Height profiles of water vapor

To simplify the final form of the equations, the following conventions are used. The “measured” radar reflectivity factor Z_m at radar frequency f and range r is defined in terms of the radar return power P_r by

where 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_m by

where k_p, k_c, and k_υ 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 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_υ) by

We define 10 log₁₀ of Z_m and Z by

Throughout the paper we assume that measurements of P_r, or equivalently Z_m, are made at frequencies ( f_l, f_c, f_u), where f_c is taken at the center of the water vapor absorption line at 22.235 GHz and ( f_l, f_u) are chosen such that f_l < f_c < f_u and

Using the water vapor model described by Ulaby et al. (1981) and Waters (1976), the values of ( f_l, f_u) that satisfy (6) are graphed in Fig. 1 as a function of the fractional bandwidth Δf, which we define by

If Δf is specified, then ( f_l, f_u) 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 ( f_l, f_u) = (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.

Taking the difference of measured reflectivity factors (dB) at the upper and lower frequencies, Z̃_m( f_u, r) − Z̃_m( f_l, r), and using (6) and the above definitions gives

The 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_l) that is approximately independent of the precipitation and cloud attenuation, we assume that

Equation (12) states that the differential attenuation arising from precipitation and cloud between frequencies ( f_c, f_l) is equal to some fraction γ of the differential attenuation between frequencies ( f_u, f_l). If the precipitation and cloud attenuation is directly proportional to frequency, 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.

Multiplying (10) by γ and subtracting (11) from the resulting equation yields

where

The bias term E₂ was discussed above where it was noted that E₂ = 0 if (13) holds. The bias term E₁ 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 E₁ ≅ 0.

To estimate A_υ( f_c, f_l) from measured data, we assume that the error terms are zero so that

Path-integrated water vapor estimates from airborne/spaceborne platforms

Before presenting an error analysis of (17), we derive equations similar to (10) and (14) by using the backscattered powers from the surface. The equations below are obviously relevant only to a down-looking geometry. On the other hand, unlike (10) and (14), they are applicable in rain-free conditions as well as raining conditions. The measured or apparent radar return power from the surface P_sm can be written as (Kozu 1995)

where σ⁰ is the normalized radar cross section of the surface (unitless) and C_s is the radar constant for surface scattering, which depends on incidence angle, antenna gain, and frequency. The function g(r_s) represents the range dependence, which is equal to r²_s for beam-limited conditions (nadir and near-nadir incidence) and r³_s for pulse-limited conditions (off-nadir incidence). The radar return power from the surface P_s, which would be measured in the absence of atmospheric attenuation, can be written as

As 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 respectively

where

One difference between the rain and surface scattering equations above is that the E_1s error depends on the nature of backscattering not from the hydrometeors, as in E₁, but from the surface. This becomes apparent if (23) is written as

where 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_l to f_u, the bracketed term in (25) is zero for any γ. Moreover, if γ is set to γ_Ray, given by (13), then it can be shown that this term is zero if σ̃⁰ changes linearly with frequency. Another difference between E₁ 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_s) 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).

Description of the simulation

To analyze errors in the estimate of differential water vapor given by (17) we construct a simple stratiform storm model derived from disdrometer-measured raindrop size distributions (Meneghini et al. 2003). For each size distribution we compute the median mass diameter D₀ (mm) and number concentration N_t (m⁻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):

where

and Γ is the gamma function given by

The 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_m 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 f = f_l, f_c, f_u are computed from the following set of equations:

where the specific attenuations (dB km⁻¹) are given by

with

Here P is the pressure (hPa), T is temperature (K), ρ_υ is the water vapor density (g m⁻³), c₀ is the speed of light (mm s⁻¹), m is the complex index of refraction of the (cloud) water, M_c is the cloud water content (g m⁻³), and σ_b( f, D) and σ_e( f, D) are respectively the backscattering and extinction cross sections (mm²) of a sphere of diameter D at frequency f. The expression for k_υ above is taken from results of Waters (1976 and Ulaby et al. (1981) for 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.

Apart from the bias terms E₁ and E₂ 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_m. For a square-law detector, where the output of each sample is proportional to the return power, the deterministic quantity Z_m (in linear units) is replaced by a Gaussian random variable with mean Z_m and standard deviation Z_mn^−1/2, where n is the number of independent samples. As an alternative, Z̃_m can be replaced by (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_c) over approximately 400 “observations” (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_l), are shown in the second panel from the top. For Figs. 2 –8, a 20% bandwidth has been chosen so that f_u = 24.694 GHz and f_l = 20.246 GHz. The magnitude of Z̃_m( f_u) − Z̃_m( f_l) 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 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_l) field shown in the panel above; as already noted, the bias terms have been exaggerated by choosing 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 D₀. 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_l) versus D₀. 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_l) shown in the middle panel of Fig. 3. Because the “shape” parameter μ of the size distribution has been fixed, Z̃( f_u) − Z̃( f_l) is a function only of D₀ 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_t as well as D₀, as seen by the variability in this quantity for fixed D₀. 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 D₀ 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 D₀ values up to about 2 mm and is slightly positive above 2 mm. This bias is the sum of E₁ and E₂ displayed in the bottom two panels of Fig. 4. The behavior of E₁ is determined almost exclusively by the median mass diameter; on the other hand, variability at a fixed D₀, indicating dependence on number concentration, can be seen in the E₂ term for large D₀ values. The results imply that if the rain estimation problem can be solved in parallel (or iteratively) with the water absorption estimate, the D₀ (and N_t) 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 Z̃( f_u) − Z̃( f_l), shown in the middle panel of Fig. 5, is a monotonically decreasing function of D₀ (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_l), shown in the top panel, is primarily determined by 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 E₂ is approximately zero while E₁ is small up to moderate D₀ values and is positive for large D₀.

In the melting layer, E₂ remains relatively small; however, E₁ is significant and changes from gate to gate. Plots of E₁ versus D₀ 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 D₀ 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_l) yields the differential specific water vapor absorption k_υ( f_c) − k_υ( f_l). To reduce the variability in A_υ( f_c, f_l), 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 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_υ 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⁻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_l) computed over the set of profiles. On the top right panel are the estimated (dashed line) and assumed mean values of k_υ( f_c) − k_υ( f_l) using the averaging procedures described above. Biases in A_υ( f_c, f_l) 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 k_υ( f_c) − k_υ( f_l) estimate reveals a large negative bias associated with scattering in the melting layer. This is directly related to the E₁ 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_l) and k_υ( f_c) − k_υ( f_l) are shown in the bottom panels. Because the assumed pressure and temperature profiles were randomized, the true values of A_υ( f_c, f_l) and k_υ( f_c) − k_υ( f_l) are also random, with standard deviations given by the solid lines. The standard deviations of the estimated profiles of A_υ( f_c, f_l) and k_υ( f_c) − k_υ( f_l) 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 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_l) at and about the melting layer produce large negative biases in ρ_υ 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.

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_υ, 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 ρ_υ 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 k_υ( f_c) − k_υ( f_l), ρ_υ, 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 n = 16 000 and 30% bandwidth, the standard deviations in ρ_υ and RH decrease to about 16% in the lowest 3 km but exhibit a maximum relative error of about 36% in the melting layer.

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.