SSM/I water vapor retrieval and radiosondes is about 5 mm for rain rates from 1 to 15 mm/h and the error is uncorrelated with the rain rate.

As expected, *T*_{U} exhibits a strong depression relative to the mean air temperature for moderate to heavy rain. This depression is due to 1) radiative scattering from large raindrops and ice and 2) the fact that most of the radiation is coming from the cold top part of the rain cloud. For the heaviest rain, the *T*_{U} depression is −10 K and −20 K for 19 and 37 GHz, respectively.

The spectral signature of the retrieved liquid water transmittance *τ*_{L} shows that the ratio of the 37-GHz to 19-GHz liquid water absorption is, on the average, about 40% lower than predicted by Mie theory for moderateto heavy rain. We attribute this difference to the beamfilling effect, which we parameterize in terms of the normalized rms variation *β* of the liquid water absorption *A*_{L}. To correct for this effect, the 37-GHz to 19-GHz liquid water absorptions are increased until the Mie ratio is realized. Globally, we find *β* ≈ 0.85, which is somewhat less than that for an exponential pdf.

In the Tropics, we find using the freezing level, which is about 5 km, to specify *H* results in tropical rain rates that appear to be too low when compared with otherrainfall climatologies. To correct the low bias, we use a value of *H* ∼ 3 km in the Tropics. This adjustment is probably compensating for two processes: 1) the existence of warm rain for which the rain layer does not extend to the freezing level and 2) very heavy rain for which the 19-GHz channels saturate. Thus *H* plays the role of the one tuning parameter in the algorithm.

Global rain rates are produced for the 1991–94 period from two SSM/Is on board the *F10* and *F11* satellites. We find that on a global basis 6% of the SSM/I observations detect measurable rain rates of *R* > 0.2 mm/h. Globally, the average rainfall over the oceans is about 2.9 mm/day, and in the Tropics (20°N–20°S) it is 3.9 mm/day. Zonal averages and global maps of the retrieved rain rates show structures that are similar to those in previously published rain climatologies (Spencer 1993; Wilheit et al. 1991). However, some differences between the SSM/I and MSU rain rates are apparent and seem to be related to nonprecipitating cloud water.

Our rain retrieval technique could probably be improved by including the SSM/I 85-GHz channels. These channels are very sensitive to radiative scattering by ice and may provide the means to better identify areas of heavy rain exceeding 15 mm/h.

There still remains the problem of absolutely calibrating the rain algorithm. The lack of good quality in situ rain measurements over the oceans has been a major source of difficulty for all satellite-based rainfall estimation techniques, and it is still not clear how to best deal with the calibration problem. Hopefully future programs such as TRMM and the Precipitation Intercomparison Project will contribute to the better calibration of rainfall derived from satellites.

## Acknowledgments

This research was supported by NASA’s Oceans Program and EOS Program under Contracts NASW-4714 and NAS5-32594. We are thankful to the Defense Meteorological Satellite Program for making the SSM/I data available to the civilian community.

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