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  • View in gallery

    Average observed summertime total precipitation (mm) over the southwest United States. The area and shading of the dots are proportional to the average seasonal mean precipitation for the years 1931–2000.

  • View in gallery

    Values of CCC (solid line) and pseudo–T2 (dashed line) at each cluster level for seasonal mean precipitation. The left y axis gives the values for the CCC and the right y axis gives the values for the pseudo–T2 criterion.

  • View in gallery

    Regionalization of interannual variations in seasonal mean summertime precipitation (see text for details). Numbers represent stations belonging to the same cluster. Number values are arbitrary. Contours represent underlying topography plotted at 1000-m intervals.

  • View in gallery

    (a)–(d) Observed variations in summertime total precipitation (mm) averaged across stations in regions 1–4 (see Fig. 3) during the period 1931–2000. Variations calculated with respect to climatological values over the same period.

  • View in gallery

    (a) Years with anomalous precipitation characteristics over region 1 (see Fig. 3). Row F indicates years with anomalous frequency of daily rainfall determined using ISM. Row C indicates years with anomalous coverage determined using K–S and t tests applied to the empirical distribution for the given year. Row I indicates years with anomalous intensities determined using K–S and t tests applied to the empirical distributions for the given year. Row T indicates years with anomalous total precipitation determined using ISM (see text for details). Up (down) arrows represent years with statistically significant positive (negative) variations in the given rainfall characteristic. No symbol indicates rainfall characteristic during the given year is statistically similar to the climatological value. (b)–(d) same as (a) except for regions 2–4.

  • View in gallery

    (a)–(d) Normalized power spectra of seasonal mean precipitation over regions 1–4 (see Fig. 3). The thick solid line represents normalized power spectra from the observed interannual time series. The thin solid line is the mean power spectra from 142 simulations of 70-yr time series returned by the stochastic chain-dependent model for the given region. The thin dotted lines are the 95% confidence intervals of the power spectra as derived from the modeled interannual time series.

  • View in gallery

    Normalized seasonal mean 500-hPa height anomalies composited for years with potentially predictable seasonal mean precipitation anomalies over region 3. See Fig. 6 for corresponding anomalous years. Normalized anomalies represent the seasonal mean gridpoint anomaly for the given year, divided by the gridpoint standard deviation of the seasonal mean anomalies for the period 1948–2000. Solid (dashed) lines represent positive (negative) anomalies. Shading indicates regions in which the normalized seasonal mean 500-hPa height anomalies are significant at the 95% confidence interval (see text for details). Dark (light) gray shading indicates positive (negative) anomalies.

  • View in gallery

    Same as Fig. 7 but for normalized seasonal mean 500-hPa height anomalies composited for years with (a) potentially predictable rainfall coverage anomalies and (b) potentially predictable rainfall intensity anomalies over region 3. See Fig. 5 for corresponding anomalous years. Because the numbers of potentially predictable events differ for each, the 95% confidence threshold differs in each map.

  • View in gallery

    (a) Seasonal mean 500-hPa heights averaged for years with positive potentially predictable precipitation event coverage anomalies over region 3. See Fig. 5 for corresponding anomalous years. Dark (light) gray shading is the same as in Fig. 8a and indicates positive (negative) normalized seasonal mean 500-hPa height anomalies are significant at the 95% confidence interval (see text for details). (b) Same as (a) but for seasonal mean 500-hPa heights averaged for years with negative potentially predictable precipitation event coverage anomalies over region 3. Shading is opposite that of (a) to indicate sign of 500-hPa anomalies during years with negative potentially predictable precipitation event coverage anomalies.

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Influence of Daily Rainfall Characteristics on Regional Summertime Precipitation over the Southwestern United States

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  • 1 Department of Geography and Environment, Boston University, Boston, Massachusetts
  • | 2 EQECAT/ABSG Consulting Inc., Oakland, California
  • | 3 Department of Geography and Environment, Boston University, Boston, Massachusetts
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Abstract

The regional variability in the summertime precipitation over the southwestern United States is studied using stochastic chain-dependent models generated from 70 yr of station-based daily precipitation observations. To begin, the spatiotemporal structure of the summertime seasonal mean precipitation over the southwestern United States is analyzed using two independent spatial cluster techniques. Four optimal clusters are identified, and their structures are robust across the techniques used. Next, regional chain-dependent models—comprising a previously dependent occurrence chain, an empirical rainfall coverage distribution, and an empirical rainfall amount distribution—are constructed over each subregime and are integrated to simulate the regional daily precipitation evolution across the summer season. Results indicate that generally less than 50% of the observed interannual variance of seasonal precipitation in a given region lies outside the regional chain-dependent models’ stochastic envelope of variability; this observed variance, which is not captured by the stochastic model, is sometimes referred to as the “potentially predictable” variance. In addition, only a small fraction of observed years (between 10% and 20% over a given subregime) contain seasonal mean precipitation anomalies that contribute to this potentially predictable variance. Further results indicate that year-to-year variations in daily rainfall coverage are the largest contributors to potentially predictable seasonal mean rainfall anomalies in most regions, whereas variations in daily rainfall frequency contribute the least. A brief analysis for one region highlights how the identification of years with potentially predictable precipitation characteristics can be used to better understand large-scale circulation patterns that modulate the underlying daily rainfall processes responsible for year-to-year variations in regional rainfall.

Corresponding author address: Jingyun Wang, 6410 Schmidt Lane, Apt. C205, El Cerrito, CA 94530. Email: jingwang@eqecat.com

Abstract

The regional variability in the summertime precipitation over the southwestern United States is studied using stochastic chain-dependent models generated from 70 yr of station-based daily precipitation observations. To begin, the spatiotemporal structure of the summertime seasonal mean precipitation over the southwestern United States is analyzed using two independent spatial cluster techniques. Four optimal clusters are identified, and their structures are robust across the techniques used. Next, regional chain-dependent models—comprising a previously dependent occurrence chain, an empirical rainfall coverage distribution, and an empirical rainfall amount distribution—are constructed over each subregime and are integrated to simulate the regional daily precipitation evolution across the summer season. Results indicate that generally less than 50% of the observed interannual variance of seasonal precipitation in a given region lies outside the regional chain-dependent models’ stochastic envelope of variability; this observed variance, which is not captured by the stochastic model, is sometimes referred to as the “potentially predictable” variance. In addition, only a small fraction of observed years (between 10% and 20% over a given subregime) contain seasonal mean precipitation anomalies that contribute to this potentially predictable variance. Further results indicate that year-to-year variations in daily rainfall coverage are the largest contributors to potentially predictable seasonal mean rainfall anomalies in most regions, whereas variations in daily rainfall frequency contribute the least. A brief analysis for one region highlights how the identification of years with potentially predictable precipitation characteristics can be used to better understand large-scale circulation patterns that modulate the underlying daily rainfall processes responsible for year-to-year variations in regional rainfall.

Corresponding author address: Jingyun Wang, 6410 Schmidt Lane, Apt. C205, El Cerrito, CA 94530. Email: jingwang@eqecat.com

1. Introduction

Summertime North American monsoon system (NAMS) precipitation over the southwestern United States has been investigated in numerous studies. Although seasonal mean precipitation in this region is generally controlled by the dynamic monsoon system, it is also under the influence of factors operating at multiple spatial and temporal scales, including large-scale atmospheric circulations (Adams and Comrie 1997; Comrie and Glenn 1998; Ellis and Hawkins 2001; Hawkins et al. 2002); the passage of midlatitude waves originating from the northern Pacific (Matthews and Kiladis 1999; Mo 2000); transient midlatitude, midtroposphere structures (Carleton 1986; Maddox et al. 1995; Anderson and Roads 2002); mesoscale convective systems (Nieto-Ferreira and Schubert 1997; Carvalho and Jones 2001); mesoscale to synoptic-scale squall lines (Cohen et al. 1995); the Madden–Julian oscillation (MJO) (e.g., Higgins et al. 2004); and Gulf of California surges (Stensrud et al. 1997; Fuller and Stensrud 2000; Douglas and Leal 2003).

It is reasonable to hypothesize that each of these components may significantly influence the NAMS precipitation only over a portion of the full monsoon region, thereby introducing spatial heterogeneity into the summertime precipitation amounts. For instance, on interannual time scales, there may be upward of nine separate seasonal mean precipitation regimes within the southern United States and northern Mexico (Comrie and Glenn 1998), whereas summertime precipitation over the core monsoon regime—centered on New Mexico, eastern Arizona, and extending along the Sierra Madre mountain ranges in Mexico—can be further subdivided into four subregions. As with interannual variability in seasonal precipitation, intraseasonal precipitation events also show distinct spatiotemporal regionalization, with at least two robust spatiotemporal precipitation patterns over the broader southwestern U.S. domain (Anderson and Roads 2002). Spatial heterogeneity in the NAMS precipitation is also suggested in previous studies of daily rainfall characteristics (Wang et al. 2006, 2007). Although the climatic precipitation over the southwestern United States displays a distinct north–south pattern with more seasonal rainfall over the southern area (Fig. 1), year-to-year variations in rainfall frequency (i.e., the number of events with measurable precipitation) indicate more variability over the northern region (Wang et al. 2006). In contrast, interannual anomalies in intensity (i.e., the amount of rainfall per event) contain an east–west dipole, with more interannual variability contained over the western part of the region (Wang et al. 2006).

The presence of spatial heterogeneity at multiple time scales raises the question of how various daily rainfall characteristics—such as rainfall frequency and intensity—influence seasonal mean rainfall variations in different regions (e.g., Higgins et al. 2007; Leibmann et al. 2008). It also raises the question of how various dynamic processes may modify these daily rainfall characteristics, and hence the seasonal mean precipitation amounts.

To begin to answer these questions, in this study we investigate the relation between year-to-year variations in daily rainfall characteristics and seasonal mean precipitation anomalies for separate regions within the southwestern United States. The paper is organized as follows: Section 2 describes the datasets used in this investigation. Section 3 studies the spatiotemporal structure of the NAMS precipitation over the southwestern United States. Section 4 analyzes the regional variability over different precipitation subregimes. Section 5 discusses the results as a step toward providing a physical explanation for variations in precipitation in these regions. Section 6 summarizes the conclusions.

2. Dataset

The daily precipitation data used here are based on the serially complete daily maximum and minimum temperatures and precipitation compiled by Eischeid et al. (2000). The latest version of this dataset comprises daily precipitation observations from at least January 1948–August 2003 for 14 317 sites in the United States, although most stations include observations prior to 1948. Summertime (1 July–30 September) precipitation observations spanning from 1931 to 2000 are analyzed in the region from 30° to 42°N and from 115° to 102°W (the southwestern United States in Fig. 1). Similar to previous studies (Wang et al. 2006, 2007), only years with full observations from 1 July to 30 September are used to determine the daily rainfall characteristics at a given station, and 78 stations are selected within the region of interest. For studies of interannual variability in seasonal mean precipitation, station-based climate mean values are used for years with missing observations; in this way, interannual time series spanning 70 yr are archived for the 78 stations in this study.

To characterize and analyze lower- and upper-air fields related to precipitation anomalies identified from the station data, daily data are taken from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) Global Reanalysis 1 (R-I) archive for the overlapping 50-plus-yr period (1948–2000). Details about this dataset—including its physics, dynamics, and numerical and computational methods—are discussed in Kalnay et al. (1996). The R-I product is available 4 times daily at 2.5° resolution (equivalent to a gridpoint spacing of approximately 250 km).

3. Regionalization of the summertime precipitation over the southwestern United States

Studies of the spatial–temporal regionalization of precipitation can enhance our understanding of the underlying meteorological and climatological processes influencing the precipitation (Gong and Richman 1995). However, many different clustering techniques have been used in the literature to analyze the spatial structure of climate datasets (Fovell and Fovell 1993; Gong and Richman 1995). The agglomeration or hierarchical clustering algorithms are generally the most popular techniques in cluster analysis (Blashfield 1976; Gong and Richman 1995). In this study, we adopt Ward’s method (Ward 1963), also termed “the minimum variance method,” to analyze the spatial structure in the NAMS precipitation because numerous studies suggest Ward’s method outperforms other hierarchical methods (Kuiper and Fisher 1975; Blashfield 1976; Mojena 1977; Gong and Richman 1995); however, Ward’s method tends to bias toward spherical clusters and assigns similar numbers of stations to individual clusters (Cormack 1971). Hence, we also use the K-mean algorithm, which is a partitioning (nonhierarchical) clustering algorithm that groups stations into a single prior-specified number of clusters by iteratively altering the members of each cluster until they optimally represent the structure of the dataset. The K-mean method has been shown to better represent the structure of precipitation anomalies compared with other nonhierarchical methods (Gong and Richman 1995).

Determination of the optimal number of clusters is also an important stage of any spatial clustering analysis. Many criteria have been proposed in the literature to determine the optimal number of clusters. However, no one criterion works for all clustering algorithms (Everitt 1979, 1980; Hartigan 1985; Bock 1985). Milligan and Cooper (1985) compared 30 methods to determine the optimal cluster level for four hierarchical clustering algorithms and argued that two of the best criteria are a transformed pseudo–T2 (Duda and Hart 1973) and a cubic clustering criterion (CCC; Sarle 1983), which we also use here. Generally, the optimal number of clusters is indicated by a local maximum in the CCC statistic and a local minimum within the decreasing trend of the pseudo–T2 statistic.

For this study, the regionalization methods are applied to the year-to-year variations in summertime total precipitation amounts across the 78 stations. The two detection criteria are first used to identify the optimal number of clusters. The CCC detects a local peak at cluster level 4, at the same level where the pseudo–T2 statistic displays a local minimum (Fig. 2), suggesting that the optimal cluster number for the interannual variations of seasonal precipitation over this region is four.

Next, the two independent spatial clustering techniques—Ward’s method and the K-mean method—are applied. Identical clusters are identified by both methods (Fig. 3). Region 1 stations are clustered along the Rocky Mountain plateau in Colorado as well as the elevated regions of Utah. Region 2 and region 3 stations are clustered mainly over the elevated plains and foothills along the eastern (region 2) and southeastern (region 3) edges of the plateau. Region 4 stations are clustered along the southwestern slope of the Rocky Mountain plateau, predominantly in Arizona. Figure 4 shows the seasonal total precipitation anomalies, averaged across stations, in each of the four regions.

We mention here that interannual rainfall anomalies in each of the four regions, as well as at each station, are not necessarily Gaussian. However, by raising the absolute precipitation amounts at each station (as opposed to the anomalies) by a fractional power (Comrie and Glenn 1998)—in this case, to the ½ power—we can produce a transformed seasonal mean precipitation dataset that has a quasi-Gaussian distribution. If we reapply the clustering methodologies using the transformed seasonal mean precipitation data, we obtain almost identical regional structures (not shown), with only one station changing its cluster designation, suggesting these four clusters are robust features of interannual variations in NAMS precipitation over this region. In addition, they qualitatively agree with previous regionalization studies of seasonal mean precipitation variations across the domain (see Comrie and Glenn 1998, their Fig. 9).

4. Regional variability in the NAMS precipitation over the southwestern United States

Earlier we identified coherent spatial structures for the seasonal mean NAMS precipitation over the southwestern United States. Now, we want to further examine the contribution of daily precipitation characteristics to variability in each region using regional chain-dependent stochastic models.

a. Regional chain-dependent stochastic models

To analyze interannual and intraseasonal variability in the NAMS precipitation at a given station, we previously adopted statistically based chain-dependent daily rainfall models (Wang et al. 2006; 2007), which treat the frequency and intensity of daily rainfall events separately (Katz 1977; Swift and Schreuder 1981; Wilks 1999). For these models, the probability of rainfall on day d is assumed to depend on the rainfall state on day d − 1 through dn, where n is termed the “chain order.” Once the model order is determined, the frequency of rainfall can be characterized by the transition probabilities, which determine the probability of a rainfall event given the previous state of the system; these transition probabilities can be estimated from the observed daily precipitation time series (discussed later) and used to simulate the occurrence of rainfall from day to day. To capture the intensity of rainfall, chain-dependent models may employ empirical (nonparametric) distributions to randomly select precipitation amounts for a given rainfall day; alternatively, they may employ theoretical (parametric) distributions, such as gamma (Katz 1977; Richardson 1981; Wilks 1989, 1992; Salvucci and Song 2000), lognormal (Swift and Schreuder 1981), and mixed exponential (Woolhiser and Roldan 1982; Foufoula-Georgiou and Lettenmaier 1987; Wilks 1998, 1999) distributions. More detailed descriptions of chain-dependent stochastic models and their use in simulating daily rainfall events can be found in Katz (1977) or Wang et al. (2006).

A well-known deficiency of these models is that they tend to underestimate the interannual variance of seasonal rainfall amounts, which is termed “overdispersion” (Katz and Parlange 1998; Wang et al. 2006). Two types of limitations may contribute to the discrepancy between observed and simulated variances (Katz and Parlange 1998): 1) the day-to-day variations in the observed rainfall are beyond the simple model’s capability; and 2) the year-to-year variations of the observed system are ignored by models using climatological (i.e., fixed) rainfall frequency and intensity parameters. The first type of deficiency can be addressed by using a higher-order chain-dependent model or intraseasonally varying parameters (which we test for in this study; see section 4b). The second type of deficiency, which arises as a result of systematic variations in the background state of the observed climate system during a given year, is the focus of this study. The observed variations that lie outside the chain-dependent model’s stochastic envelope of variability are referred to as potentially predictable variations because they cannot have been produced simply by the stochastic (i.e., random) evolution of daily precipitation events constrained by fixed daily precipitation characteristics (Singh and Kripalani 1986; Shea and Madden 1990; Gregory et al. 1993; Katz and Parlange 1998).

Historically, improvements to stochastic chain-dependent models have been made as a means of representing this potentially predictable variability, for instance, by using hidden or conditioned chains (e.g., Bardossy and Plate 1991; Zucchini and Guttorp 1991) or two-tiered models (e.g., Katz and Zheng 1999; Hansen and Mavromatis 2001). In these cases, the models were improved by accounting for nonstationary daily precipitation characteristics (Wilks and Wilby 1999). In comparison, our interest here is in identifying the years and anomalies that cannot be captured by stochastic models with fixed (i.e., stationary) daily precipitation statistics. In this sense, we do not want to fully represent the observed variability but to see how much of it arises as a result of the stochastic evolution of daily rainfall processes and how much can only arise from a systematic change in the underlying daily rainfall processes. Instead of focusing on reducing this latter source of variability, in this study we are actually interested in analyzing it.

In our previous studies, chain-dependent stochastic models have successfully identified the potentially predictable variations of the NAMS precipitation at individual stations across the southwestern United States (Wang et al. 2006, 2007). Here, we will use chain-dependent stochastic models to study the regional characteristics of NAMS precipitation over a given subregime. For the regional analyses, three seasonal variables characterize the daily rainfall in a given region—the frequency of rainfall events, the intensity of rainfall events, and the coverage of rainfall events; this last variable is important for characterizing the hydrologic state of a region (as opposed to a single station), particularly when trying to capture regional droughts (San Antonio Express-News, 21 August 1996). Here we define the coverage as the ratio of the number of stations experiencing rainfall as a fraction of all stations within a given region. In addition, we define a regional precipitation event as any day in which the average rainfall amount across all stations in a given subregime is greater than the standard measurable amount of rainfall (e.g., 0.254 mm). This regional threshold mainly filters out events in which light rainfall amounts cover a small portion of a given region, as well as most (>70%) storms that cover a single station (Wang 2007).

Similar to station-based studies that model the frequency and intensity of rainfall events separately (Wang et al. 2006, 2007), we will use regional chain-dependent models that treat the three daily precipitation variables separately. The models, described in more detail below, will represent the occurrence sequence of daily rainfall events over a given region using chain-dependent models and will represent the coverage and intensity of rainfall during these events using (separate) empirical distributions.

b. Regional variability of the daily precipitation characteristics

To study the regional variability in the frequency characteristics of daily rainfall, six chain-dependent models ordered from 0 to 5 are constructed over each cluster found in Fig. 3 (the 0 order model assumes that the daily occurrence sequence is temporally independent). The chain-dependent parameters (i.e., the transition probabilities; see Wang et al. 2006) for a given occurrence model are calculated based on observed daily regional rainfall event time series using maximum likelihood estimation (MLE). The temporal heterogeneity of all transition probabilities is investigated using χ2 tests. Test results indicate that all transition probabilities are nonstationary, both on interannual and daily time scales; however, they do not exhibit any clear statistical pattern on daily (or yearly) time scales (Wang 2007). Hence, we adopt the seasonally averaged values for all transition probabilities in the regional occurrence submodels.

To determine the optimal chain-dependent model for a given region, each of the six models are used to produce 92-day time series of daily rainfall events for the summertime season; these integrations are then repeated 10 000 separate times for each model. The simulated interannual variances in the number of regional rainfall events returned by the six chain-dependent submodels are compared with the observed variance using the overdispersion metric (Katz and Parlange 1998; Wilks 1999), which measures the fractional amount of observed variance that is unexplained by the stochastic model behavior (Table 1). All chain-dependent models reduce the overdispersion compared with the independent model (the 0 order model), suggesting the occurrence of daily precipitation over all four subregimes is an autocorrelated process. Over region 2 and region 4, a second-order chain is the optimal occurrence model and can capture all observed variance in both regions. For region 1 and region 3, the overdispersions returned by all five chain-dependent models are positive and continually decrease as the chain order increases. Further analysis indicates that the reduction of overdispersion becomes significantly less after the fourth-order-dependent chain, suggesting that increasing the model’s order further does not substantially reduce the remaining overdispersion in these two subregimes (Wang 2007). Hence, the fourth-order chain-dependent model is selected to portray the intraseasonal temporal structure over region 1 and region 3.

Next, the specific years that contribute to the overdispersion are identified for region 1 and region 3 (we do not perform a similar analysis for region 2 and region 4 because the interannual variance in rainfall frequency is completely explained by the optimal occurrence models). We first calculate the difference between the number of rainfall events in a given year and a given region with the annual-averaged value across all years within the region. By iteratively removing the observed years with the maximum absolute difference (either positive or negative), the interannual variance in rainfall frequency for the remaining years will decrease. Once the observed variance matches that in the optimal occurrence model for a given region, the removal process is stopped. Those years that are removed are that years whose frequency of rainfall cannot be captured by the optimal model and therefore have underlying precipitation characteristics that are significantly different from the climatological characteristics used within the model (discussed later). This anomaly identification process is termed the “iterative selection method” (ISM) in this study; a more detailed description of the method can be found in Wang et al. (2006). Analysis results indicate that only a small fraction of observed years (Fig. 5) actually contain anomalous numbers of regional rainfall events.

Next, we identify year-to-year changes in the statistical properties of rainfall coverage in each region. As mentioned, we will model the rainfall coverage during a given rainfall event using the observed empirical distribution for a given region. Hence, to identify years that contain statistically anomalous regional rainfall coverage distributions, we use a Kolmogorov–Smirnov (K–S) test, which is one of the most commonly used methods to compare distributions (Wilks 1995, 129–133). It uses the cumulative distribution function (CDF) curves to reject (or confirm) the null hypothesis that two datasets are drawn from the same distributions (at the 95% confidence level); here we use the K–S test to compare the empirical coverage distribution from a single year (and region) with the empirical distribution (from the same region) based on all events in all other years (thereby excluding data from the test year). We then use a standard t test—which compares the mean of the distribution from the given year against the mean of the distribution from all other years—to determine whether the given year’s rainfall events experienced anomalously low coverage or anomalously high coverage; here we only consider years in which both the t test and K–S test indicate that the mean and distribution are statistically significantly different from climatology (It should be noted that in all four regions, the K–S and t tests return identical subsets of statistically significant years.). Similarly, a K–S test is used to identify years with statistically anomalous intensity distributions, and a t test is used to identify whether the anomalous intensity during the year is high or low (Fig. 5).

c. Stochastic and observed variability in seasonal mean precipitation

To study the interannual variability of the seasonal mean precipitation over each of the regions found in Fig. 3, the optimal chain-dependent occurrence model for a given region is integrated 10 000 times to create the daily rainfall occurrence sequence for ten thousand 92-day simulation periods. The rainfall coverage on each simulated rainy day is then generated by a random selection from the observed empirical rainfall coverage distribution for the given region. Research indicates that there is a strong relationship between daily variations in regional rainfall coverage and the intensity characteristics over each region (Wang 2007). Hence, we construct four empirical rainfall intensity distributions binned upon quartiles in the CDF of observed coverage; analysis indicates that the conditioned intensity probability distribution functions (PDFs) in each region are significantly different (at the 95% confidence level) from the overall PDF for the region [after removing the test quartile (not shown)]. These multistate empirical intensity distributions are then employed to simulate the rainfall intensity for a given rainy day, conditioned on the rainfall coverage for that day. We use the empirical intensity distributions, because previous investigations indicate that the observed empirical distributions are more appropriate than function-based theoretical distributions for estimating the mean and variance of daily rainfall intensity in this region (Wang et al. 2006).

For each of the ten thousand 92-day simulations, we can then calculate the seasonal mean rainfall amounts, as well as the model-based interannual variability and overdispersion (Table 2, second column). The overdispersions in the regional chain-dependent model systems, which measure the fractional amount of observed variance that is unexplained by the stochastic model behavior, suggest that only 33%–52% of the observed variance (depending upon region) arises from systematic changes in the underlying daily precipitation characteristics during a given year and hence may be potentially predictable.

To identify the anomalous years that contribute to this overdispersion, we use the ISM technique described earlier. For each region, we iteratively remove the year with the largest (absolute) seasonal mean precipitation anomaly until the remaining observed variance matches that found in the chain-dependent model; those years that are removed during this procedure are the years whose variance lies outside the stochastic envelope of variability generated by the chain-dependent models and hence contributes to the potentially predictable variance (Fig. 5).

Overall, potentially predictable heavy-rainfall years occur more frequently than potentially predictable light-rainfall years in all four subregions. This result coincides with our previous station-based study results (Wang et al. 2006). In addition, potentially predictable anomalies occur more frequently over the eastern two subregions (region 2 and region 3)—where approximately 20% of observed years are potentially predictable—than over the western area (region 1 and region 4), where approximately 10% of the observed years are potentially predictable.

We can also use Fig. 5 to relate years with anomalous overall precipitation in a given region to those years with anomalous precipitation characteristics (in the same region and of the same sign). Results indicate that about 50% of the years with potentially predictable variations in seasonal precipitation are associated with anomalous coverage distributions, about 35% with anomalous intensity distributions, and about 25% (over region 1 and region 3) with anomalous frequency characteristics. (Because of overlap in years with anomalous daily precipitation characteristics, percentages do not have to sum to 100%.) This result suggests that although variations in all three types of precipitation characteristics (occurrence, coverage, and intensity) are contributors to the overall precipitation anomalies, the contribution of each may be different. In addition, mismatches between the four anomaly time series suggest that the contribution from any one component may be dominant during a given year.

To better quantify the contributions from year-to-year variations in precipitation characteristics to the overall precipitation variability in each region, conditioned submodels accounting for each type of anomalous characteristic are constructed over each region. For the chain-dependent occurrence submodel, separate transition probabilities are calculated for anomalously low, normal, and anomalously high event frequency years, as determined by the ISM procedure (see section 4b). Prior to the beginning of each simulation year, the hydrological state for the year is randomly determined using the observed probability of having a low, normal or high event frequency year. Then, the corresponding parameter set for that hydrological state is used to generate the daily rainfall series during that simulation year. A more detailed description of the conditioned model can be found in Wang et al. (2006). Similarly, conditioned rainfall coverage (intensity) models are constructed by finding the regional rainfall coverage (intensity) distributions for the anomalously low, normal, and anomalously high event coverage (intensity) years, as determined by the K–S test and t test. Similar to the conditioned occurrence model, at the beginning of each simulation year, the hydrological state for the year is randomly determined using the observed probability of having a low, normal, or high event coverage (intensity) year, and the corresponding event coverage (intensity) distribution for that hydrological state is used to generate the coverage (intensity) for daily rainfall events during that simulation year.

By combining one conditioned submodel for a given daily precipitation characteristic with two unconditioned submodels, year-to-year variations in a given daily precipitation characteristic are accounted for, which tends to reduce the seasonal mean precipitation overdispersion (Table 2). In general, the conditioned coverage model captures the maximum amount of observed variance in seasonal mean precipitation, and the intensity-conditioned model also substantially improves the models’ capabilities. In comparison, the frequency-conditioned regional model returns a relatively large overdispersion (for region 1 and region 3) compared to the two other conditioned models. These results suggest that although all three subprocesses can contribute to the potentially predictable precipitation variability, variations in coverage and intensity characteristics appear more significant than changes in the frequency of events.

To account for each subprocesses’ year-to-year variations, all three conditioned submodels are randomly combined and used to simulate the observed daily regional rainfall process. The fully conditioned regional models capture more than 90% of the observed variance for the seasonal mean precipitation over region 1 and region 3 (which contain interannual anomalies in all three subprocesses). For region 2 and region 4, in which only coverage and intensity anomalies are included, the fully conditioned regional model explains all the observed precipitation variance. These results confirm that potentially predictable variations in observed seasonal mean rainfall can be accounted for by the occurrence of a small subset of years with anomalous daily precipitation characteristics.

5. Discussion

Having identified specific years in which potentially predictable variations in seasonal mean precipitation occurred, it is possible to examine the physical processes that may have given rise to changes in the underlying precipitation characteristics that produced this predictability. A few notes on this type of analysis are in order.

First, as mentioned, χ2 tests indicate that although all rainfall occurrence parameters (i.e., transition probabilities) are nonstationary on interannual time scales, they do not exhibit any clear systematic variation on longer time scales. Similarly, a visual inspection of Fig. 5 does not indicate any systematic trends in potentially predictable event coverage and event intensity characteristics in any region (with the possible exception of region 1). Finally, the observed spectra for the seasonal mean precipitation (for each region) can be compared with the corresponding simulated spectra (Fig. 6). As can be seen, none of the observed spectral peaks are significantly different from those generated by a stochastic model with climatological daily rainfall characteristics. Overall, these results suggest that potentially predictable seasonal mean precipitation anomalies for these four regions are principally related to variations in daily mean rainfall characteristics during a given year, not to a systematic, low-frequency evolution of these variations.

Second, for certain regions and rainfall characteristics, our analysis indicates that it may be difficult to determine the physical processes that give rise to changes in underlying precipitation characteristics during a particular year, at least in a climatological sense. For instance, over region 4 (centered in Arizona), there are only 6 out of 70 yr that show potentially predictable seasonal mean rainfall variations, with only one negative anomaly occurring during the overlapping reanalysis period (1948–2000). Determining a “climatological” signature for low rainfall years in this region becomes problematic. Similar results hold for variations in the number and intensity of rainfall events over most regions. These findings highlight the need for the quantitative analysis of potential predictability, as performed here. Although it is always possible to “analyze” the climate signatures associated with year-to-year changes in seasonal mean precipitation and/or daily rainfall characteristics in a given region, the inclusion of all years in such an analysis (as with typical correlation/regression analyses) may simply result in the “analysis” and diagnosis of predominantly stochastic behavior.

At the same time, there are regions, and precipitation characteristics, in which potentially predictable variations occur with enough regularity that an analysis of the corresponding climate drivers is warranted. Although a complete analysis of the climatic influences upon these potentially predictable variations is beyond the scope of this paper, we briefly present results for one particular region—namely, region 3, which is traditionally considered to be a core region under the influence of the North American monsoon system (Higgins et al. 1998; Comrie and Glenn 1998).

To show the utility of isolating and analyzing the potentially predictable precipitation variations, Fig. 7 plots the seasonal mean summertime reanalysis 500-hPa height anomalies composited only for those years in which there are potentially predictable seasonal mean precipitation variations over region 3. Here the gridpoint anomalies are normalized by the standard deviation of the seasonal mean anomalies across the full time series (1948–2000). To test for significance, the years with potentially predictable seasonal mean precipitation anomalies are randomly redistributed across the reanalysis period (1948–2000), and the compositing procedure is recalculated; this process is repeated 1000 times. At each grid point, the absolute value of the 1000 anomalies is used to create a CDF; the 95th percentile value from the CDF is used to provide the 95% confidence level for that grid point. (Similar significance tests will be performed for the other potentially predictable daily rainfall characteristics; however, because each daily rainfall characteristic has a different number of potentially predictable years, the 95% confidence level will be different for each map.)

From this figure, it is apparent that regionally enhanced precipitation is associated with lower-than-normal 500-hPa heights centered over the central and southwestern United States, in agreement with previous results, indicating that interannual precipitation variations in this region are related to a weakening of the upper-air monsoon ridge (Comrie and Glenn 1998; Anderson et al. 2007). In addition, there is an indication that large-scale teleconnection features are associated with the development of the anomalous 500-hPa heights over the central/southwestern United States. The teleconnection features are particularly significant over the high-latitude North Pacific and Arctic Ocean regions, with extensions in the central extratropical and subtropical North Pacific.

We have shown here that potentially predictable variations in seasonal mean precipitation can arise from various changes in the underlying daily precipitation characteristics during a given year. The convoluted nature of the teleconnection features in the precipitation-based composites may arise from the superposition of two teleconnection features, each associated with a different anomalous precipitation characteristic. Figure 8 shows the seasonal mean summertime 500-hPa height anomalies composited only for those years in which there are potentially predictable seasonal mean coverage and intensity variations over region 3. The large-scale wavelike feature spanning the Pacific basin seen in Fig. 7 is most prominent in the 500-hPa height anomalies composited on potentially predictable coverage variations. In contrast, the intensity variations appear to be related to anomalous short-wave circulations. Importantly, both the intensity variations and the coverage variations appear to be related to lower-than-normal 500-hPa heights centered over the southwestern United States, indicating that similar upper-air circulation anomalies may contribute to both.

To better understand how these anomalous circulation patterns affect the precipitation characteristics in the given region, Fig. 9 shows the full 500-hPa heights, averaged for years in which there are larger-than-normal and smaller-than-normal potentially predictable rainfall coverage anomalies over region 3. We chose to analyze the coverage variations because they represent the main contributor to the seasonal mean precipitation variations in region 3 (see Table 2). In addition, there are too few years with higher-than-normal intensities (one total) to perform a similar analysis for potentially predictable intensity anomalies. It is apparent from this figure that the anomalous circulation features seen in Fig. 8 result in a more zonal upper-air circulation pattern during years with positive potentially predictable coverage variations. In contrast, during years with negative potentially predictable coverage variations, the anomaly pattern produces a more meridional wavelike flow in the upper-air circulation patterns. The enhanced monsoon ridge over the southwestern United States and northern Mexico during years with lower-than-normal rainfall coverage most likely acts as a blocking high, diverting synoptic-scale storm systems to the north and reducing the number entering the region of interest (region 3), thereby reducing the daily rainfall coverage distributions, and overall rainfall amounts, during these years.

6. Summary

The regional variability in the summertime (July–September) monsoon precipitation over the southwestern United States is studied using 70 yr of station-based daily precipitation observations. Two traditional spatial cluster techniques—Ward’s method and the K-means method—are used to regionalize the seasonal mean precipitation. Both techniques identify four identical cluster structures, which we then analyze using regional stochastic chain-dependent models. The regional variability in the daily rainfall frequency characteristics is examined first. For two subregions (region 2 and region 4), a second-order chain-dependent model can capture all observed variance in the number of rainfall events during a given year. For the two other regions (region 1 and region 3), a fourth-order chain-dependent model can optimally explain more than 80% of the observed variance in the number of rainfall events. Further analyses with ISM indicate that the unexplained variances in the two fourth-order regions are attributable to only about 5% of years that contain anomalous daily rainfall frequency characteristics.

The regional variability in daily rainfall coverage is simulated using the empirical distribution of rainfall coverage for each region. About 20% of observed years contain potentially predictable variations in the overall coverage for a given region (the exception being the southwestern portion of the domain (region 4), in which only about 5% of observed years have anomalous coverage characteristics). Similarly, the regional variability in daily rainfall intensity of a given region is simulated using the empirical distribution of rainfall intensity. Unlike the rainfall coverage characteristics, only about 8%–10% of the years in a given region contain statistically significant variations in their intensity distributions.

Using the three regional daily rainfall submodels in combination, the regional variability of overall precipitation is analyzed. Study results indicate that only a small fraction of observed years (i.e., between 10% and 20% over individual regions) have anomalous seasonal mean precipitation amounts that lie outside of the regional chain-dependent model’s stochastic envelope of variability. These anomalous years account for about 45% of the observed interannual variance in seasonal precipitation over a given region. Further studies using conditioned submodels, which account for year-to-year variations in a given daily rainfall characteristic—for example, rainfall occurrence, rainfall coverage, and rainfall intensity—indicate that year-to-year variations in all three types of rainfall characteristics can contribute to the potentially predictable precipitation variations during a given year. However, variations in coverage seem to have the largest influence, with the variations in rainfall occurrence having the least influence.

Lastly, we briefly analyze the large-scale background climate state during years with potentially predictable rainfall variations over the southeastern portion of the domain (region 3). Here, positive seasonal mean precipitation anomalies appear to be related to a weakening of the upper-air monsoon high, in agreement with previous results. Further analysis indicates that the weakening of the upper-air high produces a more zonal flow aloft, which most likely allows more synoptic-scale storms to intrude into the region. The increase in synoptic storms in turn would produce an increase in the rainfall coverage characteristics, and overall rainfall, during these years, as observed.

Overall, this research highlights the need to treat the seasonal mean precipitation over the southwestern United States, and the North American monsoon region in general, as spatially heterogeneous. In addition, it highlights that the systematic variations in daily precipitation characteristics, which give rise to potentially predictable seasonal mean precipitation variations, can differ from region to region, both in their influence upon the seasonal mean precipitation variations as well as the frequency of their occurrence. Finally, it highlights that only a fraction (less than 50%) of the overall variance in seasonal mean precipitation is related to these systematic variations in the underlying daily precipitation characteristics, with the rest related to stochastic (and hence unpredictable) variations in daily rainfall events over the course of the season. We argue that it is this set of years—which occurs only about 10%–20% of the time—that should to be analyzed when investigating how large-scale and regional-scale climate change processes modulate the underlying daily rainfall characteristics that influence year-to-year variations in regional rainfall.

Acknowledgments

The authors wish to thank Jon Eischeid at NOAA’s Climate Diagnostics Center for producing and providing the station-based precipitation data products. Dr. Anderson’s research was supported by a Visiting Scientist appointment to the Grantham Institute for Climate Change, administered by Imperial College of Science, Technology, and Medicine. This research was also funded by Cooperative Agreement NOAA-NA040AR431002. The views expressed here are those of the authors and do not necessarily reflect the views of NOAA. Thanks are extended to the reviewers, as well as to numerous other readers, for all their insightful and constructive comments.

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Fig. 1.
Fig. 1.

Average observed summertime total precipitation (mm) over the southwest United States. The area and shading of the dots are proportional to the average seasonal mean precipitation for the years 1931–2000.

Citation: Journal of Hydrometeorology 10, 5; 10.1175/2009JHM1104.1

Fig. 2.
Fig. 2.

Values of CCC (solid line) and pseudo–T2 (dashed line) at each cluster level for seasonal mean precipitation. The left y axis gives the values for the CCC and the right y axis gives the values for the pseudo–T2 criterion.

Citation: Journal of Hydrometeorology 10, 5; 10.1175/2009JHM1104.1

Fig. 3.
Fig. 3.

Regionalization of interannual variations in seasonal mean summertime precipitation (see text for details). Numbers represent stations belonging to the same cluster. Number values are arbitrary. Contours represent underlying topography plotted at 1000-m intervals.

Citation: Journal of Hydrometeorology 10, 5; 10.1175/2009JHM1104.1

Fig. 4.
Fig. 4.

(a)–(d) Observed variations in summertime total precipitation (mm) averaged across stations in regions 1–4 (see Fig. 3) during the period 1931–2000. Variations calculated with respect to climatological values over the same period.

Citation: Journal of Hydrometeorology 10, 5; 10.1175/2009JHM1104.1

Fig. 5.
Fig. 5.

(a) Years with anomalous precipitation characteristics over region 1 (see Fig. 3). Row F indicates years with anomalous frequency of daily rainfall determined using ISM. Row C indicates years with anomalous coverage determined using K–S and t tests applied to the empirical distribution for the given year. Row I indicates years with anomalous intensities determined using K–S and t tests applied to the empirical distributions for the given year. Row T indicates years with anomalous total precipitation determined using ISM (see text for details). Up (down) arrows represent years with statistically significant positive (negative) variations in the given rainfall characteristic. No symbol indicates rainfall characteristic during the given year is statistically similar to the climatological value. (b)–(d) same as (a) except for regions 2–4.

Citation: Journal of Hydrometeorology 10, 5; 10.1175/2009JHM1104.1

Fig. 6.
Fig. 6.

(a)–(d) Normalized power spectra of seasonal mean precipitation over regions 1–4 (see Fig. 3). The thick solid line represents normalized power spectra from the observed interannual time series. The thin solid line is the mean power spectra from 142 simulations of 70-yr time series returned by the stochastic chain-dependent model for the given region. The thin dotted lines are the 95% confidence intervals of the power spectra as derived from the modeled interannual time series.

Citation: Journal of Hydrometeorology 10, 5; 10.1175/2009JHM1104.1

Fig. 7.
Fig. 7.

Normalized seasonal mean 500-hPa height anomalies composited for years with potentially predictable seasonal mean precipitation anomalies over region 3. See Fig. 6 for corresponding anomalous years. Normalized anomalies represent the seasonal mean gridpoint anomaly for the given year, divided by the gridpoint standard deviation of the seasonal mean anomalies for the period 1948–2000. Solid (dashed) lines represent positive (negative) anomalies. Shading indicates regions in which the normalized seasonal mean 500-hPa height anomalies are significant at the 95% confidence interval (see text for details). Dark (light) gray shading indicates positive (negative) anomalies.

Citation: Journal of Hydrometeorology 10, 5; 10.1175/2009JHM1104.1

Fig. 8.
Fig. 8.

Same as Fig. 7 but for normalized seasonal mean 500-hPa height anomalies composited for years with (a) potentially predictable rainfall coverage anomalies and (b) potentially predictable rainfall intensity anomalies over region 3. See Fig. 5 for corresponding anomalous years. Because the numbers of potentially predictable events differ for each, the 95% confidence threshold differs in each map.

Citation: Journal of Hydrometeorology 10, 5; 10.1175/2009JHM1104.1

Fig. 9.
Fig. 9.

(a) Seasonal mean 500-hPa heights averaged for years with positive potentially predictable precipitation event coverage anomalies over region 3. See Fig. 5 for corresponding anomalous years. Dark (light) gray shading is the same as in Fig. 8a and indicates positive (negative) normalized seasonal mean 500-hPa height anomalies are significant at the 95% confidence interval (see text for details). (b) Same as (a) but for seasonal mean 500-hPa heights averaged for years with negative potentially predictable precipitation event coverage anomalies over region 3. Shading is opposite that of (a) to indicate sign of 500-hPa anomalies during years with negative potentially predictable precipitation event coverage anomalies.

Citation: Journal of Hydrometeorology 10, 5; 10.1175/2009JHM1104.1

Table 1.

Simulated overdispersions for the regional rainfall occurrences returned by chain-dependent models over each subregions (see Fig. 3). Simulations returned by optimal regional chain-dependent models are in boldface.

Table 1.
Table 2.

Model overdispersions for the seasonal mean precipitation over the given subregion (see Fig. 3), based upon stochastic chain-dependent models with the given conditioned daily rainfall characteristic (see text for details). The conditioned chain-dependent models with the smallest overdispersion in each region are in boldface (excluding the fully conditioned models).

Table 2.
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