In several continental regions a weekly cycle of air pollution aerosols has been observed. It is usually characterized by concentration minima on weekends (Saturday and Sunday) and maxima on weekdays (Tuesday–Friday). Several studies have associated varying aerosol concentrations with precipitation production and attempted to determine whether or not there is a corresponding weekly cycle of precipitation. Results to date have been mixed. Here we examine a 12 year national composited radar data set for evidence of weekly precipitation cycles during the warm season (June–August). Various statistical quantities are calculated and subjected to “bootstrap” testing in order to assess significance. In many parts of the United States, warm season precipitation is relatively infrequent, with a few extreme events contributing to a large percentage of the total 12 year rainfall. For this reason, the statistics are often difficult to interpret. The general area east of the Mississippi River and north of 37°N contains regions where 25%–50% daily rainfall increases are inferred for weekdays (Tuesday–Friday) relative to weekends. The statistics suggest that western Pennsylvania is the largest and most likely contiguous region to have a weekly cycle. Parts of northern Florida and southeastern coastal areas infer a reverse-phase cycle, with less rainfall during the week than on weekends. Spot checks of surface rain gauge data confirm the phase of these radar-observed anomalies in both Pennsylvania and Florida. While there are indications of a weekly cycle in other locations of the United States, the degree of confidence is considerably lower. There is a strong statistical inference of weekday rainfall maxima over a net 8% of the area examined, which is approximately twice the area of cities. Future examination of lofted aerosol content, related condensation/ice nuclei spectra, and knowledge of the convective dynamical regime are needed in order to assess how anthropogenic aerosols may affect rainfall at urban and regional scales. If radar is the primary method of observation, it is also necessary to examine the effects of variable aerosol content on the parametric relationship between rainfall rate and radar reflectivity factor. Polarimetric radar observations could also serve to verify microphysical-dynamical hypotheses regarding precipitation production.
 In many populated continental regions a weekly cycle of anthropogenic air pollution aerosols has been observed. It is usually characterized by concentration minima on weekends (Saturday and Sunday) and maxima on weekdays (Monday–Friday). Several studies in the past decade have investigated the impact of the varying aerosol concentrations on precipitation production and whether or not there is also a weekly cycle of precipitation. Results to date have been mixed, rendering anthropogenic influences on precipitation as uncertain and controversial.
 Consider first those studies finding a significant weekly cycle of precipitation. Cerveny and Balling  derived rainfall estimates from satellite data over the western North Atlantic for the period July 1991–January 1995. They observed a maximum (minimum) in rainfall on Saturday (Monday) and attributed the increase to air pollution aerosols advecting off the heavily populated eastern coastal region of the United States. The variation in rainfall from the mean was ∼15%. Bäumer and Vogel  examined 15 years of precipitation data from the German Weather Service network. They found a weekly cycle with a maximum on Saturday and a minimum on Monday. The weekly cycle was observed in both urban and rural mountain stations leading to the conclusion that anthropogenic effects were at least of order meso-alpha in scale. One of the more comprehensive studies is that of Bell et al.  in which 8 years (1998–2005) of Tropical Rainfall Measuring Mission (TRMM) satellite data were analyzed for the warm season (June–August). Computing daily rainfall amounts, they found a significant increase in rainfall over the southeastern United States during the middle of the week (Tuesday-Wednesday) and a minimum on weekends. They also attributed the midweek maximum to increases in anthropogenic air pollution aerosols. On a more local scale, Lacke et al.  compared PM2.5 aerosol concentrations to rainfall derived from Weather Surveillance Radar–1988 Doppler (WSR-88D) observations over the Atlanta, Georgia metropolitan area and found a midweek maximum in both quantities compared to Sunday.
 Other studies have refuted the presence of a weekly precipitation cycle. DeLisi et al.  examined 20 years of precipitation from seven coastal cities in the northeastern United States and found no evidence of any weekly precipitation cycle at the 95% confidence level. Their results pertained to both precipitation intensity and frequency, for precipitation at individual sites, and at all sites combined. Note that DeLisi et al.  examined all seasons of precipitation, did not consider warm season precipitation separately, and they performed the analysis in areas where Bell et al.  found little evidence of a weekly cycle. Schultz et al.  examined 42 years of precipitation records in the United States (219 stations), finding no significant dependency on day of week in any season. However, Bell and Rosenfeld  raised valid concerns about their results, particularly with regard to some of the statistical analyses employed. Using a variety of statistical tests Barmet et al.  examined 17 stations in Switzerland and found no significant 7 day cycle in precipitation, a result inconsistent with that of Bäumer and Vogel  in an adjacent region. They concluded that the disparate findings were the result of different statistical procedures.
 Given the wide range of dynamical regimes and sensitivity to the microphysical complexity of deep moist convection, diverse outcomes with respect to aerosol variability might well be anticipated. None of the U.S. rainfall studies cited herein have established the occurrence of a weekly cycle of aerosol uptake in clouds or storms, or related information on the activity spectrum of cloud condensation nuclei (CCN) and ice nuclei (IN). Herein, we examine unique radar data from a U.S. national composite archive to search for evidence of a weekly cycle in precipitation during the warm season (June–August, JJA). Regrettably, this study is purely statistical, and similarly devoid of relevant atmospheric constituent information. We present a statistical analysis of the radar-estimated rainfall together with spot checks of NOAA National Climatic Data Center (NCDC) rain gauge data to verify the sense of the largest radar-observed anomalies.
2. Data Source and Methodology
 The data used in this study are the WSI Corporation National Operational Weather Radar (NOW-rad) national composite radar reflectivity. The precise algorithm for creating the radar composite is information proprietary to WSI, but is usually described as the maximum value of radar reflectivity as measured by any WSR-88D at any height in a vertical column. The properties of the product include an ∼2 km latitude-longitude grid with 15 min temporal resolution at 5 dBZ intervals. The reflectivity values, Z, are converted to a rainfall rate R (mm h−1) using a Z-R relationship (Z = 300R1.6), which is one among a family of such relationships widely regarded as appropriate for midlatitude convection [Imai, 1960; Battan, 1973]. To reduce the data to a more manageable size, the rainfall rate data are averaged onto two separate grids having longitude-latitude grid spacings of 0.2° and 1.0°, and then integrated over 24 h (between 00:00 and 24:00 central daylight time) to obtain estimates of daily rainfall at each grid point. If a radar measurement is more than 200 km from the nearest WSR-88D it is not included in the analysis.
 It is well known that instantaneous rainfall rates, as determined from radar measurements, can easily be in error by a factor of 2 or more due to variations in the drop size distribution among several other factors. Spatial and temporal averaging (as is being done here) can significantly reduce uncertainty in rainfall estimates [Wilson and Brandes, 1979]. The main concern in this application is the relative amount of rainfall, not absolute values. A related concern is that day-of-week changes in air pollution aerosol could lead to systematic changes in the hydrometeor size distribution, which may then lead to systematic biases in radar rainfall estimation that an averaging process could not mitigate.
 Twelve years (1996–2007) of radar-estimated rainfall during the warm season (June–August) are examined for evidence of a weekly cycle. There are approximately 13 weeks each warm season, hence each grid point has about 150 samples for each day of the week. Various statistical quantities are computed and p values are determined using Monte Carlo bootstrapping and permutation methods [Delucchi and Bostrom, 2004; Efron, 1979; Manly, 1997]. Bootstrapping is one of the more robust tests, requires fewer assumptions of the data, and can be applied to a broad class of problems. In the bootstrap, random resampling is done, for example, with say the Tuesday and Saturday samples being drawn (with replacement) from the actual Tuesday and Saturday populations, respectively. The sample sizes are kept equal to the original samples and, because of the replacement procedure, a bootstrap sample may contain more than one copy of an observation and none of another. Permutation testing is similar to the bootstrap except that virtual Tuesday and Saturday samples are created by randomly drawing from the entire population without regard to the day of week. In either testing procedure the random resampling is repeated 5000 times to obtain a distribution of the statistical quantity of interest (for example the difference of the Tuesday and Saturday means) and p values are determined empirically from the area underneath the distributions.
3.1. Tuesday and Saturday Statistics
 Since Bell et al.  found a Tuesday-Wednesday rainfall maximum and a weekend minimum over the southeastern United States, the initial focus here is a comparison of the Tuesday and Saturday rainfall. Later the entire weekly cycle will be examined in more detail. Figure 1 shows the 12 year averaged Tuesday-Saturday rainfall difference and the anomalies for each day expressed as a percent from the 12 year warm season mean. Figure 1a shows a large area of generally positive differences east of the Mississippi River (∼90°W) and north of 37°N. This region is hereafter referred to as NE. The northern boundary of the NE region is loosely defined as the northern borders of Illinois, Indiana, Ohio and then stretching northeastward along the Canadian–United States border. The Atlantic coastal regions tend to have near neutral anomalies and are excluded from the NE region. For ease of computation when computing bulk statistics for NE, the northern and eastern boundaries are defined as 43°N and 75°W, respectively.
 The southeastern United States shows mixed results with the coastal areas of South Carolina, Georgia, and Florida having mostly negative differences and the Appalachians having positive differences. The Central Plains states of Kansas and Nebraska (hereafter referred to as CP and bounded by 95°W–103°W and 37°N–45°N) also contain large areas of positive differences. The only other areas of notable negative differences are the central parts of Texas and Oklahoma. In the NE and CP regions the Tuesday rainfall exceeds that of Saturday by as much as 50% or more in some local areas. The NE positive differences are due to both strong Tuesday positive and Saturday negative anomalies of order 15%–40% (Figures 1b and 1c). The positive differences of the southern Appalachians are generally due to a combination of moderate positive Tuesday and negative Saturday differences (northern Georgia, western North and South Carolina and eastern Tennessee). Other regions of the Southeast have positive differences due to either positive Tuesday anomalies (northern Mississippi and central Arkansas) or negative Saturday anomalies (southern Alabama), but not both. The positive differences over CP are in large part due to a strong negative Saturday anomaly (especially over Kansas).
 The patterns in Figure 1 exhibit broad areas of coherent positive or negative anomalies. Considering that these represent 12 year averages, it is somewhat surprising that small-scale anomalies are also common. Such variability suggests that a 12 year period of record may be heavily influenced by extreme rainfall events on local scales. Caution must be used in interpreting some of these results, so a closer look at the rainfall statistics is warranted.
 One useful method of examining the data is to rank the Tuesday and Saturday daily rainfalls from highest to lowest and examine the differences between the Tuesday-Saturday ranked pairs. Figures 2a–2d show examples of this for grid points in western Pennsylvania, eastern Nebraska, central Ohio and northern Florida. The locations of Figures 2a and 2b are in areas that have a strong Tuesday-Saturday difference. In both locations the Tuesday rainfall (red curve) is always greater than the corresponding Saturday rainfall (green curve); i.e., 100% of the Tuesday-Saturday ranked differences are positive. The tabulated statistics show that not only are the Tuesday rainfalls considerably higher than Saturday (125% greater in western Pennsylvania), the number of days with measurable rainfall and the average rainfall per event is also greater. Even when removing the top 10 rain days from the calculation (bottom statistic in Figures 2a–2d) the Tuesday averages are still considerably higher. The Ohio location (Figure 2c) is in a local area that has negative Tuesday-Saturday differences surrounded by broad areas of positive differences (Figure 1a). This is due solely to an extreme rain event on a Saturday (having a value of 16.2 mm h−1, well off the scale of the plot) causing both the average rainfall and rainfall per event on Saturday to be larger. This is in spite of the fact that Tuesday has 21 more days of measurable rainfall and 95% of the Tuesday-Saturday differences are positive. This clearly demonstrates the impact that extreme events can have on the local statistics. Figure 2d shows an example of a negative Tuesday-Saturday difference in Florida. Here 97% of the ranked pairs have a negative difference and compared to the other three locations, considerably more days with measurable rain. A longitude-latitude map of the percent occurrences of positive differences (Figure 2e) shows broad coherent areas having mostly positive differences. The largest area of 90%–100% positive differences (red areas) is in NE and CP, the same areas that have positive Tuesday-Saturday differences. The areas having mostly negative Tuesday-Saturday differences (purple areas) are comparable in significance but smaller in size, with the two prominent locations being in (1) Florida and (2) central Texas and Oklahoma.
Figure 3 shows the number of days of measurable rainfall (>0.25 mm) for Tuesday and Saturday and the Tuesday-Saturday difference in the number of days. Generally about 50–70 days (30%–50% of days) experience measurable precipitation over large parts of the United States. The exceptions are over the Rocky Mountains and Florida where rainfall is experienced over 100 days due to the diurnal heating of elevated topography and land–sea breeze interactions, respectively [Carbone and Tuttle, 2008]. Parts of interior Texas experience the fewest number of days (<30–40 days) with measurable rainfall. Over NE and CP, Tuesday has significantly more days (∼10–20 more days) of measurable rainfall than Saturday. The key points to be taken from Figures 2 and 3 are that those areas having greater rainfall on Tuesday than on Saturday are because Tuesday has both a greater number of days with rainfall and the rainfall is more intense in those areas where the number of rain days exceeds ∼50. This is similar to the findings of Bell et al. .
 A common characteristic of Figures 1–3 is that there are large coherent patterns of apparently meaningful Tuesday-Saturday differences interlaced with large amplitude perturbations of small scale, presumably due to a few extreme rainfall events. Figure 4 shows the percent of total rainfall that is due to the top 5 rain days for Tuesday and Saturday. Generally east of the Mississippi River 20%–35% of the total 12 year rainfall is due to the top 5 days while in the Central Plains the percentages increase slightly to 30%–40%. Texas has the highest percentage, with many areas of central Texas having values of 40%–55%. Scattered throughout the United States, however, are many local areas having 45%–60% of rainfall due to the top 5 days. For example, local hot spots can be seen in Minnesota, Iowa and Missouri on Tuesday and in Ohio (Figure 2c), Missouri, and Kansas on Saturday. This illustrates that extreme events can severely skew local rainfall statistics and care must be taken when interpreting point measurements.
Figure 5 shows the Tuesday-Saturday difference when removing the top 10 rainfall days at each grid point from the Tuesday and Saturday populations. While the general patterns look similar to those of Figure 1, many of the smaller-scale anomalies are removed. For example, the negative anomaly that can be seen in eastern central Ohio in Figure 1a and Figure 2c is now a positive anomaly closer in agreement with the surrounding grid points. Thus while the heavy rainfall events have a large impact on the statistics at local scales, the large-scale patterns remain intact.
3.2. Permutation and Bootstrap Testing
Figures 1 and 2 show large areas of Tuesday-Saturday differences covering several states, which intuitively appear to be significant. One method to assess the significance of these patterns is to perform random draws of the data without regard to the day of week, i.e., virtual Tuesday and Saturday samples are created by drawing from all days. During a random selection each grid point is assigned the same date of data. Figure 6 shows the virtual Tuesday-Saturday differences for the first four random draws. An important feature of Figure 6 is that random draws produce difference anomalies that are large in scale and amplitude and in many cases comparable to the actual Tuesday-Saturday differences (Figure 1a). This raises concerns about the significance of the results of Figure 1 and begs the question of why do random draws produce anomalies of such large scale. It should be noted that an average of the random draws trends toward zero as more draws are included and, by the 15th draw, the entire domain is essentially zero.
Figure 7 shows a random draw done in the same manner as Figure 6 except that data values are chosen independently at each grid point, i.e., grid points do not share the same set of randomly chosen dates. This is a subtle but important difference. It is obvious that the spatial scales of the anomalies in Figure 7 are much reduced from those of Figure 6. Doing the draws in this manner eliminates the large spatial coherence that is inherent to major rainfall events. This spatial coherence leads to the large, but fictitious perturbations seen in Figure 6. It is important to note that the magnitudes of the anomalies of Figures 6 and 7 are comparable. In both random draw procedures the same number of data points are being chosen from the same data set, allowing for the temporal variability of rain events to be retained while the temporal correlations are not preserved. The spatial variability of the virtual Tuesday-Saturday difference in Figure 7 is large and any particular grid point can differ substantially from its neighbors. This highlights some of the problems in dealing with highly skewed rainfall data and the inability to obtain representative statistics even with a 12 year data set. Finally note that the magnitudes of the anomalies are generally larger in the Central Plains and smaller along the Appalachians.
 There are a number of standard statistical tests that can be used to determine the confidence of the results (student t test, Wilcoxon rank sum, etc.). Here we use the bootstrap and permutation methods, considered to be among the more robust techniques [Delucchi and Bostrom, 2004]. As with any statistical test, care must be taken when interpreting results.
Figures 8a and 8b show examples of the distributions of average Tuesday-Saturday differences from 5000 permutation and bootstrap samples, respectively, at a specific grid point. The vertical lines indicate the zero and actual Tuesday-Saturday difference values. The permutation distribution is centered on zero (Figure 8a), to be expected since it represents random draws from all days of the week. The bootstrap samples (Figure 8b) are centered on the actual Tuesday-Saturday difference since it represents random draws from the actual Tuesday and Saturday populations. In the two examples shown the p values are calculated empirically as the area under the curve to the right of the actual Tuesday-Saturday value in Figure 8a and to the left of the zero value in Figure 8b. The p values for the two methods are nearly identical (0.003 compared to 0.004).
Figure 9 shows maps of the p values determined from the permutation and bootstrap samples. Areas of low p values (areas with high confidence where an actual Tuesday-Saturday rainfall difference of either sign exists) at the 0.05 and 0.01 levels are shown by the dark red and bright red contours, respectively. The most notable area of low p values is in western Pennsylvania where most of the area is covered by the 0.01 contour. In general the NE shows a broad area of relatively low p values of less than 0.25 with embedded smaller areas of low p values (<0.05) such as northern Illinois and Kentucky. Despite the broad area of positive Tuesday-Saturday differences in the CP only a small fraction has p values <0.05. Areas having a negative Tuesday-Saturday difference with low p values include central Texas and northern Florida. The results of the two methods are remarkably similar (compare Figure 9a to Figure 9b). Subsequent maps of p values will only be shown for those determined by the bootstrap method. Recognizing the limitations of significance tests it appears that there are not large areas where there is high confidence of a Tuesday-Saturday difference. Instead such areas are mostly confined to local regions.
3.3. Statistics for Other Days of the Week
 Thus far only the Tuesday and Saturday statistics have been examined. The attention is now turned to the other days of the week. Figure 10 shows the variation of the areas of low p values (<0.05) for those locations having positive and negative Tuesday-Saturday differences, i.e., the “day of week” minus Saturday (blue) and the “day of week” minus Sunday (green). The domain of calculation is bounded by 105°W on the west and by 45°N and 32°N on the north and south, respectively. There is an increase in the area of significant positive difference (Figure 10a) from the weekend into Monday and Tuesday before leveling off and remaining near constant Tuesday through Friday. The area of negative difference (Figure 10b) is essentially flat throughout the week with perhaps a slight decrease in the middle of the week. The positive curve shows a sustained weekday peak of ∼10%–12% of the total area. From the Saturday-Sunday and Sunday-Saturday area values of Figures 10a and 10b it is estimated that the noise level in the area values is ∼3%–4%. Thus there is a high degree of confidence that 7%–9% of the area east of the Rocky Mountains experiences a significant weekly, weekday maximum cycle.
 There are several aspects of the rainfall data that make these challenging to analyze. First, ∼40%–50% of the days have zero values (no measurable rainfall). Second, in many locations ∼10–15 days contribute to 50% or more of the total 12 year rainfall. Thus while it may seem that the population is sufficiently large (150 days for each day of the week) to obtain meaningful statistics, the sample size is relatively small. In addition, the large spatial coherence of rainfall events further complicates the analysis potentially causing erroneous anomalies of relatively large scale. The inclusion of more data points into the analysis would be expected to improve the statistics. A simple way to increase the number of data points is to include more days into the averaging process. Figure 11 shows the difference between a Tuesday-Wednesday and a Saturday-Sunday average thereby doubling the number of days in each sample. Overall the difference patterns of Figure 11a are similar to the Saturday-Tuesday-only differences of Figure 1. Figure 11b shows the p values of these results from the bootstrap samples. Compared to Figure 9, the averaging produces a significant increase in the area of the 0.05 contour. This increase is particularly evident in the NE, where close to 50% of the area has p values <0.05. Note that the inclusion of more data has little effect on the p values in the CP.
 Including an adjacent day into the averaging process raises the important issue of data independence and the temporal correlation of rainfall between adjacent days. A temporal correlation was done between rainfall on Saturday and Sunday (as well as between Tuesday and Wednesday). The results (not illustrated) reveal a few local areas where the correlation can be as high as 0.5–0.6, however, ∼90% of the domain had correlations of 0.0 to 0.25. The bootstrap samples in Figure 11b were constructed by picking pairs of consecutive days (commonly known as a blocking bootstrap) thus capturing the dependence structure of neighboring (in time) data samples. Thus it is concluded that biases introduced by data dependence is not a large issue.
 Up to this point all of the analyses have been done on a 0.2° grid, in part to illustrate the impact that the skewness of convective rainfall events can have on the statistics on local scales. Increasing the grid spacing to 1.0° would be expected to filter out many of the smaller-scale features, thereby facilitating the interpretation of results. Figures 12a, 12c, 12e, and 12g show the Tuesday-Saturday and Tuesday-Wednesday–Saturday-Sunday differences and their p values. Figures 12a, 12c, 12e, and 12g may be characterized as smoothed versions of Figures 1, 5, 9, and 11 (repeated in Figures 12b, 12d, 12f, and 12h). Many of the same features can be identified, e.g., the largest area of low p values is over western Pennsylvania with scattered smaller areas over Florida and Illinois (Figure 12c), and the inclusion of Wednesday and Sunday into the averaging expands the area of low p values over the NE (Figure 12g). In Figure 12g ∼58% of the NE is covered with p values <0.05 compared to ∼41% for the Tuesday-Saturday-only difference.
 Even though a large fraction of the differences in the NE tests with low p values, there is concern whether or not these areas are being artificially inflated because of the underlying spatial coherence of the rainfall events. This has been discussed in detail in the studies of Livezey and Chen  and Wilks  and an evaluation of the “field significance” is prudent. Using the 1° data this is evaluated by performing 500 random draws from the entire data set, thus creating a virtual Tuesday-Saturday difference where the null hypothesis would be true. Bootstrapping is then used to find the p values for each draw and a pdf of the percent area testing significant (by accident) at the p = 0.05 level is constructed. The 5% tail of the pdf determines the threshold of field significance (ao). If the percent area of the true Tuesday-Saturday difference (aT) exceeds ao, it is considered “field significant.” This is similar to Figure 5 and the related discussion of Livezey and Chen . Field significance is evaluated over the full domain and in each of the subdomains NE, CP and Florida. In the NE the Tuesday-Saturday and Tuesday-Wednesday–Saturday-Sunday aT values are 41% and 58% respectively, and the respective ao are 18.4% and 21.0%. Since the aT values are much greater than ao the NE differences are considered to be very highly field significant. In the CP the corresponding values of aT are 9.9% and 7.4% versus ao values of 17.4% and 18.4%. Thus those few areas of low p values in the CP are highly questionable and are not considered to be significant. In Florida the aT values for Tuesday-Saturday is 50% which is much greater than its ao value of 24.1%. Thus the negative Tuesday-Saturday difference in Florida should be considered highly significant. The entire domain is also considered to be field significant for both the Tuesday-Saturday (aT = 12.0% and ao = 10.0%) and the Tuesday-Wednesday–Saturday-Sunday (aT = 16.6% and ao = 11.8%) differences.
 Consider now the weekly variation of rainfall over the NE and CP regions. Figure 13 shows the averaged rainfall versus the day of the week (lined curve) over the two regions. The colored contours show the distribution of average values (pdf) from the bootstrap samples for each day. The error bars represent the bootstrap standard error. Over NE (Figure 13a) the average rainfall is minimum on the weekend, increases on Monday and continues to increase to a near constant level for Tuesday-Friday. This pattern is very similar to Figure 10a. Note also that the error bars of the Saturday and Sunday rainfall have little or no overlap with the other days of the week. Together the results from Figures 11, 12, and 13 increase the confidence that a weekly cycle is detectable over a large portion of the NE region.
 Over CP the weekly variation of rainfall is different (Figure 13b). Saturday has a strong minimum in rainfall. Sunday and Monday rainfall is quite a bit larger, thereafter building to a peak on Thursday. This pattern is less intuitive than NE results and could reflect a fundamental difference between more rural and more urbanized areas. The Saturday minimum in large part explains why there is a large decrease in the Tuesday-Saturday difference (Figure 1) when Sunday and Wednesday are included in the calculations (Figure 11a). One could characterize the CP cycle as a Thursday-Saturday step function, with Friday in transition. Considering the Thursday maximum, field significance is evaluated for Thursday-Saturday (aT = 23.5% and ao = 17.0%) and Thursday/Friday-Saturday/Sunday (aT = 2.5% and ao = 17.6%). The single day Thursday-Saturday differences are significant, but the paired days are not. Overall the confidence of a weekday-weekend difference in the CP is relatively low.
3.4. Comparisons With Surface Rainfall Observations
 A concern with using radar reflectivity measurements as estimates of rainfall is that systematic changes in the aerosol concentrations could lead to systematic changes in the coefficients of the Z-R relationship. A comprehensive analysis of surface rainfall observations and evaluation of the Z-R relationship is beyond the scope of this study. However, a few spot checks with rain gauge observations should provide some verification to the sense of the radar rainfall estimate.
 Spot checks are performed in each of the areas exhibiting the highest confidence for a weekly cycle, western Pennsylvania and northern Florida. Twelve years of surface hourly rainfall observations were obtained from NOAA/NCDC and only those stations that had a near complete record (less than 5% missing reports) are used. In western Pennsylvania and northern Florida, 4 and 16 stations, respectively, met the criteria for temporal continuity. Figure 14 shows the rainfall data in ranked format for three stations in each of the locations. At these locations nearly all of the ranked Tuesday (Saturday) observations are greater than the Saturday (Tuesday) observations in Pennsylvania (Florida). In Pennsylvania the average (for four stations) Tuesday (Saturday) rain is 0.15 (0.09) mm h−1 and the number of days of measurable rainfall is 56 (41) days. The 16 station average in Florida shows the reverse trend with average Tuesday (Saturday) rainfall values of 0.20 (0.28) mm h−1 and number of Tuesday (Saturday) with rainfall of 56 (68) days. This agrees with the sense of the radar rainfall estimates and gives confidence that such estimates are at least consistent with the correct sign of the difference.
4. Discussion and Conclusions
 The concept of a weekly cycle of rainfall due to anthropogenic influences has been controversial and fraught with uncertainty. An underlying limitation to various findings is twofold: (1) the challenging nature of purely statistical treatments given various properties of the time series and (2) inadequate physical, chemical and dynamical information associated with the populations of events, which often suggest a causal effect of one sign or the other. It is safe to say that all studies on this topic, including this one, are significantly deficient by any reasonable standard of physical science attribution. Nevertheless, collectively such studies represent useful information from which more definitive investigations can be designed and conducted.
 Roughly half of the relevant publications have found a significant weekly cycle while an equal number of studies have refuted such claims. The reasons for this controversy are several. First, most of the studies are conducted in dynamically and physically distinct regions and seasons, reason enough to anticipate statistically distinct results. Second, the statistics of rainfall events are problematic, especially during the warm season, when deep convection prevails, and relatively few very large events determine outcomes. East of the Rocky Mountains and north of the Gulf of Mexico coastal regions ∼50% of days experience no measurable rainfall. Many previous studies have relied on widely spaced in situ point measurements of rainfall. Given the characteristic scales of convective rainfall, point measurements are especially prone to sampling representativeness uncertainty, owing to spatial aliasing [Wilson and Brandes, 1979]. While temporal sampling was sparse, Bell et al.  mitigated this problem in its spatial aspects, through use of TRMM satellite data. They found a weekly cycle in the southeastern part of the United States (TRMM data coverage is limited to 40°N). It is interesting to note that both the Cerveny and Balling  study, which also relied on satellite data, and the Bell et al.  study found a maximum in rainfall on Saturday off the east coast of the United States. This is similar to the results shown herein (Figure 1) off the coasts of Georgia and North Carolina and South Carolina.
 The 12 season radar data set employed here is used to estimate daily accumulated rainfall by means of a standard Z-R relationship. While having its own limitations, this data set is comprehensive in two spatial dimensions and time. An initial examination of the Tuesday and Saturday rainfall indicated that many areas in the industrial northeastern quarter of the United States and the Kansas and Nebraska area of the Central Plains experienced ∼20% more (20% less) rainfall on Tuesday (Saturday) than the 12 year warm season mean. There were also a few areas where the opposite phase of a weekly cycle is indicated, most notably in northern Florida. Rain gauge data from the affected regions confirm the weekend/Saturday maximum. Based upon significance testing using 5000 bootstrap samples and other statistical analyses, it was concluded with a high degree of confidence that the ∼3° × 3° latitude-longitude region centered on western Pennsylvania experiences more rainfall Tuesday through Friday than on weekends. Note that the rainfall statistics in this area are generally more stable, having pdfs of reduced skewness (Figure 2a) and less spatial variability in the random draws (Figure 7) than most other locations. Comparison with rain gauge data in this region clearly confirms the sense of weekday-weekend differences as estimated by radar.
 Large portions of the NE region exhibited a tendency toward more weekday rainfall (p values <0.25). Increasing the sample averaging size to include Wednesday and Sunday (together with averaging to a 1° grid) increased the confidence level and, together with the daily pdf values from the bootstrap sample, led to a increased degree of confidence of there being a weekly cycle in the NE. The other notable area of positive Tuesday-Saturday difference is over the CP. Here the difference exhibits a particularly low value for Saturday rainfall (Figure 13b). Sunday rainfall is nearer to the averages for Monday-Wednesday. In the CP region, there are only a few small areas at the p value <0.05 level, the data are more highly skewed (Figures 2b and 4), and there is greater variability in the random draw samples (Figure 7). Evaluation of field significance led to the conclusion that a true weekday-weekend difference in this region is suspect, but cannot be completely ruled out.
 Statistically there appears to be a significant negative Tuesday-Saturday difference over central Texas. However, the authors hesitate to suggest that there is indeed a true difference since warm season rainfall in Texas is much less frequent and is often dominated by extreme events. In these situations the statistical tools at our disposal are less reliable and a longer data record is needed. Florida, on the other hand, has frequent rainfall providing twice the number of samples of most other locations. The bootstrap p value test indicates a high degree of confidence that weekend rainfall in northern Florida exceeds midweek rain.
 Overall, western Pennsylvania and northern Florida are the most likely areas where a weekly cycle of rainfall is observable and would be among the best locations to conduct a comprehensive field experiment. Such experiments could study the effects of anthropogenic aerosols, in the context of the microphysical and dynamical evolution of storms, together with total rainfall amount. These studies would benefit significantly from analyses of Doppler and polarimetric radar information.
 This study relied on using a standard Z-R relationship to derived rainfall from radar measurements. It is recognized that systematic variations in the aerosol concentration could lead to systematic changes in the hydrometeor size distribution and hence, to day of the week changes in the coefficients of the Z-R relationship. This issue was not investigated here, except for the spot checks discussed in section 3.4. Such an investigation should be pursued if additional radar-based analyses are performed. A more extensive comparison should be made between radar reflectivity factor and surface observations of daily rainfall. A mean Z-R relationship can be derived for each day of the week and examined for any systematic changes. If the coefficients exhibit significant dependence on day of week, then the daily rainfall amounts can be reevaluated using the appropriate coefficients. By doing so it is expected that greater clarity could be achieved with respect to the existence of regional weekly cycles.
 Anthropogenic influences on warm season rainfall are not necessarily limited to aerosol effects. Those that are affected by aerosols effects will not necessarily lead to the same sign or magnitude of precipitation modulation in different convective regimes, which occur under a wide range of environmental conditions. Dynamically, such regimes include ordinary isolated thunderstorms, supercells, and various categories of mesoscale convective systems. Existing statistical information needs to be stratified according to these and other factors of microphysical significance, such as cloud base temperature, its relation to depth of the planetary boundary layer and presence or absence of dry air at midlevels. The physical chain of events should be examined through process-resolving observations and models. Ground-based lidar profiling (e.g., NASA Micropulse Lidar Network (MPLNET)) has the potential to substantially clarify the lofting issue and could provide case study evidence of aerosol uptake into cumulus clouds. Aerosol effects over large regional scales are more demanding, requiring chemical weather models for transport, and the chemical evolution of pollutants as these are transported in both clear and cloudy air. A central issue in these studies should be the distinction between urbanized area effects and regional (subcontinental) effects.
 Our concluding thoughts, some of which are clearly speculative, are as follows:
 1. An extensive region, extending eastward from Chicago and St. Louis to the Appalachians, exhibits a tendency toward weekly cycle with weekday maxima. Much of this region, best known as the “rust belt,” contains traditional heavy industries and related pollution. The storms in this climate regime are known to have vigorous ice phase processes, which account for the majority of rainfall at the surface and therefore are “eligible” for the classical microphysical hypothesis advanced by many, including Bell et al. . Large parts of this region exhibit low p values (up to 58% of total area) especially when a Tuesday/Wednesday-Saturday/Sunday difference is considered.
 2. The southeastern Atlantic coastal zone and the northern Florida peninsula exhibit a tendency toward a weekly cycle with weekend maxima, an appreciable fraction of which achieves a high level of significance according to methods employed. Storms in this region and within this season have strong maritime influences in prevailing easterly flow. Generally speaking, ice phase processes are highly variable, restricted to the upper levels of clouds in this region, and have a much stronger presence of warm (coalescence) rain than similar clouds to the north and west. While pollution sources are not known to be especially abundant, it is possible that such sources are more likely to suppress coalescence rain on weekdays, which, in the absence of a vigorous ice phase, could lower the rainfall production.
 3. Coastal regions, including most of New England and mid-Atlantic states, tend toward neutrality with respect to weekly cycles, where marine ventilation via breezes may dilute the effects of anthropogenic aerosol buildup.
 4. The Central Plains weekday maximum anomaly rises to a lower level of significance but warrants further investigation into its physical significance and the nature of potential physical mechanisms. The area is mainly managed for agriculture, an activity well known for production of copious aerosol under drier conditions. However, it is unclear to what extent, if any, agricultural activity conforms to a conventional weekly cycle. It is also known that area-extensive monocultures, such as wheat, can produce enormous quantities of ice nuclei during the germination stage [Christner et al., 2008], the effects of which could be active in clouds that have relatively cold cloud base temperatures.
 5. Considerable research remains to be performed before the current mix of hypotheses, conjecture, and challenging statistical analyses can be brought to a greater point of clarity.
 We would like to express our appreciation to Wojciech Grabowski for encouraging us to begin this study. Special thanks to Thomas Bell, Randall Cerveny, and an anonymous reviewer for their helpful reviews leading to an improved manuscript. The surface rainfall observations were courtesy of NOAA/NCDC. The National Center for Atmospheric Research is sponsored by the National Science Foundation. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation.