Quantifying Drop Size Distribution Variability over Areas: Some

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Quantifying Drop Size Distribution Variability over Areas: Some Implications for Ground Validation Experiments A. R. JAMESON RJH Scientific, Inc., El Cajon, California (Manuscript received 13 April 2016, in final form 24 July 2016) ABSTRACT In previous work it was found that over a small network of disdrometers, the variability of probability size distributions (PSDs) expressed using the relative dispersion (RD; the ratio of the standard deviation to the mean) increased with the expansion of the network size. The explanation is that the network acts to integrate the Fourier transform of the spatial correlation function from smallest wavelengths to those comparable to the network size D. Consequently, as D increases, so do the variances at the different drop sizes. Thus, RD and PSD variability grow as D increases. The limits to this growth, however, were not determined quantitatively. This finding is given fuller theoretical quantitative meaning over much larger dimensions by explicitly deriving the variance contributions at all the different drop sizes as well as for a variety of moments of the PSD by using spatial radial correlation functions estimated from temporal correlations. This is justifiable when the time for each observation is short. One example is provided. The relative dispersion of the PSD is dominated by fluctuations in the occurrences of the larger drops. The RDs of the raw moments are only a few percent of the PSD. Thus, approaches attempting to estimate radial correlation functions using, say, radar measurements of moments are of limited utility, a usefulness further compromised by the distortion of the correlation function by filtering over the beam dimension. These findings present a challenge for efforts to validate remote sensing measurements by ground truth experiments using networks.

1. Introduction It is widely recognized that rain is highly variable in time and space. While a nuisance in everyday life, this variability has more profound implications with regard to weather forecasts, flooding, and soil erosion, for example. Moreover, this variability is expressed over a wide range of spatial scales from centimeters to hundreds of kilometers. While the latter dimensions are largely controlled by synoptic-scale dynamics interactions with the moisture field, meso- and microscale processes dominate spatial structures at scales less than tens of kilometers. Structures on these scales are expressions of microphysical processes and randomness, and they are the scales of interest here. Perhaps one of the most important and most frequently studied of these expressions is the drop size Denotes Open Access content.

Corresponding author address: A. R. Jameson, 5625 N. 32nd St., Arlington, VA 22207-1560. E-mail: [email protected] DOI: 10.1175/JHM-D-16-0094.1 Ó 2016 American Meteorological Society

distribution (DSD) discovered in the 1940s (Laws and Parsons 1943; Marshall and Palmer 1948; Best 1950) in part because of the central role the DSD plays in turning the first radar detection of rain into potentially useful measurements of precipitation over large areas (Fletcher 1990). The drop size distribution itself can be considered to consist of two components, the total number of drops Nt times the probability size distribution (PSD; Kostinski and Jameson 1999). This is an illuminating separation since Nt plays an ever increasingly important role in the variability of bulk parameters such as the rainfall rate for scales larger than one or a few kilometers (Jameson and Kostinski 2001; Jameson and Larsen 2016). The variability of the PSD is very important on scales from a meter to several hundreds of meters. It must be remembered, however, that the moments of the PSD are always important for estimating bulk parameters derived from the integrals over the DSD even on larger scales. While rainfall rate spectra have been studied previously (e.g., Crane 1990), detailed studies of the PSD on smaller scales have only recently garnered attention (Tapiador et al. 2010; Jaffrain and Berne 2012; Jameson

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et al. 2015a,b; Raupach and Berne 2016b). In particular, both Tapiador et al. (2010) and Jaffrain and Berne (2012) focus on bulk properties of the DSD on scales from 100 m to 1 km. Recently, Raupach and Berne (2016a) considered interpolated DSD over kilometer scales. On the other hand, Jameson et al. (2015b) studied the variability of the PSD on scales up to 100 m using a small network of optical disdrometers. Since the attention here is also on the relation between spatial scales and the PSD, it is worth briefly describing those latter results. Using the relative dispersion [RD; defined in (3)] as the measure of variability, Jameson et al. (2015b) found that RD increased both with increasing network size and with drop size. The reason given is that the network acts as a high-pass spatial filter, effectively excluding wavelengths much greater than the dimension of the network. (As an aside, it is worth noting here that this is just the opposite of what a radar beam does when it smooths out all fluctuations less than the beam dimension while including all larger-scale contributions as discussed again later. It, then, acts like a low-pass spatial filter.) But what happens on scales of 1 km and larger? Tapiador et al. (2010) and Jaffrain and Berne (2012) use moments of the PSD to describe the variability on longer scales than 100 m. It is shown below that the use of moments fails to capture the full extent of PSD variability. The difficulty, however, is that there are no networks of disdrometers extending out to several kilometers that can provide the observation required for the direct measurements of PSD variability over long scales using the approach in Jameson et al. (2015b). Instead, it is necessary to resort to a reasonable, available alternative. By using a constant translation velocity y, we transform temporal exponential correlation functions to become estimates of spatial correlation functions. This can be justified when the time required for each observation is short (i.e., on the order of 1 min or less depending upon y). That is, letting x 5 yt where t is elapsed time,    yt   t x , (1a) 5 exp 2 r(t) 5 exp 2 5 exp 2 T yT XL

Obviously, different y will yield different x and XL, but (1) will be unchanged. While one could use a ‘‘scaled x’’ so that y does not matter, it is useful to express XL explicitly. We do this knowing that, strictly speaking, such transformations are not complete because spatial and temporal correlation functions are different (Jameson et al. 2015a). That is, there are fluctuations due to advection of precipitation patterns, but even if there was no advection, there would still be temporal fluctuations. These latter contributions are ignored here. We will see below, however, that in spite of this, the results are entirely consistent with the previous findings of Jameson et al. (2015b). Consequently, while it is fair to argue about particular details, we believe that the results generally reflect what will likely be found once disdrometer observations over a several-kilometer network become available if, indeed, they ever do. Before continuing with the analyses, however, it is necessary to consider the calculations of the RD for both the PSD and for the various raw moments Dp as discussed next. One dataset will also be considered as an example.

2. Preliminary considerations First, let us consider a statistically stationary time series of counts over M size bins of drops associated with a mean number of total drops N t . Let us then consider a single drop size D so that there is a time series of the number of drops per unit interval ND. Then for that interval the estimate of the PSD for that D is PSD 5 ND /N t . For the entire time series the variance of PSD for that drop size is given by PSD 5

(PSD)2 5

2

(PSD) 5

so that its Fourier transform over an infinitely large network becomes S(k) 5

XL2 , p(1 1 k2 XL2 )

(1b)

where XL is the spatial decorrelation length, T is the temporal decorrelation time, and S is the spectral variance as a function of wavenumber k. This function is then modified by the finite size of a network as discussed later.

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[s2 (PSD) 5

ND Nt

,

2 ND

Nt

,

2

ND Nt

2

2

,

s2 (ND ) Nt

2

.

(2)

The square of the relative dispersion (RD)2 is then given by (RD)2 5

s2 (PSD) 2

PSD

2

5

s2 (PSD)N t 2 ND

5

s2 (ND ) 2

ND

.

(3)

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The variance s2 (ND ) is a spectral function given by the Wiener–Khintchine theorem (Wiener 1930; Khintchine 1934) so that the Fourier transform of the correlation function gives s2 (ND ) as a function of wavenumber k, so that (RD)2 also becomes a function of k. Treating each drop size as a statistically independent random variable, one can then define an algebraic measure of the total potential relative dispersion as a function of wavenumber over M all the drop size distributions as RDS (k) 5 i51 (RD)i (k), where RDi is the relative dispersion of the ith diameter bin. Consequently, the qualitative interpretation in Jameson et al. (2015b) can be given more quantitative meaning out to kilometer dimensions by using the correlation functions over all the drop sizes. Furthermore, the spectral dependence of various raw moments of the distributions (i.e., Dp for 1 # p # 6) can be readily calculated by using the drop counts at all sizes to produce a time series of 1-min values of the quantities and then forming the respective correlation functions that can then be Fourier transformed. We do this below as well. We discuss the data next, noting that there is nothing special about these observations. Other observations show similar correlation functions, but, of course, the details will depend upon the different meteorological conditions. What is more important is to demonstrate the procedure that can then be applied to any dataset.

3. Data and analyses As one illustration, data that have been used and described in previous studies (Jameson 2007) from a Joss– Waldvogel impact disdrometer are used here. Briefly, they were provided by Prof. Carl Ulbrich while at Clemson University. The data are associated with a weather system producing a preliminary 350-min period of steady light rain followed by a 700-min period of more convective rain. It is this latter period we use in this analysis. The correlation functions for each of 17 drop size bins are converted into estimates of the radial correlation functions that then provide information over several kilometers. Figure 1 is a plot of the 1-min values of the rainfall rate R and total number of drops (i.e., Nt) for drops in 17 range bin sizes extending from 0.35 to 3.91 mm diameter. Obviously, the data fluctuate considerably. They are, however, statistically stationary. Specifically, we apply the approach described in Anderson and Kostinski (2010, 2011) using the parameter a 5 (Nmax 1 Nmin)forward 2 (Nmax 1 Nmin)backward, where Nmax and Nmin refer to the number of new record maxima and new record minima counted going forward and backward in time. When the data are statistically stationary, a 5 0. In reality, however, there are always

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FIG. 1. A statistically stationary portion of impact disdrometer time series measurements of the rainfall rate (i.e., R) and total drop counts each minute.

statistical fluctuations with a magnitude dependent upon the sample size. Numerical calculations for 104 random 700-min size samples gives sa 5 4.6, where s denotes the standard deviation. For the rainfall rate, the observations yield a 5 4. Thus, we can assert that any deviations from stationarity are not statistically significant and can be attributed to randomness. This conclusion is important because it permits the calculation of correlation functions over the 700 min, and it allows the application of the Wiener–Khintchine theorem under the usual assumptions of ergodicity. As mentioned, the measurements are all with respect to time. Here we use a translation velocity of the system of 10 km h21. While arbitrary, it at least yields an exponential decorrelation length for the rainfall rate of 2.06 km that is quite close to the 1.9 km observed by Jaffrain and Berne (2012). In any event, while some other value can be used [see the discussion concerning (1)], this one is reasonable and sufficient as an example for looking at the variability of the PSD at longer dimensions. In Fig. 2 the correlation functions for a sample of different sizes (Fig. 2a) and for Dp (Fig. 2b) are plotted along with corresponding exponential correlation fits to the data. Examinations of other observations show similar structures. However, they will show somewhat different values since the meteorology will never repeat precisely. In all cases, though, all correlation functions show decreasing correlation with increasing lags, which many times can be fit with exponentials. In any event, for quantitative purposes, only exponentially decreasing correlation functions are considered here. They will be sufficient to illustrate many of the basic features expected for other decreasing

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FIG. 3. Plots of the decorrelation lengths (i.e., XL) as a function of the drop size (black) and as a function of the power p of the raw moment (red).

FIG. 2. (a) Plots of estimated sample radial correlation functions along with the corresponding best fit exponential function at the indicated diameters. The decorrelation distances of the exponentials reliably capture the 1/e values of the observations. (b) The estimated radial correlation functions for the raw moments and their corresponding best fit exponential functions.

correlation functions. Exponential fits then yield 1/e decorrelation lengths (i.e., XL) plotted in Fig. 3. While the decorrelation lengths of Dp are essentially a monotonic decreasing function of p, those for the different drop sizes exhibit greater variability with a peak of 1.8 km for a D of 2.26 mm. The minimum near sizes around 1 mm diameter may arise from the random breakup of drops larger than 3 mm, which would then leave 1 mm remnants (List and McFarquhar 1990) while suppressing correlation. This, in turn,

would then create a relative peak at around 2 mm because the rarer, larger drops are dominated by more random shot statistics having reduced correlations. However, this is just speculation, and other data will undoubtedly appear somewhat different. It is also not surprising that the decorrelation lengths for the Dp are shorter than for individual D because larger p will be dominated by the larger drops that in many cases tend to be associated with shorter decorrelation lengths, as illustrated in Fig. 3. As derived in (1), the Fourier transforms of these exponential correlation functions then yield the variance spectra for all D and all D p, namely, s2 (D, k) 5

XL2 (D) , fp[1 1 XL2 (D)k2 ]g

(4)

where k is the wavenumber (km21) and XL is the decorrelation length (km). Since the wavelength l is the inverse of k, we can take the network domain size as being approximately equivalent to l. The relative dispersion at each D is then given by (3), and we sum over all RD to estimate the total relative dispersion of the drop size distributions as a function of l, as illustrated in Fig. 4, where the curve is normalized by the total RD out to 10 km (essentially out to infinity). In this example, the normalized RD S reaches 95% of its total by 1.2 km and 99% by 3.1 km. This is consistent with the finding in Jameson and Larsen (2016), using a different set of

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FIG. 4. A plot of the fraction of the total RD for the observed PSDs as a function of the network size for these data and the assumed translation speed. In this example, 95% of the total variance of the PSD is captured using a network of 1.2 km dimension and 99% by 3.1 km.

observations, that the relative dispersion of the drop sizes becomes negligible beyond around 1 km. These precise values, however, will undoubtedly change with different meteorological conditions. A natural question is, what drop diameters are most responsible for the fluctuations in the PSD? Consistent with the findings of Jameson et al. (2015b) for a different rain event, the answer is well illustrated in Fig. 5, which shows that it is drops larger than 2.5 mm and especially larger than 3 mm that contribute most. At least for these data, it is like the tail wagging the dog, with 90% of the total RD from drops larger than 2.6 mm diameter. This is not too surprising given the usual rapidly decreasing mean concentrations with increasing drop size and the greater drop clustering at larger sizes [see Fig. 7 in Jameson and Kostinski (1999)]. In several past studies (Lee and Zawadzki 2005; Lee et al. 2009; Tokay and Bashor 2010; Jaffrain and Berne 2011, 2012), average raw moments of the size distribution have been used as surrogates for describing the variability of drop size distributions. While this obviously has some limitations since several PSDs can produce the same moment (Jameson et al. 2015b), we next consider this point in greater detail in Fig. 6. First, we note that the accumulated RD of the moments increases much more rapidly toward unity as the network dimension increases than does the accumulated RD of the PSD. Consequently, for any given

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FIG. 5. The contributions of the various drop sizes to the RD of the PSDs as a fraction of the total (solid) and to the accumulated total RD (dot–dashed). Note that it is sizes larger than 2.5 mm in diameter that contribute most strongly to the variabilities of the PSDs.

network dimension, the variabilities of the moments are closer to their maxima than is the variability of the PSD, so that they are not very representative of the response of the RD of the PSD to network size. Perhaps even more importantly, the total RDs of the moments are considerably smaller than is the RD of the PSD, as illustrated in Fig. 7. The reason for this is that in contrast to the PSD itself, the moments are

FIG. 6. Plots of the fraction of the total RDs for the PSDs and for the raw moments as functions of the network size. The saturation variabilities of the raw moments are achieved over much smaller network sizes than is the case for the PSDs themselves. However, the higher moments do take larger networks before stabilizing.

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FIG. 7. Plot of the ratio of the RDs of the various raw moments to the RD of the PSDs. The variabilities of the moments are poor representatives of the variability of the PSDs.

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the network, for a remote sensor the situation is reversed. That is, the variances for wavelengths less than the dimension of the footprint are suppressed so that the contributions at longer wavelengths assume everincreasing importance. This can be explored more quantitatively by considering a variable observed using point measurements to have an exponential radial correlation function with a decorrelation length XL 5 2 km, for example. This variable can be raw moments of distributions, integrated parameters such as the rainfall rate or radar reflectivity factor, or total RDs of PSDs. The footprint then defines the domain over which the remote sensor averages. Mathematically, in its most basic one-dimensional form, this is expressed as a rectangular sample box < (uniform averaging vs, say, Gaussian weighted) applied to a field of values. The resulting power spectrum for our assumed exponential correlation function for one dimension of variable V becomes S 5 F(R5V) 3 F(R5V)*

integrals over the entire distribution. Consequently, they smooth over the variability of each of the drop sizes. Hence, raw moments are not very representative of the true extent of the PSD variability. Thus, for example, if one had measurements of the radar reflectivity factor Z (Z } D6 ) for Rayleigh scattering, one would have to multiply the observed RD of Z by a factor of 164 at least in this example so that any error in the estimate would also be greatly magnified. Nevertheless, remote sensing is likely the only way to obtain measurements over large areas. As Raupach and Berne (2016a) discuss, however, point measurements are sometimes used to characterize the mean properties over areas, while at other times remote sensing measurements are used to characterize the same mean properties over areas. Raupach and Berne (2016b) emphasize that going in either of these two directions can lead to substantial differences. Since differences are related to variances, aside from algorithm deficiencies, much of it is then physically rooted in the dependencies of the variances and RDs on the variance spectra. For example, point measurements will see variances not observed by remote sensing over large domains, while remote sensors will see more of the variance contributions at longer wavelengths than will measurements over small areas while smoothing out small-scale fluctuations seen by a network. This will be illustrated later. While we have just explored how the expansion of a network size leads to increases in observed RDs over

5 [F(R) 3 F(V)] 3 [F(R) 3 F(V)]* 5 [F(R) 3 F(R)*] 3 [F(V) 3 F(V)*] so that S5

XL2 sinc2 (kL) , p2 (1 1 k2 XL2 )

(5)

where F represents the Fourier transform, the asterisk denotes complex conjugation, 5 represents the convolution operation, S is the variance spectrum, sinc 5 sin(x)/x, k is the wavenumber (;1/l), L is the dimension of the footprint or beam size, and X L is the decorrelation length. Converting k to wavelength and normalizing by the full variability, Fig. 8 illustrates the effect of the dimension of < (beam size L) on S. When the footprint size is only 1 m, nothing is lost by averaging. The instrument sees all the variability. However, as the footprint dimension increases to beyond a few hundred meters, we start to lose the variability over short wavelengths associated with subbeam size variance, so that the total observed variability loss is about 31% at 1 km and 84% at 4 km. It is no wonder that there would be discrepancies between point measurements and observations using Global Precipitation Measurement remote sensing measurements as reported in Raupach and Berne (2016b) and as discussed further in the final section of this work. It is well known that such averaging smooths out precipitation structures as well [e.g., Fig. 3 in Jameson (2015)]. Since spatial structures are the expression of

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FIG. 8. Plots of the ratios of the variabilities for different sizes of footprints (averaging boxes) normalized to the full variabilities for point measurements as functions of variance spectral wavelengths. The total variabilities decrease as the footprint size expands because of a smoothing over wavelengths smaller than the footprint size. These results are for an exponential radial correlation function having XL 5 2 km.

spatial correlations, such averaging should also alter the observed radial correlation functions. This can be

rradial (x) 5

2[e

XL / L

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FIG. 9. The radial correlation functions corresponding to the variance spectra associated with the different sizes of footprints. While a 1-m footprint reproduces the input radial correlation function, larger footprints increasingly distort the radial correlation functions, as discussed further in the text.

expressed exactly by calculating the inverse Fourier transform of (5) normalized to be unity at null separation, namely,

1 e2x/L 3 (L 2 2eXL /L L 1 e2x/L LH(XL 2 x) 1 e2XL /L LH(2XL 1 x) (XL 2 L) 1 L]

1 e(XL 1x)/L fXL sgn(XL 2 x) 1 x[22 1 sgn(2XL 1 x)] 1 XL 1 xg) , where XL is the decorrelation length, L is the footprint characteristic size, sgn is the sign function, H is the unit step function, and x is the separation distance. Examples are plotted in Fig. 9. The 1-m footprint yields the original input correlation function, but as the footprint size increases, the correlations at smaller wavelengths increase significantly, consistent with the suppression of variability through averaging. There are also lesser increases at increasing wavelengths that end up giving the correlation function a completely different appearance. The function for 4 km, for example, is best fit by a Pearson IV distribution. Hence, any attempt to use remote sensing measurements to estimate correlation functions on finer scales must try to account for these effects. Last, in order to provide one example, we consider values observed using a network smaller than a footprint of a remote sensor in a rain field under observation that

(6)

can lead to significant deviations from what the remote sensor might see. That is, for statistically homogeneous data, one would expect to see the same mean value regardless of area covered. However, a remote sensor and a network will likely see different mean values for any given single comparison just because the variance spectra produce local spatial heterogeneities filtered differently by the two measurement approaches. To give this more quantitative meaning, we use the synthesized rain field described in Jameson (2015) and replotted in Fig. 10a for the radar reflectivity factor. The field observed by 2500 nonoverlapping 1-km networks containing 16 disdrometers each would yield the smoothed field as illustrated in Fig. 10b. Now suppose there was a radar in orbit at an altitude so that it produces a 5-km wide Gaussian-weighted illumination beam and average values calculated assuming 64 statistically independent samples. Obviously,

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FIG. 10. (a) Synthesized input data field (Jameson 2015) compared to (b) the field of values that would have been produced by 2500 nonoverlapping 1-km networks of 16 disdrometers each. The smoothing effects and loss of high-frequency variance is apparent.

Fig. 11 shows that the mismatch between network dimension and footprint size leads to significant differences between the two fields, so that the meaning of a comparison of any two realizations would become more ambiguous. This is more apparent in Fig. 12, where the bias of the networks toward larger values compared to the radar values is evident. The small network often sees small-scale fluctuations and resulting enhanced mean values not seen by the radar. Therefore, one should not just plop a network down and expect to get anything very useful for ground validation of a remote sensor if no thought is given to scaling. A network size larger than

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FIG. 11. (a) As in Fig. 10b, but for data that are color scaled to match the color scale of (b) the orbiting radar centered on each 1 km network but having a 5-km beamwidth as discussed in the text. Of course, the radar misses detailed structures observed by the networks and produces lower values because of beam smoothing (variance spectral filtering).

the footprint might be particularly helpful, but for several practical reasons it is not likely to be achieved and longer wavelengths would still influence the remote sensor more than the network. Consequently, it is one of the primary motivations for generating footprint size finescale pseudonetworks of data for algorithm development either using the Bayesian, physical approach described in Jameson (2015) or by using high-resolution radar measurements as proposed by Raupach and Berne (2016b). While it will not necessarily eliminate all

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FIG. 12. Comparison of 2500 1-km network centered values compared to orbiting radar values over the data field in Fig. 10a, with the beam centered on the center of each network. Clearly, the network sees many different values from what the radar observes, which emphasizes the importance of a network at least matching the beam dimension when performing ground validation experiments.

differences between the mean values because there will likely still be slight differences between the two variance spectra, it should at least help. Fortunately, as the network size increases to several kilometers, the variance spectra approach saturation (Fig. 6), and the two approaches should become equal at least for statistically homogeneous data. In other words, the network acts as a high-pass spatial filter while the radar footprint acts like a low-pass filter. A comparison between the two for ground validation is really only valid when you get to such large dimensions (maybe 4 times the decorrelation length) that the variance is saturated and filtering is no longer important. This is not the only factor affecting such comparisons, of course, given radar side lobes and the vertical structure of the precipitation.

4. Conclusions Precipitation structures are the expressions of spatial correlation functions. Because of the Wiener– Khintchine theorem, however, they are also expressions of the variance spectra. The variance spectrum allows us to understand what we are looking at when making measurements from two different perspectives. Measurements are made either internally or externally over a finite domain having a characteristic dimension D. When collecting measurements internally, the observer witnesses the variability beginning with the smallest scales of the variance spectrum up to

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those being filtered by D. On the other hand, when viewing externally, often using remote sensors, variance spectral wavelengths less than D are suppressed, while those larger than D are retained. The network, then, acts as a high-pass spatial filter while the beam acts as a low-pass spatial filter. Ground-truth experiments trying to validate remote sensing observations therefore contain an inherent conflict because one is then trying to compare internal to external measurements, with each looking at somewhat different parts of the variance spectrum. In contrast to infinite samples, for any normal finite sample size, this is likely to lead to differences between the two mean values because of statistical fluctuations. (This is easy to imagine by comparing the running average of a curve to the values of the curve itself.) Consequently, incompatibilities and inconsistencies can arise (Raupach and Berne 2016b), and ironically, the remote sensing measurements can become a test of the network values rather than the other way around. Other conclusions from this work are that the variabilities of PSDs are increasing functions of domain size. While already noted in Jameson et al. (2015b), the qualitative interpretation using observations over a spatially limited network in that work is given full quantitative meaning here by considering a much more extended domain. In doing so, we find that the variability (RD) reaches a steady, maximum value for network sizes beyond around 1 km for one particular example presented here. Other limits will apply to other data in different meteorological conditions, of course. This is consistent with the finding that the variability of the total number of drops dominates the variability of the rainfall rate R for network sizes larger than around 1 km reported in Jameson and Larsen (2016) for their different set of observations. They found that for their data, the variability of the PSD no longer contributes significantly to the variability of R beyond about 1 km. Furthermore, the variabilities of raw moments of the PSD also increase with domain size, but they achieve stable maximum variability at much smaller domain sizes than do the RDs of the PSDs. More significantly, the RDs of moments are a small fraction of those for the PSD, not only because different PSDs can yield the same moment, but also because moments are integrals over the PSDs that suppress much of their variabilities. As p of Dp increases, the RD decreases. Thus, such moments should not be considered as representative of the RDs of the PSDs. Finally, when using external measurements from a remote sensor having a characteristic beam dimension L, the beam acts to filter out variance spectral components smaller than L. This not only reduces the total

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variance, but it also produces a distortion of the radial correlation function. Therefore, if one wishes to estimate the radial correlation function of an integral variable such as the Z or R over distances longer than can be achieved using a practical or affordable network of disdrometers, one should use a remote sensor with as small a footprint as possible in order to retain as much of the information as possible. Along the same vein, it is also noteworthy that a single disdrometer is sensitive to all the variance spectral components in time which, under the assumption of a uniform advection, can be converted into all the spatial variance spectral components in space. Acknowledgments. This work was supported by the National Science Foundation (NSF) under Grant AGS1532423 as well as by the United States Social Security Administration through my Social Security allotment. The author also appreciates the reading by Prof. A. B Kostinski, whose suggestion was very helpful. REFERENCES Anderson, A., and A. Kostinski, 2010: Reversible record breaking and variability: Temperature distributions across the globe. J. Appl. Meteor. Climatol., 49, 1681–1691, doi:10.1175/ 2010JAMC2407.1. ——, and ——, 2011: Evolution and distribution of recordbreaking high and low monthly mean temperatures. J. Appl. Meteor. Climatol., 50, 1859–1871, doi:10.1175/ JAMC-D-10-05025.1. Best, A. C., 1950: The size distribution of raindrops. Quart. J. Roy. Meteor. Soc., 76, 16–36, doi:10.1002/qj.49707632704. Crane, R. K., 1990: Space–time structure of rain rate fields. J. Geophys. Res., 95, 2011–2020, doi:10.1029/ JD095iD03p02011. Fletcher, J. O., 1990: Early developments of weather radar during World War II. Radar in Meteorology: Battan Memorial and 40th Anniversary Radar Meteorology Conference, D. Atlas, Ed., Amer. Meteor. Soc., 3–6, doi:10.1007/ 978-1-935704-15-7_1. Jaffrain, J., and A. Berne, 2011: Experimental quantification of the sampling uncertainty associated with measurements from PARSIVEL disdrometers. J. Hydrometeor., 12, 352–370, doi:10.1175/2010JHM1244.1. ——, and ——, 2012: Quantification of the small-scale spatial structure of the raindrop size distribution from a network of disdrometers. J. Appl. Meteor. Climatol., 51, 941–953, doi:10.1175/JAMC-D-11-0136.1. Jameson, A. R., 2007: A new characterization of rain and clouds: Results from a statistical inversion of count data. J. Atmos. Sci., 64, 2012–2028, doi:10.1175/JAS3950.1. ——, 2015: A Bayesian method for upsizing single disdrometer drop size counts for rain physics studies and areal applications.

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