# Improved Estimates of the Vertical Structures of Rain Using Single Frequency Doppler Radars

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}), while in a strong downdraft the slowest moving particles moving toward the radar would correspond to the downdraft velocity plus the fall speed of the smallest drops. The remaining fall speeds are then then assigned to the remaining velocity bins according to their fall speeds. While a logical approach, size sorting of raindrops in convective storms is not unusual, so that there is no guarantee that the drop size distributions are going to be sufficiently broad. In the second power-law-method, the fall speed depends upon the radar reflectivity factor, Z. This is hard to justify since Z also depends upon the drop concentration while fall speeds do not.

_{m}= C × Z

_{m}, where C is a constant associated with the characteristics of the radar, and Z

_{m}is then the measured radar reflectivity. However, Z

_{m}is also equal to ${Z}_{m}={C}_{\eta}{\displaystyle \sum N(D)\sigma (D)}$ where C

_{η}is factor related to the radar wavelength, λ, that is described below The Doppler spectrum provides the bridge between theory (Z) and measurements (Z

_{m}).

_{m}, while the calculated Z can be computed either by circularly shifting the array of drop fall speeds corresponding to each radar velocity bin with the null vertical air speed across the observed Doppler spectrum or alternatively by circularly shifting the Doppler spectrum systematically across a fixed array of drop sizes (fall speeds) to produce Z. By computing different N

_{ob}(D,w) for different w one can then search for the w that produces the correct N

_{ob}(D,w) = N(D) so that the computed Z = Z

_{m}. This is where the Doppler spectra enter, because the spectral power values can be shifted one velocity bin at a time through the entire Nyquist velocity interval of the radar until Z = Z

_{m}. The number of velocity bins shifted to achieve this result then provides the estimate of w required to derive N(D). This will become clearer in the next section.

## 2. Background

#### 2.1. Basic Considerations

^{6}dependence. Nevertheless, the deviations do not appear as severe as often thought. Using cross-sections as computed in [14] (see Appendix A.1), it is clear from Figure 1 that over the indicated range of drop sizes where σ

_{B}is in cm

^{2}and D is the drop diameter in mm,

_{B}at the largest drop sizes. Such large drops are usually a rarity compared to the prevalence of the other drop sizes, however. Other fits are possible, of course, but tests of a few alternatives show that they do not significantly alter the deduced vertical structures in the rain shown below.

_{B}, where the summation is over all of the drops in a unit volume. For a Doppler spectrum, η is then spread over the velocity bins that for our purpose here using the MRR is 64 bins distributed over a range from 0.1858 m s

^{−1}to 11.89 m s

^{−1}depending upon the drop sizes and concentrations at each velocity.

_{t}(D) is the terminal fall speed of drop of diameter D, and the summation is over all the drops. Often V

_{t}is in m s

^{−1}while D is usually expressed in cm. However, for convenience in this case, D is in mm. Using the relation in [6] (i.e., V

_{t}= −0.193 + 4.96 D − 0.904 D

^{2}+ 0.0566 D

^{3}, where D is in cm), we consider the combined term D

^{3}V

_{t}to yield the relation shown in Figure 2.

^{2}is a factor related to the microwave complex index of refraction of the scattering material and is 0.92 for water at most precipitation radar wavelengths. The measured Z is then derived from [15] and more specifically for the MRR radar by

_{m}reported by the MRR radar processor.

_{η}= 1.339 × 10

^{−2}when η has units of mm

^{2}m

^{−3}while C

_{R}= (π/6)(3.6 × 10

^{−3}) × 6.637 = 1.2510 × 10

^{−2}and R is in mm h

^{−1}. We then have

^{2.89}term over the diameter distribution of R. What is done in practice is that for a given Doppler spectrum, the distribution of R is calculated using the diameter bins corresponding to a particular spectrum. The mean D

^{2.89}is then computed, weighted by that distribution and normalized by the total R. Additionally, since $Z=\frac{1}{{C}_{\eta}}\eta $ from (4), it follows finally that

_{m}.

_{m}measured by the radar. We know that η = ∑S(v) Δv, where S(v) are the spectral power densities and Δv is the Doppler velocity increment, which for the MRR is approximately Δv = 0.1858 m s

^{−1}(=11.89 m s

^{−1}/64 bins). We also know that to the first approximation at the i

^{th}velocity bin, the measured v

_{i}= V

_{t}(D

_{i}) + w

_{t}where V

_{t}(D

_{i}) is the actual fall speed of the drop of diameter D

_{i}and w

_{t}is the true vertical air speed. To search for w

_{t}, we can add (or subtract) jΔv for j from 1 to 64. That is, for each velocity bin v

_{i}and the jth Δv

_{i}= V

_{t}(D

_{i}) + w

_{t}+ jΔv.

_{i}corresponding to V’

_{t}we can then compute a new estimate of Z using (10). We then do this for all 64 Δv so that we end up with an ensemble of calculated Z

_{j}for j from 1 to 64. By then using the observed Z

_{m}reported by the MRR, we can find the j corresponding to the Z

_{j}= Z

_{m}. We then know that this is the incremental change to all the fall speeds for all the drops because of w. However, since we also now know that we have the correct size distribution because Z = Z

_{m}, we know that the spectral fall speeds now represent their true values in still air, namely v

_{i}= V

_{t}(D

_{i}), so that from (11), 0 = w

_{t}+ jΔv, and w

_{t}= −jΔv.

^{−1}. Thus, one expects that the ratio Z/∑η(v) should vary as illustrated in Figure 3.

#### 2.2. An Example

^{−1}where the sign convention is such that negative values are downdrafts. This means that the w

_{t}above is null at j = 32. Consequently, the estimated w

_{t}= (32 − j)Δv if j is determined by a counter clockwise circular shift of the velocity bins with respect to the fixed array of drop diameters and w

_{t}= (j − 32)Δv if j is determined by a clockwise circular shift of the velocity bins.

^{−1}) is most consistent with historical Z–R relations (e.g., [11,16]) More significantly, it is also the one associated with the greatest total concentration of particles, N, as shown in Figure 5.

_{Solution}or Z

_{Sol}.

## 3. Some Results of Analyses

^{−1}. In these analyses, we used the reported Z

_{e}(referred to as Z

_{m}above). However, in order to match these Z

_{e}to the observed S(v), the reported Z

_{e}were adjusted downward, as explained further in Appendix A.2, because the attenuation is likely greater than that used in the MRR calculations, which were based upon the usually erroneous assumption that the vertical air velocities were null. The solution spaces for the two time periods analyzed below are shown in Appendix A.3.

#### 3.1. Lighter Convective Rain during Later Period of Convection

^{−1}(all times are in Universal Coordinated Time (UCT)). The outstanding features are the overall weak rainfall rates with the exception of those at the top of the figure (1.28 km) showing values in excess of 1023 mm h

^{−1}. This is a peculiar structure that is difficult to understand physically. The derived vertical airspeeds shown later suggest that rather than w = 0 assumed by the MRR, there were updrafts. Hence, the MRR would have substantially over-estimated the number of small drops required to produce the observed Z, thus leading to unrealistically large rainfall rates.

^{−1}, which seem much more consistent with convection, while the unusual layer at the top in Figure 6 is also gone for the reason just given.

_{m}appears to be uncorrelated to the MRR raw rainfall rates, which, of course, is nonsense. It must be concluded then that when looking at ‘instantaneous’ MRR data, one must account for the effect of the vertical air velocity if the rainfall rate values are to be believed.

^{−1}with a standard deviation of 0.57 m s

^{−1}), they can obviously profoundly alter the deduced drop size distributions and their integrated properties, as noted previously by [11].

#### 3.2. Convective Rain Early Period

_{e}had to be decreased to account for more attenuation, as discussed in Appendix A.2.

_{m}to be to 1054 mm h

^{−1}, a very unrealistic value.

^{−1}. Indeed, at a few locations, the vertical airspeeds at the top right at times exceed the Nyquist interval, so that there was folding of the Doppler spectra. This has been taken into account in Figure 15, which is why the vertical air speed is shown to exceed –5.94 m s

^{−1}at some locations. In this case, the mean w was 0.07 m s

^{−1}, while the standard deviation was 1.66 m s

^{−1}. Consequently, while the mean value was indeed close to zero, the standard deviation illustrates that significant deviations from this mean were occurring throughout the data. In any event, as evident from the analyses, small changes in w can have profound consequences on the deduced drop size distributions and their integral properties, such as the rainfall rate. Nevertheless, it is clear that the patterns indicate that the air velocity estimates are coherent and not just random numbers.

^{−1}up to around 6 ms

^{−1}in gusty conditions. In the later time, the speeds were steadier at around 2–3 m s

^{−1}, but the direction was fluctuating between west to north. Hence, some differences between the ground and MRR observations are to be expected, since the comparison will only be as meaningful as is the persistence of the correlations in time and height over the distance the precipitation moves.

^{−1}so that over one minute (the resolution of the data to be shown), they fall a distance of 360 m. The nominal sampling area of a 2DVD disdrometer is around 0.01 m

^{2}so that the sample volume for this size of drops would be on the order of 3.6 m

^{3}. On the other hand, the radar has a sample volume depth of 10 m and a nominal 1.5° beam width with an averaging interval of 10 s for these data. Hence, over one minute (the temporal resolution of the ground measuring devices), the radar samples over approximately 22 m

^{3}or a volume about 6 times larger than does the 2DVD. In addition, of course, the precipitation is moving say at 2 m s

^{−1}so that any comparison must assume rain coherence over around 120 m horizontally and over 20 m vertically for a volume coherence over 2.4 km

^{3}. Figure 7 and Figure 15b suggest that that can certainly but not always happen. Under that assumption, then, we compare ground observations with the near surface radar observations in Figure 19.

_{m}) along with the values after adjusting for the vertical air velocity. There is remarkable agreement between the disdrometer and R

_{w}at 4 min, and both plots show a peak at 8 min, although the R

_{w}value is significantly larger, certainly possible in convective given the 20 m height separation.

_{w}) values are remarkably similar, while the MRR raw values are unrealistic, again suggesting the importance of adjusting the MRR observations for the vertical air speed when estimating rainfall rates.

## 4. Discussion

^{−1}, that the radar reflectivity is poorly related to the mean fall speed in part because it also depends upon drop concentration and assumptions about the form of the size distribution, and that the vertical air velocity is often non-zero.

_{R}depends upon the fit to D

^{3}V

_{t}used and equals 79.936 using the fit in Figure 2. Interestingly, this relation is quite general, and it implies that Z–R relations for point observations are controlled by the diameter term.

## 5. Conclusions

^{−1}. These values are often deficient and likely produce unrealistic results. In contrast, the approach used here has the advantage that one can experiment with different values of overall attenuation that can maximize the number of solutions derived using (10) while producing much more realistic results. It must be remembered, however, that these results are only estimates that may or may not be correct, but they are likely better than just using the raw MRR values. Moreover, as a reminder, the important point here is that this approach is applicable to other radar frequencies, particularly to those without the complications that attenuation poses for the MRR radars.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Backscatter Cross-Sections Used

**Figure A1.**The ratio of the T-matrix calculated backscatter cross-sections for spherical raindrops to that expected if the scattering were Rayleigh. The inset corresponds to that reported in the MRR literature showing that the backscatter cross-sections used in this study are in agreement.

#### Appendix A.2. Corrections to MRR Z

_{a}and estimates of the equivalent reflectivity factor Z

_{e}calculated by using the drop size distributions deduced from the spectra under the assumption that the vertical air velocities were null. As shown in the paper, this is not usually a good assumption that likely contains unaccounted attenuation. A different approach is used here.

_{e}is in mm

^{6}m

^{−3}, and η is in m

^{−1}, while η is the sum over the Doppler spectrum, η = ∑S(v). A similar expression can be written for Z

_{a}as well except that it contains the unknown attenuation component, A, namely

**Figure A2.**Linear fits of the logarithms of the MRR-reported equivalent radar reflectivities, Ze, and the attenuated reflectivities vs. the logarithms of the summations over the reported Doppler spectral powers. SE is the standard error of the fits.

_{a}is affected by varying amounts of attenuation evident by the spread in the ∑S(v). While some spread may also arise from non-Rayleigh scattering, the largest effects in rain will be likely largely from attenuation.

**Figure A3.**As in Figure A1, linear fits of the logarithms of the MRR-reported equivalent radar reflectivities, Ze, and the attenuated reflectivities vs. the logarithms of the summations over the reported Doppler spectral powers but for the later period. SE is the standard error of the fits.

_{e}will mostly match Z

_{a}simply by subtraction. This can be determined by looking at the end farthest to the left when attenuation should be at a minimum, so that the shift represents some other change in the system aside from attenuation. However, at larger ∑S(v), some additional adjustment has to be included, as is evident in Figure A3, where the separation between the Z

_{a}and the Z

_{e}lines increase with increasing ∑S(v), likely because of increasing attenuation. Thus, for the later set of data, Z

_{e}only had to be mostly adjusted downward to get solutions using (10), as Figure A3 illustrates.

_{a}and the Z

_{e}as a measure of the characteristic A. This turns out to be equivalent to the shift between the Z

_{a}and the Z

_{e}fits plus the combined standard errors given in Figure A2, which are used to provide a measure of this spread caused by the attenuation. This procedure produces solutions using (10) throughout most of the data field. The final resulting fit for these data is illustrated in Figure A4. It is close to theoretical expectations as given in (A2).

**Figure A4.**Resulting relation between Z

_{e}and ∑S(v) after the adjustments for attenuation to the Ze in Figure A1, as described in the text.

#### Appendix A.3. Examples of Solution Time–Height Profiles

**Figure A5.**Plots of the solution spaces using (10) for the earlier time period (

**a**) and the later time period (

**b**). Solutions were found throughout except at the locations in black where there were no back-scattered powers.

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**Figure 1.**A plot of the T-matrix computed [14] backscatter cross-sections σ

_{B}of spherical raindrops of size D (appropriate for vertical pointing observations) corresponding to the Micro-Rain Radar (MRR). The validity of these cross-sections is illustrated in Figure A4 in Appendix A. While there is a slight roll-off at the very largest D, these drops rarely contribute much to most rainfall rates or even Z, so that these deviations are likely usually inconsequential.

**Figure 2.**The volume flux of raindrops falling at terminal velocity, V

_{t}, as a function of spherical drop size.

**Figure 3.**Histogram of the ratio of the measured radar reflectivity to the sum of the power densities over the Doppler spectrum showing variable rather than a constant value.

**Figure 4.**An example of the application of the approach described in the text. The solid line represents the theoretical calculated radar reflectivity (vertical axis) using (10), while the dashed line represents the MRR measured value. There are two solutions for the vertical air velocity, w, one of which is the preferred, as discussed in the text.

**Figure 5.**A plot of the difference between the measured (Z

_{m}) and the calculated Z (Z

_{calc}), illustrating that the likely correct solution (Z

_{Solution}) is associated with large values of the total number of drops as appears to always be the case in this study.

**Figure 6.**The contour plot of the time–height values of the log of the measured rainfall rates, R

_{m}, reported by the MRR processor, computed assuming that there is no vertical air velocity. Note the weak value except near the top of the figure. Times are UCT, and a log scale is used to highlight the weak structure.

**Figure 7.**The time–height plot of air velocity corrected rainfall rate (R

_{w}) corresponding to Figure 6. Note the dramatic vertical shafts of rain that are now apparent as well as the disappearance of the spurious values at the top compared to the previous figure. Note also the general significant increase in the values of R.

**Figure 8.**The time–height plot of the relative dispersion = σ

_{Rw}/R

_{w}corresponding to Figure 7. Note that in this instance, throughout most of the domain the relative dispersion is unity or smaller.

**Figure 9.**(

**a**) The time–height plot of the measured MRR measured radar reflectivity for this time period and (

**b**) the calculated solution values (Z

_{Solution}). Note the vertical structure in Z as reflected by the rainfall rates in Figure 7. The added jitter in (

**b**) is due to the quantization of the velocity bins. The black denotes no power observed.

**Figure 10.**Histograms of the MRR measured (subscript Obs, solid bars) and of the solution (subscript Sol, red empty bars) radar reflectivities in Figure 8 showing the excellent matching, suggesting the general validity of the solutions.

**Figure 11.**Z–R correlations derived (

**a**) for the rainfall rates adjusted for vertical air speed and (

**b**) those from the MRR processor, as discussed further in the text.

**Figure 12.**The air vertical velocities, w, derived using the approach described in the text. Note the coherent temporal and spatial structure of the winds with weak updrafts and downdrafts in the center and stronger downdrafts on each side. The velocity magnitudes are also consistent with the observations of others elsewhere, as discussed in the text.

**Figure 13.**The time–height plots of (

**a**) the observed radar reflectivity Z and (

**b**) Z

_{Sol}from the solutions for the earlier, more convective time period on this day. Finer structure is apparent in (

**b**), but the values are nearly identical on the whole, as indicated in the next figure. Black denotes no usable power observations after adjustment.

**Figure 14.**Histograms of the MRR measured (Obs) and solution radar reflectivities in Figure 13 showing the excellent matching suggesting the general validity of the solutions.

**Figure 15.**Time–height plots of the rainfall rates (

**a**) from the MRR processor (R

_{raw}) and (

**b**) those after accounting for vertical air speed (R

_{Sol}). Compared to (

**a**), note the significantly larger values throughout (

**b**), which are consistent with the structure of Z in Figure 13. Black denotes no power.

**Figure 17.**The air vertical velocities derived using the approach described in the text. Note the coherent but smaller scale temporal and spatial structure of the winds compared to those found in the later time period (Figure 11) likely reflecting the more convective nature of these data. Spectral folding occurred in a few locations leading to the enhanced velocity scale. The large black areas are due to a lack of power, while smaller ones are where no solutions were found without having to further process the data.

**Figure 18.**Z–R correlations derived (

**a**) for the measured rainfall rates (R

_{m}) from the MRR processor and (

**b**) those corrected for vertical air speed (R

_{w}), as discussed further in the text.

**Figure 19.**Comparisons of the MRR measured (black lines) average one-minute rainfall rates (R

_{m}) at 20 m AGL and of the air velocity adjusted (red lines) rainfall rates (R

_{w}) with the nearby 2DVD estimated rainfall (blue lines), as discussed in greater detail in the text for (

**a**) the more convective time period and (

**b**) for the later time period. Note the poor performance of the measured MRR values in (

**b**).

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**MDPI and ACS Style**

Jameson, A.R.; Larsen, M.L.; Wolff, D.B.
Improved Estimates of the Vertical Structures of Rain Using Single Frequency Doppler Radars. *Atmosphere* **2021**, *12*, 699.
https://doi.org/10.3390/atmos12060699

**AMA Style**

Jameson AR, Larsen ML, Wolff DB.
Improved Estimates of the Vertical Structures of Rain Using Single Frequency Doppler Radars. *Atmosphere*. 2021; 12(6):699.
https://doi.org/10.3390/atmos12060699

**Chicago/Turabian Style**

Jameson, Arthur R., Michael L. Larsen, and David B. Wolff.
2021. "Improved Estimates of the Vertical Structures of Rain Using Single Frequency Doppler Radars" *Atmosphere* 12, no. 6: 699.
https://doi.org/10.3390/atmos12060699