## Introduction

Many remote sensing techniques are being used experimentally or operationally to estimate atmospheric water vapor. These include radiometers at microwave (Westwater et al. 2001; Solheim et al. 1998; Schultz et al. 1993; Schlüssel and Bauer 1993), millimeter-wave (Rosenkranz 2001; Wang et al. 1995; Wilheit 1990) and IR (Schmetz and Turpeinen 1988; Hagen et al. 2004) wavelengths; Raman lidar and differential absorption lidar (DIAL) (Whiteman 2003; Behrendt et al. 2002; Brassington 1982); and GPS-based techniques (Bevis et al. 1992; Alber et al. 1997). With a few possible exceptions (e.g., Liljegren 2004), most of the instruments and retrieval methods are not applicable in the presence of rain. In fact, the measurement of water vapor in rain is difficult even with conventional rawinsondes. Nevertheless, its estimation is important in microphysical studies and in the evaluation of cloud models.

Although little work has been done on water vapor estimation using radar, an exception is the work of Tian et al. (2004) who have analyzed dual-frequency (10 and 94 GHz) airborne Doppler radar data. By deriving the hydrometeor size distribution from the Doppler velocities and then modifying the radar reflectivity factors *Z* to account for Mie scattering and hydrometeor attenuation, the attenuation from cloud and gases can be inferred from the difference in the modified *Z* values at the two frequencies. In this paper, a three-frequency radar is studied for which one of the frequencies is taken at the 22.235-GHz line center with the others chosen at a lower and higher frequency with equal water vapor absorption coefficients.

Equations for estimates of water vapor absorption and precipitation and cloud attenuation are given in section 2, followed by a simulation and error analysis of the water vapor absorption and density estimates in section 3. Discussions on the algorithm and radar implementation are given in section 4.

## Equations for precipitation and water vapor path attenuation

### Height profiles of water vapor

*Z*at radar frequency

_{m}*f*and range

*r*is defined in terms of the radar return power

*P*bywhere

_{r}*C*is the radar constant and |

*K*|

_{w}^{2}is the dielectric factor of water, which, by convention, is taken to be equal to its approximate value (0.93) for frequencies between 3 and 10 GHz and for temperatures between 0° and 20°C (Battan 1973). The unattenuated radar reflectivity factor, or simply radar reflectivity factor,

*Z*is related to

*Z*bywhere

_{m}*k*,

_{p}*k*, and

_{c}*k*are the specific attenuations from precipitation, cloud water, and water vapor, respectively, and where the precipitation may include rain, snow, and mixed-phase hydrometeors. Contributions from oxygen and cloud ice should be added to (2). If the units of

_{υ}*k*are taken to be decibels per kilometer, then

*c*= 0.2 × ln10; if the units of

*k*are in inverse kilometers, then

*c*= 2. In this paper, we use the former. It is also convenient to define two-way differential attenuations (dB) to range

*r*for the precipitation and cloud (

*A*

_{pc}) and for the water vapor (

*A*) byWe define 10 log

_{υ}_{10}of

*Z*and

_{m}*Z*byThroughout the paper we assume that measurements of

*P*, or equivalently

_{r}*Z*, are made at frequencies (

_{m}*f*,

_{l}*f*,

_{c}*f*), where

_{u}*f*is taken at the center of the water vapor absorption line at 22.235 GHz and (

_{c}*f*,

_{l}*f*) are chosen such that

_{u}*f*<

_{l}*f*<

_{c}*f*andUsing the water vapor model described by Ulaby et al. (1981) and Waters (1976), the values of (

_{u}*f*,

_{l}*f*) that satisfy (6) are graphed in Fig. 1 as a function of the fractional bandwidth Δ

_{u}*f*, which we define byIf Δ

*f*is specified, then (

*f*,

_{l}*f*) can be found from the following approximations:For the calculations given later in the paper, percent bandwidths of 10%, 20%, and 30% are used, where the frequency pairs (GHz) are given respectively by (

_{u}*f*,

_{l}*f*) = (21.248, 23.420), (20.246, 24.694), and (19.409, 26.079). Note also that for percent bandwidths of 20% or less a “differential frequency” implementation of the radar might be feasible. By this term, we mean that a single radar antenna and transceiver would provide measurements of the radar returns at the three frequencies. We will return to this issue later in the paper.

_{u}*Z̃*(

_{m}*f*,

_{u}*r*) −

*Z̃*(

_{m}*f*,

_{l}*r*), and using (6) and the above definitions givesThe difference between the measured reflectivity factors (dB) at the center and lower frequencies,

*Z̃*(

_{m}*f*,

_{c}*r*) −

*Z̃*(

_{m}*f*,

_{l}*r*), can be written similarly:To obtain an estimate of the two-way differential water vapor absorption

*A*(

_{υ}*f*,

_{c}*f*) that is approximately independent of the precipitation and cloud attenuation, we assume thatEquation (12) states that the differential attenuation arising from precipitation and cloud between frequencies (

_{l}*f*,

_{c}*f*) is equal to some fraction

_{l}*γ*of the differential attenuation between frequencies (

*f*,

_{u}*f*). If the precipitation and cloud attenuation is directly proportional to frequency,

_{l}*γ*is simply the ratio of frequency differences:and (12) is satisfied exactly. However, because of the frequency dependence of the dielectric constant of water and non-Rayleigh scattering,

*γ*depends on the hydrometeor size distribution so that (12) will generally be incorrect. As will be shown in the results below, the magnitude of the error depends on the nature of the backscattering medium (rain, snow, or mixed phase), the parameters of the hydrometeor size distribution, the fractional bandwidth of the radar frequencies that are used, and the choice of

*γ*.

*γ*and subtracting (11) from the resulting equation yieldswhereThe bias term

*E*

_{2}was discussed above where it was noted that

*E*

_{2}= 0 if (13) holds. The bias term

*E*

_{1}is a function of the reflectivity factors at the three frequencies. Unlike the measured reflectivity factors on the left-hand side of (14), they are not measurable and are not easily estimated unless the precipitation retrieval problem is solved in parallel. Similar considerations apply to this term as apply to (12): if the backscattering is Rayleigh, the reflectivity factors are approximately frequency independent and

*E*

_{1}≅ 0.

*A*(

_{υ}*f*,

_{c}*f*) from measured data, we assume that the error terms are zero so that

_{l}### Path-integrated water vapor estimates from airborne/spaceborne platforms

*P*

_{sm}can be written as (Kozu 1995)where

*σ*

^{0}is the normalized radar cross section of the surface (unitless) and

*C*is the radar constant for surface scattering, which depends on incidence angle, antenna gain, and frequency. The function

_{s}*g*(

*r*) represents the range dependence, which is equal to

_{s}*r*

^{ 2}

_{s}for beam-limited conditions (nadir and near-nadir incidence) and

*r*

^{ 3}

_{s}for pulse-limited conditions (off-nadir incidence). The radar return power from the surface

*P*, which would be measured in the absence of atmospheric attenuation, can be written asAs before, we define the following quantities:A comparison of (18)–(20) with (1), (2), and (5) shows that for the case of surface scattering (10) and (14) become respectivelywhereOne difference between the rain and surface scattering equations above is that the

_{s}*E*

_{1s}error depends on the nature of backscattering not from the hydrometeors, as in

*E*

_{1}, but from the surface. This becomes apparent if (23) is written aswhere the notation “+ · · ·” is used to indicate additional terms that are functions of the radar constants and range. Because these terms are known, or presumed to be known, they can be accounted for in an offset term. A comparison of the bracketed term in (25) with (15) shows that the approximation for the surface scattering is analogous to that used for the precipitation, with the normalized surface cross sections replacing the radar reflectivity factors. If the surface cross section is constant over the frequency span from

*f*to

_{l}*f*, the bracketed term in (25) is zero for any

_{u}*γ*. Moreover, if

*γ*is set to

*γ*

_{Ray}, given by (13), then it can be shown that this term is zero if

*σ̃*

^{0}changes linearly with frequency. Another difference between

*E*

_{1}and

*E*

_{1s}is that an estimate for the latter quantity can be obtained from surface scattering measurements in regions of low path-integrated water vapor. This is analogous to the “surface-reference technique,” in which measurements of the surface return in rain-free areas are used to estimate path attenuation in the presence of rain (Tian et al. 2002; Meneghini et al. 2004). In particular, an estimate for the differential path attenuation from cloud and hydrometeors follows directly from (21), where the term

*P̃*(

_{s}*f*,

_{u}*r*) −

_{s}*P̃*(

_{s}*f*,

_{l}*r*) is approximated by surface returns measured in clear conditions. Because the main subject of the paper is the feasibility of height-profiled water vapor estimates in rain, we focus the error analysis on the range-dependent estimate of differential water vapor attenuation given by (17).

_{s}## Error analysis of water vapor profiling retrievals

### Description of the simulation

*D*

_{0}(mm) and number concentration

*N*(m

_{t}^{−3}). To complete the specification, we assume that the drop diameter distribution

*N*(

*D*) (mm

^{−1}m

^{−3}) follows a gamma distribution with a fixed shape parameter

*μ*equal to 2 (Ulbrich 1983):whereand Γ is the gamma function given byThe storm structure is assumed to be composed of a layer of snow from 4 to 5 km above the surface, a transition region of mixed-phase hydrometeors from about 3.5 to 4 km, and a rain layer from about 3.5 km to the surface. The fractional meltwater in the melting layer is prescribed as a function of the initial snow size, mass density, and distance below the 0° isotherm (Yokoyama and Tanaka 1984). The effective dielectric constant of the mixed-phase particles is given by the effective medium approximation (Bohren and Battan 1980). For the results shown here, the snow mass density is fixed and is equal to 0.2 g cm

^{−3}. The location and density of the cloud water are also included although this contribution is not critical to the behavior of the water vapor retrievals. The temperature lapse rate is taken to be constant and is equal to 6°C km

^{−1}so that, with the 0°C isotherm at 4 km, the surface temperature is 24°C. To focus on the bias and random errors of the estimate in this set of examples (Figs. 2 –9), the relative humidity is taken to be constant from the surface to a height of 5 km and is equal to 80%. From this simple storm model, approximately 400 range profiles of

*Z*at the three frequencies can be generated, where each profile is derived from measured raindrop size distribution parameters and consists of the returns from the snow, mixed-phase, and liquid hydrometeors and includes effects of absorption from cloud water, water vapor, and molecular oxygen. Note also that the range resolution is taken to be 125 m so that the profile consists of 40 gates or bins. The simulated radar measurements at

_{m}*f*=

*f*,

_{l}*f*,

_{c}*f*are computed from the following set of equations:where the specific attenuations (dB km

_{u}^{−1}) are given bywithHere

*P*is the pressure (hPa),

*T*is temperature (K),

*ρ*is the water vapor density (g m

_{υ}^{−3}),

*c*

_{0}is the speed of light (mm s

^{−1}),

*m*is the complex index of refraction of the (cloud) water,

*M*is the cloud water content (g m

_{c}^{−3}), and

*σ*(

_{b}*f*,

*D*) and

*σ*(

_{e}*f*,

*D*) are respectively the backscattering and extinction cross sections (mm

^{2}) of a sphere of diameter

*D*at frequency

*f*. The expression for

*k*above is taken from results of Waters (1976 and Ulaby et al. (1981) for

_{υ}*f*< 100 GHz. In (32) the frequency is to be specified in gigahertz; in all of the other equations the frequency is to be expressed in hertz. Following Ulaby et al. (1981), the contribution from molecular oxygen is also included.

The error analysis consists of computing the right-hand side of (17) using (29)––(35) and comparing the estimated two-way differential vapor absorption profile with the assumed profile as calculated from (4), (32), and (35). To compute the right-hand side of (17), a value of *γ* is required. Although the approximation from (13) can be used, it was found by trial and error that somewhat smaller values reduce the bias. For the results shown in the paper, we use *γ* = 0.44 (10% bandwidth), *γ* = 0.42 (20%), and *γ* = 0.39 (30%) instead of the results from (13), which give *γ*_{Ray} = 0.455 (10% bandwidth), *γ*_{Ray} = 0.447 (20%), and *γ*_{Ray} = 0.424 (30%). An exception to these assumptions are the cases presented in Fig. 2 and Fig. 8, explained below, in which we have chosen *γ* = 0.39 for the 20% bandwidth case to emphasize the bias in the estimate; that is, the bias is qualitatively the same but smaller when the value *γ* = 0.42 is used.

*E*

_{1}and

*E*

_{2}described in the previous section, an important additional source of error arises from the finite number of independent samples used to estimate the radar return power and

*Z*. For a square-law detector, where the output of each sample is proportional to the return power, the deterministic quantity

_{m}*Z*(in linear units) is replaced by a Gaussian random variable with mean

_{m}*Z*and standard deviation

_{m}*Z*

_{m}n^{−1/2}, where

*n*is the number of independent samples. As an alternative,

*Z̃*can be replaced by (

_{m}*Z̃*+

_{m}*g*), where

*g*is approximated by a zero-mean Gaussian with variance (Doviak and Zrnić 1993; Bringi et al. 1983)For Figs. 2 –8,

*n*is taken to be a large number (64 000) so that the bias terms can be examined without large background variability from finite sampling.

### Error estimate of differential water vapor attenuation

The top panel of Fig. 2 shows simulations of *Z̃ _{m}*(

*f*) over approximately 400 “observations” (

_{c}*x*axis) over the 5-km storm height (

*y*axis). The results clearly show a bright band, corresponding to returns from the partially melted snow, in the ranges just below the 0°C isotherm at 4 km. Because the radar is assumed to be above the storm viewing along nadir, effects of attenuation are evident by the reduction in the measured reflectivity factor as the penetration depth increases. Results for the measured differential reflectivity for the upper and lower frequencies,

*Z̃*(

_{m}*f*) −

_{u}*Z̃*(

_{m}*f*), are shown in the second panel from the top. For Figs. 2 –8, a 20% bandwidth has been chosen so that

_{l}*f*= 24.694 GHz and

_{u}*f*= 20.246 GHz. The magnitude of

_{l}*Z̃*(

_{m}*f*) −

_{u}*Z̃*(

_{m}*f*) depends on the drop size distribution, the phase of the hydrometeors (solid, liquid, or partially melted), and the cumulative attenuation by precipitation and cloud. However, it is independent of water vapor because

_{l}*k*(

_{υ}*f*) =

_{u}*k*(

_{υ}*f*).

_{l}The bottom two panels of Fig. 2 show the estimated and assumed values of the two-way differential water vapor absorption *A _{υ}*(

*f*,

_{c}*f*;

_{l}*r*), where the estimated value is computed from the right-hand side of (17). Despite the large number of independent samples (64 000), variations caused by the finite number of samples are still evident. Superimposed on the fluctuations are biases that are seen to be well correlated with decreases in the

*Z̃*(

_{m}*f*) −

_{u}*Z̃*(

_{m}*f*) field shown in the panel above; as already noted, the bias terms have been exaggerated by choosing

_{l}*γ*= 0.39 for this plot.

To understand how the bias in *A _{υ}*(

*f*,

_{c}*f*;

_{l}*r*) depends on the parameters of the precipitation, it is useful to examine some of the relevant quantities as functions of the median mass diameter

*D*

_{0}. The results in Figs. 3 and 4 are taken from the lowest gate, just above the surface, so that the attenuations are approximately the same as the total path attenuations. Shown in the top panel of Fig. 3 are values of

*Z̃*(

_{m}*f*) −

_{u}*Z̃*(

_{m}*f*) versus

_{l}*D*

_{0}. Variations in this quantity originate from two sources: non-Rayleigh scattering and attenuation effects from the precipitation and cloud. The non-Rayleigh source of variability can be seen in the results for

*Z̃*(

*f*) −

_{u}*Z̃*(

*f*) shown in the middle panel of Fig. 3. Because the “shape” parameter

_{l}*μ*of the size distribution has been fixed,

*Z̃*(

*f*) −

_{u}*Z̃*(

*f*) is a function only of

_{l}*D*

_{0}and is independent of the number concentration. A scatterplot of the differential path attenuation from cloud and precipitation

*A*

_{pc}(

*f*,

_{u}*f*;

_{l}*r*) is shown in the bottom panel. Note that this quantity depends on

*N*as well as

_{t}*D*

_{0}, as seen by the variability in this quantity for fixed

*D*

_{0}. Because the cloud liquid water was taken to be zero for this simulation, the attenuation arises entirely from the hydrometeors. The addition of an integrated cloud water content of 1 kg m

^{−2}for all cases yields an increase in

*A*

_{pc}(

*f*,

_{u}*f*;

_{l}*r*) of about 0.4 dB (not shown) but has a negligible effect on the bias errors shown in the lower two panels of Fig. 4.

The top panel of Fig. 4 shows the estimated (times signs) and assumed values (solid line) of the differential water vapor absorption *A _{υ}*(

*f*,

_{c}*f*;

_{l}*r*). The variability in the estimates for constant

*D*

_{0}is the result of finite sampling and reduces to zero as

*n*goes to infinity. In addition to the variability there is a bias that is negative for

*D*

_{0}values up to about 2 mm and is slightly positive above 2 mm. This bias is the sum of

*E*

_{1}and

*E*

_{2}displayed in the bottom two panels of Fig. 4. The behavior of

*E*

_{1}is determined almost exclusively by the median mass diameter; on the other hand, variability at a fixed

*D*

_{0}, indicating dependence on number concentration, can be seen in the

*E*

_{2}term for large

*D*

_{0}values. The results imply that if the rain estimation problem can be solved in parallel (or iteratively) with the water absorption estimate, the

*D*

_{0}(and

*N*) values, as estimated from the precipitation algorithms can, in principle, be used to correct for biases in the water vapor retrieval. Although a quantitative description of algorithms of this type is beyond the scope of the paper, a qualitative description of a possible approach is outlined in section 4. The results in Figs. 3 and 4 are taken from the range gate in rain at the bottom of the column. For the case shown in Figs. 5 and 6, the range is taken to be just above the melting layer at a snow depth of about 0.9 km. Comparison of the results of Fig. 5 and Fig. 3 show a number of obvious differences between the two cases. The differential reflectivity factor in snow

_{t}*Z̃*(

*f*) −

_{u}*Z̃*(

*f*), shown in the middle panel of Fig. 5, is a monotonically decreasing function of

_{l}*D*

_{0}(the median mass diameter of the melted snow); moreover, because the attenuation through snow is small (bottom panel), the behavior of the measured differential reflectivity factor

*Z̃*(

_{m}*f*) −

_{u}*Z̃*(

_{m}*f*), shown in the top panel, is primarily determined by

_{l}*Z̃*(

*f*) −

_{u}*Z̃*(

*f*).

_{l}The estimated and true values of the water vapor absorption to this range are shown in the top panel of Fig. 6. Notice that the magnitude of the fluctuations is essentially the same as before but, because the water vapor absorption to this range is small, the relative error caused by the finite sampling is much higher. On the other hand, the biases in the estimate are small. This can be seen in the bottom panels where *E*_{2} is approximately zero while *E*_{1} is small up to moderate *D*_{0} values and is positive for large *D*_{0}.

In the melting layer, *E*_{2} remains relatively small; however, *E*_{1} is significant and changes from gate to gate. Plots of *E*_{1} versus *D*_{0} are shown in Fig. 7 for four range gates within the melting layer. For gates 13 and 15, the sign of the bias is negative for small *D*_{0} and positive for larger values. As will be shown later, this bias has a strong effect on the water vapor retrievals in and about the melting layer.

Before looking at statistics of the *A _{υ}*(

*f*,

_{c}*f*;

_{l}*r*) estimates, it is instructive to view individual retrievals. In Fig. 8, five arbitrarily chosen profiles of

*A*(

_{υ}*f*,

_{c}*f*;

_{l}*r*), from the results of Fig. 2, are represented by the thin solid lines. The true (assumed) profile is represented by the heavy solid line. The results for the nearly 400 profiles are indicated by the plus signs; that is, at each range gate, approximately 400 data points are displayed corresponding to the estimated values of

*A*(

_{υ}*f*,

_{c}*f*;

_{l}*r*) at range

*r*. The behavior of the individual profiles indicate that, even for a very large number of independent samples, the fluctuations in

*A*(

_{υ}*f*,

_{c}*f*;

_{l}*r*) with range can be substantial. Computing the differential specific absorption without smoothing in range or further averaging in space or time would clearly lead to large errors.

The results shown in Fig. 8 are from the single simulation shown in Fig. 2. To obtain estimates of the mean and standard deviation as a function of range, 200 simulations of the retrievals were made. Note that from realization to realization the set of drop size distributions is fixed and only the *Z̃ _{m}* are changed to simulate the effects of finite sampling. The results are shown in Fig. 9 for bandwidths of 10%, 20%, and 30%. Unlike the previous results, a more realistic number of independent samples (

*n*= 4000) is used. For each panel, the mean (times signs) and 2 times the standard deviation of the estimate (vertical height of error bar), as well as the true differential absorption

*A*(

_{υ}*f*,

_{c}*f*;

_{l}*r*), are plotted as a function of range. As in Fig. 8, gate 40 represents the range gate just above the surface.

The results show that the standard deviation is approximately independent of range so that as the path absorption increases with radar range, the fractional standard deviation of the estimate decreases. In snow, the bias in *A _{υ}*(

*f*,

_{c}*f*;

_{l}*r*) is small. In the rain, however, the bias for the 30% case (Fig. 9, top) is relatively large. The bias decreases in going to smaller bandwidths so that the 10% case (Fig. 9, bottom) shows the smallest bias. On the other hand, the fractional standard deviation is largest for the 10% case and decreases in going to 20% and again to 30% bandwidth.

### Error estimates of water vapor retrievals

A range derivative of the differential path attenuation *A _{υ}*(

*f*,

_{c}*f*) yields the differential specific water vapor absorption

_{l}*k*(

_{υ}*f*) −

_{c}*k*(

_{υ}*f*). To reduce the variability in

_{l}*A*(

_{υ}*f*,

_{c}*f*), a five-gate moving average is done, followed by a differencing over successive five-gate intervals. After normalizing by the range difference, this procedure gives

_{l}*k*(

_{υ}*f*,

_{c}*r*) −

_{j}*k*(

_{υ}*f*,

_{l}*r*);

_{j}*j*= 3, . . . , 38, where estimates at the two lowermost and uppermost range gates are discarded. From the estimate of differential specific absorption and model temperature (with lapse rate as before of 6°C km

^{−1}) and pressure profiles, the water vapor density and relative humidity are computed. The “true” temperature and pressure profiles are taken to be Gaussian random variables with a mean given by the model profiles with standard deviations of 1 K and 2 hPa, respectively, as derived from the results of 200 soundings in the South China Sea. Note that the true

*ρ*field is computed from the assumed RH and randomly varying temperature and pressure profiles. The simulated storm parameters are similar to those used previously with the exceptions that the integrated cloud water content is taken to be 1 kg m

_{υ}^{−2}and the relative humidity is assumed to be 100% above the 0° isotherm, decreasing linearly to 70% at the surface. Results are shown only for the 20% bandwidth case with the number of samples

*n*equal to 16 000 samples. For

*n*= 64 000, the rms errors in the water vapor and RH retrievals in the lowest 3 km are approximately halved; for

*n*= 4000, the errors are approximately doubled.

Shown in the top left panel of Fig. 10 are the estimated (dashed line) and assumed (solid line) mean values of *A _{υ}*(

*f*,

_{c}*f*) computed over the set of profiles. On the top right panel are the estimated (dashed line) and assumed mean values of

_{l}*k*(

_{υ}*f*) −

_{c}*k*(

_{υ}*f*) using the averaging procedures described above. Biases in

_{l}*A*(

_{υ}*f*,

_{c}*f*) can be seen to increase just below the melting layer at 4 km and assume nearly a constant value from about 3 km down to the surface. Examination of the

_{l}*k*(

_{υ}*f*) −

_{c}*k*(

_{υ}*f*) estimate reveals a large negative bias associated with scattering in the melting layer. This is directly related to the

_{l}*E*

_{1}bias term shown in Fig. 7. Because of the need to smooth the data, however, the influence of this error extends into the snow above and the rain below. Standard deviations of

*A*(

_{υ}*f*,

_{c}*f*) and

_{l}*k*(

_{υ}*f*) −

_{c}*k*(

_{υ}*f*) are shown in the bottom panels. Because the assumed pressure and temperature profiles were randomized, the true values of

_{l}*A*(

_{υ}*f*,

_{c}*f*) and

_{l}*k*(

_{υ}*f*) −

_{c}*k*(

_{υ}*f*) are also random, with standard deviations given by the solid lines. The standard deviations of the estimated profiles of

_{l}*A*(

_{υ}*f*,

_{c}*f*) and

_{l}*k*(

_{υ}*f*) −

_{c}*k*(

_{υ}*f*) are given by the dashed lines in the lower panels; the greater variability in estimated profiles is caused by the finite number of samples. As noted above, increasing

_{l}*n*by a factor of 4 reduces the estimated standard deviations by about a factor of 2, and vice versa.

Corresponding statistics for the water vapor density *ρ _{υ}* and RH are shown in Fig. 11, where it can be seen that negative biases in

*k*(

_{υ}*f*) −

_{c}*k*(

_{υ}*f*) at and about the melting layer produce large negative biases in

_{l}*ρ*and RH over the same range. Although the vertical extent of the bias can be reduced by choosing an averaging interval of fewer than five gates, this leads to higher standard deviations in the estimates. As in the previous plot, the standard deviations of the true and estimated fields are shown in the lower panels.

_{υ}The rms errors for *ρ _{υ}* and RH are presented in the top panels of Fig. 12, and the normalized errors are shown in the bottom panels, that is, the rms error divided by the true value. For

*ρ*, the relative errors in the bottom 3 km are between 20% and 28%; above 3 km, the errors become as large as 32%. Similar accuracies can be seen in RH. If the assumed temperature and pressure are taken to be equal to the model values, the relative errors decrease by about 4%. Increasing the sample number to 64 000 decreases the relative errors in

_{υ}*ρ*and RH to between 12% and 16% in the lowest 3 km. In the melting layer, however, the maximum relative error is about 25%. Increasing the bandwidth to 30% decreases the standard deviations in

_{υ}*k*(

_{υ}*f*) −

_{c}*k*(

_{υ}*f*),

_{l}*ρ*, and RH in the lowest 3 km but also produces higher negative biases and relative errors in and about the melting layer. For the case of

_{υ}*n*= 16 000 and 30% bandwidth, the standard deviations in

*ρ*and RH decrease to about 16% in the lowest 3 km but exhibit a maximum relative error of about 36% in the melting layer.

_{υ}More sophisticated averaging and smoothing techniques might serve to reduce the rms errors. As already pointed out, information on the median mass diameter of the hydrometeors, particularly in the melting layer, would serve to reduce the bias. Further investigations and, in the end, an analysis of measured data will be needed to address these issues.

## Discussion and summary

The approach used here has some similarities to the differential absorption lidar technique in that it uses frequencies on and off line center to estimate the strength of absorption. The fact that precipitation is the background scattering medium, however, implies that the differential attenuation by hydrometeors can easily be as large as water vapor absorption; moreover, non-Rayleigh backscattering effects at these frequencies can be comparable in magnitude to attenuation and absorption. By the use of three frequencies, we can take advantage of the fact that differential attenuation from precipitation and cloud is approximately an odd function with respect to the center frequency while the differential water vapor absorption is approximately an even function.

In deriving an expression for the differential vapor absorption, the critical assumption is that the differential attenuation from cloud and precipitation between the upper and lower frequencies can be expressed as a fraction of the differential attenuation between the center and lower frequencies. Results of the simulations show that this assumption leads to biases in the estimate of water vapor density that become larger as the bandwidth increases. Although the biases are small in snow, they are particularly strong in the melting layer and lead to large negative biases in the vapor density estimates in and around the melting layer.

Although the focus of the paper is estimation of water vapor, the approach also offers the potential of precipitation estimation. Because of the choice of lower and upper frequencies, the differential measured reflectivity factor *Z̃ _{m}*(

*f*) −

_{u}*Z̃*(

_{m}*f*) is a function only of the characteristics of the precipitation and cloud and is independent of water vapor. If the differential path attenuation from cloud and precipitation can be estimated, then

_{l}*Z̃*(

*f*) −

_{u}*Z̃*(

*f*) follows directly from (10). It is clear from the results of Fig. 3 and Fig. 5, however, that from

_{l}*Z̃*(

*f*) −

_{u}*Z̃*(

*f*) an estimate of the

_{l}*D*

_{0}can be obtained. Moreover, an estimate of the number concentration

*N*can be obtained from the radar equation (1). To start the procedure requires an initial or estimated path attenuation. One such estimate can be obtained from (21) by measuring the surface return powers in clear regions. However, as recently shown by Mardiana et al. (2004), the equations also can be solved iteratively without an independent path-attenuation estimate. In either case, the procedure yields estimates of the hydrometeor size distribution parameters in range. The

_{t}*D*

_{0}and

*N*values can be used, in turn, to improve the estimate of the differential water vapor absorption by providing estimates of the bias terms

_{t}*E*

_{1}and

*E*

_{2}. A drawback to the procedure is that the equation for

*N*, as derived from the radar equation, is a function

_{t}*k*(

_{υ}*f*) or

_{l}*k*(

_{υ}*f*). Although this term is usually small relative to the hydrometeor attenuation for the 20% and 30% bandwidth cases, it represents an additional error source in the precipitation retrieval problem. In principle, just as the hydrometeor size distribution parameters can be used to correct for biases in the water vapor retrieval, the water vapor retrieval can be used to account for this contribution in the precipitation retrieval. Whether iterating between solutions to the precipitation and water vapor equations will provide stable solutions is not clear, however.

_{u}Another way to view the technique is as a variation of the differential-frequency implementation. In this approach, a single antenna and transceiver are used to transmit and receive signals at more than one frequency. However, this requires a wideband power amplifier and a wideband antenna. Wideband power amplifiers with bandwidths up to 20% are now available in some frequency bands (J. Carswell 2004, personal communication). For broadband solid-state amplifiers, with high duty cycles but low peak powers, pulse-compression techniques can be used to achieve a fine range resolution. Averaging the data to a coarser vertical resolution may provide a sufficient number of independent samples to make the measurement technique feasible without excessive space or time averaging. One other way of increasing the effective number of independent samples is by the use of data whitening (Koivunen and Kostinski 1999; Torres and Zrnić 2003). Another requirement of the radar would be well-matched beamwidths at the three frequencies. Because the total differential path absorption for a 20% bandwidth is on the order of 1 dB, any mismatches in the radar resolution volumes will have a strong effect on accuracy, particularly in convective rain where vertical and horizontal gradients in the reflectivity field can be large. Horn-lens and parabolic antennas are inherently broad band and should be capable of good performance over a 20% bandwidth. Nevertheless, detailed calculations would be needed to assess the degree of beam matching needed relative to the gradients in the reflectivity field.

## Acknowledgments

We thank Professor Ramesh C. Srivastava for suggesting the use of multiple frequencies for estimates of water vapor.

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