## 1. Introduction

The close connection between dual-polarization and dual-wavelength radar algorithms can be understood from similarities in the two sets of measurements. Typically, two parameters of the hydrometeor size distribution are estimated from two independent radar reflectivity factor measurements. In the dual-polarization case, the independent data are the copolarized radar reflectivity factors at horizontal and vertical polarizations. The dual-wavelength radar provides radar reflectivity factors at two wavelengths at the same polarization state.

A further similarity between the two situations is that the measured or apparent reflectivity factors must be corrected for attenuation before the estimation of the size distribution parameters can be made. The attenuation correction can proceed either in the forward direction, with increasing radar range, or in the backward direction, starting from a final gate and progressing inward toward the radar. However, the forward-going solutions tend to be unstable because the attenuation out to the range of interest becomes “large” in some sense. This is analogous to the case of a single attenuating-wavelength radar where the forward solution to the Hitschfeld–Bordan (Hitschfeld and Bordan 1954) equation becomes unstable as the attenuation increases. To circumvent this problem, the equations can be expressed in a form that includes an independent estimate of path attenuation. For the dual-polarization radar, it has been shown that a measurement of the differential phase between horizontal and vertical polarizations (Testud et al. 2000; Bringi et al. 2001; Matrosov et al. 2002, 2005) provides a good estimate of path attenuation at the two polarizations. For airborne and spaceborne dual-wavelength measurements, the surface reference technique can be used to estimate path attenuation at both wavelengths.

In this paper, two types of attenuation correction procedures are formulated for application to dual-wavelength and dual-polarization weather radar data. In the integral equation approach, which has been used in the analysis of dual-wavelength airborne radar data, the parameters of the drop size distribution (DSD) are used to adjust the path attenuation to the adjacent range gate, which, in turn, is used to correct the measured reflectivity factors at that gate. In this way, a recursion procedure is defined. The most notable use of the *kZ* (specific attenuation–radar reflectivity factor) parameterization approach was provided by Hitschfeld and Bordan (1954) who analyzed the radar equation for a single attenuating-wavelength radar. Modification of this estimate to include an independent path attenuation constraint has led to its application in the analysis of airborne and spaceborne radar data. An important recent development is the application of methods of this type to polarimetric data. The approach has not, however, been widely used for the analysis of dual-wavelength radar data.

The primary objective of the paper is to make clear the relationships between the polarimetric and dual-wavelength equations in the presence of attenuation for both the integral equation and *kZ* parameterization approaches. We begin by writing the integral equations for the median mass diameter *D*_{0} and number concentration *N _{t}* that are applicable to both dual-polarization and dual-wavelength radar returns for the initial value and final value cases. This is followed by a similar development for the

*kZ*parameterization. For this case, however, only the final value version is discussed. Simulations of the retrievals are presented for the case of an X-band polarimetric radar with an emphasis on the differences between the backward solutions using the integral equation and

*kZ*parameterization.

## 2. Integral equations

*r*and frequency

*f*,

*Z*

_{m,pp}(

*r*,

*f*) (mm

^{6}m

^{−3}), when the transmit and receive polarization are along the direction

*p*, can be defined in terms of the radar return power

*P*

_{r,pp}(

*r*,

*f*) by

*C*is the radar constant and |

*K*|

_{w}^{2}is the dielectric factor of water, which, by the convention used here, 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 nonattenuated effective radar reflectivity factor, or simply radar reflectivity factor

*Z*, is related to

*Z*by

_{m}*k*,

_{r}*k*, and

_{c}*k*are the specific attenuations from precipitation, cloud water, and water vapor, respectively, and where the precipitation may include rain, snow, and mixed-phase hydrometeors. Throughout the paper, the copolarized return powers are denoted by the subscripts

_{v}*pp*, where the first subscript represents the polarization state of the transmitted wave and the second the polarization state of the received signal. Because only horizontal h and vertical v copolarized signals are considered,

*pp*is equal either to hh or vv. Because the cloud and water vapor attenuation are polarization independently, we omit subscripts on these quantities. The units of

*k*are taken to be decibels per kilometer so that

*c*= 0.2 ln(10) = 0.46. To further simplify the equations, we let

*r*

_{1}to

*r*

_{2}is

*A*(0,

_{pp}*r*;

*f*) =

*A*(

_{pp}*r*,

*f*) is used. It should be noted that the

*n*th gate, at range

*r*, is taken as the final gate of the path so that the total path-integrated attenuation (PIA) is

_{n}*A*(

_{pp}*r*,

_{n}*f*). Next, the raindrop diameter distribution

*N*(

*D*,

*s*) (m

^{−3}mm

^{−1}) is expressed as the product of the particle number concentration

*N*(m

_{t}^{−3}), and a normalized size distribution

*n*(

*D*) (mm

^{−1}),

*μ*are functions of the radar range. In the numerical results presented later, we use the median mass diameter

*D*

_{0}(mm), related to Λ,

*μ*, by (Ulbrich 1983)

*N*:

_{t}*c*=

_{Z}*c*

_{0}

^{4}/(

*f*

^{4}

*π*

^{5}|

*K*|

_{w}^{2}) and

*c*= 4.343 × 10

_{e}^{−3}, and where

*c*

_{0}is the speed of light. Note that

*k*

_{r,pp}=

*N*

_{t}I_{e,pp}and

*Z̃*=

_{pp}*I*

_{b,pp}+ 10 log

*N*.

_{t}*D*

_{0}and

*N*for the dual-polarization case can be obtained by writing

_{t}*Z̃*

_{m,pp}(

*r*,

*f*) and

*Z̃*

_{m,hh}(

*r*,

*f*) −

*Z̃*

_{m,vv}(

*r*,

*f*) in terms of

*D*

_{0}and

*N*and expressing the path attenuation to range

_{t}*r*in the form (Meneghini et al. 1992)

*r*to

*r*is written in terms of the DSD parameters at the range gates within this range interval. This is the essence of the backward integral equation approach, where path attenuation to range

_{n}*r*is found from an estimate of the total path attenuation and the DSD parameters in the range from

*r*to

*r*. The equations can be written in the form

_{n}*δ*is a difference operator, defined in the case of dual-polarization radar by

*δX*≡

*δ*≡

_{p}X*X*

_{hh}−

*X*

_{vv}and in the case of a dual-frequency radar by

*δX*≡

*δ*≡

_{f}X*X*(

*f*

_{1}) −

*X*(

*f*

_{2}). Note that in the case of the polarization radar,

*δ*(

*A*+

_{c}*A*) =

_{v}*δ*(

_{p}*A*+

_{c}*A*) = 0. Using the definition of

_{v}*I*given by (9), the usual differential reflectivity (dB) with respect to polarization is

_{b}*δI*=

_{b}*Z*

_{dr}, while the dual-frequency ratio (dB) is

*δI*= DFR.

_{b}*h*

_{1},

*h*

_{2}should be changed to

*h*

_{1}and

*h*

_{2}in (18) and (19) are identical. However, if an independent estimate is made of the precipitation attenuation only,

*A*(

_{r}*r*), then contributions from cloud water and water vapor to range

_{n}*r*must be added, as in (18). If an independent estimate is made of total attenuation, then contributions from cloud water and water vapor over the range interval (

*r*,

*r*) must be subtracted, as in (19). For example, in polarimetric applications, the differential phase is well correlated with the differential attenuation from precipitation, but will be unaffected by cloud or water vapor attenuation. In contrast, for dual-wavelength airborne or spaceborne applications, if the surface reference is taken in a rain- and cloud-free environment with low water vapor, the decrease in the surface return in the presence of precipitation can be associated with the total path attenuation.

_{n}If the path attenuation is independently measured, then it can be seen from (16) and (17) that the range profiles of *D*_{0} and *N _{t}* can be obtained by starting at the far range

*r*, continuing inward toward the radar. At

_{n}*r*=

*r*, the integrals appearing in (16) and (17) are zero, so that

_{n}*D*

_{0}can be found by numerically solving the equation

*g*

_{1}(

*D*

_{0},

*μ*) =

*h*

_{1}; once

*D*

_{0}is determined, it is substituted into (17) to give

*N*. Proceeding to the (

_{t}*n*− 1)th gate, the values of

*D*

_{0}and

*N*from the

_{t}*n*th gate are substituted into the integrals in (16) and (17); because the right-hand side of (16) is determined,

*D*

_{0}can be solved numerically. Substituting this into (17) gives

*h*

_{2}and

*N*. The recursion continues in this way until the full path is traversed.

_{t}In solving the equations numerically, the discrete forms of (16) and (17) take the form of nonlinear algebraic equations for *D*_{0} and *N _{t}* that can be solved by Broyden’s method (Press et al. 1992). For example, at the final gate, both (16) and (17) are functions of

*D*

_{0}(

*r*) and

_{n}*N*(

_{t}*r*) if the contributions from the last gate are included. However, if the attenuation per range gate is small, the approximate and general procedures yield nearly identical results. It should also be pointed out that in some cases, such as the dual-wavelength radar returns in rain or mixed-phase hydrometeors, there can be more than one value of

_{n}*D*

_{0}that satisfies (16). Procedures exist to reduce the ambiguities but not eliminate them entirely (Liao and Meneghini 2005). Moreover, because there are only two equations, the “shape” parameter

*μ*must either be fixed or expressed as a function of the other DSD parameters (Zhang et al. 2001; Seifert 2005).

*g*

_{1},

*g*

_{2},

*f*

_{1},

*f*

_{2}are given by (18) and

*r*] for the forward (left-hand side) and backward (right-hand side) integral equations:

*r*in the forward equations is expressed in terms of the size distribution parameters obtained from prior gates between the range of interest and the radar. Despite the similarities between (16)–(18) and (21)–(23), the two formulations show significant differences in performance, as will be shown in section 4.

## 3. Equations based on the *kZ* parameterization

*α*adjustment” solution (Meneghini et al. 1983; Iguchi and Meneghini 1994), where, taking 10 log of (19) of Iguchi and Meneghini (1994), gives (

*r*≤

*r*)

_{n}*β*must be taken as constant in range, whereas

*α*is allowed to vary in range. An example of this is a path along which distinct regions of frozen, mixed-phase, and liquid precipitation are present. However, if

*α*can be considered constant with range, then (24) is independent of this parameter.

*α*along the path, (24) can also be obtained by taking the expression for

*k*from the final value solution [(24) of Testud et al. (2000)], integrating it from 0 to

*r*and using the definition of

*Z*:

_{m}*A*(

*r*) can be estimated by the differential phase shift over the path ΔΦ

_{n}_{dp}(°) by using a relationship between

*k*and

*κ*

_{dp}(° km

^{−1}) (Testud et al. 2000; Matrosov et al. 2002; Anagnostou et al. 2004). In the case of a linear

*k*–

*κ*

_{dp}relationship:

*A*(

*r*) instead of ΔΦ

_{n}_{dp}. In deriving equations similar to those obtained for the integral equation approach, it is convenient to write (27) in the form

*Z̃*

_{m,pp}(

*r*,

*f*) and

*Z̃*

_{m,hh}(

*r*,

*f*) −

*Z̃*

_{m,vv}(

*r*,

*f*) in terms of

*D*

_{0}and

*N*:

_{t}*Ñ*≡ 10 log

_{t}*N*.

_{t}*kZ*backward formulation, equations that include dual-frequency and dual-polarization cases can be written in a form similar to (16) and (17):

*g*

_{1},

*g*

_{2},

*h*

_{1}, and

*h*

_{2}are given by (18). In the dual-frequency radar case, the last term in (38) becomes

*r*to

*r*. In the (backward) integral equation approach, the interval attenuations are expressed as functions of the DSD parameters obtained from previous steps in the recursion. In the

_{n}*kZ*formulation, the attenuations are estimated by means of the

*kZ*parameterization using path attenuation and measured radar reflectivity factors in the range from

*r*to

*r*. In both of these backward-going solutions, the contributions are subtracted from the total path attenuation to obtain an estimate of attenuation to range

_{n}*r*.

It is worth pointing out that forward equations based on the *kZ* parameterization can be obtained directly from the Hitschfeld–Bordan equation, and can be written in a form similar to that of (38) and (39). However, the estimates for *N _{t}* and

*D*

_{0}based on these equations are highly sensitive to attenuation and errors in the various parameters, so that these equations will not be considered here.

## 4. Comparisons of formulations for an X-band polarimetric radar

To illustrate some aspects of the solutions to the equations based on the integral and *kZ* formulations, we construct a simulation for an X-band polarimetric radar using disdrometer-measured raindrop size distributions as the input data. We assume a 50-km path consisting of 250 gates with 0.2-km range resolution. In the general case, a sequence of 250 one-minute-averaged DSDs provides the particle number concentration and median mass diameter at each range gate. However, to better understand the behavior of the equations, the DSD parameters, and therefore the rain rates, are assumed to be constant in range. Assuming the Beard and Chuang (1987) shape–size relationship and a fixed *μ* value along the path, the simulated range profiles of the measured radar reflectivity factors at the two polarizations are calculated, that is, [*Z*_{m,hh}(*r*_{j}), *Z*_{m,vv}(*r*_{j})]; *j* = 1, . . . , 250. To further simplify the discussion, we assume infinite signal-to-noise ratios without fluctuations in the *Z* fields from finite sampling. Moreover, the shape–size relationship is assumed to be exact and without raindrop canting.

*kZ*approaches reduce to the same set of equations at the

*n*th gate. In particular, (12) and (34) at

*r*=

*r*reduce to

_{n}*Z*

_{dr}=

*I*

_{b,hh}−

*I*

_{b,vv}is unbiased. However, if

*μ*>

*μ*, where the

_{T}*T*subscript denotes the true or input value, then it can be seen from the top left-hand plot in Fig. 1 that

*D*

_{0}>

*D*

_{0,T}. Also, because

*I*,

_{b}*I*,

_{e}*δI*are derived from the estimated median mass diameter, these quantities will also be positively biased as can be seen from an inspection of the

_{e}*I*,

_{b}*I*, and

_{e}*δI*plots in Fig. 1. From (43) it follows that an overestimate in

_{e}*I*, in the absence of other errors, yields an underestimate in

_{b}*N*. The opposite behavior occurs if

_{t}*μ*<

*μ*. These relationships can be summarized by the following inequalities:

_{T}*μ*are unbiased, then

*D*

_{0},

*I*are unbiased. In this case the bias in

_{b}*Ñ*is determined solely by the bias in the quantity

_{t}*Z̃*+

_{m}*A*(

*r*).

_{n}Although the two formulations yield the same results at the final gate, the estimates generally exhibit significant differences with range. Figures 2 and 3 show the range dependence of rain rate (Fig. 2) and *D*_{0} (Fig. 3) estimates for *μ* = (0,2,6), where *μ _{T}* = 2. In all cases, the rain rate is derived from the estimated (

*N*,

_{t}*D*

_{0}) and assumed

*μ*value. Input values of rain rate, path attenuation, and differential path attenuation are

*R*= 12.5 mm h

_{T}^{–1};

*A*

_{hh,T}(

*r*) = 33 dB;

_{n}*A*

_{hh,T}(

*r*) −

_{n}*A*

_{vv,T}(

*r*) = 5.6 dB. In each figure, the results from the

_{n}*kZ*parameterization [(34) and (35)] are shown in the upper panel, and results from the integral equations [(12) and (13)] are shown in the lower panel. Unless stated otherwise, all results are obtained from a backward recursion. To understand why the results from the

*kZ*parameterization are range independent, note that (34) and (35) are identical to (42) and (43) except for the attenuation correction terms involving

*Q̃*. But, these terms are determined solely by the path attenuations and the measured radar reflectivity factors, and are independent of the DSD parameters derived at other range gates. As will be seen in subsequent examples, only errors in the path attenuation yield range dependencies in the

*kZ*attenuation correction method.

One other feature of Fig. 2 and subsequent results is that even in the absence of errors, the estimated quantity (in this case rain rate) differs from the input value. There are two reasons for this. The first is that *D*_{0} is estimated by means of a third-order polynomial in *Z*_{dr}. Although the fit is fairly accurate, small errors in *D*_{0} can translate into larger errors in rain rate. A second source of error arises in estimating *I _{b}*,

*I*, and

_{e}*δI*from

_{e}*D*

_{0}via lookup tables and a linear interpolation. (For the

*kZ*parameterization only the

*I*−

_{b}*D*

_{0}relationship is needed.) Improvements in accuracy can be made by replacing the

*D*

_{0}−

*Z*

_{dr}fit with a lookup table and using more finely sampled lookup tables for estimating

*I*,

_{b}*I*, and

_{e}*δI*.

_{e}For the integral equations, the *D*_{0} estimated from (12) is used to determine both *I _{b}* and

*I*that are then used in (13). Moreover, the

_{e}*N*obtained in (13), along with the

_{t}*D*

_{0}from previous steps, determines the differential attenuation term given by the last term in (12). This strong linkage between the equations usually produces a negative feedback where the biases are reduced in magnitude when progressing toward ranges closer to the radar. To see this in detail in the present case, consider the

*μ*= 6 example. As noted above, because

*μ*>

*μ*,

_{T}*D*

_{0}is overestimated and

*N*is underestimated at the

_{t}*n*th gate. At the (

*n*− 1)th gate, the negatively biased

*N*value, along with the positively biased

_{t}*δI*term, is used to determine the differential attenuation term in (12). Initially, at the far ranges, this produces a somewhat larger value than the true value and therefore a smaller value for the right-hand side of (12). This yields, in turn, a slight decrease in

_{e}*D*

_{0}from its value at the

*n*th gate. From Fig. 1, a decrease in

*D*

_{0}produces a decrease in

*I*and, in accordance with (13), a reduction in the negative bias of

_{b}*N*. For the curves in the bottom panel of Fig. 2, an inspection of the numerical results shows that feedback becomes slightly positive for ranges less than 18.4 km for

_{t}*μ*= 6 and less than 19.2 km for

*μ*= 0.

Negative feedback in the integral equation solutions also occurs for offset errors in *Z*. For the example shown in Figs. 4 –6, the bias in the differential reflectivity factor is assumed to zero, but with offsets in *Z̃*_{hh}, *Z̃*_{vv} of +2, −2, or 0. The input data for these examples are *R _{T}* = 10.5 mm h

^{–1};

*A*

_{hh,T}(

*r*) = 17.5 dB;

_{n}*A*

_{hh,T}(

*r*) −

_{n}*A*

_{vv,T}(

*r*) = 2.2 dB. The range dependences of

_{n}*R*,

*D*

_{0}, and

*Ñ*are shown in Figs. 4 –6, respectively. At the final gate, and for both solutions,

_{t}*D*

_{0}is unbiased while the biases in

*N*are determined by the biases in the

_{t}*Z̃*

_{hh},

*Z̃*

_{vv}terms. In the

*kZ*approach,

*D*

_{0}remains unbiased over the full range and the biases in

*R*and

*N*remain constant with range. For the integral equation solution, in the negatively biased

_{t}*Z*case, the underestimate in

*N*produces an underestimate in the differential attenuation term ∫

_{t}*in (12), which produces an overestimate in*

^{rn}_{r}N_{t}δI_{e}ds*D*

_{0}as shown in Fig. 5. This overestimate in

*D*

_{0}produces an overestimate in

*I*, which, according to (13), reduces the bias in

_{b}*N*. For positive

_{t}*Z̃*

_{hh},

*Z̃*

_{vv}biases, the effects are reversed; in particular, a positive bias in

*Z̃*

_{hh},

*Z̃*

_{vv}produces an identical bias in

*Ñ*. This leads to underestimates in

_{t}*D*

_{0}and

*I*that reduce the positive bias in

_{b}*Ñ*.

_{t}An inspection of (12) and (13) shows that a bias in *Z̃*_{hh}(*r*) has the same effect on the solutions as does an identical bias in *A*_{hh}(*r _{n}*). Likewise, a bias in

*Z̃*

_{hh}(

*r*) −

*Z̃*

_{vv}(

*r*) is equivalent to a bias in

*A*

_{hh}(

*r*) −

_{n}*A*

_{vv}(

*r*). This equivalence does not hold for the

_{n}*kZ*formulation. As seen in the previous example, offsets in

*Z̃*

_{hh}(

*r*) produce

*kZ*-derived solutions that are constant in range. Offsets in

*A*

_{hh}(

*r*), however, produce range-dependent solutions. Moreover, unlike the integral equation method, the

_{n}*kZ*-based results converge to the input values as the radar range goes to zero. This behavior is shown in Fig. 7, where the

*kZ*solutions are shown for

*R*,

*D*

_{0}, and

*N*for offsets in

_{t}*A*

_{hh}(

*r*),

_{n}*A*

_{vv}(

*r*) of 2, −2, and 0 dB. The integral equation results are displayed in the lower panels of Figs. 4 –6 because they are identical to the

_{n}*Z*-offset case. To understand the behavior of the

*kZ*solutions in this case, note that the log(

*Q̃*) and

*δ*log(

*Q̃*) terms in (34) and (35) are functions of the path attenuations that play the same role as the corresponding terms in the integral equations. In particular, these terms function as negative feedback, reducing the magnitude of the biases as the radar range decreases. Unlike the integral equations, however, the

*kZ*formulation yields exact solutions in the absence of other errors. This follows from (36) by noting that as

*r*→ 0,

*Q̃*= 10

_{pp}^{0.1βpp}

*A*(

_{pp}*r*), so that the last term in (35) exactly cancels the quantity

_{n}*A*(

*r*). In a similar way, the last term in (34) cancels the quantity

_{n}*A*

_{hh}(

*r*) −

_{n}*A*

_{vv}(

*r*) so that the equations reduce to the dual-polarization equations in the absence of attenuation.

_{n}In the examples given, moderate rain rates were used. Similar characteristics of the solutions are observed at lighter rain rates and path attenuations. However, because of the nonlinear nature of the equations, the qualitative behavior of the solution can change abruptly as the rain rate and path attenuation increase. For the rain-rate estimates shown in Fig. 8, values of *μ* = [0, 2, 6] are assumed, where *μ _{T}* = 2. In this case, the input rain rate, path attenuation, and differential path attenuation are given by

*R*= 21.5 mm h

_{T}^{–1};

*A*

_{hh,T}(

*r*) = 65 dB;

_{n}*A*

_{hh,T}(

*r*) −

_{n}*A*

_{vv,T}(

*r*) = 12 dB. Recall that the same assumptions regarding

_{n}*μ*are used for the results in the lighter rain-rate case shown in Fig. 2. Except for evidence of numerical instabilities in the

*kZ*formulation at the far ranges (Fig. 8, top panel), the results are qualitatively similar to those in Fig. 2 (top panel). In contrast, the integral equation solution, shown in the bottom panel of Fig. 8, exhibits a slight oscillatory behavior so that in moving toward the radar from the far range the error first decreases, attaining a minimum at about 36 km, and then begins gradually to increase, attaining a maximum at 19.4 km for

*μ*= 0 and a maximum at 8.6 km for

*μ*= 6. At closer ranges, the error once more decreases.

*A*

_{hh}and

*A*

_{vv}are either both positively or negatively biased by 2 dB. The

*kZ*solution is qualitatively similar to that shown in Fig. 7 (top panel) except for numerical instabilities at the far ranges. However, unlike the result shown in the bottom panel of Fig. 4, the integral equation solution for rain rate (as well as for

*D*

_{0}and

*N*) exhibits a damped oscillatory behavior about the true value. As the rain rate and path attenuation are further increased, the integral equations continue to yield damped oscillatory behavior, but with an increase in amplitude and frequency. For the

_{t}*kZ*solution, numerical instabilities restrict the solution to ranges near the radar. The reason for the instability can be seen from (36) where, for large values of attenuation,

*S*(

_{pp}*r*) rapidly approaches

*S*(

_{pp}*r*) as

_{n}*r*increases, so that

*Q̃*is computed from the difference of large quantities that are nearly equal in magnitude. An improvement in stability can be obtained by writing (36) in the form

*S*(

_{pp}*r*) −

_{n}*S*(

_{pp}*r*) rapidly approaches zero as

*r*→

*r*, so that

_{n}*Q̃*is computed from the multiplication of a very large and a very small number.

To investigate the oscillatory nature of the integral equation solutions for large path attenuation, we use phase-state diagrams (Fig. 10) in which the (*D*_{0}, *N _{t}*) values as a function of radar range are represented by a trajectory in (

*D*

_{0},

*N*) space. The input values of (

_{t}*D*

_{0},

*N*), assumed constant in range, is represented by an “X” and the range-dependent solution to the integral equation are represented by a curve that begins at 50 km (indicated by the box around the point), which spirals in a counterclockwise direction toward the input value. For the upper panels in Fig. 10, errors in

_{t}*Z*

_{hh},

*Z*

_{vv}of 2 dB are assumed; for the lower panels errors in

*Z*

_{hh},

*Z*

_{vv}of −2 dB are assumed. In moving from left to right, the input

*D*

_{0}increases from 2.1 (left) to 2.4 (middle) to 2.7 (right) mm. For all cases, the assumed value of

*N*is taken to be 600 m

_{t}^{−3}. The results show that as

*D*

_{0}and path attenuation increase, the amplitude and frequency of the oscillations increase. On the other hand, as the radar range decreases, the amplitude is damped and the solution spirals in toward the input values. As pointed out earlier, at the farthest range (50 km), the value of

*D*

_{0}is unbiased while the bias in

*N*is determined by the bias in the radar reflectivity factors. Although our focus is on the behavior of the dual-polarization equations, it is worth noting that for the dual-wavelength integral equations, oscillatory solutions do not occur and that the error decreases uniformly with decreasing range. This appears to result from the differences in sign between the

_{t}*δI*,

_{b}*δI*for the two situations; in particular, for the dual-polarization data, these quantities are of the same sign (as seen by the data in upper and lower left-hand panels of Fig. 1), whereas in the dual-wavelength tables for 13.6 and 35.5 GHz,

_{e}*δI*,

_{b}*δI*are of opposite sign for

_{e}*D*

_{0}greater than about 1 mm.

Although the backward formulations are generally preferable because of their more robust behavior, some features of the forward integral equations are worth noting. As mentioned in section 2, for the backward recursion, the interval attenuation [*r*, *r _{n}*] is subtracted from the total attenuation, while in the forward case it is simply the interval attenuation [0,

*r*] that is used in the equations. In the backward recursion, the bias errors are usually reduced by negative feedback; in the forward recursion, however, the bias errors from this term are usually amplified by positive feedback. This difference between the formulations leads to behavior in which errors in the forward solution grow rapidly with increasing attenuation. On the other hand, because the method does not require an estimate of path attenuation, it can have better accuracies than the backward solutions for either light rain rates or when the independent estimate of path attenuation is inaccurate. In Figs. 11 –12 estimates of

*R*,

*D*

_{0}, and

*N*are shown for the forward and backward integral equations, respectively, for the case of calibration errors in

_{t}*Z*of 0 and ±2 dB. Even in this relatively light rain-rate case [

*R*= 4.1 mm/h;

_{T}*A*

_{hh,T}(

*r*) = 5 dB;

_{n}*A*

_{hh,T}(

*r*) −

_{n}*A*

_{vv,T}(

*r*) = 0.6 dB], instabilities in the forward estimates can be seen, particularly in the

_{n}*D*

_{0}estimates shown in the center panel of Fig. 11. On the other hand, in the absence of errors in

*Z*but with errors in the path attenuations, the forward estimates (given by the zero offset case in Fig. 11) would generally be more accurate than those from either of the backward recursion methods.

As noted by Iguchi (2005), the backward integral equations without path attenuation constraints (Mardiana et al. 2004) are mathematically the same as the forward integral equations given here. For light rain-rate cases, this has been verified using the present simulation. Although the solutions diverge as the rain rate increases, this can be attributed to numerical instabilities in both formulations. For the modified backward iterative procedure of Rose and Chandrasekar (2006), we expect a similar equivalence to an appropriately modified forward procedure. However, this has not been checked.

## 5. Discussion and summary

Integral equations for the parameters of the particle size distribution have several useful features in that they explicitly include path attenuation constraints and provide attenuation correction in terms of the particle size distribution parameters determined in earlier steps (range gates) of the procedure. Because the dual-wavelength and dual-polarization radar data are governed by essentially the same equations, a common theoretical framework is provided by which errors in the retrievals can be assessed. This should be beneficial to the proposed Global Precipitation Measurement Mission (Iguchi et al. 2002) where quantities derived from a dual-wavelength spaceborne radar can be expected to be compared with similar quantities derived from ground-based dual-polarization radars. Making good use of these data will depend on an understanding of the inherent errors in both spaceborne and ground-based algorithms.

By using the *kZ* parameterization, similar sets of equations applicable to dual-wavelength and dual-polarization radars can be derived. For the polarization radar, these equations are similar in content to those derived by Testud et al. (2000) and Bringi et al. (2001), and recently analyzed by Gorgucci and Chandrasekar (2005). As illustrated in the examples of section 4, despite differences, the two formulations function in a somewhat similar manner. Advantages of the integral equation approach were noted in cases of errors in the shape parameter *μ* or in *Z*. On the other hand, the *kZ* formulation was seen to be more accurate than the integral equation solution in the presence of errors in path attenuation.

In a study comparing what is here called the *kZ* formulation with an attenuation correction obtained directly from the differential phase estimate (Matrosov et al. 2002), Gorgucci and Chandrasekar (2005) concluded that neither approach was best in all cases. A similar conclusion can be drawn for the *kZ* and integral equation approaches, implying that for polarimetric data at attenuating wavelengths, comparisons among the three approaches should be useful as a diagnostic tool. Comparisons of results from *kZ* and integral equation formulations should also be useful for dual-wavelength data.

It is worth noting that apart from the integral equation and *kZ* parameterization formulations, other dual-wavelength techniques have been proposed (e.g., Marzoug and Amayenc 1994; Adhikari and Nakamura 2003; Grecu and Anagnostou 2004; Iguchi 2005). In view of the close relationship between dual-wavelength and dual-polarization algorithms, some of these formulations may also be applicable to both types of data.

## REFERENCES

Adhikari, N. B., and Nakamura K. , 2003: Simulation-based analysis of rainrate estimation errors in dual-wavelength precipitation radar from space.

,*Radio Sci.***38****.**1066, doi:10.1029/2002RS002775.Anagnostou, E. N., Anagnostou M. N. , Krajewski W. F. , Kruger A. , and Miriovsky B. J. , 2004: High-resolution rainfall estimation from X-band polarimetric radar measurements.

,*J. Hydrometeor.***5****,**110–128.Battan, L. J., 1973:

*Radar Observation of the Atmosphere*. University of Chicago Press, 324 pp.Beard, K. V., and Chuang C. , 1987: A new model for the equilibrium shape of raindrops.

,*J. Atmos. Sci.***44****,**1509–1524.Bringi, V. N., and Chandrasekar V. , 2001:

*Polarimetric Doppler Weather Radar: Principles and Applications*. Cambridge University Press, 636 pp.Bringi, V. N., Keenan T. D. , and Chandrasekar V. , 2001: Correcting C-band radar reflectivity and differential reflectivity data for rain attenuation: A self-consistent method with constraints.

,*IEEE Trans. Geosci. Remote Sens.***39****,**1906–1915.Gorgucci, E., and Chandrasekar V. , 2005: Evaluation of attenuation correction methodology for dual-polarization radars: Application to X-band systems.

,*J. Atmos. Oceanic Technol.***22****,**1195–1206.Grecu, M., and Anagnostou E. N. , 2004: A differential attenuation based algorithm for estimating precipitation from dual-wavelength spaceborne radar.

,*Can. J. Remote Sens.***30****,**697–705.Hitschfeld, W., and Bordan J. , 1954: Errors inherent in the radar measurement of rainfall at attenuating wavelengths.

,*J. Meteor.***11****,**58–67.Iguchi, T., 2005: A possible algorithm for the dual-frequency radar on the Global Precipitation Mission.

*Proc. 32d Conf. on Radar Meteorology*, Albuquerque, NM, Amer. Meteor. Soc., CD-ROM, 5R.4.Iguchi, T., and Meneghini R. , 1994: Intercomparison of single-frequency methods for retrieving a vertical rain profile from airborne or spaceborne radar data.

,*J. Atmos. Oceanic Technol.***11****,**1507–1516.Iguchi, T., Oki R. , Smith E. A. , and Furuhama Y. , 2002: Global Precipitation Measurement program and the development of dual-frequency precipitation radar.

,*J. Commun. Res. Lab. (Japan)***49****,**37–45.Liao, L., and Meneghini R. , 2005: A study of air/space-borne dual-wavelength radar for estimates of rain profiles.

,*Adv. Atmos. Sci.***22****,**841–851.Lim, S., and Chandrasekar V. , 2006: A dual-polarization rain profiling algorithm.

,*IEEE Trans. Geosci. Remote Sens.***44****,**1011–1021.Mardiana, R., Iguchi T. , and Takahashi N. , 2004: A dual-frequency rain profiling algorithm without the use of the surface reference technique.

,*IEEE Trans. Geosci. Remote Sens.***42****,**2214–2225.Marzoug, M., and Amayenc P. , 1991: Improved range-profiling algorithm of rainfall rate from a spaceborne radar with path-integrated attenuation constraint.

,*IEEE Trans. Geosci. Remote Sens.***29****,**584–592.Marzoug, M., and Amayenc P. , 1994: A class of single- and dual-frequency algorithms for rain-rate profiling from a spaceborne radar. Part I: Principle and tests from numerical simulations.

,*J. Atmos. Oceanic Technol.***11****,**1480–1506.Matrosov, S. Y., Clark K. A. , Martner B. E. , and Tokay A. , 2002: X-band polarimetric radar measurements of rainfall.

,*J. Appl. Meteor.***41****,**941–952.Matrosov, S. Y., Kingsmill D. E. , Martner B. E. , and Ralph F. M. , 2005: The utility of X-band polarimetric radar for quantitative estimates of rainfall parameters.

,*J. Hydrometeor.***6****,**248–262.Meneghini, R., Eckerman J. , and Atlas D. , 1983: Determination of rain rate from a space-borne radar using measurements of total attenuation.

,*IEEE Trans. Geosci. Remote Sens.***21****,**34–43.Meneghini, R., Kozu T. , Kumagai H. , and Boncyk W. C. , 1992: A study of rain estimation methods from space using dual-wavelength radar measurements at near-nadir incidence over ocean.

,*J. Atmos. Oceanic Technol.***9****,**364–382.Press, W. H., Teukolsky S. A. , Vetterling W. T. , and Flannery B. P. , 1992:

*Numerical Recipes in FORTRAN*. 2d ed. Cambridge University Press, 933 pp.Rose, C. R., and Chandrasekar V. , 2006: Extension of GPM dual-frequency iterative retrieval method with DSD-profile constraint.

,*IEEE Trans. Geosci. Remote Sens.***44****,**328–335.Seifert, A., 2005: On the shape–size relation of drop size distributions in convective rain.

,*J. Appl. Meteor.***44****,**1146–1151.Testud, J., Le Bouar E. , Obligis E. , and Ali-Mehenni M. , 2000: The rain profiling algorithm applied to polarimetric weather radar.

,*J. Atmos. Oceanic Technol.***17****,**332–356.Ulbrich, C. W., 1983: Natural variations in the analytical form of the raindrop size distribution.

,*J. Climate Appl. Meteor.***22****,**1764–1775.Zhang, G., Vivekanandan J. , and Brandes E. , 2001: A method for estimating rain rate and drop size distribution from polarimetric radar measurements.

,*IEEE Trans. Geosci. Remote Sens.***39****,**830–841.

Rain-rate estimates from the (top) *kZ* and (bottom) integral equation formulations for three values of the assumed shape parameter with input value, *μ _{T}* = 2.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Rain-rate estimates from the (top) *kZ* and (bottom) integral equation formulations for three values of the assumed shape parameter with input value, *μ _{T}* = 2.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Rain-rate estimates from the (top) *kZ* and (bottom) integral equation formulations for three values of the assumed shape parameter with input value, *μ _{T}* = 2.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of median mass diameter from the (top) *kZ* and (bottom) integral equation formulations for three values of the assumed shape parameter, with input value *μ _{T}* = 2.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of median mass diameter from the (top) *kZ* and (bottom) integral equation formulations for three values of the assumed shape parameter, with input value *μ _{T}* = 2.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of median mass diameter from the (top) *kZ* and (bottom) integral equation formulations for three values of the assumed shape parameter, with input value *μ _{T}* = 2.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Rain-rate estimates from the (top) *kZ* and (bottom) integral equation formulations for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Rain-rate estimates from the (top) *kZ* and (bottom) integral equation formulations for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Rain-rate estimates from the (top) *kZ* and (bottom) integral equation formulations for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of median mass diameter from the (top) *kZ* and (bottom) integral equation formulations for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of median mass diameter from the (top) *kZ* and (bottom) integral equation formulations for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of median mass diameter from the (top) *kZ* and (bottom) integral equation formulations for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of particle number concentration from the (top) *kZ* and (bottom) integral equation formulations for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of particle number concentration from the (top) *kZ* and (bottom) integral equation formulations for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of particle number concentration from the (top) *kZ* and (bottom) integral equation formulations for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of (top) rain rate, (middle) median mass diameter, and (bottom) number concentration using the *kZ* formulation for errors in the path attenuations.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of (top) rain rate, (middle) median mass diameter, and (bottom) number concentration using the *kZ* formulation for errors in the path attenuations.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of (top) rain rate, (middle) median mass diameter, and (bottom) number concentration using the *kZ* formulation for errors in the path attenuations.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

*kZ* and (bottom) integral equation formulations for three values of the assumed shape parameter with input value, *μ _{T}* = 2.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

*kZ* and (bottom) integral equation formulations for three values of the assumed shape parameter with input value, *μ _{T}* = 2.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

*kZ* and (bottom) integral equation formulations for three values of the assumed shape parameter with input value, *μ _{T}* = 2.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Rain-rate estimates from the (top) *kZ* and (bottom) integral equation formulations for errors in the path attenuations.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Rain-rate estimates from the (top) *kZ* and (bottom) integral equation formulations for errors in the path attenuations.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Rain-rate estimates from the (top) *kZ* and (bottom) integral equation formulations for errors in the path attenuations.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Solutions of integral equation for (*D*_{0}, *N _{t}*). Solution at 50 km from the radar is enclosed by box; counterclockwise trajectory from this point represents decreasing radar range. Input (

*D*

_{0},

*N*) value is represented by “X.” (top) Solutions for +2 dB offset in

_{t}*Z*

_{hh},

*Z*

_{vv}; (bottom) solutions for −2 dB offset in

*Z*

_{hh},

*Z*

_{vv}. Input values of

*D*

_{0}from (left) 2.1, (middle) 2.4, and (right) 2.7 mm.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Solutions of integral equation for (*D*_{0}, *N _{t}*). Solution at 50 km from the radar is enclosed by box; counterclockwise trajectory from this point represents decreasing radar range. Input (

*D*

_{0},

*N*) value is represented by “X.” (top) Solutions for +2 dB offset in

_{t}*Z*

_{hh},

*Z*

_{vv}; (bottom) solutions for −2 dB offset in

*Z*

_{hh},

*Z*

_{vv}. Input values of

*D*

_{0}from (left) 2.1, (middle) 2.4, and (right) 2.7 mm.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Solutions of integral equation for (*D*_{0}, *N _{t}*). Solution at 50 km from the radar is enclosed by box; counterclockwise trajectory from this point represents decreasing radar range. Input (

*D*

_{0},

*N*) value is represented by “X.” (top) Solutions for +2 dB offset in

_{t}*Z*

_{hh},

*Z*

_{vv}; (bottom) solutions for −2 dB offset in

*Z*

_{hh},

*Z*

_{vv}. Input values of

*D*

_{0}from (left) 2.1, (middle) 2.4, and (right) 2.7 mm.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of (top) rain rate, (middle) median mass diameter, and (bottom) number concentration using the forward integral equation formulation for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of (top) rain rate, (middle) median mass diameter, and (bottom) number concentration using the forward integral equation formulation for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of (top) rain rate, (middle) median mass diameter, and (bottom) number concentration using the forward integral equation formulation for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of (top) rain rate, (middle) median mass diameter, and (bottom) number concentration using the backward integral equation formulation for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of (top) rain rate, (middle) median mass diameter, and (bottom) number concentration using the backward integral equation formulation for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1

Estimates of (top) rain rate, (middle) median mass diameter, and (bottom) number concentration using the backward integral equation formulation for offset errors in the radar reflectivity factors.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2005.1