Spatial and Temporal Network Sampling Effects on the Correlation and. 1. Variance Structures of ... VA 22207-1560 U.S.A.. 57. Email: arjatrjhsci@verizon.net.
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Spatial and Temporal Network Sampling Effects on the Correlation and Variance Structures of Rain Observations A. R. JAMESON RJH Scientific, Inc., El Cajon, California (Manuscript received 6 June 2016, in final form 17 October 2016) ABSTRACT Network observations are affected by the length of the temporal interval over which measurements are combined as well as by the size of the network. When the observation interval is small, only network size matters. Networks then act as high-pass filters that distort both the spatial correlation function rr and, consequently, the variance spectrum. For an exponentially decreasing rr, a method is presented for returning the observed spatial correlation to its original, intrinsic value. This can be accomplished for other forms of rr. When the observation interval becomes large, however, advection enhances the contributions from longer wavelengths, leading to a distortion of rr and the associated variance spectrum. However, there is no known way to correct for this effect, which means that the observation interval should be kept as small as possible in order to measure the spatial correlation correctly. Finally, it is shown that, in contrast to network measurements, remote sensing instruments act as low-pass filters, thus complicating comparisons between the two sets of observations. It is shown that when the network-observed spatial correlation function can be corrected to become a good estimate of the intrinsic spatial correlation function, the Fourier transform of this function (variance spectrum) can then be spatially low-pass filtered in a manner appropriate for the remote sensor. If desired, this filtered field can then be Fourier transformed to yield the spatial correlation function relevant to the remote sensor. The network and simulations of the remote sensor observations can then be compared to better understand the physics of differences between the two set of observations.
1. Introduction Rainfall is a very complex and variable medium. Consequently, there are at least hundreds of papers written describing and modeling its variability in space (e.g., Jaffrain et al. 2011; Lee et al. 2009, 2007; Schleiss et al. 2012; Tapiador et al. 2010; Tokay and Öztürk 2012; Crane 1990; Krajewski et al. 2003; Habib et al. 2001; Kundu and Bell 2003; Seed et al. 1999; Zawadzki 1973; Ciach and Krajewski 2006; Jameson et al. 2015b; Jameson and Larsen 2016a,b) and in time (Crane 1990; Kundu and Bell 2003; Paulson 2002; Schleiss et al. 2012; Steiner and Smith 2004; Ciach and Krajewski 2006; Jameson et al. 2015a; Zawadzki 1973). Many of these papers discuss both spatial and temporal scaling with resolution, particularly with regard to applications to
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Corresponding author e-mail: A. R. Jameson, arjatrjhsci@ verizon.net DOI: 10.1175/JHM-D-16-0129.1 Ó 2017 American Meteorological Society
remote sensing measurements (Brommundt and Bárdossy 2007; Morrissey 1991; Habib and Krajewski 2002; Raupach and Berne 2016a). Others (Jaffrain and Berne 2012a; Raupach and Berne 2016a) attempt to collect detailed observations over dimensions consistent with the footprint size of radar or satellite instruments for direct comparison with observations by those sensors. Whatever the motivation, a common thread is that of measurements. In all of these studies, however, the process of measurement is treated as a passive act, even though most investigators are quite aware that the scales both in time and space over which measurements are collected lead to different results (e.g., Ciach and Krajewski 2006). The focus in this work is not on the nature of rain, but on how the measurements themselves can lead to some of the observations that might be falsely attributed to being a characteristic of the rain itself. A particularly interesting example of this is presented in the very informative work of Ciach and Krajewski (2006, their Fig. 6), which illustrates just how the radial correlation function observed over an approximately 4-km 2D grid of rain
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gauges increases as the temporal period of observations increased from 5 min to 3 h. While they attributed this to the complex spatial–temporal structure of rain, we show below that the most likely alternative explanation arises solely from the measurement process. There have been a few hints at the importance of the measurement process in recent work. For example, Jameson et al. (2015b) found that the increase in the variability of drop size distributions over a small network of disdrometers could be explained qualitatively as arising from a filtering effect of the network itself, although they provided no quantitative description of how this would work. Similarly, Jameson and Larsen (2016b) argue that the same effect leads to the observed increase in the variability of the rainfall rate with increasing characteristic dimension D of a network, again without a comprehensive, quantitative description of this filtering effect. In neither of these works was there a discussion of temporal effects that are now considered below as part of this expansion on that work. The central objective here is to point out that observations over limited spaces but brief times represent highpass filtering of the information. This will be given more precise meaning below. It will also be discussed how attempts to use measurements over a network for direct comparison with observations by a remote sensor are confounded because remote sensor footprints act like low-pass filters so that comparing the two represents a fundamental incompatibility both in spatial correlation structures and in their respective variance spectra. This, then, is a challenge to ground validation efforts of some space missions for observing Earth. We begin below with the development of a quantitative description of the filtering effect of a network. The effect of temporal filtering will then be presented. These filters are then applied to data having assumed exponentially decreasing spatial and temporal correlation functions often observed in practice. When applying these filters, it is assumed that the temporal and spatial structures are fundamentally statistically and physically independent. That is, even if rain were not moving, it would exhibit fluctuations in time, and conversely, if it were not fluctuating in time, it would exhibit fluctuations at a point because of the movement of the rain field. In most cases, of course, the variability at a point or over a network is a D 5 2
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FIG. 1. Fractional amplitude contribution of different wavelengths as functions of the network length scaled by the characteristic network dimension (i.e., D) showing that longer and longer wavelengths contribute less and less to the variance over the network.
combination of both the temporal and spatial fluctuations (Jameson et al. 2015a).
2. Preliminary considerations To calculate how the network characteristic dimension affects measurements, first consider the spatial wavelength spectrum interacting with a grid. The qualitative discussions in Jameson et al. (2015b) and Jameson and Larsen (2016b) describing a grid as a ‘‘porous’’ or ‘‘soft’’ filter are now given explicit quantitative meaning in this work. As Fig. 1 illustrates, a network feels the full amplitude of a wavelength l only for l # 4D. For l . 4D, less and less of the full amplitude impacts the network. In other words, a network actually feels the full effects of all wavelengths less than 4 times its size with decreasing contributions from wavelengths longer than 4D. Since the variance is proportional to jlj2, it follows after replacing the wavelength with its inverse, namely, the spatial wavenumber k, that the results illustrated in Fig. 1 can be expressed explicitly as a wavenumber filter illustrated in Fig. 2, where D2 is the magnitude of the filter. The expression in Fig. 2 can also be rewritten in a continuum mode as
k k k 2 2pk 2 UnitStep k 2 sin 1 1 2 UnitStep(k) 2 UnitStep k 2 , UnitStep k 1 D D D D (1)
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FIG. 2. Expression for the low-pass filter corresponding to Fig. 1, where k is the wavenumber and D is the network size.
transform pair when the data are spatially statistically homogeneous or temporally statistically stationary. While certainly not always the case, there are instances of rain as shown in Jameson (2016) for a several-hourlong event and in Jameson et al. (2015a) for two 2-h-long periods of convective and stratiform rain. Moreover, there is evidence that sometimes statistically heterogeneous rain can be partitioned into several additive statistically homogeneous components (Jameson 2007) or intervals so that each could be treated separately in the manner described in this work. Finally, statistical inhomogeneity often appears as trends. In that case, the trends can be removed using one of several standard techniques so that the data would then reveal their underlying statistically homogeneous characteristics. Therefore, the existence of statistically heterogeneous rain is not necessarily a serious obstacle to the conclusions reached below. Thus, the Fourier transform equivalent expressions corresponding to (2) are S(k) 5
where UnitStep denotes the Heaviside step function. This is the filter that is then multiplied by the unfiltered variance spectrum to yield the filtered variance spectrum as discussed later. While there are potentially a wide variety of forms for the spatial correlation function, throughout this work we will only consider exponentially decreasing correlation functions as have been often found in observations (Zawadzki 1973; Jameson 2016). The discussions presented below can also be applied to other forms as well, of course. That is, for radial correlation functions we have rr 5 exp(2r/L) rt 5 exp(2t/T) ,
and
S(v) 5
where rr is the spatial radial correlation function where r is the separation distance between two locations and L is the correlation distance. For time, the equivalent expression is the temporal correlation function at a location rt, where t is the time between a pair of observations and T is the correlation time. Jaffrain and Berne (2012b) reported a stretched exponential spatial correlation function when using measurements over a mountainous, developed terrain in Switzerland. This will lead to some differences as discussed below and in Jameson and Larsen (2016b). The Weiner–Khintchine theorem (Wiener 1930; Khintchine 1934) shows that the correlation function and the variance (power) spectrum are a Fourier
and
2p(1 1 L2 k2 ) T2 2p(1 1 T2 v2 )
,
(3)
where k is the wavenumber, v is the frequency, and S is the variance (power) spectrum. When the expression in Fig. 2 is applied to S(k) in (3), the effect of network filtering is obvious in Fig. 3. Because the smaller wavenumbers play a lesser role, we anticipate that a network will see a more rapid decorrelation than that of the unfiltered or intrinsic field. This will be illustrated in a moment, but for completeness the filtered S(k) becomes k S(k) 5 UnitStep k 1 D 2p(1 1 L2 k2 ) k 2pk sin2 2 UnitStep k 2 D D k , 1 1 2 UnitStep(k) 2 UnitStep k 2 D (4) L2
(2)
L2
where UnitStep denotes the Heaviside step function. In the temporal domain, the addition of several 1-min observations into a longer block of time of length T is equivalent to a low-pass filter because it suppresses all frequencies greater than 1/T. In standard Fourier theory, this is represented by a rectangle of width T having a unit area. Its filtering effect on frequencies is given by the Fourier transform of this rectangle,
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FIG. 3. An example of the intrinsic variance spectrum and that filtered by a 1-km network for a 1/e decorrelation distance of 2 km for an exponentially decreasing spatial correlation function.
FIG. 4. The filtered and unfiltered frequency spectrum for a 30-min observation period and decorrelation time T 5 10 min and where v is the frequency.
namely, the well-known sinc function, sin(v)/v, where v is the frequency. Figure 4 illustrates the filtering effect for T 5 30 min and for T 5 10 min. As T increases the filter narrows, which, of course, is what happens to the variance of the sample mean as the number of independent samples increases. Again, for completeness we have
one is mostly affected by what is within an area (the network) while the other is affected only by what lies outside the area (beam footprint). This leads to differences in the correlation functions and in the variance spectra. Thus, there is an inherent difficulty when attempting to compare these two different kinds of observations. It is shown below that this difference is further complicated if the network is used so that each observation is over an extended interval. Yet, the central objective of ground validation experiments, particularly with regard to spaceborne instruments, is just such a comparison. These points will be explored further below.
S(v) 5
T2 sinc2 (Tv) p2 (1 1 T2 v2 )
.
(5)
In two dimensions, this same kind of low-pass filtering is occurring over the footprint of a remote sensor. A stationary footprint, however, cannot detect structures associated with wavelengths less than the beam dimension B. Rather, any signal fluctuations on these smaller scales are the result of the reshuffling of the relative spatial positions of the scatterers during sampling and not because of any effects of substructures (Jameson and Kostinski 1996). Consequently, there is no way to measure the subbeam variance for a stationary beam so that variances associated with wavelengths less than B play no role in the variance spectrum or associated correlation function of the remote sensor observations. Therefore, for remote sensors, it is appropriate to simply introduce a rectangular sampling box into wavenumber space, setting the contributions from all wavenumbers greater than 2p/B to zero while retaining all smaller wavenumbers. In terms of wavelengths, we will only retain those wavelengths greater than B. Obviously, then, there are significant differences between the filtering by a remote sensor and by a network;
3. Some examples From the discussion above, it is apparent that both spatial and temporal effects of the measurement process must be considered. First, we look at the spatial component. In these examples, we assume that the observation interval is no more than a minute or two so that temporal effects are very small.
a. Examples of spatial effects As in (2), we use an exponentially decreasing spatial correlation function having L 5 2 km. The corresponding variance spectrum (3) is then modified using (4). The Fourier transform of this output then yields the radial correlation function rr that would be observed by networks of different sizes. These results are illustrated in Fig. 5. The rr are well represented by exponential fits, although other fits are possible as well. Using these fits, it
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FIG. 5. The filtering effect on the radial (spatial) correlation function (i.e., rr) for various sizes of networks.
is clear that L is significantly altered from the intrinsic 2 km depending upon D, so that for D 5 0.5 km the observed L would only be 1.47 km while for D 5 4 km, it would be 1.83 km. A 1-km network such as that of Jaffrain and Berne (2012b) would underestimate L by 21%, while a 0.1-km network such as that of Jameson et al. (2015a) would only see about 57% of L. The relation between the intrinsic or actual Li and the observed Lo in kilometers can be captured in a simple relation that is valid to the 0.9999 correlation level for 0.01 km # D , ‘ as 2 1 Li 5 Lo1/2 1 D21/2 2p 2 1 21/2 1/2 Lo 5 Li 2 D 2p
or (6)
provided that the intrinsic correlation function is a decreasing exponential function, of course. This is illustrated in Fig. 6, clearly showing that the observed correlation distance is underestimated by measurements using a network. One of the objectives of research of rain using networks is the comparison with what a remote sensor would measure (e.g., Tokay and Bashor 2010; Wolff et al. 2005; Datta et al. 2003; Jaffrain and Berne 2012b; Raupach and Berne 2016b). As discussed above, a remote sensor is essentially a low-pass filter while a network is more of a high-pass filter. To see how this affects the observations, we again consider an intrinsic exponentially decreasing radial correlation function rr with L 5 2 km. Because of the Fourier
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FIG. 6. The relation between the networks measured correlation lengths of an exponentially decaying spatial correlation function and the true or intrinsic values as given in (6). Finite domains reduce the observed correlation length from their intrinsic values.
transform equivalence of the variance spectrum and the spatial correlation function, we only consider what happens to rr. Figure 7 illustrates what the radial correlation function a 1-km network would measure compared to what a 1-km radar beam having a uniform illumination function would measure. Because of the difference in the filtering, the radar would measure an L of nearly 4.23 km or more than twice the intrinsic spatial correlation length of the rain, while the network would observe L 5 1.57 km. Of course, an astute observer realizes that the radar would see a different rr and would, therefore, apply the filtering effect of the beam to the network observations. The result is shown by the dash–dotted line in Fig. 7, where the correlation length hypothesized to be observed by the radar would now be closer to 3.42 km. Furthermore, if one assumes or knows that the intrinsic spatial correlation is a decreasing exponential, one can use (6) to estimate Li . One could then apply the beam filter effect to precisely derive what the remote sensor would see with regard the spatial correlation function and hence with regard to the proper variance spectrum. In the process one would, of course, also derive the correct intrinsic spatial correlation. These are indicated by the red lines in Fig. 7. Of course, this requires observation times of at most a few minutes so as to minimize time effects discussed below. Moreover, other conversions similar to (6) would have to be derived for other forms of the spatial correlation functions.
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FIG. 7. The intrinsic (subscript i) and the observed (subscript o) rr for a 1-km network as well as that observed by a remote sensor having a 1-km beam. The dotted red line denotes the corrected observed rr while the dashed red line denotes the expected correlation function of the 1 km beam using the corrected rr. The dash– dotted line is the expected rr for a 1-km beam had there been no correction to the observed rr.
This illustration highlights the importance of using as small a beamwidth as possible when attempting to use radar observations as a surrogate to direct network rain measurements as in Raupach and Berne (2016a,b). It also highlights the extreme difficulty in using network observations for studying large remote sensor footprints that are most sensitive to the largest wavelengths. The variance structures in the two sets of observations will be considerably different, and, as illustrated later, these differences are likely to lead to significant deviations between estimated mean quantities, particularly when the ensemble sizes of samples are small. Potential methods for compensating for these differences are discussed at the end of this work. Aside from these spatial considerations, what a network measures will also depend upon how long the sample period lasts. That is, often 1-min measurements are collected but are then combined over a longer time period T in order to reduce the effects of fluctuations. Next, we consider what happens to estimates of rr when T increases both with and without advection. The latter is important because it tends to offset some of the filtering effects of the network, but, unfortunately, in an unknown amount.
FIG. 8. Examples of the filtering effects on an intrinsic temporal exponential correlation function (i.e., rt) having a decorrelation time T as the interval T increases.
to approximately 12, 14, and 24 min for T 5 5, 10, and 30 min, respectively, while the shape of the correlation function transitions by T 5 30 min to a stretched exponential of the form exp(2at 2), where a is a constant. However, network observations are often used to estimate the spatial structure of the rain, not just its temporal characteristics. The question becomes, then, how does this temporal sampling affect the observed spatial structure? To address this, it must be noted that time and space are connected through motion, that is, the velocities of medium that we will refer to loosely as advection whether it be mean air motion or the motion of a precipitating system. Here as a zeroth-order approximation consistent with the Taylor frozen turbulence hypothesis and in order to avoid distracting complexity, we just use a mean advection velocity of 2 m s21, realizing, of course, that there is likely a distribution of advection velocities. This assertion is consistent with the finding of Zawadzki (1973) of a constant mean advection in one rain event. Specifically, we consider a D 5 4 km network similar to that of Ciach and Krajewski (2006). Assume we collect measurements, say, every minute and that for these 1-min values we observe an exponentially decreasing temporal correlation function. We then combine them over a time interval T. The variance spectrum will then be
b. Examples of temporal sample size effects Accumulating observations over time (averaging) can profoundly alter the correlation and, therefore, the variance structure in time. As illustrated in Fig. 8, the network decorrelation times increase from the intrinsic value of 10 min
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S(v) 5
T2
[sinc(vT)]2 . p p(1 1 T2 v2 )
(7)
Since v is the number of cycles moving across a location at a rate, we can set v 5 2pky, where y is a mean
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advection velocity and k is the wavenumber. Advection, therefore, transforms the frequency spectrum into a spatial spectrum. In addition, of course, we have the
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variance spectrum associated with the spatial correlation function observed by a network of size D so that the wavenumber variance spectrum becomes
[sinc(2pkyT)]2 L2 k 1 UnitStep k 1 S(k) 5 D p p[1 1 T2 (2pky)2 ] 2p(1 1 L2 k2 ) k k 2 2pk sin 1 1 2 UnitStep(k) 2 UnitStep k 2 , 2 UnitStep k 2 D D D T2
where y is the mean advection velocity equal to 2 m s21 in the example and the other parameters are defined above. We can calculate (8) numerically. However, if one were willing to replace the filter from Fig. 2 by the more crude approximation of a spatial box filter similar to (7), (8) then becomes T2
[sinc(2pkyT)]2 p p[1 1 T2 (2pky)2 ]
S(k) 5
1
L2
[sinc(kD)]2 . p p(1 1 L2 k2 )
(9)
Unlike (8), the Fourier transform of (9) then yields a closed-form solution for rr using the Weiner–Khintchine theorem. Doing so somewhat underestimates the contributions by the longer wavelengths, but for a large network such as D 5 4 km with L 5 2 km, the differences in rr will be relatively small, as Fig. 6 suggests. Assuming these conditions, the results are illustrated in Fig. 9 for T 5 1, 5, 10, 30, and 60 min, recalling that the effects would be slightly even more enhanced had we used (8) rather than (9). Clearly, the spatial correlation function is increasingly modified as T increases, just as was observed by Ciach and Krajewski (2006) and presented in their Fig. 6 with a flattening and increase in the observed spatial correlation function with increasing T. While they interpreted their observations as reflecting some unknown temporal–spatial characteristic of the rain, these results suggest that their observations were just as likely the result of the temporal sampling effects dominated by advection. These investigators are not alone in ignoring potential sampling effects both spatially and temporally, particularly when combined with advection (e.g., Jameson et al. 2015a; Jaffrain and Berne 2012b; Tokay et al. 2014). Clearly, care must be taken when attributing findings solely to the properties of the rain itself. Thus far, there is also no obvious way for correcting for most of the effects of advection except avoidance.
4. Discussion The measurement of the spatial structure of rain requires using pairs of instruments usually defining some
(8)
kind of network. It is then assumed that what is measured reflects characteristics of the rain itself. But such observations occur in both time and space. It is shown here that the finite spatial dimensions act as a high-pass filter so that a network only sees the full brunt of wavelengths less than 4 times larger than the characteristic network dimension. For longer wavelengths the contribution decreases in a manner described by the filter developed in this work. This affects the variance spectrum and, therefore, the observed correlation function so that it no longer represents the intrinsic or true correlation function (or variance spectrum). Often measurements are combined in time in order to achieve acceptable statistical reliability such as, for example, counting a sufficient number of drops. Many current detectors provide 1-min observations that exclude any significant temporal effects (Tapiador et al. 2010; Jaffrain and Berne 2012b; Jameson et al. 2015a,b). However, these are subsequently often combined over an observation interval of length T. Doing so, however, means that the time interval acts like a low-pass filter that suppresses contributions from those times less than T or, to put it an alternative way, it suppresses high frequencies while allowing for those smaller than 1/T. Hence, temporal filtering and network spatial filtering act in opposite directions. While it is tempting, therefore, to propose using a T greater than the smallest increment, the effects of doing so are unpredictable. That is, because precipitation advects over time, usually this can lead to enhanced but unknown temporal effects on the spatial correlation function. In general, it appears best to use as small an observation period T as possible. Finally, it was noted that the sampling volume of remote sensors also acts as a low-pass filter. This complicates attempts to use network observations as direct validation tools for remote sensing estimates of rainfall variables, for example. At the very least, the network dimension should be at least 1/4 that of the beam dimension, and even then those measurements will be associated with different variance spectra as illustrated in Fig. 10. While a network suppresses smaller wavenumbers associated with wavelengths longer than 4D, a radar beam suppresses the
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FIG. 10. The intrinsic variance spectrum (black) associated with an exponentially decaying radial correlation function having a correlation length of L 5 2 km. The green line denotes the variance spectrum that a 1-km observation network would experience, while the red line denotes that associated with corresponding measurements by a uniformly illuminated radar beam of dimension 1 km. As discussed in the text, the variance spectra observed by the network and by the radar differ significantly.
having a 1-km uniformly illuminated radar beam centered on each of the disdrometer networks. As one would anticipate from Fig. 10, the network sees fluctuations not seen by the radar leading to deviations from the mean
FIG. 9. (a) The advection (velocity) effects for various observation intervals for an intrinsic exponential temporal correlation function having T 5 10 min and a mean advection velocity of 2 m s21 combined with the intrinsic exponential spatial correlation function having L 5 2 km as observed using a 4-km network. Clearly, the advection distorts the intrinsic spatial correlation function (;T 5 1). (b) Figure 6 reprinted from Ciach and Krajewski (2006) with permission from Elsevier and the authors mimics the findings in (a).
larger wavenumbers (smaller wavelengths) while retaining the influence of wavelengths larger than the beamwidth. Clearly, the fluctuations will differ for the different sets of measurements, particularly since radar observations always include sample averaging in which the particle spatial reshuffling significantly further suppresses small-scale fluctuations. Such differences may, therefore, even affect deduced mean values as illustrated in Fig. 11. Using a set of simulated observations as discussed and illustrated in Jameson (2016, Fig. 10), 2500 values of the mean radar reflectivity factor Z were computed over 1-km networks consisting of 16 disdrometers and for a radar
FIG. 11. A comparison of 2500 mean values of the radar reflectivity factor (i.e., Z) as seen by a 1-km observation network and a radar having a 1-km uniform illuminated beam centered on the center of each of the corresponding networks as discussed in the text. The differences can be attributed to the differences in the variance spectra illustrated in Fig. 10.
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values observed by the radar. What can be done to overcome such differences? That is, how can one determine the fluctuation spectrum one would expect to be seen by the radar one is interested in the validation experiment? If one knows (or models) the form of the spatial correlation function for a particular variable of interest based upon observations over short intervals (so as to avoid contamination from the effects of advection), one can then determine the intrinsic spatial correlation length in a manner illustrated in Fig. 7. This function can then be Fourier transformed to estimate the intrinsic variance spectrum. Next, assuming one knows the radar beam illumination function, its spatial Fourier transform can then be used to yield the variance spectrum the radar should have seen by multiplying this transform with the justderived intrinsic variance spectrum. One can then generate simulated radar observations and sets of mean values for direct comparisons between the network observations and what the radar should have seen. Such comparisons can be used to adjust algorithms, for example. The advantage of this approach is that no other instrumentation is required. On the other hand, there is another more direct approach of Raupach and Berne (2016a), who use very fine resolution radar measurements in order to construct a field of estimates that a larger beam radar can then sample. Of course, this approach requires having such an instrument at one’s disposal, which is usually not the case. Acknowledgments. The author was funded by the National Science Foundation (NSF) under Grant AGS1532423 and partially supported by the United States Social Security Administration through my Social Security allotment. REFERENCES Brommundt, J., and A. Bárdossy, 2007: Spatial correlation of radar and gauge precipitation data in high temporal resolution. Adv. Geosci., 10, 103–109, doi:10.5194/adgeo-10-103-2007. Ciach, G. J., and W. F. Krajewski, 2006: Analysis and modeling of spatial correlation structure in small-scale rainfall in central Oklahoma. Adv. Water Resour., 29, 1450–1463, doi:10.1016/ j.advwatres.2005.11.003. Crane, R. K., 1990: Space–time structure of rain rate fields. J. Geophys. Res., 95, 2011–2020, doi:10.1029/JD095iD03p02011. Datta, S., W. L. Jones, B. Roy, and A. Tokay, 2003: Spatial variability of surface rainfall as observed from TRMM field campaign data. J. Appl. Meteor., 42, 598–610, doi:10.1175/ 1520-0450(2003)042,0598:SVOSRA.2.0.CO;2. Habib, E., and W. F. Krajewski, 2002: Uncertainty analysis of the TRMM ground-validation radar-rainfall products: Application to the TEFLUN-B field campaign. J. Appl. Meteor., 41, 558–572, doi:10.1175/1520-0450(2002)041,0558: UAOTTG.2.0.CO;2. ——, ——, and G. J. Ciach, 2001: Estimation of rainfall interstation correlation. J. Hydrometeor., 2, 621–629, doi:10.1175/ 1525-7541(2001)002,0621:EORIC.2.0.CO;2.
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