Orographic influences on the multiscale statistical properties of precipitation

Authors


Abstract

[1] A case study consisting of three consecutive orographic thunderstorms that occurred on 27 June 1995 in the Blue Ridge Mountains of Virginia is examined from the perspective of relating the surrounding meteorological forcings and underlying orography to the multiscale structure of the rainfall fields. The statistical framework for this multiscale characterization is cascade based and offers a parsimonious parameterization, which can be used in future studies for the purposes of stochastically downscaling rainfall fields. Sequences of radar-derived rainfall maps provide data with which to characterize the multiscale statistical structure and variability of the rainfall. In this case study, rainfall falling at higher topographic elevations was shown to be more intermittent and more organized than rainfall at lower elevations. This trend is contrary to previous studies analyzing the multiscale structure of orographic rainfall and is argued to be the direct result of differing meteorological factors for this type of storm such as the presence of warm rain processes and leeside orographic forcing.

1. Introduction

[2] Hydrologic predictions often require precipitation fields at scales different than those that are physically and/or computationally feasible with the current generation of mesoscale and climate-scale atmospheric models. Therefore upscaling and downscaling techniques are needed to transfer the precipitation fields from one scale to another. In order to develop upscaling and downscaling schemes it is necessary to first characterize the space-time organization and structure of the process (i.e., precipitation). Substantial progress has been made on characterizing the space-time organization of midwestern convective systems and tropical rainfall [Kumar and Foufoula-Georgiou, 1993a, 1993b; Venugopal et al., 1999a, 1999b; Harris and Foufoula-Georgiou, 2001; Harris et al., 2001]. Based on these findings, statistical/dynamical models of precipitation downscaling have been developed [Perica and Foufoula-Georgiou, 1996a, 1996b; Venugopal et al., 1999a]. Statistical-based downscaling gains additional usefulness when the scaling parameters can be linked to changes in the meteorological processes, as in the work by Over and Gupta [1994, 1996] who linked scaling parameters to synoptic scale weather and Perica and Foufoula-Georgiou [1996a, 1996b] who linked scaling parameters to the convective available potential energy (CAPE) of the prestorm environment. With meteorological linkages in the scaling parameters, the precipitation downscaling model can be dynamically coupled into hydrologic-atmospheric prediction models [e.g., Nykanen et al., 2001].

[3] Space-time analysis of orographic precipitation has received much less attention due to the complexities of orographic influences on precipitation processes. Harris et al. [1996] and Purdy et al. [2001] were two of the first studies to explore the multiscale properties of orographic precipitation. Both studies focused primarily on the temporal variability of rainfall by using high density networks of high temporal resolution rain gages along transects in the Southern Alps of New Zealand. The study by Harris et al. [1996] found that for a system moving warm, moist air eastward from the Tasman Sea toward the main divide of the Southern Alps, rainfall time series became less intermittent and less organized as one increased in elevation along a mountain transect. Purdy et al. [2001] furthered the study by Harris et al. [1996] by focusing on the meteorological processes explaining these results. More importantly, Purdy et al. [2001] decomposed a specific three day event into periods of differing atmospheric stability and found time series characteristics differing from the general results of Harris et al. [1996]. This led to linkages of significant changes in rainfall multiscale variability to the stability of the atmosphere in addition to changes in variability as a result of location relative to the underlying orography.

[4] This paper builds on the work of Harris et al. [1996] and Purdy et al. [2001] by using multiscale statistical analysis to characterize the organization of a series of three convective orographic rainfall events that occurred on 27 June 1995 in the Blue Ridge Mountains of Virginia. The multiscale statistical analysis is applied to the two-dimensional (2-D) spatial rainfall patterns, rather than to 1-D time series, and parameter estimates are analyzed over time. Focus is placed on linking changes in the multiscale parameters with the orographic influences on the rainfall process. Of particular interest, this paper will show that the relation between multiscale parameters and the underlying orography found in this study contradict those found by Harris et al. [1996] and Purdy et al. [2001]. This contradiction of findings is argued to be a direct cause of the pronounced differences in the meteorology surrounding the different cases, and exemplifies the need for careful consideration of meteorological influences on multiscale properties of rainfall, if one is to use multiscale frameworks for stochastic downscaling purposes.

2. Event and Data

[5] The Rapidan River basin and southern Virginia storms of 27 June 1995 were chosen as a case study for their pronounced influence of orography and extensive documentation available in the literature. These catastrophic rainfall events occurred on the eastern slope of the Blue Ridge Mountains of Virginia. Three consecutive storm systems (later referred to as storms 1, 2 and 3) passed within the Sterling, Virginia WSR-88D area of coverage as shown in Figure 1. The meteorological conditions and associated orographic forcings that produced and enhanced this series of storms is described in detail by Smith et al. [1996a], Landel et al. [1999], and Pontrelli et al. [1999]. Section 2.1 provides a brief summary of the storm environment while section 2.2 is devoted to the specific meteorological conditions that are similar and different from the Southern Alps of New Zealand events studied by Harris et al. [1996] and Purdy et al. [2001]. The data used in this study is summarized in section 2.3.

Figure 1.

Tracks of the 27 June 1995 Rapidan River basin and southern Virginia storms by surface rainfall centroid locations. Times shown corresponds to the beginning and end of the storm tracks using the WSR-88D Sterling, Virginia, data for 27 June 1995.

2.1. Overview of Storm Environment

[6] The environment on 27 June 1995 was characterized by a synoptic high pressure system over the New England states, low pressure system over the east-central states, and a pronounced easterly flow which fed cool, moist Atlantic air into the eastern slopes of the Appalachians. Synoptic maps, soundings, and radar images along with extensive discussion of the storm environment are given by Pontrelli et al. [1999] and Smith et al. [1996a]. At 0600 UTC, a cold front pushed southward into northeastern Virginia and triggered a convective system (the Piedmont storm) in the prefrontal air over the Virginia Piedmont plains (mideastern portion of Figure 1). The Piedmont storm, here called storm 1, was the largest of the three systems on 27 June 1995 and moved along with the cold front through the lower Rapidan River basin producing the first wave of flash flood warnings for Madison County.

[7] A second convective system (the Rapidan storm or Madison County storm) was triggered just northeast of Madison County at approximately 1200 UTC. Pontrelli et al. [1999] characterized the first storm as prefrontal and this second storm as postfrontal with the layer of low-level easterlies growing stronger and deeper. The conditionally unstable environment of the Rapidan storm, here called storm 2, was characterized by a persistent low-level easterly jet, southerly mid-level winds, westerly upper level winds, and a near-saturated atmospheric column to approximately 6 km above ground level. The orographic upward forcing of the Blue Ridge Mountains lifted the moist easterly winds which sustained and strengthened the convection while the light upper level southwesterly winds resulted in small net storm motion. The storm moved quickly into the Rapidan River basin. It then slowed and intensified producing extreme, heavy rainfall for approximately 6 hours. At 1800 UTC, the Rapidan storm moved around the ridge divide producing hail and then dissipated two hours later.

[8] A third storm, here called storm 3, developed around 1900 UTC along the vicinity of the cold front which had now progressed to just south of the North Carolina-Virginia border and drifted into the Blue Ridge Mountains of southern Virginia where the environment was unstable [Pontrelli et al., 1999]. Combined frontal convergence and orographic lift triggered convection in a zone centered over the mountains. The system remained quasi-stationary over the Blue Ridge Mountains of southern Virginia producing heavy rainfall and flooding into the morning of 29 June 1995.

[9] The Rapidan storm (storm 2) was characterized by Smith et al. [1996a], Landel et al. [1999], and Pontrelli et al. [1999] as having a low-echo centroid, meaning the highest reflectivity values were located in the warm portion of the cloud below the 0°C level (which occurred at approximately 4 km altitude). Low-echo centroids are identified in these references as being an important feature of catastrophic orographic precipitation as it reflects the importance of warm rainfall processes and is an indicator of high efficiency in precipitation formation. Storm 2 was dominated by warm rainfall processes and saturated air aloft, the importance of which is further explored by Landel et al. [1999] and Pontrelli et al. [1999] who provide a comparison of the Piedmont storm (storm 1) with the Rapidan storm (storm 2). Although storm 2 was smaller than storm 1, it moved slower and had higher peak intensities. Smith et al. [1996a] describes the motion of storm 2 as propagation rather than advection, meaning that the net movement was controlled by the location and growth of new storm cells rather than by wind direction at any level. This propagation mechanism was noted to enhance the spatial and temporal variability of the rainfall at the scales of cell growth. The combination of slow storm motion with large rain rates produced record flooding with devastating landslides and debris flows. Storm total accumulations exceeded 600 mm in a 6-hr period, placing the Rapidan storm on the envelope curve for flood discharge for the United States east of the Mississippi River [Smith et al., 1996a].

2.2. Comparison to Storm Environments of Other Orographic Events

[10] The pronounced influence of orography is a feature common to many catastrophic storms in the United States (see Smith et al. [1996a] for references). Also, prevailing southwesterly winds aloft with moist low-level easterlies resulting in significant leeside upslope convection is a common feature of many devastating precipitation events along the Front Range of the Rocky Mountains and the eastern slope of the Appalachians [e.g., Landel et al., 1999; Pontrelli et al., 1999]. Thus the Rapidan River basin and southern Virginia storms of 27 June 1995 provide a good case study to examine the influences of orography on the spatial organization and multiscale properties of rainfall.

[11] As mentioned in the introduction, this study builds on the work by Harris et al. [1996] and Purdy et al. [2001], adopting essentially the same multiscale statistical analysis framework. Thus it is of considerable value to compare results between the case studies and this is presented in section 4. Here, the similarities and differences in meteorological conditions for the various events will be considered.

[12] The study by Purdy et al. [2001] documents a storm occurring 5–9 November 1994 on the west (windward) side of the Southern Alps of New Zealand. Like the storms in this study, the Southern Alps event was forced by moist, low level upslope flow. It also featured initial prefrontal conditions of conditional instability followed by a postfrontal period with an unstable air mass. This change in the stability of the atmosphere resulted in marked changes in the rainfall time series variability particularly in the lowlands versus alpine region.

[13] The Southern Alps of New Zealand case contrasts the case study in this paper in two important details. First, the Southern Alps event was windward-side (not leeside) orographic precipitation. In other words, there was little directional shear as both high and low level winds were more or less westerly. Second, vertically pointing radar (VPR) data in the Southern Alps showed the presence of a bright band which indicated cold rain processes aloft. Purdy et al. [2001] employed a diagnostic modeling approach using the Regional Atmospheric Modeling System (RAMS) and found that regions of descending flow aloft existed just upwind of the main divide. Based on these findings, it was hypothesized that this descent of saturated air created a suppression mechanism that stabilized the alpine air. This was further supported by observation that convective elements that developed in the lowlands and were advected upslope by westerly winds into the foothills did not propagate to the alpine region. Accumulation maps derived from horizontally scanning radar showed lateral spreading of individual cell tracks and that rainfall near the main divide was incessant and generally void of the extreme fluctuations and intermittency that one would associate with convective rainfall [Purdy et al., 2001]. This sharply contrasts the present case study (particularly storm 2) where leeside convection intensified with increasing topographic elevation. As will be shown later, these contrasting observations resulted in opposing trends in rainfall statistics with increasing topographic elevation.

2.3. Data

[14] One-kilometer horizontally gridded rainfall derived from the Sterling, Virginia WSR-88D radar are used in this study to characterize the multiscale spatial organization of the 27 June 1995 storms. The radar is located at latitude 38.98°N and longitude 77.48°W, which is approximately 30 km east and 11 km north of the upper right (northeast) corner of Figure 1. The close range of the Sterling, Virginia radar to the 100 × 100 km2 area of investigation allowed for the data to be processed at the 1-km resolution. Details on the processing of WSR-88D radar data to derive rain rates along with additional references are given by Smith et al. [1996b] and Baeck and Smith [1998]. The gridded rainfall maps of 100 × 100 km2 in dimension, formed a sequence of rainfall patterns at 6 min time intervals, allowing one to follow the temporal changes in the multiscale parameters as the centroids of the storms moved in relation to the orography. One-kilometer digital elevation model (DEM) data from the USGS EROS Data Center was used to track the topographic elevation at the location of the surface rainfall centroids.

[15] Even radar data of the best quality is subject to artifacts and these are mentioned briefly in the context of multiscale analysis. Radar processing artifacts were considered by Krajewski et al. [1996] and touched upon by Harris et al. [1997]; however, further study is needed to better quantify the effects on multiscaling parameter estimates. It is worthy to note for this study that radar data noise (electrical, digital quantization, and sampling noise) tends to be averaged out better at short distances from the radar, which is one reason for limiting the analyzed rainfall data to ranges of approximately 100 km.

3. Multiscale Statistical Analysis

[16] Multiscale statistical analysis methods in the hydrometeorological sciences have been actively developed over the last 15 years. Theoretical treatments are given by Schertzer and Lovejoy [1987] and Gupta and Waymire [1993] and follow from multiplicative cascade theory. For analysis oriented papers focusing primarily on parameter estimation methods and analysis results the reader is referred to papers such as Davis et al. [1994, 1996] and Harris et al. [1996, 1997, 2001], among others.

[17] A brief description of the methodology of multiscale statistical analysis as well as some insight into the physical/geometrical interpretation of the multiscale parameters are illustrated here using one rainfall pattern from each of the three storms of 27 June 1995 as examples. Figure 2 shows the spatial rainfall patterns for 27 June 1995 at 0802 UTC (storm 1), 1602 UTC (storm 2), and 2200 UTC (storm 3). The analysis was performed over the entire 100 × 100 km2 area and was not done separately for each storm's sub-area of concentration. This area of investigation captures the spatial extent of the three storms, and further enlargement of the area would not change the parameter estimates as it would only add surrounding nonrainy (zero) pixels which do not contribute to the multiscale statistical analysis.

Figure 2.

Rainfall for the 27 June 1995 Rapidan River basin and southern Virginia storms at (a) 0802 UTC, (b) 1602 UTC, and (c) 2200 UTC representing storms 1, 2, and 3, respectively.

[18] The essence of the framework is that for the purposes of both characterization and downscaling, the 2-D rain fields are seen as smoothed or spatially averaged self-similar multiplicative cascades. Cascades in 2-D are simple constructs whereby one takes a unit area, divides the area into four quarters of equal rainfall, and multiplies each of the four rainfall values by a different random multiplicative weight. Each quarter is then further divided in four and the rainfall is multiplied again by random weights and so on. Despite their simplicity of construction, cascades form constructs with rich statistical properties such as long range correlations, intermittency, and extreme fluctuations, of the type observed in a wide range of geophysical phenomena including rainfall. It is the simplicity of cascade construction combined with their ability to mimic the variability observed in rainfall that make cascades attractive tools for downscaling purposes.

[19] Another reason for adopting this framework is that the multiscale structure of the fields can be described by three parameters governing the variability over a wide range of scales. Furthermore, the three parameters can be interpreted to quantify physical characteristics of the fields such as smoothness, intermittency, and spikiness. Of the three parameters, two characterize the multiplicative cascade (that would ultimately be used in a cascade-based downscaling procedure), and one determines the extent of spatial integration to transform a cascade (too rough to resemble rainfall) to a smoother field having the same statistical structure as observed rainfall. The two parameters characterizing the underlying cascade are simply the parameters that characterize the distribution of the random multiplicative weights used in a cascade construction process and the analysis amounts to estimating and parameterizing this distribution. The parameters and their estimation are described briefly here. Further details can be found in the references listed throughout this section.

[20] Before proceeding with the analysis, it is beneficial to first provide more precise definitions of what is meant by smoothness, organization, intermittency and spikiness in the context of this paper. Smoothness (or organization) relates to the degree of correlation in the rainfall fields. Intermittency characterizes the heterogeneity of fluctuations and degree of variability in the rainfall intensities, whereas spikiness depicts the rarity of extreme fluctuations in the field. The relationship between intermittency and organization can be seen by comparing stratiform versus convective rainfall. Both systems can have similar levels of organization, but the scales across which this organization extends will vary. Convective rainfall tends to be highly organized at scales of individual storm cells, whereas stratiform rainfall is well organized across larger scales. The intermittency is generally much higher for convective than stratiform rainfall. Therefore the combined interpretation is that rainfall fields tend to become more intermittent and more organized as the system becomes more convective.

3.1. Fourier Spectral Analysis

[21] Fourier power spectral analysis is usually the first step in any scaling analysis. Although this step does not yield any one of the three parameters, it does verify the presence of scaling in a field and identifies the scaling range. A log-log linear relationship in the Fourier power spectra over a finite range of scales, as shown in Figure 3, is a characteristic of all scaling fields and is necessary for meaningful multiscale analysis. In other words, a scaling power spectrum obeys

equation image

where the magnitude of the negative slope, β, computed from Figure 3 over the scaling range (2 km to ∼20 km) is the power spectral exponent. For 2-D fields, P(k) is computed via a 2-D fast Fourier transform (FFT) [e.g., Press et al., 1992] of the original 2-D field R(x, y) to obtain the Fourier transform equation image(kx, ky). equation image is then multiplied by its complex conjugate to obtain the 2-D power spectrum P(kx, ky) which is then averaged over all angles about kx = ky = 0, to obtain P(k), where equation image. This is often referred to as the isotropic power spectrum as any anisotropy present in the 2-D power spectrum is averaged out for graphical purposes. Fourier power spectral analysis performed in this way provides insight into the field's suitability for multiscale analysis and identifies the scale range over which scaling holds.

Figure 3.

Fourier power spectra for the 27 June 1995 Rapidan River basin and southern Virginia storms. The spectra have been offset for clarity.

[22] β is an indicator of the smoothness of the precipitation fields, with higher β indicating a smoother, more organized field [e.g., Harris et al., 1996]. As mentioned above, we use smoothness and organization synonymously here and to see how β relates to these qualities, one can imagine two extremes. A random uncorrelated (i.e., white noise) field is unsmoothed, disorganized and has β = 0. On the other hand, a correlated, highly organized field would have a power spectrum which falls off rapidly with increasing frequency (high β).

[23] The value of β is also important for the next stages of the analysis described below. The theory shows [Menabde et al., 1997] that self-similar cascades cannot produce fields with β greater than the dimension of the field, D (i.e., D = 2). Thus if β > D, as is usually the case in rainfall fields, the field must be seen as one representative of a smoothed (e.g., spatially averaged) self-similar cascade. The smoothing of the underlying multiplicative cascade is performed so as to preserve the scaling nature of the power spectrum by applying a power law filter. The implications of this from an analysis perspective is that in order to obtain the two parameters of the multiplicative cascade, the field must first be differentiated to remove this smoothing. This is akin to the method of removing nonstationarity in time series analysis by taking differences. Again this is done in a fashion consistent with the scaling nature of the field by applying a power law filter. This is equivalent to fractional-differentiation and has its roots in fractional-calculus (see Harris et al. [2001] for a brief description and further references).

3.2. Structure Function Analysis

[24] While the power spectral exponent, β, does not give the parameter determining the degree of fractional differentiation, it is related to the one that does. To obtain this parameter, a structure function (akin to the variogram) analysis is carried out. Note that if β ≤ 2 then one would not carry out this step as the field can be represented as an unsmoothed or pure multiplicative cascade. For this analysis step a first-order generalized structure function is computed,

equation image

for the field R, where x, y denote spatial coordinates and lx and ly denote lags in the x and y directions, respectively. S1(lx, ly) is azimuthally averaged about lag equation image is then tested for scaling or log-log linearity,

equation image

where, H, is known as the Hurst exponent. An example is shown in Figure 4. H is the slope of the q = 1 generalized structure function within the finite scaling range of 1 to ∼20 km. Like β, H too is an indicator of smoothness, but as alluded to above, this is also the degree of fractional differentiation one would apply to the original field before estimating the cascade parameters. Alternatively, once one obtains the cascade parameters, H indicates the degree of smoothing (fractional integration) one would use in a downscaling procedure (as in the work of Harris et al. [2003]).

Figure 4.

First-order (q = 1) generalized structure function for the 27 June 1995 Rapidan River basin and southern Virginia storms.

[25] Fractional differentiation and integration is most easily carried out in Fourier space [e.g., Schertzer and Lovejoy, 1987; Davis et al., 1996]. If equation image(k) is the Fourier transform of the (real valued) generic field R, then the inverse Fourier transform of equation image(k)∣kH, H > 0 corresponds to fractional differentiation, while the inverse Fourier transform of equation image(k)∣kH, H > 0 corresponds to fractional integration. It is worthy to note that while H defines and is therefore a sensitive indicator of the degree of smoothness in a field [Harris et al., 2003], the results of the moment-scale analysis described below are not overly sensitive to the degree of fractional differentiation performed prior to the analysis. This empirical observation is a desirable trait of the multiscaling analysis procedures.

3.3. Moment-Scale Analysis

[26] The final step in the multiscale analysis is the moment-scale analysis performed by computing standard statistical moments of the fractionally differentiated field for multiple scales. This yields the remaining two parameters characterizing the field which correspond to the cascade parameters one would use in a downscaling scheme. Let us refer to the fractionally differentiated field as φ,

equation image

[27] The moment-scale analysis involves a range of averaging scales, r (higher r implies lower scale),

equation image

where φr represents fractionally differentiated field values at scale r, q is the order of the moment, and 〈…〉 denotes the average over all the pixels of scale r in an image. Scaling of the moments occurs when

equation image

where K(q) is the moment scaling exponent function that, in practice, is estimated by log-log linear regressions of the qth moment of ∣φr∣ versus r (as shown in Figure 5), and repeated for a range of q. Clearly, K(1) = 0 since the unconditional mean of the entire field is scale independent. Examples are shown in Figure 6. In theory, the entire K(q) curve is necessary to fully characterize the moment scaling properties of the field. In practice, however, it is useful to parameterize this field and there is a rich literature, which defines different analytical forms of the function K(q) for multiplicative cascades with certain random generators for the multiplicative weights [e.g., Gupta and Waymire, 1993]. The formulation for the derivation for many of these analytic forms of K(q) for a specific random generator, lies in Taubian theorems of probability theory [e.g., Feller, 1966]. In this work, the cascade model adopted has log-stable distributed weights, and produces what is also referred to as universal multifractals [e.g., Lovejoy and Schertzer, 1995a, 1995b; Tessier et al., 1993; Pecknold et al., 1993; Wilson et al., 1991]. The subject of stable distributions is in itself enormous [e.g., Samorodnitsky and Taqqu, 1994], and the derivation is beyond the scope of this work.

Figure 5.

Moment scaling for the fractionally differentiated 2-D rainfall fields for q (moment) values of (a) q = 0.5, (b) q = 2.0, and (c) q = 4.0 for the 27 June 1995 Rapidan River basin and southern Virginia storms.

Figure 6.

K(q) curves for the 27 June 1995 Rapidan River basin and southern Virginia storms. The C1 parameter values are estimated as 0.26, 0.29, and 0.32 for 0802 UTC, 1602 UTC, and 2200 UTC, respectively. The α parameter values are estimated as 1.22, 1.14, and 1.10 for 0802 UTC, 1602 UTC, and 2200 UTC, respectively.

[28] For a log-stable cascade, K(q) is a function of two parameters essentially representing the width of the log-stable distribution and the shape parameter given by the Levy index, α (see Harris [1998, appendix] for a concise derivation),

equation image

C1 is referred to as the intermittency parameter and α is the Levy index, (α = 2 indicates a lognormal cascade). C1 and α are estimated here via nonlinear regression using (7), although other methods exist, such as the double trace moment (DTM) technique [e.g., Lovejoy and Schertzer, 1995a].

[29] Important to this study and any characterization study are the physical/geometrical interpretations of C1 and α. C1 increases as the intermittency of the field increases, while α increases as the field becomes more spiky. The reason for the spikiness increasing with α is that the narrower tails of high α fields mean extreme fluctuations are rare, so that when they occur, they stand out as spikes. For excellent visualizations of these characteristics the reader is referred to Pecknold et al. [1993].

4. Results

[30] The spatial organization of the 27 June 1995 storms on the eastern slope of the Blue Ridge Mountains in Virginia has been characterized using the 2-D multiscale statistical analysis methodology discussed in the previous section. The multiscale parameters have been analyzed over time and focus has been placed on linking changes in multiscale parameters with orographic influences on the rainfall. The multiscale properties found for this case study are presented in section 4.1 and compared to those found for orographic rainfall in the Southern Alps of New Zealand by Harris et al. [1996] and Purdy et al. [2001] in section 4.2.

[31] Before performing the multiscale analysis, a few basic statistical measures were computed as a preliminary step in characterizing the space-time structure of the storms. The average magnitude and spatial coverage area of rainfall exceeding 0 mm/hr (nonzero) and 20 mm/hr are shown in Figure 7. The traditional statistical measures of nonzero rainfall intensity and size (see Figures 7a and 7c) emphasize storm 1 as being the most severe of the three storm systems. However, it was storm 2 that produced heavy rainfall and record flooding in the Rapidan River basin. Storm 2 begins to be highlighted when thresholds are used to condition the basic statistical measures (see Figures 7b and 7d). It can be seen from Figure 7 that basic statistical measures tend to favor larger storm systems over smaller but potentially more severe systems.

Figure 7.

Traditional statistical measures for the 27 June 1995 Rapidan River basin and southern Virginia storms.

4.1. Multiscale Properties

[32] The multiscale statistical analysis and the resulting multiscale parameters highlight different features of the 27 June 1995 storms than the basic measures of Figure 7. As described in section 3, the spatial organization, intermittency, and scale dependent variability of rainfall can be assessed through multiscale analysis. The multiscale parameters β, H, C1, and α for the 27 June 1995 storm systems were computed over the entire 100 × 100 km2 area using the WSR-88D derived 1-km horizontally gridded rainfall. The time series shown in Figure 8 was formed by repeating the 2-D multiscale computations for a sequence of 6-min interval rainfall patterns throughout the duration of the three storms. An 18-min moving average was used in plotting the multiscale statistical parameters, where each parameter value was averaged with that from the previous (6 min earlier) and following (6 min later) rainfall fields.

Figure 8.

Multiscale statistical parameters for the 27 June 1995 Rapidan River basin and southern Virginia storms. The parameters have been plotted as 18-min moving averages, where each parameter value was averaged with that from the previous (6 min earlier) and following (6 min later) rainfall fields.

[33] The finite range of scales under which log-log linear relationships were found for computing β from the Fourier power spectra (see Figure 3) varied somewhat between the three storms. The scaling range used to compute β during storm 1 was the entire range of observable scales. During storm 2, the scaling range for β was 1 to ∼64 km until 1630 UTC and 1 to ∼42 km thereafter. The β scaling range was reduced to 1 to ∼21 km during storm 3. The finite range of scales used to compute H from the first-order generalized structure function (see Figure 4) was 1 to ∼20 km for all three storms. The K(q) values for storms 1, 2, and 3 were computed using the first five points of the q-order moment scaling (see Figure 5) and the C1 and α multiscale parameters where estimated by nonlinear regression to the K(q) curves (Figure 6) fitted using q ≤ 3. This is common practice in K(q) regressions since for large q, the moment scaling is dominated by the single largest value in the field yielding linear K(q). At small q (i.e., 0 ≤ q < 1), the K(q) curve is sensitive to noise and discretization in the data. The effects of both small and large q on fitting the K(q) curve are well documented by Harris et al. [1997].

[34] The smoothness or organization parameter β (see Figure 8a) shows that the rainfall fields are relatively well organized during both storms 1 and 2, with a higher β during the second storm representing more organization in the rain field structure. Purdy et al. [2001] provided a physical interpretation of high β being linked to the existence of individual rain cells. With this interpretation, the increase in β between storm 1 and storm 2 can be physically explained by the storm environment (see details in section 2). Storm 1 was characterized as a convective system with large spatial coverage being advected along with the cold front. The movement of storm 2 was controlled by propagation through the location and growth of new storm cells. This sustained growth and decay of individual rain cells explains the higher β during storm 2. The decreasing trend in β during storm 3 indicates that the storm environment was less organized. The scaling parameter H shown in Figure 8b clearly depicts the temporal evolution of the three storm systems. Furthermore, it highlights the significance of storm 2 with higher H values. This echoes the trend shown by β, but shows it more clearly as H is a first-order (q = 1) measure and so less noisy than the second-order (q = 2) measure, β.

[35] The intermittency parameter C1 (see Figure 8c) characterizes the higher intermittency and clustering of peak rain rate values in the rainfall fields during storms 2 and 3 than those during storm 1. Consistent with interpretations of Purdy et al. [2001], the lower C1 and β in storm 1 indicates a widespread background rainfall feature (embedded convection), whereas the higher C1 values in storms 2 and 3 indicate more focus of these storms on intermittent convective elements. The sustained propagation and growth of individual rain cells in storm 2 and unstable air in storm 3 would lead one to expect higher intermittency in rainfall. As a result of the highly sensitive nature of the shape parameter α (which is an indicator of spikiness or how extreme the fluctuations are) shown in Figure 8d, α was found to be highly variable for all three storms and thus of limited use. However, rainfall in storms 2 and 3 did show higher peaks in α.

[36] The purpose of this study was not just to characterize the multiscale parameters of spatial rainfall patterns over time, but also to assess if the temporal evolution of these parameters is related to the orographic influences on the rainfall. Using the surface rainfall centroid to track storm movement, Figure 1 shows the geographical location of the storm track overlaid on the topographic elevation of the region. The surface rainfall centroids were estimated separately for storms 1, 2, and 3 using storm sub-areas. The storm track derived in this study very closely matched that found by Smith et al. [1996a] for the Rapidan storm (storm 2). The topographic elevation associated with the geographic location at the surface rainfall centroid was derived using 1-km DEM data. It can be seen visually from Figure 1 and quantitatively from Figure 9 that the topographic elevation, and consequently the orographic influence on the rainfall, changes over time.

Figure 9.

Topographic elevation of the surface rainfall centroid for the 27 June 1995 Rapidan River basin and southern Virginia storms. The topographic elevation at the rainfall centroid locations has been plotted as 18-min moving average, where each elevation was averaged with that from the previous (6 min earlier) and following (6 min later) rainfall fields.

[37] Comparison of Figure 8c with Figure 9 (overlaid in Figure 10) reveals very similar trends in the multiscale intermittency parameter C1 with topographic elevation of the rainfall centroid over time. The correlation between C1 and orography observed in Figure 10 is quantified in Figure 11. A correlation coefficient of 0.7 was found between the topographic elevation of the surface rainfall centroid and the multiscale parameter C1. Even though the R2 for this correlation is somewhat low, Figure 11 provides evidence of orographic influence on the spatial organization of these rainfall events. However, the results of increasing C1 with topographic elevation are in contrast to those found by other studies (referenced below). The different trends in the multiscaling parameters are investigated by comparing the differences in storm environment and meteorological forcings.

Figure 10.

Topographic ELEV trend versus C1 trend for the 27 June 1995 Rapidan River basin and southern Virginia storms. The C1 parameter and topographic elevation have been plotted as 18-min moving averages, where each C1 value and elevation was averaged with that from the previous (6 min earlier) and following (6 min later) rainfall fields.

Figure 11.

Correlation of C1 to topographic ELEV for the 27 June 1995 Rapidan River basin and southern Virginia storms. The actual C1 and topographic elevation pairs were plotted here and used in computing the correlation rather than the 18-min moving average values (as was shown in Figures 810).

4.2. Comparison to Multiscale Properties of Other Orographic Events

[38] The results of this study that are in contrast with those of Harris et al. [1996] and Purdy et al. [2001] are summarized as follows: an increasing trend in the multiscale parameter C1 with increasing topographic elevation, a decreasing trend in scaling range with increasing topographic elevation, and a positive correlation between C1 and β for storms 1 and 2 with a negative correlation between C1 and β for storm 3. Whereas, Harris et al. [1996] and Purdy et al. [2001] found increasing scaling range and decreasing C1 and β (the latter two of which were always positively correlated) with increasing topographic elevation from the lowlands through the foothills and unto the alpine region.

[39] Since the results of the studies referenced above comprise only a couple events which all occurred at the same location, the Southern Alps of New Zealand, it is difficult to draw any strong conclusions by comparing to the results presented in section 4.1 of this study. However, some insight into the differing trends in the multiscale properties of this case study with those that were found in the Southern Alps events can be gained by comparing several key differences in the environment and meteorological conditions of the different locations and events (see section 2 for more details).

[40] The Southern Alps of New Zealand storms were all windward-side events, meaning that the prevailing upper level winds were westerly and the upslope low-level winds were also westerly. In the Blue Ridge Mountains of Virginia storms, the prevailing winds aloft were southwesterly but the upslope low-level winds were easterly making it a leeside orographic event. This is a most notable difference, but difficult to directly relate to the multiscale parameter trends without further case studies to consider.

[41] The contrast in trends in the scaling range with orography is more related to the meteorological environment present at that elevation during the various events than it is to the topographic elevation itself. During the Southern Alps event, the atmospheric stability increased with orography from unstable in the lowlands to stable in the alpine and the rainfall process changed from convective in the lowlands to mixed in the foothills and stratiform in the alpine. Since the spatial extent of rainfall is typically larger in stable, stratiform than unstable, convective systems, it is expected that the scaling range would increase from the lowlands to the alpine region. Storm 1 was characterized as a convective system with a widespread background rainfall feature (embedded convection), whereas storms 2 and 3 were more focused on convective elements with storm 3 also being more unstable. Again, it is expected that the scaling range would decrease from storm 1 to storm 3 since the spatial extent of these systems was decreasing.

[42] As discussed in section 2.2 and by Purdy et al. [2001], cold rain and snow processes aloft in the alpine region combined with a suppression mechanism stabilized the atmosphere upwind of the main divide in the Southern Alps event. This was a key feature in preventing the propagation of convective elements from the lowlands to the alpine region. Descending saturated air from the alpine region converging with the upslope lowland flow caused lateral spreading and mixed rainfall processes in the foothills. Hence, the intermittency (C1) and organization (β) of the rainfall decreased with increasing topographic elevation along the transect. No evidence of a suppression mechanism with stabilization of alpine air or cold rainfall processes with snow aloft existed in the 27 June 1995 events in the Blue Ridge Mountains. Pontrelli et al. [1999] discusses the absence of moist downdrafts (descending flow) in the Rapidan storm and relates it to the near ground cloud base, moist troposphere, and warm rainfall processes. Due to the very weak or absent downdrafts, the terrain was the primary focusing mechanism for storm motion and the upslope movement and propagation of convective elements were not suppressed. In fact, the convection was sustained and strengthened by the persistent low-level easterly jet directing moist air upslope. As a result, the intermittency parameter C1 increased with rising topographic elevation.

[43] Higher β (organization) and higher C1 (intermittency) parameter values were found in this study, which generally indicates more convective structure in the 27 June 1995 storms in Virginia than the Southern Alps events. The β values found here varied from approximately 2.2 to 3.0, and the C1 values increased from approximately 0.23 to 0.45 with increasing topographic elevation. Harris et al. [1996] found average β values of 1.5 to 0.95 which increased with topographic elevation and much lower C1 values which decreased from 0.192 to 0.040 for the lowlands to near the main divide. Purdy et al. [2001] found a similar β range for both the prefrontal and postfrontal periods, and found postfrontal C1 values which decreased from 0.28 to 0.042 with increasing topographic elevation. The previous studies by Harris et al. [1996] and Purdy et al. [2001] used temporal rainfall measurements, whereas this study used a temporal sequence of spatial rainfall measurements. Only an assumption akin to that of “frozen turbulence” in the fluid dynamical literature (where a direct relation between temporal and spatial statistics is made possible by assuming that the spatial structure simply advects past a point observer) would allow a direct comparison of temporal statistics to spatial statistics. As such an assumption is debatable, great caution must be taken in drawing any conclusions from comparing actual parameter values between these studies. Nevertheless, the relative trends in parameters may be directly compared, and is where the emphasis is placed in this study. For example, greater intermittency in the time series will almost always be accompanied by greater intermittency in space.

[44] The positive correlation between C1 and β (or H) found during storms 1 and 2 is consistent with the results of Purdy et al. [2001]. However, in this study a decreasing β was found with an increasing C1 during storm 3. The physical interpretation of high C1 with low β is a field which is comparatively more intermittent, yet spatially less organized. Pontrelli et al. [1999] describes the storm 3 environment as unstable with strong frontal convergence coupled with orographic lift producing a favored zone for triggering convection that was centered over the mountains. In storm 1, the convective cells advected with a cold front away from the upslope. In storm 2 and the Southern Alps events, the convective cells were triggered in the lowlands and forced upslope by low-level winds. The initiation of convection centered over the mountains is a plausible explanation for the increased level of intermittency accompanied by decreased organization found in storm 3. This observation of rainfall characterized by relatively high C1 and low β has not been observed before and the hypothesis of triggered convection over the mountains being the cause of such statistics requires further case studies with similar meteorological and orographic influences.

5. Conclusions

[45] The key findings of this study are two-fold. First, multiscaling behavior was found to hold in the 2-D spatial patterns of orographic rainfall for the Rapidan River basin and southern Virginia storms of 27 June 1995. This provides additional evidence to the studies by Harris et al. [1996] and Purdy et al. [2001] on the multiscale organization of orographic precipitation. The usefulness of multiscale statistical analysis in characterizing the spatial organization of orographic precipitation and providing insight beyond what can be gained from basic statistical measures was also shown. Establishing the presence of scaling over a range of scales from at least 20 km down to 1 km is an essential step in the primary incentive for this study, which is the application of cascade-based downscaling methods (as developed by Harris et al. [2003]) to stochastically downscale orographic rainfall that could be obtained from model forecasts or satellite estimates. Forecasts and satellite estimates of rainfall are examples of fields usually provided at resolutions of around 20 km, whereas many hydrological forecast models and land-atmosphere models benefit greatly from much higher resolution estimates of rainfall [e.g., Winchell et al., 1998; Nykanen et al., 2001]. However, unlike many applications of stochastic downscaling such as that by Harris et al. [2003], orographic rainfall is complicated by significant trends governing changes in the multiscaling (downscaling) parameters with changes in topographic elevation. It is a detailed consideration of these trends that provides the second key finding of this study.

[46] This second finding was that the relation between multiscale parameters and the underlying orography found in this study contradicts those found by previous studies [Harris et al., 1996; Purdy et al., 2001]. This contradiction of findings is argued to be a direct cause of the pronounced differences in the meteorology surrounding the different cases. Thus pointing to the importance of distinguishing between windward-side and leeside orographic forcing and the presence of cold rain versus warm rain processes. More generally, this indicates the absence of any universal trend between the multiscaling statistical structure and the underlying orography.

[47] In light of the absence of any general trend in multiscaling structure with orography, it is clear that future research is needed to further characterize orographic precipitation. Moreover, this must be done with meticulous attention paid to the surrounding meteorological environment, thereby making additional case studies of orographic rainfall with detailed meteorological data essential. Only then can the multiscale statistical parameters be related to the orographic influences on the space-time organization of the precipitation, and used for downscaling orographic precipitation. Currently, studies are underway focusing on other cases of leeside orographic precipitation where warm rain processes dominate. These studies are essential to corroborate the findings of this paper. After that, the search for cases of orographic precipitation in yet again entirely different meteorological conditions will continue to be of interest, as this study exemplifies the need for careful consideration of the meteorological influences on the multiscale statistical parameters in any development of multiscale frameworks for stochastic downscaling of orographic precipitation.

Acknowledgments

[48] The authors wish to thank James A. Smith and Mary Lynn Baeck at Princeton University for providing the rainfall data. This research was funded by Michigan Technological University.

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