## 1. Introduction

Dual-wavelength ratio (DWR) techniques, which make use of the ratio of reflectivity measurements from two collocated radars operating at different frequencies, have long been used to remotely detect rainfall rate and hail in convective storms (Atlas 1954; Eccles and Mueller 1971). More recently, DWR methods have been applied to measurements from millimeter radar pairs, such as K_{a} and W bands, X and K_{a} bands, or S and K_{a} bands, for the remote sensing of cloud microphysical quantities such as cloud liquid water content (LWC) and drop size estimates (Gosset and Sauvageot 1992; Vivekanandan et al. 1999a, 2001, 2004; Hogan et al. 2005). If effective, this capability would offer the prospect of remotely monitoring cloud microphysical characteristics, providing data useful for understanding meteorological processes, assimilation into numerical weather prediction (NWP) models, validating NWP forecasts and satellite-based retrievals, identifying conditions appropriate for weather modification efforts, and diagnosing icing conditions hazardous to aviation. Unfortunately, attempts to apply the DWR technique in practice have generally been unsuccessful in obtaining realistic, high-resolution LWC profiles; instead, published results typically exhibit total-path LWC estimates at fixed ranges, range-resolved results for individual beams only, or smoothed 2D depictions that do not contain the small-scale features and inhomogeneities seen in raw reflectivity data. The purpose of the present paper is to examine in detail the sources of error responsible for contaminating the DWR retrievals of LWC. This exposition suggests considerations for designing dual-wavelength radar systems and for processing DWR data in ways that may increase their usefulness for LWC and drop size retrieval, or for hail detection and rain-rate estimation at longer wavelengths. For instance, the importance of closely matching the radar illumination volumes was a key consideration in the design of the National Center for Atmospheric Research (NCAR) S-PolKa system (Vivekanandan et al. 2004).

The authors were motivated to undertake this study after analyzing radar data collected in April 1999 during the Mount Washington Icing Sensors Project (MWISP; Ryerson et al. 2000) in New Hampshire. These included data from the University of Massachusetts’s Cloud Profiling Radar System (CPRS; Lohmeier et al. 1997), which utilizes K_{a}- and W-band wavelengths with a common dish. In several cases of interest, the K_{a}- and W-band CPRS radar data were collected at range gate spacings of 30 and 75 m, respectively. Using an interpolation of the W-band data onto the K_{a}-band’s 30-m range gates, a straightforward application the DWR technique described in the following section generated unacceptably poor results. Figure 1 depicts results of an analysis of CPRS data from 1620–1635 UTC 15 April 1999, when the radars were oriented at an 18° elevation angle. The retrieved values of LWC obtained between slant ranges of 1 and 4 km, where both radar signals appear good, vary between −10.24 and +11.24 g m^{−3} in an approximately Gaussian distribution with mean 0.35 and standard deviation 1.75 g m^{−3}. The negative LWC values are obviously physically impossible, and the large positive values are unrealistically high. However, the spurious values do not appear to be the results of random noise alone, since independent adjacent beams frequently contain correlated negative or large positive values. These artifacts—time-range areas of related spurious values—most often appear in regions where the radar reflectivity changes rapidly, a clue that they are deterministic features of the radar measurement or data processing. Identifying the causes of these and other less obvious errors could lead to methods to mitigate their effects. The treatment presented here complements that presented in Hogan et al. (2005) by presenting error formulas for individual realizations, rather than RMS error analyses that apply to an ensemble of cases, and by dealing directly and quantitatively with measurement volume mismatches.

## 2. Theoretical basis of the DWR technique

The DWR method for retrieving LWC profiles is based on the observation that liquid water attenuates radar signals differently depending on their wavelengths (Doviak and Zrnić 1993), and hence is sometimes also called the differential attenuation method. Under small-drop liquid conditions and after adjusting for gas attenuation, the attenuation of each wavelength is proportional to the LWC; thus, dividing the range derivative of the measured reflectivity difference by the difference in attenuation coefficients yields the LWC.

*r*km from a radar having wavelength

*λ*, denoted

*Z*(

_{λ}*r*), depends on the atmospheric reflectivity

*Z*(

*r*) in the radar illumination volume and the two-way, roundtrip atmospheric absorption of the intervening medium,

*a*

_{abs}(

*λ, r*). This relation may be written where reflectivity has units of mm

^{6}m

^{−3}and

*a*

_{abs}(

*λ, r*) is in dB km

^{−1}. The atmospheric absorption coefficient is a function of the wavelength and the state of the atmosphere at range

*r*; it may be split into the contributions from gasses and LWC: where

*T*denotes temperature,

*P*pressure,

*H*relative humidity, and

*L*LWC. Replacing the radar wavelength

*λ*in (1) successively with

*λ*

_{1}and

*λ*

_{2}and subtracting produces a relation between the DWR and the gas and liquid water absorption: Here

*a*(

_{G}*λ, r*) is the two-way gas absorption coefficient in dB km

^{−1},

*a*(

_{L}*λ, r*) is the two-way liquid water absorption coefficient in dB km

^{−1}(g m

^{−3})

^{−1}, and

*L*(

*r*) is LWC in g m

^{−3}. When water drops are sufficiently small to be approximately spherical and are much smaller than

*λ*, then (Bohren and Huffman 1983). Here

*ρ*is the density of water,

_{w}*λ*has the same units as

*r*, and

*m*(

*λ*,

*T*) is the complex index of refraction of water at temperature

*T*for wavelength

*λ*. The index of refraction may be computed using formulas in Ray (1972) or Liebe et al. (1991). However, laboratory measurements involving supercooled water are technically challenging, and significant differences have been observed in values for

*T*< 0° found in the literature (Westwater et al. 2001).

*a*(

_{G}*λ, r*), may be well approximated as where

*a*

_{H2O},

*a*

_{O2}, and

*a*

_{N2}signify the absorption coefficients of water vapor, oxygen, and nitrogen, respectively, in dB km

^{−1}. Formulas for computing these coefficients may be found in Janssen (1993), Schwartz (1998), and Rosenkranz (1998, 1999.

*A*(

_{G}*λ*

_{1},

*λ*

_{2},

*r*) =

*a*(

_{G}*λ*

_{2},

*r*) −

*a*(

_{G}*λ*

_{1},

*r*) and the two-way differential liquid absorption coefficient

*A*(

_{L}*λ*

_{1},

*λ*

_{2},

*r*) =

*a*(

_{L}*λ*

_{2},

*r*) −

*a*(

_{L}*λ*

_{1},

*r*) and taking the derivative of both sides of (3) with respect to range yields Equation (6) is the basis of the DWR technique for retrieving LWC. It states that, when adjusted for gas absorption, the range derivative of the DWR is proportional to the LWC, the constant of proportionality being the differential absorption of the two wavelengths.

## 3. Error analysis

*A*and

_{L}*A*; errors in the measured DWR caused by Mie scattering, measurement volume mismatches, and measurement noise, including nonmeteorological signals; and errors in the numerical estimation of the range derivative using DWR values at discrete range gates. Inserting terms for each of these errors and using hats for estimated or measured quantities, (6) becomes Here

_{G}*Â*(

_{L}*λ*

_{1},

*λ*

_{2},

*r*) =

*A*(

_{L}*λ*

_{1},

*λ*

_{2},

*r*) +

*E*(

_{AL}*r*) denotes the estimated differential liquid absorption coefficient for the two wavelengths at range

*r*, having error

*E*(

_{AL}*r*);

*Â*(

_{G}*λ*

_{1},

*λ*

_{2},

*r*) =

*A*(

_{G}*λ*

_{1},

*λ*

_{2},

*r*) +

*E*(

_{AG}*r*) signifies the estimated differential gas absorption coefficient with error

*E*(

_{AG}*r*);

*D̂*[dB

_{r}*Z*

_{λ1}(

*r*) − dB

*Z*

_{λ2}(

*r*)] = (∂/∂

*r*)[dB

*Z*

_{λ1}(

*r*) − dB

*Z*

_{λ2}(

*r*)] +

*E*(

_{D̂r}*r*) denotes a discrete derivative of the true DWR via a linear operator (e.g., a finite difference) having error

*E*(

_{D̂r}*r*);

*Ẑ*(

_{λ}*r*) represents the radar-measured reflectivity for wavelength

*λ*at range

*r*;

*E*

_{Mie},

*E*

_{geom}, and

*E*

_{noise}represent errors in the DWR due to Mie scattering, geometric errors, and measurement noise, respectively, which are defined more precisely below; and the estimated LWC value,

*L̂*(

*r*), is defined in analogy to (6) via It follows from (7) that the error in

*L̂*(

*r*) is given by Equation (9) shows that the error in

*L̂*has two distinct constituents: the discrete derivative of the measurement noise, the discrete differentiation operator error, the derivatives of the Mie scattering and geometric mismatch errors, and the differential gas absorption error each cause an error in the LWC estimate proportional to the true differential liquid absorption coefficient, while the error in the differential liquid absorption coefficient produces a relative error, that is, an error proportional to the value of

*L̂*. These various sources of error are discussed in detail below.

### a. Differential absorption factors

Errors in the values of the differential absorption factors *A _{L}* and

*A*may be caused by uncertainties in temperature, pressure, and vapor content. Values of the liquid absorption coefficients

_{G}*a*for various frequencies and temperatures may be computed via the formula given in (4) and are shown in Fig. 2 for a range of temperatures. Figure 2 also depicts the dependence on temperature of the differential absorption

_{L}*A*for the CPRS K

_{G}_{a}- and W-band, National Oceanic and Atmospheric Administration (NOAA) X- and K

_{a}-band, and S-PolKa S- and K

_{a}-band pairs. Comparing these values, it is clear that the latter frequency pairs are substantially more sensitive to uncertainty in temperature. Finally, Fig. 2 shows a plot of the relative error in DWR LWC,

*E*/

_{L}*A*, due to a 1° error in the temperature used to compute

_{L}*A*. Between −20° and +20°C, a temperature error of 4°C would cause approximately 10% error in retrieved LWC for the X- and K

_{L}_{a}- or S- and K

_{a}-band pairs, but at most 5% error for the K

_{a}- and W-band pair. Hence, this source of error is one of the factors that should be considered in designing a dual-wavelength system. Another important factor is the magnitude of the differential attenuation (Reehorst and Koenig 2004; Hogan et al. 2005); Fig. 2 suggests that the CPRS K

_{a}- and W-band radar pair is at least 2 to over 5 times more sensitive to LWC than the X- and K

_{a}- or S- and K

_{a}-band radars for temperatures between −20° and +20°C.

The gas absorption factor, *a _{G}*, increases monotonically with increasing temperature, pressure, and relative humidity, with absorption values rising with decreasing wavelengths. At the K

_{a}-band frequency 33.12 GHz,

*a*ranges from 0.026 dB km

_{G}^{−1}at

*T*= −20°C,

*P*= 500 hPa, and

*H*= 80% to 0.41 dB km

^{−1}at

*T*= 20°C,

*P*= 1000 hPa, and

*H*= 100%. For S band (2.809 GHz), X band (9.34 GHz), and W band (94.92 GHz), values of

*a*range from 0.0041 to 0.019, 0.0061 to 0.041, and 0.063 to 2.10 dB km

_{G}^{−1}, respectively, over the same domain. The gas absorption factor for S band is smallest and also least variable, in a relative sense, while that of W band is the largest and most variable. Figure 3 shows the dependence of the differential attenuation

*A*on temperature for the three radar pairs at pressures of 1000 and 500 hPa. The K

_{G}_{a}- and W-band pair has the highest values, with somewhat smaller values at 500 hPa than at 1000 hPa; values for all three pairs increase rapidly with temperature. Figure 3 also shows a plot of the error in retrieved DWR LWC,

*E*/

_{G}*A*, due to a 1° error in the temperature used to compute

_{L}*A*. For the K

_{G}_{a}- and W-band pair at a pressure of 1000 hPa, a 4°C error in temperature would cause an error of 0.015 g m

^{−3}at 0°C and nearly 0.08 g m

^{−3}at 20°C; for S and K

_{a}band at 500 hPa, the error would be 0.004 g m

^{−3}at 0°C and more than 0.06 g m

^{−3}at 20°C. Finally, Fig. 3 depicts the error in retrieved DWR LWC that would result from using a value of 100% relative humidity when the true value is 80%. This comparison is motivated by the fact that, while clouds may be assumed to be at or very near saturation, lower relative humidity may occur in precipitation. For the K

_{a}- and W-band pair at a pressure of 1000 hPa, the error is just over 0.01 g m

^{−3}at 0°C and 0.07 g m

^{−3}at 20°C; for S and K

_{a}band at 500 hPa, the error would be 0.006 g m

^{−3}at 0°C and more than 0.05 g m

^{−3}at 20°C. Sensitivity to temperature and relative humidity is greater at higher pressures and temperatures, which is fortunate since these atmospheric quantities can be most accurately estimated near the surface based on ground measurements. For supercooled liquid water conditions that might cause aircraft icing hazards, the effect of errors in

*A*is generally small. Nevertheless, it is worth noting that bias due to

_{G}*E*/

_{G}*A*can be sufficient to cause negative retrievals in regions of low LWC.

_{L}Bounds on the LWC errors caused by uncertainty in the differential absorption factors may be estimated in an operational system. If temperature, pressure, and relative humidity estimates are accompanied by error bars ±*ɛ _{T}*(

*r*), ±

*ɛ*(

_{P}*r*), ±

*ɛ*(

_{H}*r*), respectively, the uncertainty in retrieved LWC can be estimated from a linearization of the error function or by computing LWC using values of

*A*and

_{L}*A*at each vertex of the cube defined by [

_{G}*T̂*(

*r*) ± ɛ

_{T}(

*r*),

*P̂*(

*r*) ± ɛ

_{P}(

*r*),

*Ĥ*(

*r*) ± ɛ

_{H}(

*r*)]. Moreover, if probability distributions for the

*T*,

*P*, and

*H*errors are known, these could be sampled to create a distribution of retrieved LWC values.

### b. Reflectivity measurement errors

#### 1) Mie scattering

*D*<

*λ*/16 or so. As a drop diameter exceeds

*λ*/16, it begins to enter the Mie scattering regime, and its measured reflectivity decreases as the signal is scattered disproportionately in the forward direction. Eventually, the signal attenuation will also begin to increase due to the increased scattering (Gosset and Sauvageot 1992). The Mie scattering error,

*E*

_{Mie}, may be defined as the difference between the DWR and the DWR that would be obtained from the same LWC under Rayleigh scattering conditions: where

*Z*

^{(R)}

_{λ}(

*r*) represents the attenuated Rayleigh reflectivity for wavelength

*λ*at range

*r*. From (9), the bias in retrieved LWC caused by Mie scattering is given by (∂/∂

*r*)[

*E*

_{Mie}(

*r*)/

*A*(

_{L}*λ*

_{1},

*λ*

_{2},

*r*)]. If the radar beams pass through a region having

*E*

_{Mie}> 0, this bias will be positive at the near side [where

*E*

_{Mie}(

*r*) is increasing] and negative at the far side, resulting in a pair of anomalously high and low LWC retrievals having magnitudes related to the degree of Mie scattering.

Typically, liquid drops that constitute the bulk of cloud LWC and absorption are less than 40 *μ*m, whereas radar reflectivity is mainly due to ice and drizzle that are larger than 50 *μ*m. Hence, it is important to select dual-wavelength radar parameters such that they are sensitive to absorption due to LWC and also relatively unaffected by Mie scattering. For the S-, X-, K_{a}-, and W-band frequencies of 2.809, 9.34, 34, and 94.92 GHz, the wavelengths *λ* = 106.4, 32.0, 8.8, and 3.150 mm, and *λ*/16 = 6.65, 2.0, 0.55, and 0.20 mm, respectively. Thus, the DWR from a radar pair including K_{a} or W band will be most easily contaminated by Mie scattering, whereas S band will rarely experience Mie scattering. Since large ice crystals are frequently present in supercooled clouds, Mie scattering at W band presents a significant challenge for using high radar frequencies in such cases. However, if the radars are directed at a slant range and at least one has polarimetric capability, it may be possible to identify regions containing large drops or ice by examining the linear depolarization ratio (LDR), radar reflectivity, and differential reflectivity, *Z*_{DR} (Herzegh and Jameson 1992; Vivekanandan et al. 1999a, 1999b). In the case depicted in Fig. 1, LDR values are generally below −30, suggesting that little Mie scattering is present at W band. If the radars are pointing vertically or at a sufficiently high elevation angle, Doppler velocities can also be utilized for this purpose (Hogan et al. 2005). Alternatively, a three-wavelength method may be used to separate Mie scattering effects from attenuation, allowing unambiguous retrieval of LWC even when Mie scattering is present (Gaussiat et al. 2003).

#### 2) Radar measurement geometry

*Z*(

_{λ}*r*) cannot be measured at a point; instead, a radar effectively averages

*Z*over an illumination volume centered at the measurement point. To better describe the radar measurement in three dimensions,

_{λ}*Z*(

_{λ}*r*) may be written as

*Z*(

_{λ}**x**+

**r**), where

*r*= ||

**r**|| and the radar signal traverses the path from

**x**to

**r**, that is,

**r**−

**x**. The radar-averaged reflectivity at range

*r*may be written as Here

*I*(

_{n}**r**,

**) denotes the illumination function at displacement**

*ρ***from its center, which is**

*ρ***r**from the radar, normalized so that ∫

_{ℜ}

^{3}

*I*(

_{n}**r**,

**)**

*ρ**d*= 1 (see appendix C). The reflectivity error due to the volumetric averaging inherent in the radar measurement is The volumetric averaging error (in dB) is

**ρ***E*

_{V;λ}(

**x**+

**r**) = dB

*(*Z

_{λ}**x**+

**r**) − dB

*Z*(

_{λ}**x**+

**r**). If

*Z*(

_{λ}**x**+

**r**+

**) is uniform over the region where the illumination function is nonzero, then the bracketed terms in (12) cancel each other and it follows that**

*ρ**(*Z

_{λ}**x**+

**r**) =

*Z*(

_{λ}**x**+

**r**) and

*E*

_{V;λ}(

**x**+

**r**) = 0. If the illumination function is symmetric about its center—generally a good approximation at large ranges—then this result also holds when

*Z*is linear about the center of the pulse volume, or even simply antisymmetric with respect to the center. When reflectivity is highly variable, significant attenuation is present, or the illumination volume is large,

_{λ}*Z*will likely not satisfy these conditions, and

_{λ}*E*

_{V;λ}(

**x**+

**r**) may be nontrivial.

*E*

_{interp}(

*r*) and assuming that the two radars are positioned at

**x**

_{1}and

**x**

_{2}with beam directions along unit vectors

**r̂**

_{1}and

**r̂**

_{2}, respectively, while the intended profile has origin

**x**and beam direction along

**r̂**, the geometric error may written This expression assumes that the displacement between the radars is perpendicular to the intended profile so that their range measurements are consistent; if this is not the case, the appropriate range increment should be added to one radar’s values of

*r*. In atmospheric conditions for which there is a refractivity gradient across the radar beams, the signals will not travel in the straight lines presupposed by (13) and, in fact, radars with identical orientations may experience distinct signal paths for different wavelengths. However, substantial beam bending is unlikely except for very low elevation angles, so this complication shall be ignored. Different propagation delays due to refractive index differences should be on the order of a few millimeters or less for any atmospheric conditions and hence may also be safely ignored.

Equation (9) shows that the geometric error causes a bias in the retrieved LWC having magnitude [(∂/∂*r*)*E*_{geom}(*r*)]/*A _{L}*(

*λ*

_{1},

*λ*

_{2},

*r*). The contributions to this bias from the various terms in (13) are considered in reverse order.

##### (i) Displaced or misaligned beams

**x**, and both beams lie along its direction,

**r̂**, then the final two bracketed differences in (13) become zero. Otherwise, the situation may be that depicted in Fig. 4 (top). Suppose that the intended profile is along the beam of the leftmost radar, so that

**x**=

**x**

_{1},

**r̂**=

**r̂**

_{1}, and dB

*Z*

_{λ1}(

**x**+

**r̂**) − dB

*Z*

_{λ1}(

**x**

_{1}+

**r̂**

_{1}) = 0. The contribution to the derivative of

*E*

_{geom}due to the displacement of the radars is then This term may have significant magnitude if either reflectivity or attenuation along the two beams differ. For example, as an isolated concentration of liquid water represented by the moving cloudlet in Fig. 4 passes into the first beam, the reflectivity and attenuation increase in the ranges of intersection while remaining unchanged for the second beam. As a result, dB

*Z*

_{λ2}(

**x**

_{1}+

*r*

**r̂**

_{1}) − dB

*Z*

_{λ2}(

**x**

_{2}+

*r*

**r̂**

_{2}) becomes large for the ranges containing the cloudlet, with a positive range derivative at the near side, a negative derivative at the far side, and a slight negative trend in between due to attenuation. The result is a strong positive bias in the retrieved LWC for the near ranges of the intersection, coupled with a strong negative bias on the far side. A more detailed analysis of this error may be found in appendix A, where it is also shown that averaging the retrieved LWC over time and range may provide some mitigation.

##### (ii) Misregistration and interpolation

Geometric errors may arise when two radars do not have the same range gates, are displaced so that the registered ranges do not correspond, or make measurements at different times. While cross-correlation analysis may be helpful in identifying temporal or spatial shifts, and resampling may be used place the data on comparable grids, interpolation or averaging may blur features and create artifacts in the DWR derivative. These registration and interpolation errors, together denoted *E*_{interp}, create a bias in the retrieved LWC equal to (∂/∂*r*)*E*_{interp}(*r*).

An instructive example is found in the MWISP CPRS radar data depicted in Fig. 1. When the 75-m W-band data are interpolated onto the K_{a}-band’s 30-m range gates, sharp features in the K_{a}-band reflectivity field are blurred in the W-band data, causing biases in the DWR near their edges, as depicted in Fig. 5 for a single beam. A careful inspection of these data also reveal that features in the W-band reflectivity consistently appear about 40 m closer to the radar origin than the same features in the K_{a}-band data. This shift could be due a problem with the radar signal processing or data recording, or to very strong radial advection coupled with a mismatch of measurement times. The shift creates several “hump” artifacts in the DWR where a positive slope is matched by a subsequent negative one, for example, near 3.2 and 3.6 km in Fig. 5. Shifting the ranges by 40 m before interpolating and computing the LWC significantly reduces the paired LWC overestimates and negative retrievals for this beam, as does smoothing using a Gaussian kernel to resample the K_{a}-band data to 75 m.

##### (iii) Volumetric averaging and illumination volume mismatch

*E*

_{V;λ1}(

**x**

_{1}+

*r*

**r̂**

_{1}) −

*E*

_{V;λ2}(

**x**

_{2}+

*r*

**r̂**

_{2}), and creates a bias in the retrieved LWC measurement equal to As mentioned previously, each

*E*

_{V;λ}term is small when the local (attenuated) reflectivity field is nearly linear; otherwise, they depend on attenuation and the variability of the reflectivity field. This is demonstrated analytically in appendix B, where a relationship between

*E*

_{V;λ}and the Hessian of the reflectivity field at the illumination volume center is derived. The range derivative in (15) may have significant magnitude when the reflectivity field is nonlinear and the illumination volumes are poorly matched. This may occur, for instance, in the case of mismatched beamwidths illustrated in Fig. 4. Errors in dual-wavelength detection of hail due to such mismatches were identified by Rinehart and Tuttle (1982, 1984).

Simulations were performed using 3D “clouds” to investigate the geometric errors due to illumination volume mismatches. The first cloud consists of several Gaussian concentrations of liquid water having peak values of 1 g m^{−3}; a cross section is shown in Fig. 6. The Gaussians in the first two columns have a standard deviation of 200 m in all three dimensions, with centers in the plane swept out by the radar and 200 m behind the plane, respectively. The next two columns are similar but with a standard deviation of 100 m. The Gaussians in the fifth and sixth column have centers in the plane and standard deviations (*σ _{x}* = 200 m,

*σ*= 50 m,

_{y}*σ*= 500 m) and (

_{z}*σ*= 200 m,

_{x}*σ*= 500 m,

_{y}*σ*= 50 m), respectively. Simulations were performed for the CPRS K

_{z}_{a}- and W-band radar pair (beamwidths 0.5° and 0.18°), and the NOAA X- and K

_{a}-band pair (beamwidths 0.5° and 0.9°);

*A*was set to zero, and the temperature was fixed at

_{G}*T*= −10°C. The idealized radar simulation described in appendix C was used for each radar pair to obtain reflectivity measurements as the LWC field was translated perpendicularly to the radar beams at 20 m s

^{−1}.

The CPRS results shown in Fig. 7 used illumination volume lengths and range gate spacings fixed at 30 m to focus on the effect of the mismatched beamwidths. The artifacts resulting from the beam-mismatch may be understood as follows. As a cloudlet penetrates the wider beam, *E _{V}*

_{;}

_{λ1}(

**x**+

**r**) becomes large for the ranges of intersection, while

*E*

_{V}_{;}

_{λ2}(

**x**+

**r**) remains small. The derivative in (15) is positive for the near ranges of the intersection and negative for the far side, resulting in paired positive and negative bias in the measured LWC; if the higher frequency radar had the larger beam, these signs would be reversed. If the cloudlet is small with respect to the beamwidths and penetrates farther into the two beams, then

*E*

_{V}_{;}

_{λ2}(

**x**+

**r**) eventually grows as well and

*E*

_{V}_{;}

_{λ1}(

**x**+

**r**) −

*E*

_{V}_{;}

_{λ2}(

**x**+

**r**) diminishes. If the cloudlet is sufficiently small, the value of

*E*

_{V}_{;}

_{λ1}(

**x**+

**r**) may turn negative as the reflectivity at the center of the pulse volume exceeds the average in the illumination volume. When this occurs,

*E*

_{V}_{;}

_{λ1}(

**x**+

**r**) −

*E*

_{V}_{;}

_{λ2}(

**x**+

**r**) turns negative within the region of intersection, and results in paired negative and positive biases to retrieved LWC. As the cloudlet exits the two beams, the situation reverses.

The artifacts for the NOAA radar pair in Fig. 7 have nearly the same appearance, but their magnitude is about 7 times larger because the difference in their beamwidths is larger and the differential absorption coefficient in the denominator of (9) is more than 3 times smaller. The artifacts have the greatest magnitude for the narrowest liquid water concentrations, which represent larger inhomogeneity and hence, larger Hessians in (B2). For both radar pairs, the simulated artifacts’ magnitudes increase with range as the span of the beams becomes larger and the inhomogeneity in reflectivity across the beams grows.

If the retrieved LWC values are averaged over the entire domain or over any of the artifacts in the first four columns of Fig. 7, their mean value is found to be very close (within 1%) to the mean value obtained by performing the same averaging on the true LWC values shown in Fig. 6. This result suggests that averaging over both time and range may reduce the error due to illumination volume mismatch. The result does not hold for the elongated artifacts of the fifth column, however, where the difference between the true and retrieved averages is about 30%. This discrepancy is because the absolute width of the radar beam broadens significantly over the range of the artifact, so the underestimates at the farther ranges have substantially greater magnitude than their paired overestimates at the near edge. This phenomenon was also observed for the symmetric Gaussians when a radial component was added to the advection, resulting in a diagonal elongation of the artifacts. Note that smoothing the retrieved LWC field using a sliding range–time window—a common technique—may not remove spurious and unrealistic values when the window does not completely cover the paired over- and underestimates. More sophisticated methods may be required to optimally handle the contamination. Of course, utilizing a dual-wavelength system with collocated radars having identical illumination volumes that measure along identical profiles with equal gate locations would also minimize the difficulty.

Similar artifacts in retrieved LWC arise for mismatched illumination volume lengths, as illustrated by the simulation results shown in Fig. 8 for which the actual illumination volume lengths (30 m for the CPRS K_{a}-band radar and 75 m for the W-band and NOAA radars) were used. It is evident from comparing these results with Fig. 7 that the effect of the mismatched illumination volume lengths is the most significant factor for contaminating the CPRS retrieval, while the mismatched beamwidths are significant for the NOAA radar pair. The artifacts’ magnitude is largely independent of range for illumination volume length mismatch, while it increases with range when caused by beamwidth mismatch. Referring back to Fig. 1, this observation suggests that the most significant source of contamination in the CPRS MWISP data was the mismatched illumination volume lengths.

A somewhat less artificial cloud was produced by performing circular convolution of a 3D Gaussian, *σ* = 85 m, with a 3D random field having grid spacing of 5 m, creating the correlated random LWC fields shown in Fig. 9. Simulated measurements were again performed for the CPRS K_{a}- and W-band pair and the NOAA X- and K_{a}-band pair. As in the processing of the MWISP CPRS data, the CPRS W-band reflectivity data were interpolated from 75 m to the K_{a}-band’s 30-m range spacing before the DWR method was applied. Results shown in Fig. 10 for both radar pairs again reveal numerous artifacts of varying magnitudes.

To perform a statistical analysis of illumination volume mismatch artifacts, 200 correlated random LWC fields were generated and measurements simulated using this methodology. In addition, averaging was performed for each beam over a sliding 300-m range interval. For the CPRS radar pair, two processing methods were used: interpolating the W-band data to 30-m gate spacing, and convolving a Gaussian smoothing kernel with the K_{a}-band reflectivity data (in linear scale) to resample it to 75-m spacing. RMS error and mean absolute (MA) error of the retrieved LWC are depicted in Fig. 11 as a function of range; mean biases were less than 0.02 g m^{−3} for all methods over the ranges shown. Note that the MA errors are significantly smaller, suggesting that much of the RMS error is due to a small number of very large outliers. The linearly interpolated CPRS data at 30-m spacing showed the worst performance, with an RMS error of about 2 g m^{−3} (MA error 0.8 g m^{−3}) and a slight increase with range consistent with the error being due predominantly to pulse volume length mismatch and interpolation error. It was also the most favorably improved by range averaging: the RMS and MA errors declined by a factor of about 4 with averaging over 10 range gates, presumably because over- and underestimates due to pulse volume length mismatches and interpolation tend to cancel out locally. The NOAA pair exhibited very low error at close ranges with an approximately linear increase with range up to an RMS error of 1.3 g m^{−3} (MA error of 0.56 g m^{−3}) at 5 km, consistent with the error being due to beamwidth mismatch. Averaging the retrieved LWC over four range gates reduced the RMS error by a factor of about 2. The Gaussian resampled CPRS data at 75-m range gate spacing showed the best performance, with low error at near ranges and a more moderate increase with range, rising to an RMS error of 1.1 g m^{−3} (MA error of 0.21 g m^{−3}). This result indicates that resampling removed most of the effect of the mismatched pulse volume lengths, but errors remain because of the mismatched beamwidths. Averaging over four range gates again reduced the RMS error by a factor of 2, and produced an MA error of, at most, 0.12 g m^{−3}, a value at which the retrievals might be marginally useful. Gaussian smoothing or averaging across beams (i.e., temporal or azimuthal) might similarly reduce the effect of beamwidth mismatches, though the width of the Gaussian would likely need to increase in range and be tuned to the rate of advection or radar scan rate to achieve optimal results.

Of course, cloud liquid water does not often present itself in either the small discrete clouds or correlated random fields treated here, so no claim is made that these results are quantitatively realistic. It is well known that clouds frequently exhibit small-scale patchiness in drop size distributions and, hence, reflectivity and LWC (Jameson and Kostinski 2000). The simulation results suggest the likely form of illumination volume mismatch artifacts due to such inhomogeneities: matched pairs of high-biased LWC values and negative ones, just like those observed in Fig. 1. Further simulations involving more realistic atmospheres—for instance, LWC distributions whose statistics match those of cases measured in situ by aircraft, similar to the approach taken by Hogan and Illingworth (1999), or model clouds like those described in Jones et al. (1997) but with higher resolution—may be useful in obtaining quantitative estimates of the error caused by mismatched illumination volumes for realistic scenarios.

### c. Discrete differentiation error

*D̂*. The discrete differentiation error is

_{r}*E*(

_{D̂r}*r*)=

*D̂*[

_{r}*f*(

*r*)] − (∂/∂

*r*)

*f*(

*r*), where

*f*(

*r*) = dB

*Z*

_{λ1}(

*r*) − dB

*Z*

_{λ2}(

*r*); it produces an error in retrieved LWC equal to

*E*(

_{D̂r}*r*)/

*A*(

_{L}*λ*

_{1},

*λ*

_{2},

*r*). One discrete differentiation operator is the finite central difference where the evaluation point

*r*may represent a radar range gate and Δ

*r*the range gate spacing. The differentiation error for this operator obtained from a Taylor series expansion of

*f*around

*r*is

*E*(

_{D̂r}*r*) ≈ ⅙(Δ

*r*)

^{2}(∂

^{3}/∂

*r*

^{3})

*f*(

*r*). This error is small when either Δ

*r*is small or

*f*has small values of cubic and higher odd derivatives at

*r*. For example, if Δ

*r*= 0.075 km, the coefficient of the error term is 9.4 × 10

^{−4}km

^{−1}, while if Δ

*r*= 0.030 km, it is 1.5 × 10

^{−4}km

^{−1}. The error is substantially worse for the one-sided finite-difference estimate,

*D̂*[

_{r}*f*(

*r*)] = [

*f*(

*r*+ Δ

*r*) −

*f*(

*r*)]/Δ

*r*: for this operator,

*E*(

_{D̂r}*r*) ≈ ½Δ

*r*(∂

^{2}/∂

*r*

^{2})

*f*(

*r*). In addition to the fact that the second derivative of

*f*is likely greater than the cubic, the coefficient is much greater as well, 3.75 × 10

^{−2}km

^{−1}for Δ

*r*= 0.075 km and 1.5 × 10

^{−2}km

^{−1}for Δ

*r*= 0.030 km. Although it is difficult to estimate typical values of the higher-order derivatives of the true two-way differential attenuation due to the discrete sampling and smoothing inherent in radar measurements, these considerations suggest that the choice of differentiation operator may be significant.

*f*, the division by a small value of Δ

*r*in (16) may amplify it and produce significant error in the computed derivative. In this case, it is frequently useful to average many independent finite differences; for instance, one might use the operator where

*n*is a positive integer. (This approach is essentially equivalent to range-averaging retrieved LWC.) The error for this operator is

*E*(

_{D̂r}*r*) ≈ ⅙[(1/

*n*)Σ

^{n}

_{k=1}

*k*

^{2}](Δ

*r*)

^{2}(∂

^{3}/∂

*r*

^{3})

*f*(

*r*). For example, if

*n*= 10, the coefficient is 3.6 × 10

^{−2}km

^{−1}for Δ

*r*= 0.075 km and 5.8 × 10

^{−3}km

^{−1}for Δ

*r*= 0.030 km, substantially larger than for the central difference (16). Whether this loss of accuracy is acceptable may depend on the attributes and random errors in

*f*, as discussed further below.

### d. Measurement noise

*D̂*[

_{r}*E*

_{noise}(

*r*)], where Measurements dB

*Ẑ*are affected by errors in radar calibration due to uncertainty in the antenna/receiver gain, radome absorption, transmitter power fluctuation, thermal noise, receiver bandwidth loss, or other factors; however, because this error is the same for all range gates, it will not affect the DWR derivative or the derived LWC—an attractive feature of the DWR technique. Contamination due to receiver saturation, ground clutter, or a method used to compensate for clutter could produce spurious reflectivity measurements, and should be identified and censored to maintain the integrity of the DWR. The remaining measurement noise is mean-zero random noise,

_{λ}*Ẽ*

_{noise}(

*r*) =

*E*

_{noise}(

*r*) − 〈

*E*

_{noise}(

*r*)〉, where the angle brackets 〈 · 〉 represent the expected value over all possible realizations of the radar measurement. For instance, Hogan et al. (2005) derived an equation that relates the RMS error in measured reflectivity to the number of pulses and range gates, range gate spacing, pulse repetition time, spectral width, and signal-to-noise ratio. Assuming

*D̂*is a linear operator, 〈

_{r}*D̂*(

_{r}*Ẽ*

_{noise}(

*r*))〉 =

*D̂*(〈

_{r}*Ẽ*

_{noise}(

*r*)〉) =

*D̂*(0) = 0 and var[

_{r}*D̂*(

_{r}*Ẽ*

_{noise}(

*r*))] = 〈

*D*(

_{r}*Ẽ*

_{noise}(

*r*)

^{2})〉. For example, if

*D̂*is the differentiation operator defined by (17) and the values of

_{r}*Ẽ*

_{noise}(

*r*) are independent for distinct range gates,

*r*, then If, additionally, the values of

*Ẽ*

_{noise}(

*r*) are identically distributed, then (19) becomes To reduce the effect of random noise, a large value of

*n*should be chosen, but this choice dramatically increases the discrete differentiation error, as shown in the previous section. This noise/resolution trade-off is a fundamental paradox that limits the accuracy of the DWR technique.

## 4. Conclusions

The present paper has described the DWR method for remotely sensing cloud LWC and elucidated a number of limitations of this technique. Factors responsible for contaminating LWC retrievals include errors in the computed values of the differential absorption coefficients *A _{L}* and

*A*due to uncertainty in temperature, pressure, or relative humidity; errors in the measured DWR caused by Mie scattering; artifacts in the DWR due to illumination volume mismatches; reflectivity measurement errors, including both random noise and nonmeteorological signals; and errors in the numerical estimation of the DWR range derivative. Errors in

_{G}*A*and

_{G}*A*cause absolute and relative errors, respectively. The error in DWR LWC due to uncertainty in

_{L}*A*is generally quite small, especially at low temperatures, but could cause negative retrievals in regions of low LWC. The

_{G}*A*values for K

_{L}_{a}and W bands are substantially less sensitive to errors in temperature than X- and K

_{a}-band or S- and K

_{a}-band pairs, particularly at low temperatures. Furthermore, since the LWC retrieval error is inversely proportional to

*A*, the much larger values of

_{L}*A*for the K

_{L}_{a}- and W-band pair suggests that it has better potential than either the S- and K

_{a}-band or X- and K

_{a}-band pairs to accurately detect LWC. Unfortunately, W band is the wavelength most susceptible to Mie scattering due to large drops or ice crystals; while use of polarimetric variables may help identify and censor regions of Mie scattering, this limitation could affect a K

_{a}- and W-band pair’s usefulness. Because W band is strongly attenuated by both liquid water and gasses, the large radar power required to penetrate deeply into clouds may further limit its practicality.

In the MWISP deployments, the CPRS radar pair had significant illumination volume length mismatches, while the NOAA radar pair had a substantial beamwidth mismatch. Simulated measurements of artificial “clouds” demonstrated that these illumination volume mismatches could generate artifacts in DWR that severely contaminate LWC retrievals. Statistics computed from a large number of simulations demonstrated that averaging over range was effective in mitigating the error due to illumination volume mismatches. Moreover, resampling the high-resolution K_{a}-band data to the coarser W-band gate spacing using a Gaussian smoothing kernel substantially removed the error due to mismatched pulse volume lengths. Similar averaging over time or azimuth could reduce the effect of beamwidth mismatches, though the Gaussian smoothing kernel would have to vary in range and be adapted to the velocity of the LWC field across the beam. Finally, an analysis of discrete differentiation operators showed that a central difference operator is substantially more accurate than a one-sided difference. Smoothing operators needed to average away measurement noise inherently reduce the accuracy of the derivative estimate, and hence the resulting LWC retrieval. Thus, matching illumination volumes as closely as possible and reducing measurement noise are critically important for achieving accurate, high-resolution LWC retrievals in a dual-wavelength system.

## Acknowledgments

This research is in response to requirements and funding by the Federal Aviation Administration (FAA). The views expressed are those of the authors and do not necessarily represent the official policy or position of the FAA.

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## APPENDIX A

### Further Analysis of the Error due to Displaced Radars

*L*(·) denotes LWC at the given position, and

*a*and

_{G}*a*are represented similarly. If

_{L}*a*and

_{G}*a*are equal along the two profiles, that is,

_{L}*a*(

_{G}*λ*

_{2},

**x**

_{0,1}+

*r*

**x̂**

_{1}) =

*a*(

_{G}*λ*

_{2},

**x**

_{0,2}+

*r*

**x̂**

_{2}) and

*a*(

_{L}*λ*

_{2},

**x**

_{0,1}+

*r*

**x̂**

_{1}) =

*a*(

_{L}*λ*

_{2},

**x**

_{0,2}+

*r*

**x̂**

_{2}) ≡

*a*(

_{L}*λ*

_{2},

*r*), then the right-hand side of (A1) becomes Suppose that dB

*Z*(

**x**

_{0,1}+

*r*

**x̂**

_{1}) − dB

*Z*(

**x**

_{0,2}+

*r*

**x̂**

_{2}) is constant in

*r*. It follows from (14) that the error in the LWC estimate,

*L̂*, due to the radar displacement is [

*L*(

**x**

_{0,2}+

*r*

**x̂**

_{2}) −

*L*(

**x**

_{0,1}+

*r*

**x̂**

_{1})]

*a*(

_{L}*λ*

_{2},

*r*)/

*A*(

_{L}*λ*

_{1},

*λ*

_{2},

*r*); that is, it is proportional to the difference in LWC along the two profiles.

*a*is equal along both profiles, and in the neighborhood of a range

_{G}*r*

_{0}the LWC along the first radar’s beam has a Gaussian distribution

*L*(

**x**

_{0,1}+

*r*

**x̂**

_{1}) =

*L*

_{0}

*e*

^{−(r−r0)2/(2σ2)}while

*L*(

**x**

_{0,2}+

*r*

**x̂**

_{2}) is small and constant and that

*Z*=

*κL*for a positive constant

^{2}*κ*. Then This expression confirms the qualitative analysis of the advecting Gaussian cloudlet presented in the text: for

*r*<

*r*

_{0},

*L̂*(

*r*) could be biased high; at

*r*=

*r*

_{0},

*L̂*(

*r*) is negatively biased; and for

*r*>

*r*

_{0},

*L̂*(

*r*) < 0. In addition, the magnitude of the first term on the right-hand side of (A3) increases as the Gaussian’s width,

*σ*, decreases. Averaging over range would remove the error due to the first term, but would still leave a negative bias due to the second.

Finally, consider the case in which the liquid water densities along the two beams switch values over time, as would be the case if the cloudlet passes first through one beam and then the second. This would interchange the values of dB*Z*(**x**_{0,1} + *r***x̂**_{1}) with dB*Z*(**x**_{0,2} + *r***x̂**_{2}) and *L*(**x**_{0,1} + *r***x̂**_{1}) with *L*(**x**_{0,2} + *r***x̂**_{2}) in (A2), thereby simply reversing the sign. Thus, if the retrieved LWC profiles for these two times were averaged, the errors caused by displaced radars would cancel each other out. This result confirms the potential value of averaging liquid water estimates in time or azimuth as well as in range.

## APPENDIX B

### Further Analysis of the Volumetric Averaging Error

*E*

_{V;λ}(

**x**

_{0}+

*r*

**x̂**) = dB

*(*Z

_{λ}**x**

_{0}+

*r*

**x̂**) − dB

*Z*(

_{λ}**x**

_{0}+

*r*

**x̂**), where dB

*(*Z

_{λ}**x**

_{0}+

*r*

**x̂**) is defined by (11). Approximating the attenuated reflectivity by the first three terms of its Taylor series expansion about the measurement point and assuming that

*I*(

_{n}*r*

**x̂**,

**) is symmetric in**

*ρ***—a good approximation for large**

*ρ**r*, where the relative spread of the beam is small—this error may be written where

**∇**

*Z*(

_{λ}**x**

_{0}+

*r*

**x̂**) denotes the gradient of

*Z*at

_{λ}**x**

_{0}+

*r*

**x̂**, 𝗛

_{Zλ}(

**x**

_{0}+

*r*

**x̂**) represents the Hessian matrix of

*Z*at

_{λ}**x**

_{0}+

*r*

**x̂**(the 3 × 3 matrix of its second-order partial derivatives), and the superscript T indicates the transpose operator. Note that

*Z*may not even be smooth in reality, so (B1) may be more suggestive of a source of error than it is precise; nevertheless, it confirms that the radar measurement error due to volumetric averaging is a function of the higher-order derivatives of the attenuated reflectivity field. This error will therefore be greatest when

_{λ}*Z*is nonlinear within the illumination volume. Although this effect will likely be most significant when

_{λ}*Z*itself is nonlinear, it is also affected by signal attenuation. Finally, if the illumination volume is small, the integrand on the right-hand side of (B1) is small, and hence

*E*

_{V;λ}(

**x**

_{0}+

*r*

**x̂**) is small; if the illumination volume is large, the reverse could be true. Moreover, if 𝗛

_{Zλ}(

**x**

_{0}+

*r*

**x̂**) is determined mostly by

*Z*(i.e., the reflectivity variation is much higher than the attenuation), the DWR error, will be predominantly a function of the difference in illumination functions, and may be substantial if they are mismatched.

## APPENDIX C

### Radar Simulation Methodology

The methodology for the idealized radar simulations consisted of two parts: the creation of a very simple model of atmospheric LWC and reflectivity, and the simulation of the radar measurement of the (attenuated) reflectivity.

Rather than attempting to utilize a physically realistic cloud model, reflectivity was obtained from a sample LWC field via the relation *Z* = 100 *L*^{2}, where *L* denotes the LWC. The coefficient 100 is arbitrary; drop size distributions measured by aircraft exhibit coefficients ranging from less than 0.1 to more than 20 000 (Williams et al. 2002).

*Z*, along a large number of uniformly spaced rays throughout the radar’s effective beamwidth according to (1) and (2), with

_{λ}*a*set to zero for simplicity. A simulated measurement then consists of weighting the values around a given range gate according to (11). From the center of the radar illumination volume at range

_{G}*r*

_{0}, the illumination function at range

*r*and angular displacement

*ϕ*may be written as (Doviak and Zrnić 1993, p. 80). For a Gaussian transfer function, the range-weighting function

*W*may be well approximated by (Doviak and Zrnić 1993, p. 79), where erf denotes the error function,

*B*

_{6}is the 6-dB frequency bandwidth of the amplitude transfer function,

*c*is the speed of light, and

*τ*is the pulse transmission time. The angular power density

*f*depends on the radar geometry and reflector illumination function, but for many radars is given approximately by where

*J*

_{2}[·] denotes the Bessel function of the first kind with

*ν*= 2 and

*ϕ*

_{1}is the 3-dB beamwidth, in radians (Doviak and Zrnić 1993, p. 34). Given a set of points {

**x**

*= (*

_{i}*r*,

_{i}*θ*,

_{i}*ϕ*)} surrounding

_{i}*r*

_{0}

**x̂**in discrete volumes {Δ

*V*}, the simulated reflectivity measurement is then obtained via When

_{i}*Z*is a smooth function and the number of points in the sum is large and covers an adequate region of space, (C4) is a good approximation to the integral in (11). If a simulation of the radar’s measurement noise is desired, appropriate random values may be added to the simulated reflectivity measurements, but the simulations in the present paper were performed without noise.

_{λ}