## 1. Background and introduction

One of the goals of rain retrievals using measurements from a dual-frequency radar is to estimate two parameters of the raindrop size distribution (RSD) at each range gate. Apart from the important microphysical information that these parameters provide, they can be used to infer bulk parameters of the rain such as rain rate and liquid water content. For satellite-based measurements, the approach offers the possibility of linking microphysical processes to different climatological regimes, changes in surface type (land versus ocean), orography, and diurnal and seasonal variations over the globe. The basic idea that underlies the class of methods based on the dual-frequency ratio (DFR) is that the stronger non-Rayleigh backscattering at the higher frequency provides information on a characteristic size parameter of the distribution. In particular, the ratio of radar reflectivity factors at the two frequencies is independent of the number concentration of hydrometeors so that this ratio depends only on a characteristic size parameter of the distribution and, in the gamma parameterization of the RSD, the shape factor *μ*.

The basic equations for this approach were given in work related to the analysis of measurements from a dual-frequency airborne radar operating at X band (10 GHz) and Ka band (35 GHz) (Meneghini et al. 1992, 1997). Iguchi and Meneghini (1995), Iguchi (2005), and Iguchi and Haddad (2020) showed that the idea could be formulated as coupled nonlinear differential equations for parameters related to the number concentration and characteristic diameter of the raindrop distribution. Mardiana et al. (2004) and Rose and Chandrasekhar (2006a,b) explored iterative solutions to the equations that do not depend on an external constraint of path-integrated attenuation (PIA). Liao and Meneghini (2005) and Adhikari et al. (2007) compared several constrained and unconstrained solutions to the problem. Seto and Iguchi (2011) provided a detailed analysis of the ambiguities that can arise in both the forward and iterative backward solutions. More recent work includes that of Gorgucci and Baldini (2016), who investigated a self-consistent numerical method and its application to data from the Microwave Rain Radar, and Le and Chandrasekar (2013), in which linear models of *N _{w}* and

*D*in rain are used to circumvent the dual-value problem described below and where the method was applied to data obtained from the JPL dual-frequency airborne radar. Seto et al. (2013) proposed a formulation that depends on the DFR and the Hitschfeld and Bordan (1954) method. The approach was later supplemented by estimates of PIA from the surface reference technique (SRT) (Seto and Iguchi 2015). In both studies, the goal was to develop an algorithm applicable to measurements from the Dual-Frequency Precipitation Radar (DPR) on board the GPM satellite (Hou et al. 2008, 2014).

_{m}One of the principal sources of error in the DFR method is the double-valued nature of the DFR–*D _{m}* relationship, where DFR = 10 log

_{10}[

*Z*(

*f*

_{1})/

*Z*(

*f*

_{2})] = dB

*Z*(

*f*

_{1}) − dB

*Z*(

*f*

_{2}),

*f*

_{1}and

*f*

_{2}are the lower and upper frequencies of the radar,

*Z*is the effective radar reflectivity factor, dB

*Z*= 10 log

_{10}(

*Z*), and

*D*is the mass-weighted diameter defined as the ratio of the fourth to third moment of the RSD. Unlike the situation in snow where the relationship between the DFR and a characteristic size parameter of the distribution is monotonic (Matrosov 1992, 1998; Meneghini and Kumagai 1994), in rain and for negative values of DFR, there are two solutions of the equation: DFR =

_{m}*g*(

*D*); that is, there are two values of

_{m}*D*that satisfy the equation:

_{m}*D*=

_{m}*g*

^{−1}(DFR). Moreover, it is sometimes the case that the estimated DFR is smaller than the theoretical value so that no solution for

*D*is available. A modified DFR, DFR′, can be defined as

_{m}*ν*∼ 0.6 to minimize the region of ambiguity (Liao and Meneghini 2019b); a drawback of the strategy, however, is that DFR′ depends on the particle number concentration, thereby sacrificing one of the main advantages of DFR.

Our approach here is to consider multiple solutions (i.e., multiple range profiles of the raindrop size distribution parameters and rainfall rate) that the double-valued DFR–*D _{m}* relationship generates and to select the “best” solution either by using an external constraint such as the path-integrated attenuation or, in the case of the backward formulation, by imposing a self-consistency condition.

Although our focus is on the DFR approach, which is an analytical method that seeks to solve a set of equations, the operational algorithms for the DPR alone or combined with radiometer data are more accurately classified as optimal estimation methods (Seto et al. 2021; Iguchi et al. 2020; Grecu et al. 2016). Earlier work along these lines includes Marzoug and Amayenc (1994), Haddad et al. (1997), Marzano et al. (1999), and Grecu et al. (2011). These methods tend to be more robust with respect to the many error sources that exist. The DFR approach is subject not only to the DFR–*D _{m}* ambiguity but often to greater instabilities than the optimal estimation techniques. Analytical methods do have advantages, however, by dispensing with some of the assumptions used in the optimal estimation approaches thereby providing, in some cases, more accurate information on the raindrop size distribution and insight into relationships among the radar measurements and microphysical parameters.

One other relevant study is that of Iguchi and Meneghini (2022), who use a differential form of the dual-frequency equations to study their stability with respect to changes in the initial conditions. The study also proposes a new method of retrieval that employs both the DFR and the ratio of specific attenuations, dFk, *k*(*f*_{1})/*k*(*f*_{2}). This approach mitigates the DFR–*D _{m}* ambiguity when the attenuation within the bin is sufficiently large. The relationship between this method and the approach taken here is the subject of ongoing investigation.

## 2. Basic considerations and equations

### a. Dual-frequency equations

*D*(mm) is the equivalent drop diameter of an oblate spheroid,

*D*(mm) is the mass-weighted diameter [fourth over the third moment of

_{m}*N*(

*D*)],

*N*(m

_{w}^{−3}mm

^{−1}) is the normalized intercept parameter, and

*μ*is the shape parameter. As the effective radar reflectivity factor,

*Z*(mm

^{6}m

^{−3}), specific attenuation,

*k*(dB km

^{−1}), and rain rate,

*R*(mm h

^{−1}), are directly proportional to

*N*, these quantities can be written

_{w}*f*=

_{i}*f*

_{1}or

*f*

_{2}denote the lower and upper frequencies and are here taken to be the DPR frequencies of 13.6 and 35.5 GHz. Explicitly

*σ*(mm

_{b}^{2}) and

*σ*(mm

_{e}^{2}) are the backscattering and extinction cross sections,

*c*(m s

^{−1}) is the speed of light, and

*υ*(

*D*) (m s

^{−1}) is the drop fall speed expressed as a function of the particle diameter. The height dependence of

*υ*(

*D*) is ignored in the simulations presented. Although

*σ*and

_{b}*σ*depend on temperature, the variations are small and will be neglected; the lookup tables used in the retrievals have been calculated for a temperature of 10°C.

_{e}*Z*, which is related to

_{m}*Z*and

*k*by

*Z*= 10 log

_{10}(

*Z*) and dB

*Z*= 10 log

_{m}_{10}(

*Z*) will be used in the following equations.

_{m}It is important to note that the integrals *I _{b}*,

*I*, and

_{e}*I*can be computed at the outset as a function of

_{R}*D*, for each value of

_{m}*μ*. We assume that the drops are oblate spheroids with rotational axes along the vertical and a shape–size relationship given by Thurai et al. (2007) and that the viewing angle of the radar is near nadir. The scattering cross sections are computed by means of the T-matrix method (Mishchenko and Travis 1998). In large part, the retrievals amount to the repeated use of the lookup tables in concert with the set of measured radar reflectivity factors and path attenuations.

*N*it provides an estimate of

_{w}*D*if

_{m}*μ*is fixed or related to

*D*. The DFR–

_{m}*D*relationship for rain is shown in Fig. 1 for several values of

_{m}*μ*where the Ku-band frequency is taken to be 13.6 GHz and the Ka band 35.5 GHz. It can be seen from the figure that for DFR < 0, and a fixed value of

*μ*, two values of

*D*yield the same DFR. As in previous work, we can simply take the larger of the two roots. This, however, leads to instabilities in the retrievals particularly at lighter rain rates. Use of both roots also provides more flexibility in the equations when the underlying RSD does not follow the gamma form of the distribution.

_{m}The approach taken here is to generate all valid solutions to the equations and then choose one among them based primarily on the assumption that independent estimates of the path attenuation are available at both frequencies. As described later in the paper, for the backward recursion, a consistency condition between the input and output path attenuations can also be used to select a solution.

To get an idea of what is meant by a “valid” solution, consider the extreme case where DFR_{min} < DFR < 0 at each of the *n* gates that comprise the path from the storm top to the surface. At each range gate, two values of *D _{m}* will be consistent with the measured DFR so that a total of 2

*valid solutions can be generated. At the opposite extreme is the case where DFR > 0 at all gates; in this case, a single*

^{n}*D*exists for each DFR so that only a single profile/solution is possible. Most cases will be intermediate to these extremes with the number of solutions between 1 and 2

_{m}*. It is also worth noting that the set of solutions depends both on the way the equations are solved (forward versus backward) and the*

^{n}*μ*value that is assumed. For example, the solution set using the forward or initial-value equations, described below, with

*μ*= 3, will generally be different than the solution set using the backward or final-value equations with

*μ*= 6, or, in fact, with

*μ*= 3.

Before discussing the generation of multiple solutions, we first describe the basic solution procedure. Because of the way the multiple solutions are generated, it is sufficient to assume that only a single root exists to the equation *D _{m}* =

*g*

^{−1}(DFR). For DFR < 0, this can be either the upper or lower root; for DFR > 0, only a single solution, considered the upper root, exists.

*n*range gates are written in the form {

*Z*(

_{m}*f*

_{1},

*r*),

_{j}*Z*(

_{m}*f*

_{2},

*r*),

_{j}*j*= 1, …,

*n*}. At the storm top,

*r*=

*r*

_{1}, and as a first approximation, the

*Z*are taken to be equal to the attenuation-corrected value (

_{m}*Z*=

*Z*) so that DFR(

_{m}*r*

_{1}) = dB

*Z*(

_{m}*f*

_{1},

*r*

_{1}) − dB

*Z*(

_{m}*f*

_{2},

*r*

_{1}). From this, an estimate of

*D*is obtained. Next,

_{m}*D*along with

_{m}*Z*and an assumed value of

*μ*yield an estimate of

*N*. Note that the

_{w}*Z*and

*I*from either frequency give the same result. We choose

_{b}*f*

_{1}so that, from (3),

*D*and

_{m}*N*,

_{w}*k*(

*f*

_{1}) and

*k*(

*f*

_{2}) can be obtained from (4). With the estimates of the specific attenuations, the estimated

*Z*can be updated to dB

*Z*(

*f*) = dB

_{i}*Z*(

_{m}*f*) + 2

_{i}*εhk*(

*f*), where

_{i}*h*is the range resolution and

*ε*is taken to be 0.5. The iteration can be continued until the

*Z*values effectively become constant.

Having determined the values of *Z*(*f*_{1}), *Z*(*f*_{2}), *k*(*f*_{1}), *k*(*f*_{2}), *D _{m}*, and

*N*at the first gate, we can proceed to the second gate using the initial approximation dB

_{w}*Z*(

*f*,

_{i}*r*

_{2}) = dB

*Z*(

_{m}*f*,

_{i}*r*

_{2}) + 2

*hk*(

*f*,

_{i}*r*

_{1}) to estimate DFR at this gate and the corresponding value of

*D*. As in the first gate,

_{m}*D*is used along with

_{m}*Z*to estimate

*N*; next,

_{w}*D*and

_{m}*N*yield

_{w}*k*(

*f*

_{1},

*r*

_{2}) and

*k*(

*f*

_{2},

*r*

_{2}). Updated estimates of dB

*Z*can then be written: dB

*Z*(

*f*,

_{i}*r*

_{2}) = dB

*Z*(

_{m}*f*,

_{i}*r*

_{2}) + 2

*hk*(

*f*,

_{i}*r*

_{1}) + 2

*hεk*(

*f*,

_{i}*r*

_{2}). This procedure can be iterated until stable estimates of the quantities are attained. For the calculations we have taken the number of iterations to be six. A change in this iteration number does sometimes modify the results but the changes are negligible with respect to the minimum-error and standard solutions, which are discussed below, and the differences between these solutions and the true profiles.

*j*th gate (

*j*≤

*n*) and for

*i*= 1 and 2, representing the Ku- and Ka-band frequencies, the equations become

*I*denotes the iteration number, which begins at 1 and where

*k*

_{I}_{=0}= 0. For example, for the second iterate,

*I*= 2, the specific attenuations from the first iteration, at gate

*j*, are used to update the

*Z*estimates. Note that the summation term in (12) is taken to be zero for

*j*= 1.

*n*th) gate and ascends along the path to the first gate. In this case, however, the dB

*Z*(

*f*,

_{i}*r*) (

_{n}*i*= 1, 2) are taken to be equal to the sum of dB

*Z*(

_{m}*f*,

_{i}*r*) and

_{n}*A*(

_{m}*f*), where

_{i}*A*(

_{m}*f*) are the independently measured values of the path attenuation, obtained from methods such as the SRT (e.g., Meneghini et al. 2021). The attenuations associated with each gate are successively subtracted from this sum so that at the

_{i}*j*th gate, the equations for dB

*Z*(

*f*) become

_{i}*D*, etc., remain the same. Although the change in the equations from forward to backward is minor, the impact on the solutions is significant where the backward equations usually show a higher degree of stability as the path attenuations increase. The reason for this is that the path attenuations are bounded by the measured path attenuations

_{m}*A*(

_{m}*f*), whereas the path attenuations obtained from (12) can increase without limit.

_{i}### b. Generation of multiple solutions

We are interested in generating all valid solutions of both the initial [(12)–(17)] and final-value [(18), (13)–(17)] equations. One way to do this is to define an integer that runs consecutively from 0 to (2* ^{n}* − 1), where

*n*is the number of range gates that comprise the down-looking path along which precipitation is viewed by the radar. For each of these values, the decimal number is converted to a binary which can be interpreted as an instruction to take either the lower root (when the binary digit is 1) or upper root (when the binary digit is 0) of the equation

*D*=

_{m}*g*

^{−1}(DFR). If the instruction calls for the lower root but only the upper root exists (i.e., DFR > 0), then the instruction is considered invalid and discarded.

Consider the case where *n* = 20 so that the counting variable, *j*, will range from 0 to 2^{20} − 1. For example, for *j* = 2^{10} − 1 = 1023, or, in binary, 00000000000111111111, the instruction is interpreted as taking the upper root for the first 11 gates of the path, starting from the storm top, followed by taking the lower root for the last 9 gates. If the DFR over any of the last nine gates is greater than 0, however, the lower root does not exist and the instruction is considered invalid, i.e., a valid solution to the equations for this instruction does not exist. It is worth noting that valid instructions/solutions for the forward-going problem are typically not the same as valid instructions/solutions for the backward-going problem. Since the solution procedure is somewhat like a random walk, the number of solutions for the forward procedure tend to grow and diverge in magnitude as the recursion descends the column; for the backward problem, a similar proliferation of solutions can occur as the recursion ascends the column. Of course, the exact number of solutions depends on the input data. It should also be noted that, because of numerical instabilities and because of differences between the gamma model and disdrometer-measured DSD’s, which will be used to test the method, the true or input instruction often will not be chosen as the “best” solution. In fact, it is sometimes the case that the input instruction yields no solution for either the forward or backward recursion.

In some cases, DFR(*μ*) < DFR_{min}(*μ*), that is, the DFR at a particular range gate is smaller than the theoretical minimum for the particular *μ* value that has been assumed. When this happens we can either discard the instruction or set *D _{m}* to

*D*for which DFR = DFR

_{m}_{min}(

*μ*). For the results shown below, we have used the first option since the discarded solutions rarely include a candidate for the “best” solution as defined below.

To make these ideas more concrete, consider an example of a path consisting of three range gates and that, for the retrieval, we assume *μ* = 3. For *μ* = 3, *T* = 10°C the value of *D _{m}* at which DFR is minimum (−1.15 dB) is 1.03 mm and the value of

*D*above which the DFR is positive is 1.45 mm (Fig. 1) so that if

_{m}*D*is less than 1.03 mm, the lower root of the equation

_{m}*D*=

_{m}*g*

^{−1}(DFR) should be chosen and if

*D*is greater than this value, the upper root should be used. Assume further that the true values of

_{m}*D*at these gates are [0.4, 1.1, 1.6] mm so that the input or true instruction is [1, 0, 0], i.e., the lower root is to be chosen at the first gate and the upper root is to be chosen for the second and third gates. If the estimated values of DFR are negative for the first two gates and positive for the third then four valid instructions would exist out of a total of eight: {[0, 0, 0], [0, 1, 0], [1, 0, 0], [1, 1, 0]}. Note that all instructions ending in “1” are eliminated because “1” requires choosing the lower root; however, since DFR > 0 at this gate, the lower root does not exist. If the DFR were negative at all gates, but greater than DFR

_{m}_{min}, then the number of valid instructions would be eight whereas if the DFR were positive at all gates only a single valid instruction, [0, 0, 0], would exist. It is worth noting that the DFR values cannot be determined prior to the retrieval because they depend on the attenuation corrections which, in turn, depend on the choice of roots.

*q*, the specific attenuations at each range gate, when summed over all gates, yield the (two-way) PIA at both frequencies given by

*q*denotes the

*q*th valid solution/instruction and

*i*indicates the frequency

*f*

_{1}or

*f*

_{2}. Note that corresponding to each pair of PIAs, {

*A*(

_{q}*f*

_{1}),

*A*(

_{q}*f*

_{2})}, are profiles of {

*D*(

_{mq}*r*),

_{j}*N*(

_{wq}*r*),

_{j}*R*(

_{q}*r*),

_{j}*j*= 1, …,

*n*}. Moreover, for the backward solution, the assumed or measured PIAs,

*A*(

_{m}*f*), used in (18) generally will not match those derived from the retrieved DSD parameters given by (19). Modifying the input PIAs to match the output PIAs is similar to the Mardiana et al. (2004) approach and will be described in section 3b.

_{i}*N*valid solutions, we can define an error based on the root-mean-square (RMS) difference between the path attenuations at Ku and Ka band derived from the DSD parameters along the path, as defined in (19), and independent measurements of these quantities. We add to this a term that penalizes solutions with highly fluctuating values of

_{S}*D*along the path. This penalty can be viewed as enforcing continuity of the

_{m}*D*profiles. The error for each valid solution, denoted by the subscript

_{m}*q*, is defined by the following equation:

*s*

_{Ku}= 1 and

*s*= 0. Since the attenuation at Ka band is typically about 6 times larger than that at Ku band, the scale factor

_{N}*s*

_{Ku}can be used to give the two terms approximately equal weights by taking

*s*

_{Ku}= 6

^{2}. The final term in (20) consists of a product of the scale factor

*s*and the quantity

_{N}*N*

_{trans}which is the number of 0–1 and 1–0 transitions in the instruction. In the example given earlier, with the instruction 00000000000111111111,

*N*

_{trans}= 1, whereas for the instruction 10101010101010101010,

*N*

_{trans}= 19 and equal to the maximum number of transitions for

*n*= 20. Nominally,

*s*is set to 0.1; a change to 0 or 0.2 typically produces only minor changes in the profiles. Nevertheless, exceptions to this rule exist.

_{N}As already noted, the solution sets that are generated for the forward and backward equations will generally be different, although in many cases, there is a significant overlap. The solution sets will also change for different values of the parameter *μ*. For the examples in the paper, *μ* is set to 3. Changing this to 6 usually has a minor impact on the solutions. However, a significant loss of accuracy usually occurs if *μ* is set to 0.

Note finally that the single solution generated in previous work uses only the upper root of the *D _{m}*–DFR relationship; in the context of this paper, this corresponds to an instruction consisting of all zeros (bigger roots). In section 3, we will distinguish this “standard solution” from the minimum-error solution given by (20). In some cases, these are identical; in other cases, they differ but both exist; in other cases, only the minimum-error solution exists.

### c. The R–D_{m} method

To assess the behavior of the DFR approach, it is instructive to compare its performance to that of the *R*–*D _{m}* method which is used in the operational version of the DPR retrieval algorithm. A comprehensive description of the method has been given by Seto et al. (2021). Our intention here is to give a simplified version of the method with focus solely on the rain retrieval problem.

*R*–

*D*relationship with an adjustable parameter

_{m}*N*–

_{w}*D*relationship that can be written:

_{m}*N*and

_{w}*D*is usually expressed in the form of a power law but the expression in (22) is sufficient for present purposes. Next, we use the equation that expresses the measured reflectivity factor

_{m}*Z*in terms of the true reflectivity factor

_{m}*Z*and an attenuation factor. This is given by (9). Taking 10log

_{10}of both sides of this equation and substituting (3) and (4) gives

*D*,

_{m}*μ*, and the adjustable parameter

*μ*,

*f*, (24) can be solved recursively either in the forward or backward direction. For the backward solution, the integral in (24) is written as the difference of integrals from (0,

*r*) and (

_{n}*r*,

*r*), where

_{n}*r*denotes the range to the last gate. Recognizing that the first integral represents the measured path attenuation

_{n}*A*(

_{m}*f*), then (24) becomes

Beginning at the first gate, and for a fixed value of *D _{m}*;

*R*and

*N*then follow from (21) and (22), respectively. From

_{w}*N*and

_{w}*D*, the specific attenuation,

_{m}*k*(

*f*), can be computed from (4). Having determined these values at the first gate, the procedure can be continued to the second gate and so on until the full profiles of (

*D*,

_{m}*N*,

_{w}*R*) are obtained for a given

*Z*profiles at Ka band, as estimated by the DSD parameters, match the measured values. It is worth noting that for the dual-frequency case, the attenuation constraint is the differential path attenuation as derived from the dual-frequency SRT because this quantity typically can be obtained more accurately than estimates of attenuation from the single-frequency version of the method.

_{m}## 3. Results

In this section, we present examples from the forward/backward recursion and the *R*–*D _{m}* method. For the results here, the measured or input path attenuations are taken to be equal to the true values. The shape parameter

*μ*is fixed at 3. In evaluating (20) we choose

*s*

_{Ku}= 1 and

*s*= 0.1.

_{N}The input profiles of the measured reflectivity factor *Z _{m}* at Ku and Ka band are generated from a sequence of 20 temporally consecutive RSDs which are taken to represent the RSD along a 5-km rain column with 0.25-km range gates. A set of 500 profiles were produced from 5-min-averaged RSDs. These data are compilations of measurements made by the PARSIVEL2 disdrometers during the Iowa Flood Studies (IFloodS) field experiment in eastern Iowa from 1 May to 15 June 2013, the Midlatitude Continental Convective Clouds Experiment (MC3E) from 22 April–6 June 2011 near Lamont, Oklahoma (Jensen et al. 2016), and measurements at the NASA Wallops Flight Facility, located in Wallops Island, Virginia, from May 2013 to February 2014 (Liao et al. 2014).

From each RSD, *D _{m}*,

*N*,

_{w}*R*,

*k*(

*f*),

_{i}*Z*(

*f*),

_{i}*Z*(

_{m}*f*), and

_{i}*A*(

*f*),

_{i}*i*= 1, 2, are computed. The two-way path attenuation,

*A*(

*f*), is equal to the sum of the

_{i}*k*(

*f*) multiplied by twice the gate spacing: 2 × 0.25 km. The inputs to the retrievals are the measured radar reflectivity factor profiles

_{i}*Z*(

_{m}*f*;

_{i}*r*),

_{j}*i*= 1, 2;

*j*= 1, …, 20, and the measured path attenuations

*A*(

_{m}*f*). As noted above, for the results in 3a, the true and measured path attenuations are taken to be equal.

_{i}### a. General results

Figure 2 shows an example of a moderate rain case where the *Z _{m}* profiles (in dB) are given by the dashed–dotted dark blue lines for Ku band on the left and Ka band on the right. The true reflectivity-factor profiles

*Z*derived from the input RSDs are given by the black lines. As noted in the previous section, multiple solutions are generated for the forward recursion (top plots) and backward recursion (bottom plots). For this example, the number of forward solutions

*N*is 20 while the number of backward solutions,

_{F}*N*, is 4. For each of the

_{B}*N*and

_{F}*N*solutions, range profiles of

_{B}*Z*(

*f*),

_{i}*k*(

*f*),

_{i}*D*, log

_{m}*N*, and

_{w}*R*, as well as path attenuations, are generated. (Since all logarithms are taken to the base 10, the subscript “10” is suppressed.) The solution that yields the smallest error, as defined by (20), is taken as the minimum error result and represented in the panels by the heavy dotted red lines. The remaining estimated

*Z*profiles are represented by the light blue solid lines. These profiles are visible only in the Ka-band forward

*Z*plots (top right) since the four backward profiles are nearly identical (bottom plots) as are the 20 profiles for the forward Ku-band results (top left). Notice that the extent of the light blue lines in the upper-right panel increases toward the bottom of the column, indicating that the solutions tend to diverge as the forward recursion descends the path. It can be seen in this case that most of the estimated

*Z*profiles overestimate the true

*Z*profile.

dB*Z _{m}* and dB

*Z*range profiles at (left) Ku band and (right) Ka band. (top) Forward recursion and (bottom) backward recursion results. Blue dashed–dotted lines represent the dB

*Z*input data while the black lines represent the true dB

_{m}*Z*profiles. The heavy dotted red lines represent the estimated dB

*Z*profile from the minimum-error solution. The light blue lines visible in the top-right panel represent the dB

*Z*(Ka) profiles from the solutions to the forward recursion.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

dB*Z _{m}* and dB

*Z*range profiles at (left) Ku band and (right) Ka band. (top) Forward recursion and (bottom) backward recursion results. Blue dashed–dotted lines represent the dB

*Z*input data while the black lines represent the true dB

_{m}*Z*profiles. The heavy dotted red lines represent the estimated dB

*Z*profile from the minimum-error solution. The light blue lines visible in the top-right panel represent the dB

*Z*(Ka) profiles from the solutions to the forward recursion.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

dB*Z _{m}* and dB

*Z*range profiles at (left) Ku band and (right) Ka band. (top) Forward recursion and (bottom) backward recursion results. Blue dashed–dotted lines represent the dB

*Z*input data while the black lines represent the true dB

_{m}*Z*profiles. The heavy dotted red lines represent the estimated dB

*Z*profile from the minimum-error solution. The light blue lines visible in the top-right panel represent the dB

*Z*(Ka) profiles from the solutions to the forward recursion.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

The behavior of the backward profiles can be understood by noting that these solutions are constrained to be equal to dB*Z _{m}*(

*f*) +

_{i}*A*(

_{m}*f*) at the lowest gate; since

_{i}*A*(

_{m}*f*) is taken to be equal to the true value, the sum is equal to the true value, dB

_{i}*Z*(

*f*). Although the backward solutions sometimes diverge higher in the column, in this case, the four solutions give nearly the same profile. One other point worth making is that for this case, the standard solution exists for both the forward and backward recursion. As noted earlier, the standard solution is obtained by taking only the bigger root of the equation:

_{i}*D*=

_{m}*g*

^{−1}(DFR) at all range gates. An absence of the standard solution indicates that the choice of the larger root results in DFR becoming less than DFR

_{min}at some point along the path so that the solution is rejected. (For the forward solutions, the other reason for the absence of the standard solution is that the attenuations increase without bound. This is not the case for this example.)

The forward recursion results for this case are shown in Fig. 3. The path attenuations at Ku and Ka band are shown in the top-left panel where the small black circles represent the PIAs for the 20 forward solutions. The true PIA(Ku, Ka) is represented by the plus sign while the PIA corresponding to the minimum-error solution is given by the blue circle, which for this case is nearly coincident with the true PIA. The PIA corresponding to the standard solution is represented by the red circle.

Results from the forward recursion for the *Z _{m}* input profiles shown in Fig. 2. (top left) The 20 small black circles represent the PIA estimates from the 20 solutions. The blue circle and the plus sign represent the estimated and true path attenuations, respectively. The red circle represents the PIA from the standard solution. The heavy blue dotted lines in the remaining panels show the selected (minimum error) profiles of

*D*, log

_{m}*N*, and

_{w}*R*while the red curves represent the results from the standard solution. The black lines represent the true profiles while the light blue solid lines represent the profiles for the remaining 18 solutions.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results from the forward recursion for the *Z _{m}* input profiles shown in Fig. 2. (top left) The 20 small black circles represent the PIA estimates from the 20 solutions. The blue circle and the plus sign represent the estimated and true path attenuations, respectively. The red circle represents the PIA from the standard solution. The heavy blue dotted lines in the remaining panels show the selected (minimum error) profiles of

*D*, log

_{m}*N*, and

_{w}*R*while the red curves represent the results from the standard solution. The black lines represent the true profiles while the light blue solid lines represent the profiles for the remaining 18 solutions.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results from the forward recursion for the *Z _{m}* input profiles shown in Fig. 2. (top left) The 20 small black circles represent the PIA estimates from the 20 solutions. The blue circle and the plus sign represent the estimated and true path attenuations, respectively. The red circle represents the PIA from the standard solution. The heavy blue dotted lines in the remaining panels show the selected (minimum error) profiles of

*D*, log

_{m}*N*, and

_{w}*R*while the red curves represent the results from the standard solution. The black lines represent the true profiles while the light blue solid lines represent the profiles for the remaining 18 solutions.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

The other panels in Fig. 3 show the results for *D _{m}* (top right), log

*N*(bottom left), and

_{w}*R*(bottom right). In all cases, the black lines represent the true profiles as computed directly from the set of input raindrop size distributions. The heavy dotted blue lines represent the minimum-error solution from (20), while the red dashed lines represent the results from the standard solution. When it exists, as in this case, the standard solution yields the smallest PIA, the highest

*D*and lowest log

_{m}*N*and

_{w}*R*profiles. In this case the standard solution provides more accurate results than the minimum-error solution. In particular, the underestimates of

*D*and overestimates in log

_{m}*N*and

_{w}*R*at a height of 1.5 km that are seen in the minimum-error solution (blue) are absent in the standard solution (red), which tends to follow the input (black) profile well. What the error condition in (20) provides is the DSD-generated PIA that is “closest” to the input PIA. Satisfaction of this condition often, but not always, leads to

*D*, log

_{m}*N*, and

_{w}*R*profiles close to the input values.

The set of light blue lines in the panels represent the profiles generated by the remaining 18 profiles and define the extent of *D _{m}*, log

*N*, and

_{w}*R*profiles that are generated from the input

*Z*profiles for the initial value problem. The largest PIA in the upper-left panel is associated with the lowest

_{m}*D*and highest

_{m}*N*and

_{w}*R*profiles whereas the smallest PIA, which in this case is the standard solution, corresponds to the highest

*D*and lowest log

_{m}*N*and

_{w}*R*profiles. These correspondences are a direct consequence of the retrieval where the bigger

*D*root yields a smaller log

_{m}*N*to satisfy the

_{w}*Z*constraint. The higher

*D*–lower log

_{m}*N*combination yields smaller values of specific attenuation so that the path integrated attenuations are lower. Conversely, the lower

_{w}*D*–higher log

_{m}*N*combination yields larger values of the specific attenuation and higher path attenuations.

_{w}The PIA bias problem for the forward recursion can be explained by the PIA results in the top-left panel of Fig. 3. The allowable solutions for the forward or initial-value problem depend only on the *Z _{m}*(

*f*) profiles and are independent of the PIA constraint or choice of

_{i}*s*and

_{N}*s*

_{Ku}in (20). If the measured PIA,

*s*, or

_{N}*s*

_{Ku}is changed, the choice of solution generally will change. This situation is shown schematically in Fig. 4 where for values of the measured PIA,

*A*, above the line U, the solution associated with the highest attenuations will be selected and for values of

_{m}*A*

_{meas}below the line L, the solution associated with the lowest attenuations will be chosen. For some profiles, the PIA solution set cannot be approximated by points along a straight line; nevertheless, the idea that the procedure selects the solution “closest,” as defined by the minimum of (20), to the measured/input PIA remains valid.

Schematic of solution choice for the forward recursion. The solution set is represented by the points along a straight line in the *A*(Ku)–*A*(Ka) plane. For measured path attenuations * A_{m}* below the line L, e.g.,

*(1) and*

__A___{m}*(2), the solution with the smallest path attenuations is chosen; for*

__A___{m}*above the line U, e.g.,*

__A___{m}*(3), the solution with the highest path attenuations is chosen. For intermediate values of*

__A___{m}*, e.g.,*

__A___{m}*(4), the solution “closest” to*

__A___{m}*, as defined by (20), is chosen.*

__A___{m}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Schematic of solution choice for the forward recursion. The solution set is represented by the points along a straight line in the *A*(Ku)–*A*(Ka) plane. For measured path attenuations * A_{m}* below the line L, e.g.,

*(1) and*

__A___{m}*(2), the solution with the smallest path attenuations is chosen; for*

__A___{m}*above the line U, e.g.,*

__A___{m}*(3), the solution with the highest path attenuations is chosen. For intermediate values of*

__A___{m}*, e.g.,*

__A___{m}*(4), the solution “closest” to*

__A___{m}*, as defined by (20), is chosen.*

__A___{m}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Schematic of solution choice for the forward recursion. The solution set is represented by the points along a straight line in the *A*(Ku)–*A*(Ka) plane. For measured path attenuations * A_{m}* below the line L, e.g.,

*(1) and*

__A___{m}*(2), the solution with the smallest path attenuations is chosen; for*

__A___{m}*above the line U, e.g.,*

__A___{m}*(3), the solution with the highest path attenuations is chosen. For intermediate values of*

__A___{m}*, e.g.,*

__A___{m}*(4), the solution “closest” to*

__A___{m}*, as defined by (20), is chosen.*

__A___{m}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results for the backward recursion for this case are shown in Fig. 5 using the same format as in Fig. 3. In this case, the 4 solutions are tightly clustered in PIA space and the minimum error and the standard solutions are the same. Despite some discrepancies between the true and retrieved data in this case, the profiles are generally accurate. It is worth noting that the minimum error solution is not the solution closest to the measured/true PIA. This is a consequence of the fact that *s _{N}* in (20) has been set to 0.1. By setting

*s*to 0, the minimum error solution moves to the black circle closest to the true PIA. This results in slightly modified

_{N}*D*, log

_{m}*N*, and

_{w}*R*profiles and differences between the standard and minimum error solutions.

Results from the backward recursion for the *Z _{m}* input profiles shown in Fig. 2. (top left) The 4 small black circles represent the PIA estimates from the four solutions where the plus sign and the blue circle represent the PIA of true/input path attenuation and the path attenuation of the selected (minimum error) solution. The heavy blue dotted line in the remaining panels show the corresponding minimum error profiles of

*D*, log

_{m}*N*, and

_{w}*R*. The red lines represent the standard solution results which in this case are identical to the minimum error results. The black lines in the

*D*, log

_{m}*N*, and

_{w}*R*panels represent the true values; the light blue solid lines represent the set of profiles for the remaining three solution sets.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results from the backward recursion for the *Z _{m}* input profiles shown in Fig. 2. (top left) The 4 small black circles represent the PIA estimates from the four solutions where the plus sign and the blue circle represent the PIA of true/input path attenuation and the path attenuation of the selected (minimum error) solution. The heavy blue dotted line in the remaining panels show the corresponding minimum error profiles of

*D*, log

_{m}*N*, and

_{w}*R*. The red lines represent the standard solution results which in this case are identical to the minimum error results. The black lines in the

*D*, log

_{m}*N*, and

_{w}*R*panels represent the true values; the light blue solid lines represent the set of profiles for the remaining three solution sets.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results from the backward recursion for the *Z _{m}* input profiles shown in Fig. 2. (top left) The 4 small black circles represent the PIA estimates from the four solutions where the plus sign and the blue circle represent the PIA of true/input path attenuation and the path attenuation of the selected (minimum error) solution. The heavy blue dotted line in the remaining panels show the corresponding minimum error profiles of

*D*, log

_{m}*N*, and

_{w}*R*. The red lines represent the standard solution results which in this case are identical to the minimum error results. The black lines in the

*D*, log

_{m}*N*, and

_{w}*R*panels represent the true values; the light blue solid lines represent the set of profiles for the remaining three solution sets.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

*R*–

*D*results for this profile are shown in Fig. 6 where the retrievals can be seen to be somewhat less accurate than the backward estimates of Fig. 5. To quantify this, we compute the correlations and RMS errors between the true and estimated profiles for the forward, backward and

_{m}*R*–

*D*approaches for

_{m}*R*, log

*N*, and

_{w}*D*, respectively. Specifically, we use the notation

_{m}*ρ*= (

_{M}*c*

_{1},

*c*

_{2},

*c*

_{3}) to denote the correlation coefficients for method “M” between the true and estimated rain rate profiles,

*c*

_{1}; the true and estimated log

*N*profiles,

_{w}*c*

_{2}; and the true and estimated

*D*profiles,

_{m}*c*

_{3}. The method “M” can refer to the backward recursion, “B,” forward recursion, “F,” or the

*R*–

*D*method, “RD.” A similar notation is used for the RMS error. For the profiles shown in Figs. 3, 5, and 6, we obtain

_{m}*N*obtained from the

_{w}*R*–

*D*method will be discussed below.

_{m}Results of the *R*–*D _{m}* method for the input data shown in Fig. 2. (top left) The true (plus sign) and estimated (blue circle) path attenuations. The true

*D*, log

_{m}*N*, and

_{w}*R*profiles shown by the black lines are identical to those in Figs. 3 and 5. Results from the method are shown by the heavy dotted blue lines.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results of the *R*–*D _{m}* method for the input data shown in Fig. 2. (top left) The true (plus sign) and estimated (blue circle) path attenuations. The true

*D*, log

_{m}*N*, and

_{w}*R*profiles shown by the black lines are identical to those in Figs. 3 and 5. Results from the method are shown by the heavy dotted blue lines.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results of the *R*–*D _{m}* method for the input data shown in Fig. 2. (top left) The true (plus sign) and estimated (blue circle) path attenuations. The true

*D*, log

_{m}*N*, and

_{w}*R*profiles shown by the black lines are identical to those in Figs. 3 and 5. Results from the method are shown by the heavy dotted blue lines.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

A second example of *Z _{m}*,

*Z*profiles is shown in Fig. 7. In this case, the

*Z*,

_{m}*Z*, rain rates, and path attenuations are somewhat larger and a separation between

*Z*and

_{m}*Z*can be seen at both Ku and Ka band. For the forward recursion results shown in the top panels for Ku (left) and Ka (right), there are 49 solutions (

*N*= 49) which include the standard solution. The minimum-error

_{F}*Z*profiles are shown by the heavy red dotted lines which are close to the true profiles (black lines). The light blue lines in the upper-right panel (Ka band, forward) represent the 48 remaining profiles of

*Z*and show a clear divergence of some of the solutions especially near the bottom of the column. The number of backward solutions in this case is only 3 where all solutions yield corrected

*Z*profiles nearly identical to the true

*Z*profile.

dB*Z _{m}* and dB

*Z*range profiles for (left) Ku band and (right) Ka band. (top) Forward recursion and (bottom) backward recursion results. Blue dashed–dotted lines represent the dB

*Z*input data while the black lines represent the true dB

_{m}*Z*profiles. The heavy dotted red lines represent the estimated dB

*Z*profiles from the minimum error solution. The light blue lines visible in the top-right panel represent the

*Z*(Ka) profiles from the solutions to the forward recursion.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

dB*Z _{m}* and dB

*Z*range profiles for (left) Ku band and (right) Ka band. (top) Forward recursion and (bottom) backward recursion results. Blue dashed–dotted lines represent the dB

*Z*input data while the black lines represent the true dB

_{m}*Z*profiles. The heavy dotted red lines represent the estimated dB

*Z*profiles from the minimum error solution. The light blue lines visible in the top-right panel represent the

*Z*(Ka) profiles from the solutions to the forward recursion.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

dB*Z _{m}* and dB

*Z*range profiles for (left) Ku band and (right) Ka band. (top) Forward recursion and (bottom) backward recursion results. Blue dashed–dotted lines represent the dB

*Z*input data while the black lines represent the true dB

_{m}*Z*profiles. The heavy dotted red lines represent the estimated dB

*Z*profiles from the minimum error solution. The light blue lines visible in the top-right panel represent the

*Z*(Ka) profiles from the solutions to the forward recursion.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

The forward recursion results for this case are shown in Fig. 8. As in Figs. 3 and 5, the upper-left panel shows the PIAs corresponding to the standard solution (red circle), the minimum error solution (blue circle), and the true value (“+”). Of the 49 solutions, represented by the small black circles, 30 solutions yield Ka-band path attenuations greater than 14.5 dB and fall outside the plotting area. Profiles of *D _{m}*, log

*N*, and

_{w}*R*, represented by the heavy blue dotted lines, correspond to the minimum-error solution and show discrepancies relative to the true profiles shown in black. For the standard solution, given by the red dashed lines, the errors are smaller than for the minimum-error solution because of the absence of the discontinuity in the

*D*, log

_{m}*N*, and

_{w}*R*profiles around 2 km that appear in the minimum-error profiles. The light blue solid curves that represent the profiles associated with the other solutions show that the errors in

*D*, log

_{m}*N*, and

_{w}*R*can become large when the estimated PIA is positively biased, leading to underestimates in

*D*and overestimates in log

_{m}*N*and

_{w}*R*.

Results from the forward recursion for the *Z _{m}* input profiles shown in Fig. 7. (top left) The small black circles represent the PIA estimates from a portion of the 49 solutions where plus sign represents the true/input path attenuations. The blue circle represents the path attenuations corresponding to the minimum error solution while the red circle represents the path attenuations corresponding to the standard solution. The heavy blue dotted lines in the

*D*, log

_{m}*N*, and

_{w}*R*panels show the corresponding minimum-error profiles of

*D*, log

_{m}*N*, and

_{w}*R*. Profiles corresponding to the standard solution are represented by the red dashed lines. Black lines represent the true values while the light blue solid lines represent the profiles for the remaining solutions.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results from the forward recursion for the *Z _{m}* input profiles shown in Fig. 7. (top left) The small black circles represent the PIA estimates from a portion of the 49 solutions where plus sign represents the true/input path attenuations. The blue circle represents the path attenuations corresponding to the minimum error solution while the red circle represents the path attenuations corresponding to the standard solution. The heavy blue dotted lines in the

*D*, log

_{m}*N*, and

_{w}*R*panels show the corresponding minimum-error profiles of

*D*, log

_{m}*N*, and

_{w}*R*. Profiles corresponding to the standard solution are represented by the red dashed lines. Black lines represent the true values while the light blue solid lines represent the profiles for the remaining solutions.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results from the forward recursion for the *Z _{m}* input profiles shown in Fig. 7. (top left) The small black circles represent the PIA estimates from a portion of the 49 solutions where plus sign represents the true/input path attenuations. The blue circle represents the path attenuations corresponding to the minimum error solution while the red circle represents the path attenuations corresponding to the standard solution. The heavy blue dotted lines in the

*D*, log

_{m}*N*, and

_{w}*R*panels show the corresponding minimum-error profiles of

*D*, log

_{m}*N*, and

_{w}*R*. Profiles corresponding to the standard solution are represented by the red dashed lines. Black lines represent the true values while the light blue solid lines represent the profiles for the remaining solutions.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

The backward recursion results are given in Fig. 9. Unlike the forward solutions, the path attenuations, corresponding to the three backward solutions, and represented by the small black circles in the top-left panel, are tightly clustered with variations in the Ka-band PIA less than 0.5 dB. In contrast to the forward case, the standard solution does not exist for the backward so the corresponding results (red circle and red lines) are not available. As in previous plots, the blue circle represents the minimum-error solution. As in the forward case, the minimum-error solution for backward shows a discontinuity in the *D _{m}*, log

*N*, and

_{w}*R*profiles at the height of 2 km. Results from the

*R*–

*D*method are shown in Fig. 10. Although the log

_{m}*N*profile shows deviations from the input profile at lower heights, the

_{w}*D*and

_{m}*R*yields relatively accurate results throughout the profile.

Results from the backward recursion for the *Z _{m}* input profiles shown in Fig. 7. (top left) The small black circles represent the PIA estimates from the 3 solutions where the plus sign represents the true path attenuations. The blue circle represents PIA of the minimum-variance solution. The heavy blue dotted line in the

*D*, log

_{m}*N*, and

_{w}*R*panels show the corresponding profiles of

*D*, log

_{m}*N*, and

_{w}*R*. The black lines represent the true values while the light blue solid lines represent the profiles corresponding to the remaining solutions. The standard solution (red circle and red lines) does not exist for this case.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results from the backward recursion for the *Z _{m}* input profiles shown in Fig. 7. (top left) The small black circles represent the PIA estimates from the 3 solutions where the plus sign represents the true path attenuations. The blue circle represents PIA of the minimum-variance solution. The heavy blue dotted line in the

*D*, log

_{m}*N*, and

_{w}*R*panels show the corresponding profiles of

*D*, log

_{m}*N*, and

_{w}*R*. The black lines represent the true values while the light blue solid lines represent the profiles corresponding to the remaining solutions. The standard solution (red circle and red lines) does not exist for this case.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results from the backward recursion for the *Z _{m}* input profiles shown in Fig. 7. (top left) The small black circles represent the PIA estimates from the 3 solutions where the plus sign represents the true path attenuations. The blue circle represents PIA of the minimum-variance solution. The heavy blue dotted line in the

*D*, log

_{m}*N*, and

_{w}*R*panels show the corresponding profiles of

*D*, log

_{m}*N*, and

_{w}*R*. The black lines represent the true values while the light blue solid lines represent the profiles corresponding to the remaining solutions. The standard solution (red circle and red lines) does not exist for this case.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results of the *R*–*D _{m}* method for the input data shown in Fig. 7. (top left) True (plus sign) and estimated (blue circle) path attenuations. The

*D*, log

_{m}*N*, and

_{w}*R*profiles, shown in black, represent the true profiles and are identical to those shown in Figs. 8 and 9. Results from the method are represented by the heavy dotted blue lines.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results of the *R*–*D _{m}* method for the input data shown in Fig. 7. (top left) True (plus sign) and estimated (blue circle) path attenuations. The

*D*, log

_{m}*N*, and

_{w}*R*profiles, shown in black, represent the true profiles and are identical to those shown in Figs. 8 and 9. Results from the method are represented by the heavy dotted blue lines.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Results of the *R*–*D _{m}* method for the input data shown in Fig. 7. (top left) True (plus sign) and estimated (blue circle) path attenuations. The

*D*, log

_{m}*N*, and

_{w}*R*profiles, shown in black, represent the true profiles and are identical to those shown in Figs. 8 and 9. Results from the method are represented by the heavy dotted blue lines.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

Note that the *R*–*D _{m}* results for this case are more accurate than either the forward or backward recursion with correlations in

*R*and

*D*of 0.85 and 0.93, respectively, in contrast to those from the backward recursion given by 0.79 and 0.91, respectively, and lower values from the forward. The RMS errors from the

_{m}*R*–

*D*method for

_{m}*R*, log

*N*, and

_{w}*D*are also smaller than those obtained from the forward and backward recursions. It is only in the correlation of log

_{m}*N*where the results from the

_{w}*R*–

*D*are poorer than those from the backward and forward recursions.

_{m}The reason for the low correlations in log*N _{w}* for the

*R*–

*D*method seems to arise from the fact that the

_{m}*N*–

_{w}*D*relationship is fixed to within a constant. Although this constant is adjusted by using constraints obtained from the PIA and consistency between the Ku- and Ka-band data, the exponent in this relationship is not adjustable so that the method is unable to track deviations from a fixed-slope linear log

_{m}*N*–log

_{w}*D*relationship. An additional example of this is shown in Fig. 11 where the result from the backward recursion (

_{m}*N*= 6) is compared with that from the

_{B}*R*–

*D*method. It is worth noting that, unlike the previous cases, the standard solution for this profile exists neither for the forward nor the backward recursion. On the left-hand plots of Fig. 11, the backward-derived profiles of

_{m}*D*, log

_{m}*N*, and

_{w}*R*are generally close to the true profiles despite some fluctuations. For the

*R*–

*D*result on the right, however, the retrieved log

_{m}*N*profile fails to track the decrease in log

_{w}*N*in the 3–4-km height range, which results in an overestimation of the rain rate over this range.

_{w}(left) Backward recursion and (right) *R*–*D _{m}* method for a nonuniform log

*N*range profile.

_{w}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

(left) Backward recursion and (right) *R*–*D _{m}* method for a nonuniform log

*N*range profile.

_{w}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

(left) Backward recursion and (right) *R*–*D _{m}* method for a nonuniform log

*N*range profile.

_{w}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

*R*–

*D*method shows no correlation (−0.06) in log

_{m}*N*, the RMS error (0.39) is essentially the same as that of the forward (0.39) and backward (0.37) recursions and yields an accurate mean estimate of this parameter over the profile.

_{w}### b. Backward recursion with a consistency constraint

In the foregoing examples, the measured PIA was taken to be equal to the true PIA. When this assumption is removed, the accuracy of the retrievals usually degrades. As noted in the previous section, the effect of PIA biases on the forward recursion is relatively easy to understand as the solutions are independent of the PIA constraint; that is, the PIA has no effect on the set of solutions and is used only in selecting the “best” solution. This is not the case with the backward recursion where the estimated/measured path attenuations are embedded in the basic Eq. (18).

A study of the PIA bias effect on the backward solutions brings to light the contradictory situation in which the PIA associated with the measured or input PIA, i.e., the *A _{m}* used in (18), does not match the PIA computed from the estimated drop size distribution parameters. We do, however, often find a good match between the output and input PIA when the input PIA is at or close to the true PIA. This suggests that a solution can be found with the backward recursion if we adjust the input PIA so that it matches the output or DSD-generated PIA.

This idea is reminiscent of that proposed by Mardiana et al. (2004) where an iteration is performed so that a somewhat similar kind of consistency is achieved between the input and output data. A detailed analysis of the conditions under which the backward iterative method converges to the correct solution has been given by Seto and Iguchi (2011). The present study differs from the Mardiana et al. approach in two respects. In earlier work, only the bigger-root solution was considered so that for each input PIA only a single solution, at most, is obtained. In the present work, multiple solutions (including the no-solution case) can be generated for each input PIA. To handle this situation, we use (20) to obtain the minimum error solution for each input. For *N* such PIA inputs, we select the minimum of these *N* minima to obtain the result. This is the minimum error with respect to changes in the input PIA and the minimum error among the multiple solutions generated for each input PIA.

A second difference from the Mardiana et al. approach is that an iteration is not used but rather, a simple search in PIA space about the nominally true value. In practice, estimates of the PIA can be obtained with the SRT (Meneghini et al. 2021), the Hitschfeld–Bordan (Hitschfeld and Bordan 1954) equations, or the dual-frequency method (Durden 2018). For this study, we assume a region in PIA space centered about the true PIA and search over a ±1-dB neighborhood about this point. For example, if, for a given profile, the true or SRT-estimated PIA is * A_{T}* = [

*A*(Ku),

_{T}*A*(Ka)] = (1, 6) dB, we step through values in

_{T}*space from 0 to 2 dB for the Ku-band input path attenuation and from 5 to 7 dB for the Ka-band path attenuation. We choose in particular, the following set of input attenuations:*

__A___{m}*i*= −10, −9,…, 0, 1, …10;

*j*= −10, −9, …, 0, 1, …10, where Δ

_{Ku}= Δ

_{Ka}= 0.1 dB. In other words,

*A*is assumed to take on 21 × 21 = 441 equally spaced values within ±1 dB of the true path attenuations. For each of these input PIAs, multiple solutions are generated, from which the minimum error solution, based on (20), is selected. The final step is simply to choose the smallest error among these 441 results. As in the single-input case, each PIA has associated with it profiles of

_{m}*D*, log

_{m}*N*, and

_{w}*R*so that the final profiles will be those corresponding to the minimum-error solution.

Four examples of the outputs from this procedure are shown in Figs. 12–15. In each example, the upper-left panel shows a plot of the minimum error along the *z* axis, given by (20), for various input path attenuations as shown on the *x* axis (Ku-band input PIA) and *y* axis (Ka-band input PIA). In Fig. 12, which uses the same input *Z _{m}* profiles shown in Fig. 2, the input Ku-band PIA ranges from 0 to 2 dB while the input Ka-band PIA ranges from 6.5 to 8.5 dB. (Although negative path attenuations are not physically meaningful, we have chosen to keep them to simplify the presentation of the results. Moreover, solutions to the equations can exist even with negative path attenuations.) At the center of the search region is the true path attenuation indicated by the red circle. The minimum-error solution is marked by the blue circle. The

*D*, log

_{m}*N*, and

_{w}*R*profiles associated with the minimum-error solution are in blue, those associated with the true PIA are in red while the input/true profiles, derived from the set of input raindrop size distributions, are in black. The minimum-error PIA in Fig. 12 is displaced from the true PIA by −1 dB in Ka band and by −0.1 dB in Ku band. This deviation from the true PIA usually produces errors in the profiles. In this case, however, the

*D*, log

_{m}*N*, and

_{w}*R*profiles obtained from the true path attenuation (red) differ only slightly from the results obtained from the consistency condition (blue). It should be noted that use of the true PIA, while giving slightly more accurate profiles, shows differences with the profiles in black. Results from the remaining 439 minimum variance solutions for the remaining input PIAs are represented by the light blue lines. The interpretation of these is different than the previous curves, since in this case, the lines represent the minimum error curves from the 439 set of input PIAs. These results show explicitly the error in the solutions when the input PIAs are displaced from the true value by offsets in the Ku- and Ka-band path attenuations of up to ±1 dB.

(top left) Error (*z* axis) between measured/input and output path attenuations for measured PIAs within ±1 dB of the true PIA, indicated by the red circle. The PIA corresponding to the minimum error solution is represented by the blue circle. *D _{m}*, log(

*N*), and

_{w}*R*profiles corresponding to the true PIA and minimum error PIA are given by the heavy red and blue lines, respectively. The true values derived from the input raindrop size distributions are shown by the black lines. Light blue lines represent the corresponding profiles from the remaining 439 PIA inputs.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

(top left) Error (*z* axis) between measured/input and output path attenuations for measured PIAs within ±1 dB of the true PIA, indicated by the red circle. The PIA corresponding to the minimum error solution is represented by the blue circle. *D _{m}*, log(

*N*), and

_{w}*R*profiles corresponding to the true PIA and minimum error PIA are given by the heavy red and blue lines, respectively. The true values derived from the input raindrop size distributions are shown by the black lines. Light blue lines represent the corresponding profiles from the remaining 439 PIA inputs.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

(top left) Error (*z* axis) between measured/input and output path attenuations for measured PIAs within ±1 dB of the true PIA, indicated by the red circle. The PIA corresponding to the minimum error solution is represented by the blue circle. *D _{m}*, log(

*N*), and

_{w}*R*profiles corresponding to the true PIA and minimum error PIA are given by the heavy red and blue lines, respectively. The true values derived from the input raindrop size distributions are shown by the black lines. Light blue lines represent the corresponding profiles from the remaining 439 PIA inputs.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

As in Fig. 12, but for a case where the true PIA and minimum error PIA are nearly identical.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

As in Fig. 12, but for a case where the true PIA and minimum error PIA are nearly identical.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

As in Fig. 12, but for a case where the true PIA and minimum error PIA are nearly identical.

Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

As in Fig. 12, but for a case where the input *D _{m}* profile is in the upper-root region of the DFR–

*D*curve.

_{m}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

As in Fig. 12, but for a case where the input *D _{m}* profile is in the upper-root region of the DFR–

*D*curve.

_{m}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

As in Fig. 12, but for a case where the input *D _{m}* profile is in the upper-root region of the DFR–

*D*curve.

_{m}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

(left) As in Fig. 12, but for a case where the majority of the *D _{m}* profile is in the lower-root region of the DFR–

*D*curve. (right)

_{m}*R*–

*D*results assuming that the measured and true PIAs are the same.

_{m}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

(left) As in Fig. 12, but for a case where the majority of the *D _{m}* profile is in the lower-root region of the DFR–

*D*curve. (right)

_{m}*R*–

*D*results assuming that the measured and true PIAs are the same.

_{m}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

(left) As in Fig. 12, but for a case where the majority of the *D _{m}* profile is in the lower-root region of the DFR–

*D*curve. (right)

_{m}*R*–

*D*results assuming that the measured and true PIAs are the same.

_{m}Citation: Journal of Atmospheric and Oceanic Technology 39, 9; 10.1175/JTECH-D-22-0002.1

A second example is shown in Fig. 13. In this case, the minimum error solution has a Ku-band path attenuation equal to the true value and a Ka-band path attenuation offset from the true value by 0.1 dB. Results in the other panels show only black and red lines because the red and blue lines, corresponding to the true input PIA and those from the consistency condition, are effectively identical. This case is fairly typical of other results that show that deviations from the true *D _{m}* and log

*N*profiles tend to cancel for the rain rate estimates, which in this case exhibits a high degree of accuracy.

_{w}For the case shown in Fig. 14 the *D _{m}* values are in the region where only the higher root of the equation

*D*=

_{m}*g*

^{−1}(DFR) exists. In this case, we obtain a single solution for each input PIA and the range of solutions for all 441 inputs show only a small amount of variability. This indicates that the profiles are relatively independent of the input PIA when the value is taken within ±1 dB of the true value. The forward recursion in this case yields no solution using either the true or the minimum variance PIA as input. This occurs frequently when the path attenuations are large.

Figure 15 shows a profile exhibiting the opposite kind of behavior where a large portion of the *D _{m}* profile is in the lower root region of the

*D*–DFR relationship. Whereas the true input PIA, represented by the red dashed lines, gives somewhat reasonable results, despite fluctuations in

_{m}*D*and log

_{m}*N*, the minimum variance solution shown by the heavy blue lines exhibits significant errors in the

_{w}*D*and log

_{m}*N*profiles. This example shows that the consistency condition can sometimes fail by the choosing the wrong root—in this case, the upper rather than the lower root. In contrast, the results from the

_{w}*R*–

*D*method shown in Fig. 15b, which is insensitive to the DFR–

_{m}*D*ambiguity, yields accurate estimates of

_{m}*D*, log

_{m}*N*, and

_{w}*R*. For this result, we have assumed that the measured differential path attenuation is equal to the true value.

## 4. Discussion and summary

By accounting for both roots of the *D _{m}* =

*g*

^{−1}(DFR) equation, multiple profiles of

*D*, log

_{m}*N*, and

_{w}*R*are obtained from the dual-frequency radar equations. For each profile, DSD-generated path attenuations are retrieved and compared with externally measured estimates,

*A*(

_{m}*f*

_{1}),

*A*(

_{m}*f*

_{2}) of the same quantities. If the

*A*are accurate, then the choice of that pair of DSD-generated path attenuations “closest,” as defined by the minimum of (20), to the

_{m}*A*yield profiles of

_{m}*D*, log

_{m}*N*, and

_{w}*R*that usually are accurate. Exceptions to this rule occur, however, particularly with the forward recursion at high path attenuations and for both forward and backward approaches when the majority of the

*D*profile resides in the lower branch of the

_{m}*D*–DFR curve.

_{m}Although the input set of raindrop size distributions are taken from measured disdrometer data, the assumption of a gamma distribution with *μ* = 3 often yields accurate results and the retrieval results of the paper have been obtained with this assumption. Tests of the retrievals with *μ* = 6 often yield comparable results but with slightly worse accuracy. On the other hand, the assumption of *μ* = 0 leads to a significant loss of accuracy and cases where no solution exists.

For the forward recursion or initial-value problem, the solution set is independent of the externally measured path attenuations, *A _{m}*, or the factors

*s*

_{Ku}and

*s*in (20). These quantities come into play only in selecting the “best” solution which can be thought of as choosing that DSD-generated path attenuation pair closest to the

_{N}*A*. Although examples can be found where the multiple solutions are tightly clustered, ensuring that all the solutions give approximately the same set of microphysical profiles, more generally, biases in the

_{m}*A*can lead to significant errors in the results. In fact, even when the

_{m}*A*are equal to the true values, the forward recursion often fails or becomes highly inaccurate when the path attenuations become large.

_{m}For the backward recursion, the situation is different because the *A _{m}* are incorporated into the final-value equation. Accurate microphysical and rain rate retrievals can usually be obtained if the

*A*are taken as the true values. For biases in

_{m}*A*, trial and error computations show that accurate retrievals usually can be made if the input

_{m}*A*are modified to match the DSD-generated output attenuations. In implementing this procedure, we have limited the search area to within a ±1 dB neighborhood of the true PIA. In practice, since the true PIAs are unknown, this would need to be replaced by proxy values, such as from the SRT, using a search area large enough to accommodate uncertainties in the estimates at the two frequencies.

_{m}As noted earlier, this procedure is similar to that of Mardiana et al. (2004). There are several differences, however. Mardiana et al. employed an iterative procedure beginning with zero Ku- and Ka-band path attenuations. However, because of the existence of multiple local minima for many profiles, where an iteration might fail to locate the global minimum, we have used a simple search procedure in the vicinity of the true or nominally true PIA. A second difference is that, like all earlier work, the Mardiana et al. study used only the bigger-root solution of the *D _{m}* =

*g*

^{−1}(DFR) equation. As we allow for multiple solutions, this requires selecting the minimum error solution for each input and then finding the minimum among these minima to obtain the final set of profiles and PIAs. While the consistency condition works well in many cases, examples such as that in Fig. 15a show that profiles in which

*D*is predominantly in the lower branch of the

_{m}*D*–DFR curve can give erroneous results.

_{m}A simplified form of version 6 of the *R*–*D _{m}* method (Seto et al. 2021) was implemented to identify advantages and disadvantages of the generalized DFR approach. In general, the

*R*–

*D*method works well and often provides results comparable to or with better accuracy than the DFR approach. Exceptions occur, however, for profiles that show nonuniformities in the range profiles of

_{m}*N*. Because of the constraint introduced by use of an

_{w}*N*–

_{w}*D*relationship, the method tends to produce profiles of

_{m}*N*that do not fully capture the variations in range. This translates into errors in the rain rate profile. It is worth noting that the dual-frequency version of

_{w}*R*–

*D*is sensitive to errors in the differential path attenuation,

_{m}*δA*, and a similar kind of consistency condition imposed on the input/output path attenuations might be applicable to the

*R*–

*D*backward-recursion equation, given by (25), to mitigate this dependence.

_{m}Although the DFR approach as described here explicitly accounts for the double-valued nature of the DFR–*D _{m}* relationship, it appears that the PIA alone, or the use of a consistency condition for the backward recursion, is not always sufficient to identify an accurate solution. These failed cases tend to be confined to profiles with small path attenuation, low

*D*, and relatively uniform

_{m}*N*profiles where the

_{w}*R*–

*D*method works well if the differential path attenuation can be accurately specified. These cases, nevertheless, raise the question as to whether additional constraints on the DFR approach might be needed, such as a restriction on the allowable variations in the

_{m}*N*–

_{w}*D*relationship along the profile. Indeed, comparing the effective

_{m}*N*–

_{w}*D*relationships obtained from the DFR and the

_{m}*R*–

*D*methods may help explain differences in the retrievals, either because of an overly constrained

_{m}*N*–

_{w}*D*in the

_{m}*R*–

*D*method or the lack of an

_{m}*N*–

_{w}*D*constraint in the DFR approach.

_{m}The appeal of the DFR approach is that, apart from the use of a gamma raindrop size distribution, it is independent of a priori assumptions or empirical relationships. Although the ambiguity in the DFR–*D _{m}* relationship can be handled analytically, finding an accurate solution among the multiplicity of solutions is not always possible even when an accurate PIA estimate is available or when a consistency condition is imposed. Nevertheless, the method provides insight into the characteristics of the solutions, their relationship to external PIA constraints, and the accuracy of existing operational methods.

## Acknowledgments.

We wish to thank Dr. Shinta Seto of Nagasaki University for his valuable comments. This work is funded under NASA’s Precipitation Measurement Mission (PMM) Grant NNH18ZDA001N‐PMMST.

## Data availability statement.

Information on the disdrometer data can be found at https://gpm-gv.gsfc.nasa.gov/Disdrometer. DSD data used to generate the simulated profiles and the simulated profiles themselves will be provided by the authors upon request.

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