a. Local temporal correlations

One goal of the study is to examine the spatial variability of the persistence of precipitation. The dichotomous (rain/no rain) character of precipitation suggests its occurrence can be approximated as a two-state Markov process; that is, the transition probabilities that control the next state of the system depend only on the current state (Wilks 1995). This assumption allows us to compute the serial correlation between rain rates from the time series separated by n time steps at any location (r_n) from the 1-h lag autocorrelation (r₁):

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The number of lags (hours) at which the correlation drops to e⁻¹ represents the e-folding time T_e. Although we are not focusing on assessing the strength of the Markov model to represent conditional probabilities associated with hourly precipitation, we briefly offer an analysis of how well the Markov model fits empirically derived histograms. Probability of raining at a fixed location, at subsequent n time steps τ, can be conditioned on only the first time step t₀:

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or on the of chain of all the intermediate time steps from

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The Markov model approximates condition (3) as

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Condition (2) corresponds to the autocorrelation at n lags, independent of precipitation occurrence at the intermediate lags of 1 to n − 1. Thus it would asymptote toward the climatologic frequency with increasing lag. Condition (3) measures the likelihood of consecutive hours with rain and would asymptote toward zero with time, and it can be considered a subset of Eq. (2). Comparisons between the empirically estimated conditional probabilities (2) and (3) and the Markov approximation (4) are shown in Fig. 1. The curves represent domain-averaged values for the theoretical (solid lines) and empirical (dash lines) correlation functions at different lags, while the thin bars (red and blue) represent their standard deviations across the entire domain. The thick bars (green) in the upper panels give the standard deviations of the difference between left- and right-hand sides of (4) across the entire domain. Since consecutive hours with precipitation are of great hydrologic interest, it is clear that Eq. (3) is the suitable form. In this case, because the theoretical curve exceeds the empirical one by 1% and lies well within the sample standard deviation of the difference field, we conclude the Markov assumption is an acceptable and highly accurate approximation to the raw probabilities.

The computations are done for both seasons and for three thresholds: 0 mm h⁻¹ or trace (not shown), 1 mm h⁻¹ or measurable precipitation, and 5 mm h⁻¹, which we assume represents the lower bound for predominately convective precipitation (e.g., Baldwin et al. 2005). Thresholds higher than 5 mm h⁻¹ were not examined owing to small sample size.

Stage IV precipitation rates of less than 1 mm h⁻¹ should be viewed with caution owing to the inherent uncertainties associated with converting low reflectivity to rainfall rates, especially in regions where there is paucity of gauges. Whether these light rainfall intensities are accurate or primarily denote an artifact of the radar algorithm is difficult to determine. Westrick et al. (1999) showed that a limited number of scans at low levels (below the melting level) can contribute to considerable precipitation underestimation, making the use of the resulting very low precipitation rates questionable. According to the stage IV multisensor analysis procedure, a radar–rain gauge bias correction is applied to mitigate radar Z/R errors (http://www.nws.noaa.gov/oh/hrl/papers/wsr88/tac_nov02.pdf), but the uniformity of its effectiveness for different seasons, regions, and synoptic events must be questioned. Consideration of these issues led to our decision to not include and interpret the results from trace events.

b. Spatial correlations

1) Contemporaneous spatial correlations

A simple way to quantify spatial coherence is by computing conditional probabilities. Conditional probabilities are generated over a 512 km × 512 km domain (128 × 128 4-km pixels), where the probability P at an arbitrary location and time is conditioned on the event occurring simultaneously at the center of the domain, termed the anchor point, or

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where (x₀, y₀) is the anchor point, (x, y) is any point within the 512 km × 512 km domain, and t is any time during which the anchor point receives rain.

To estimate the e-folding distance under the assumption of isotropy, the average value of (5) was computed along annuli at different radial distances from the anchor point. The radial distance at which this average value decreased to 1/e is defined as e-folding distance, d:

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where ΔR = 4 km, and P(r, θ) is the representation of (5) in polar coordinates.

The analysis was done for both cold and warm seasons and for two thresholds: 1 and 5 mm h⁻¹. The e-folding distance was estimated for four different grid box sizes: 4 km × 4 km (the nominal stage IV pixel), 8 km × 8 km, 16 km × 16 km, and 32 km × 32 km to quantify how the decorrelation scale varies with respect to spatial averaging. That is, the original data are aggregated into 8 km × 8 km pixels (the average of four adjacent stage IV pixels), 16 km × 16 km (16 stage IV pixels), and 32 km × 32 km (64 stage IV pixels) and then each of these coarse pixels became an anchor point around which conditional probabilities across a 512 km × 512 km window are computed in the same fashion as for 4 km.

2) Time-lag spatial correlations

Conditional probability maps can be generated at different time lags τ :

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The calculation is performed over the whole region (east of 105°W). Every stage IV pixel is used as an anchor point for time lags of τ = ±1 h, from which mean system propagation characteristics can be estimated. (Computation of the propagation is described in section 3c.)

c. Anisotropy measures

Anisotropic characteristics of the precipitation patterns were defined by Fourier decomposition. At various radii from the anchor point, circular arcs defined by 8-km-thick annular rings are derived from (5):

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Then the 1D Fourier transform was computed in the azimuthal direction along the resulting annular regions of (8).

An example is shown in the middle panel of Fig. 2 for a pixel over the southeast United States. The amplitude and phase of the first harmonic provide a measure of the asymmetry of the rainfall distribution about the anchor point; the amplitude and phase of the second harmonic are related to the eccentricity and the orientation of the major axis, respectively (see appendix). The amplitude squared determines the variance explained by the harmonic. For the example shown in Fig. 2b, the first harmonic points toward the SE, and the dilatation axis for the second harmonic is oriented approximately SW–NE.

The mean translation of the precipitation pattern over a 2-h interval can be estimated from the +1-h and −1-h time-lagged patterns of (7). A correlation matrix between the +1-h time-lagged pattern of (7) and the 0-h lag pattern of (5) was computed for an arbitrary anchor point (x₀, y₀) by shifting (5) in increments of 4 km in both the x and y directions:

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where ρ(x₀ + mΔx, y₀ + nΔy, t + τ) is the pattern correlation function for (m, n) lags in the (x, y) directions, [P(x, y, t)|P(x₀, y₀, t) = 1] is the conditional probability distribution within the 512 km × 512 km domain, that is, Eq. (7), and the bracket operator is the spatial correlation function. The point (x₀ + MΔx, y₀ + NΔy) is where the correlation (9) is maximized:

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The number of lags over which the optimization is computed varies between [−20, +20] for both m and n, a sufficient distance to ensure that |M| < 20 and |N| < 20, and of course Δx = Δy = 4 km.

We defined the displacement as the vector difference between the anchor point (x₀, y₀) and the point (x_τ+1, y_τ+1) = (x₀ + MΔx, y₀ + NΔy, τ + 1). The procedure was repeated for −1-h time lag, time τ − 1. The upper and lower panels of Fig. 2 show center locations at −1- and +1-h lags, respectively. The translation over the 2-h period τ − 1 to τ + 1 was computed as a vector from point (x_τ−1, y_τ−1) and point (x_τ+1, y_τ+1). The mean propagation velocity was estimated by dividing the translational vector by 2 h and centering it on the anchor point (x₀, y₀). A single pass of a nine-point smoother (Shapiro 1970) was applied to the three patterns of (5) and (7) prior to correlation estimate to mitigate the deleterious impact of small-scale variability. The displacement vector and the orientation of the elongation axis can be examined to glean the relationship between the mean system propagation and anisotropic distribution of precipitation (e.g., Doswell et al. 1996, their Figs. 3 and 4). The example of Fig. 2 yields a 2-h displacement vector of ∼128 km (∼18 m s⁻¹ or ∼35 kt) toward the northwest, an orientation slightly (∼10°) to the east of the elongation axis.

a. Sample climatology

Before proceeding to the main results, it is of interest to estimate how representative our 4-yr sample is of long-term climatology of precipitation. For that reason, we show distributions of the sample frequencies of hourly precipitation exceeding the two thresholds (Fig. 3). The maps are based on stage IV analyses for all hours of the day, and thus diurnal variations are ignored.

The 1-mm threshold during the winter (Fig. 3a) displays a band of enhanced occurrence along a SW–NE-oriented band from east Texas to the central Appalachians Mountains. Maximum values along the band exceed 5%. The distribution for 5-mm wintertime threshold (Fig. 3b) is similar to the 1-mm pattern with maximum frequencies approaching 1.5%. The 1-mm summertime distribution (Fig. 3c) shows maxima over the interior of the Florida peninsula and the panhandle region of the Gulf Coast. Other maxima tend to be clustered over the higher terrain of the Appalachians. Peak values for 1 mm approach 6%. Maxima for the heavy threshold are restricted to Florida and the Gulf Coast, and they exceed 3% over Florida. The distribution of the 5-mm threshold closely resembles the 10-yr climatology of cloud-to-ground lightning density of Orville and Huffines (2001; Fig. 3).

Note that sample sizes can be inferred directly from frequencies in Fig. 3. The total number of precipitation events at each pixel can be obtained by multiplying the frequencies of Fig. 3 by the total number of hours in four seasons (90 days or 8640 h). The total number of events varies from a low value of ∼50 at the higher threshold to a high value of ∼600 at the lower one.

b. Temporal correlations for 4-km pixel

Figures 4 and 5 show T_e distributions for the winter and summer seasons and the different thresholds (1 mm, 5 mm). The estimates of the e-folding times were derived under the assumption of a Markov process as previously noted; the computation is done at every fourth 4-km pixel spanning the entire domain considered.

The estimates of T_e for the wintertime 1 mm h⁻¹ (Fig. 4a) category exceed 2 h over a widespread region of the Gulf States and Southeast. The 5-mm category during winter (Fig. 4b) also shows longest times over the same region, but unlike the light category there are virtually no events in the NW sector of the domain. The distribution is consistent with synoptic experience and the notion that locations with consecutive hours of heavy convection are confined to the Gulf Coast region, near the primary source region for moisture, and they rarely occur north of the Ohio River during winter. The 1-mm threshold contains several regions where the e-folding times are longer than 3 h: a band that largely overlaps with the 5-mm one, a secondary parallel band from Oklahoma to the northern Ohio River valley, and a third distinct region centered over southeast New York. The secondary band likely contains major contributions from precipitation events that occur north of the wintertime storm track, and we believe that the separation between the two bands is likely an artifact of our small 4-yr sample. The third region is associated with preponderance of baroclinic cyclones along the East Coast and surface easterlies ahead of the low.

The e-folding estimates are much shorter during summer (Figs. 5a and 5b). Maximum T_e values of up to 2 h occur along an arc from central Texas into eastern Kansas. This region can experience long-lived outbreaks of organized convection associated with mesoscale convective systems (MCS) and mesoscale convective complexes (MCC; Maddox 1983), conditions under which localized heavy precipitation (5 mm h⁻¹) can persist several hours. Long e-folding times are also evident over southern New England for 1 mm h⁻¹. We doubt that this is a robust feature of the climate, but rather a consequence of the unusually wet summer of 2003, which represents a fourth of our data. (It was the wettest summer to date for 13 states east of Mississippi River; see http://www.ncdc.noaa.gov/img/climate/research/2003/aug/06-08Statewideprank_pg.gif.) Over most of the summer domain, however, T_e values are less than 2 h for 1 mm h⁻¹ and 1 h for 5 mm h⁻¹, which reflects a predominance of short-lived airmass thunderstorms in these areas. These results are not surprising, as the summer season possesses a preponderance of convective events with shorter lifetimes compared to longer-lasting and more widespread winter events.

It is evident that even more frequent than hourly sampling is necessary to properly resolve e-folding times shorter than 1 h. Such information would be particularly valuable for quantifying extremely heavy precipitation, say 25 mm h⁻¹ or higher, or the particularly intense magnitudes that are often associated with local flash flooding. In fact, information at a sampling frequency significantly shorter than the eddy turnover time for cumulus convection might be necessary. This suggests that a similar climatology using ∼5-min analyses, and over several decades to sample many events, would be a valuable (but arduous) endeavor.

c. Isotropic statistics for 4-km pixel

As discussed in section 3b(1), the characteristic spatial scale of precipitation can be easily quantified under the assumption of isotropy. Values of e-folding distance d were computed for two thresholds of 1 and 5 mm h⁻¹ and both seasons.

The winter distributions of d for the nominal 4-km stage IV pixel (Figs. 6a,b) are marked by long isotropic e-folding distances over the Southeast where maximum values exceed 160 km for 1 mm h⁻¹ over several states. The large spatial scale of precipitation is associated with the passage of wave cyclones and the frontal systems that are able to tap into Gulf of Mexico moisture. Wave cyclones that develop along the coastline would be particularly adept at producing widespread precipitation over the Southeast. The e-folding distances shorten to less than 50 km within a distance of 1000 km to the Northwest, such as over the upper Mississippi River valley and Great Lakes region. The statistics reflect that fact that the region is farther from the primary source of moisture, the Gulf of Mexico, and the saturation vapor pressure is lower in a colder climate, which would yield smaller rainfall rates for the same vertical velocity. The statistics of light precipitation in the colder climates are undoubtedly affected by the poor detection of snowfall too. The Weather Surveillance Radar-1988 Doppler (WSR-88D) radars have difficulty detecting precipitation when the raindrops are very small or when the precipitation is in the form of snow; under these conditions precipitation estimates tend to be underestimated (Westrick et al. 1999). The general pattern of d is similar as the threshold is increased to 5 mm h⁻¹ (Fig. 6b). However, the events are more tightly confined to the Southeast and the maximum decorrelation scale drops to ∼80 km. The geographic restriction and shorter scales reflect the convective nature of heavy precipitation in warmer climates.

The contrast between the winter and summer (Figs. 6c,d) statistics is drastic. The plains states contain the longest decorrelation scales, with d values for 1 mm h⁻¹ exceeding 64 km from central Texas to the upper Mississippi Valley. On the other hand, d is half as large (∼30 km) over the Southeast. The same pattern of regional enhancement is also evident at 5 mm h⁻¹, but the magnitudes of d typically are reduced by a factor of 2. The larger spatial scales in the vicinity of 95°W coincide with the region where mesoscale convective systems and larger mesoscale convective complexes preferentially occur (Maddox 1983; Fritsch et al. 1986; Kane et al. 1987). Statistics over the rest of the domain reflect a stronger influence from smaller-scale airmass thunderstorms.

As noted in section 4a, the high decorrelation values at 1 mm h⁻¹ over southern New England to New Jersey are believed to be an artifact of the record rainfalls during the 2003 summer and are not considered reproducible.

While the isotropic assumption simplifies computation and physical interpretation, it is clear that uncertainty introduced by this method would be more pronounced in areas where the spatial structure of hourly precipitation distributions is more asymmetric. To quantify the impact of asymmetry, we show in Fig. 7 the standard deviations of pixels that lie inside the annular region for the 1-mm threshold. A standard deviation of ∼0.04 would correspond to a “noise” level of just over 10%, a level that might be considered an upper bound for isotropy being an acceptable assumption. The figure reveals that values greatly exceed 10% over the 99% of the domain during the winter. Values typically run a factor of 2 smaller during the summer, but they still exceed 10% over most of the domain. This simple analysis indicates that isotropy is clearly a better assumption during the summer than the winter, but it also suggests that anisotropic considerations should not be ignored in either season.

d. Anisotropic statistics for 4-km pixel

Fourier decomposition (section 3c) was used to quantify the anisotropic nature of precipitation fields. The decomposition at each anchor point was computed at a radius R_c defined by the isotropic decorrelation scale d of Fig. 6: R_c = d ± ΔR and for annular ring of 8-km width (ΔR = 4 km). The calculation was found to be insensitive to modest changes in the annular thickness (8 km versus 16 km). The fraction of total variance explained by the first two harmonics provides a measure of the fidelity of this simple model for asymmetry (first harmonic) and elongation (second harmonic). Typically 60%–90% of the variance is contained in the first two harmonics over the majority of the domain (72% during the summer, 85% during the winter), but in limited areas it does dip below 30% where isotropy becomes a more justifiable assumption. The fraction of total variance explained by the second harmonic variance provides a measure of how good the elliptical model is in an aggregate sense.

Figures 8 and 9 give the seasonal distributions of the eccentricity for the two thresholds. The eccentricity runs higher during the winter (Fig. 8) than the summer (Fig. 9). Average wintertime eccentricities are ∼0.7, which corresponds to a minor to major axis ratio of ∼0.7, and summertime values are ∼0.5, which corresponds to a minor to major axis ratio of ∼0.8. The greater elongation during the winter likely represents the modulating effect of the more frequent occurrence of strong baroclinic wave cyclones and concomitant frontal zones. There appears to be a tendency for the 5 mm h⁻¹ convective threshold to be more elongated than the measurable threshold, especially during the winter when the 1 mm h⁻¹ rate is a better discriminator of stratiform events than during the summer. The spatial variability of the eccentricity distribution during the summer seems higher than the winter, with the spatial variance being ∼50% larger in the summer, but we are not clear whether this difference reflects actual physical differences between summer (stronger cumuliform contribution) and winter (more stratiform contribution) precipitation or a sampling fluctuation.

The orientation of the major axis at each anchor point is indicated by the blue line segments in Figs. 8 and 9. A widespread preference for SW–NE elongation is evident during each season and for both thresholds. The one region of major discrepancy is located between 105° and 102°W, immediately to the east of the Rocky Mountains, where warm season systems are aligned more N–S. This is also a region of strong elongation (eccentricities 0.8 to 0.9 in Fig. 9). The signal is consistent with convection systems, in a statistical sense, being aligned parallel to the N–S mountain ranges to the west and undergoing a transition as they propagate eastward (section 4d). Although accurate detection would be more problematic west of ∼100°W (Maddox et al. 2002) and the sample size is small (cf. Fig. 3), our estimate is on the order of 50–100 nonindependent events during the 4-yr sample; synoptic experience leads us to believe that the difference in orientation and elongation probably denotes a real feature of the summer precipitation climatology for the western plains.

e. Mean propagation characteristics for 4-km pixel

The in situ temporal correlations of Figs. 4 and 5 provide no direct information about the direction of movement, speed, or the Lagrangian time scale. Although the organized precipitation structures may last for many hours to a several days, in situ statistics for a 4 km × 4 km pixel are a sign that individual cells appear to be short-lived in a local, Eulerian sense.

The black arrows of Figs. 8 and 9 give the estimated mean propagation vector, computed from (9) for lags τ = ±1 h [sections 3b(2) and 3c]. Mean propagation speeds during the winter exceed 20–25 m s⁻¹ over most of the Southeast for both thresholds. The mean movement is usually situated in a slightly more East–West direction than the dilatation axis (blue segments in figures), which indicates an angle between line movement and orientation that is much less then 45° in most regions. The relationship over the Southeast is consistent with SW–NE-oriented cold fronts, squall lines, and rainbands moving in an ENE direction. The mean propagation speed during the warm season is a much slower value of 8–15 m s⁻¹, a value that agrees with the meridionally averaged radar reflectivity statistics of Carbone et al. (2002). The average speed is ∼20% faster for the 5 mm h⁻¹ category (9.4 m s⁻¹) than for 1 mm h⁻¹ (7.9 m s⁻¹). The preferred direction of movement is also more zonal during the summer. This means the propagation vector and dilatation axis are more skewed during the convective warm season, with the angle typically ranging from 45° to 90°.

f. Spatial averaging

Since a major goal is to evaluate model forecasts and climate simulations, it is of interest to examine comparable statistics for coarser resolutions that are consistent with those of weather and regional climate models. For sake of brevity, only in situ e-folding times and isotropic e-folding distances are discussed here. Before illustrating those results, we first show distributions of the frequency of occurrence, computed from hourly precipitation rates over 16 km ×16 km (4 × 4 = 16 total stage IV pixels) and 32 km × 32 km (8 × 8 = 64 total stage IV pixels) grid boxes, in Figs. 10 and 11. The winter frequencies do not vary appreciably as the size of the grid box is increased from 4 to 32 km. This is related to the characteristic large spatial scale of wintertime precipitation, when d typically runs much longer than 32 km (Figs. 6a and 6b) for both thresholds. On the other hand, summer frequencies behave differently. The 5-mm threshold shows that frequency steadily decreases as grid box increases, a result consistent with the spatial scale d on the 4-km grid for heavy precipitation rates being much shorter than 32 km in most regions of the CONUS (Fig. 6d). The frequency for the 1-mm threshold clearly decreases as the mesh expands from 4 to 16 km, but the frequency increases slightly from 16 to 32 km because the expanded mesh size greatly raises the probability of including an isolated region (a few 4-km pixels) with very heavy warm season precipitation (≫5 mm h⁻¹) of sufficient intensity to fulfill a 1-mm criterion.

Seasonal distributions of e-folding times for the two larger grid meshes are displayed in Figs. 12 and 13. Comparison of Figs. 12 and 13 with their 4-km counterparts (Figs. 4 and 5) reveals very similar spatial distributions. However, subtle differences in the local magnitudes exist as the averaging scale increases. Table 1, which shows the domain-average values of T_e as a function of grid size, indicates that T_e tends to increase as the grid size lengthens, with the biggest percentage jump occurring between 16 and 32 km. The statistics suggest that event persistence increases as the area over which precipitation is collected expands, but only slightly.

The seasonal distributions of e-folding distance, computed for hourly accumulations that exceeded the 1 and 5 mm h⁻¹ criteria on a 16- and 32-km mesh, are depicted in Figs. 14 and 15, respectively. Although the stage IV data extend beyond the CONUS east of 105°W, caution should be employed when interpreting the values at the boundaries, as up to half of the 512 km by 512 km window (e.g., Fig. 2) over which the d values are computed is over water or data sparse zones, where there are no or few gauges. In such regions the precipitation estimate is in essence based solely on radar reflectivity. The d values gradually increase as the grid size increases from 4 to 32 km (cf. Figs. 6, 14 and 15). The increase from 4 to 32 km typically runs ∼10%, except during summer when it is ∼50% larger for the heavy convective threshold (Table 2). We also examined the sensitivity of the d estimates to the order of the averaging operator by comparing the results of Figs. 14 and 15 to the 4-km results of Fig. 6 averaged over a corresponding larger mesh. Except for the summer time 5 mm h⁻¹ threshold, variations between the two methods are small (≤10%) and of varying sign, implying that the averaging procedure is approximately commutative for the hourly accumulation period and grid lengths examined. Although the order of the averaging operator is not important for our 1-h sampling interval and behaves as a linear system would, it is not clear whether this behavior would hold steadfast as the accumulation interval or grid mesh change to more extreme values.

When comparing the statistics for different grid sizes, it is important to bear in mind that the events fulfilling the 1 or 5 mm h⁻¹ criterion can decrease rapidly as the area increases. Moreover, precipitation events on the different grid meshes are not independent: hours during which a coarser grid satisfies the rate criterion must contain at least one 4 km × 4 km pixel (and usually many more than one) that also satisfies the same criterion, and hours during which a 5 mm h⁻¹ event occurs inside a grid box are a subset of 1 mm h⁻¹ of the events. This interdependence undoubtedly reduces the degrees of freedom significantly.