Regional Simulation of Summertime Precipitation over the Southwestern United States

Bruce T. Anderson Department of Geography, Boston University, Boston, Massachusetts

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John O. Roads Scripps Institution of Oceanography, La Jolla, California

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Abstract

Using results taken from a finescale (25 km), regional modeling simulation for the summer of 1999, along with contemporaneous daily surface observations, synoptic variations in summertime precipitation over the southwestern United States are described and analyzed. Two separate techniques for characterizing and evaluating large-scale summertime precipitation patterns within the observed and simulated systems are presented; in addition, these evaluation/characterization techniques are used to analyze the hydrologic forcings associated with observed and simulated modes of rainfall variability. Overall, two robust spatiotemporal precipitation patterns are identified involving 1) precipitation over the western portion of the Rocky Mountain plateau centered on eastern Arizona and southern Utah, and 2) precipitation located over the eastern portion of the plateau and the elevated orography of eastern New Mexico and southern Colorado. Time series associated with these two precipitation regimes are correlated with low-level and midlevel circulation patterns in order to investigate the related large-scale environmental conditions. It is found that for both regimes intraseasonal precipitation is related to the intrusion of midtroposphere, midlatitude low-pressure anomalies over the southwestern United States, resulting in synoptic-scale shifts in the position of the climatological midtroposphere monsoon ridge. The interaction between the resultant midtroposphere pressure fields and the quasi-stationary monsoon surface pressures found over the Rocky Mountain plateau during the summertime produce large-scale vertical velocities consistent with the observed and simulated rainfall patterns associated with each regime.

Corresponding author address: Dr. Bruce T. Anderson, Department of Geography, Boston University, 675 Commonwealth Avenue, Boston, MA 07215-1401. Email: brucea@bu.edu

Abstract

Using results taken from a finescale (25 km), regional modeling simulation for the summer of 1999, along with contemporaneous daily surface observations, synoptic variations in summertime precipitation over the southwestern United States are described and analyzed. Two separate techniques for characterizing and evaluating large-scale summertime precipitation patterns within the observed and simulated systems are presented; in addition, these evaluation/characterization techniques are used to analyze the hydrologic forcings associated with observed and simulated modes of rainfall variability. Overall, two robust spatiotemporal precipitation patterns are identified involving 1) precipitation over the western portion of the Rocky Mountain plateau centered on eastern Arizona and southern Utah, and 2) precipitation located over the eastern portion of the plateau and the elevated orography of eastern New Mexico and southern Colorado. Time series associated with these two precipitation regimes are correlated with low-level and midlevel circulation patterns in order to investigate the related large-scale environmental conditions. It is found that for both regimes intraseasonal precipitation is related to the intrusion of midtroposphere, midlatitude low-pressure anomalies over the southwestern United States, resulting in synoptic-scale shifts in the position of the climatological midtroposphere monsoon ridge. The interaction between the resultant midtroposphere pressure fields and the quasi-stationary monsoon surface pressures found over the Rocky Mountain plateau during the summertime produce large-scale vertical velocities consistent with the observed and simulated rainfall patterns associated with each regime.

Corresponding author address: Dr. Bruce T. Anderson, Department of Geography, Boston University, 675 Commonwealth Avenue, Boston, MA 07215-1401. Email: brucea@bu.edu

1. Introduction

Summertime precipitation over the southwestern United States has received much attention from the hydrologic and atmospheric community over the past years. In particular, the spatial, temporal, and spectral aspects of precipitation have been investigated on numerous scales using surface and satellite-derived observations of rainfall, outgoing longwave radiation, and lightning, as well as large-scale and finescale reanalysis and model simulation data (e.g., Stensrud et al. 1995; Schmitz and Mullen 1996; Petersen and Rutledge 1998; Comrie and Glenn 1998; Berg et al. 2000). Early work found that much of the interannual and intraseasonal variability in precipitation could be related to the position of the midtropospheric (500 mb) subtropical ridge, which connects the Bermuda and east Pacific highs (Moore et al. 1989; Carleton et al. 1990; Maddox et al. 1995; Higgins et al. 1998). On interannual timescales, a northward migration of this ridge to the four-corners region of Arizona, New Mexico, Utah, and Colorado is strongly correlated with wetter summers over the southwestern United States, whereas a southern migration to a location above northwest Mexico is correlated with somewhat drier conditions (Carleton et al. 1990; Comrie and Glenn 1998).

Intraseasonally, this base state is modified by the intrusion of midtroposphere lows that lead to monsoon “bursts” in the region (Carleton 1986). It has been suggested that these low pressure systems advect cold air over the southwestern United States, which destabilizes the atmosphere and results in more intense convective activity (Carleton 1986). In addition, Maddox et al. (1995) examined the relationship between severe summertime thunderstorms over Arizona and variability in the 500-mb pressure fields and described three distinct synoptic patterns that resulted in severe weather for the region, two of which involved the northern migration of the subtropical ridge. In both cases, however, it appears that summertime precipitation in this region is strongly modulated by large-scale midlatitude forcings. For instance, it has been shown that the statistical spectral peak for intraseasonal precipitation anomalies over Arizona, along with those over the Great Plains region, is 12–18 days (Mullen et al. 1998; Mo 2000); this period seems to be set by the advection timescale associated with the passage of midlatitude waves originating from the northern Pacific (Kiladis 1999; Mo 2000), further suggesting an atmospheric link relating precipitation anomalies over the southwestern United States to midlatitude, midtroposphere dynamics.

Interestingly however, with the exception of a few of these studies (e.g., Comrie and Glenn 1998), almost all treat the southwestern United States as a homogeneous region for precipitation; those that look at variability over smaller spatial scales still usually only examine one particular area of interest (e.g., southeastern Arizona, Mullen et al. 1998). No study as yet appears to have looked at regionalization of the precipitation over the southwestern United States on intraseasonal timescales to see if the spatial patterns of precipitation truly vary in phase. Based upon the seasonal cycle of the climatological rainfall, there appear to be upwards of nine separate precipitation regimes over the southern United States and northern Mexico; it has also been shown that significantly different 500-mb patterns are associated with interannual precipitation variability in various regimes (Comrie and Glenn 1998). We were therefore interested in whether similar heterogeneity exists on intraseasonal timescales. For example, what are the spatial patterns of precipitating events? What are their temporal variations? How do they relate to variations in the dynamic and hydrodynamic fields?

To examine these questions, we use the National Centers for Environmental Prediction's (NCEP's) Regional Spectral Model. Previous research has shown that only finescale numerical models appear capable of reproducing the horizontal and temporal structures of the summertime dynamic and hydrodynamic fields in this region. In particular, global analyses from NCEP's predecessor, the National Meteorological Center (NMC), and the European Centre for Medium-Range Weather Forecasts (ECMWF), show persistent low-level northwesterly winds over the Gulf of California (Stensrud et al. 1995; Schmitz and Mullen 1996), in disagreement with observations that indicate prevalent southerly winds over much of the Gulf and northwestern Mexico (Badan-Dangon et al. 1991; Douglas 1995). In addition, NCEP's Nested-Grid Model and Eta Model (both with 80-km resolution) have difficulty reproducing synoptic variations in precipitation and low-level winds over Arizona (Dunn and Horel 1994a,b). However, the Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model, run at 25-km resolution, does significantly better at reproducing the wind fields, as well as low-level moisture, convection, and total rainfall (Stensrud et al. 1995). Additional simulations using NCEP's regional spectral modeling system at 10 km × 20-km resolution also capture the synoptic and diurnal variability in low-level winds over the Gulf of California and northwest Mexico (Anderson et al. 2000a,b).

In this paper, NCEP's Regional Spectral Model is now used to investigate the intraseasonal variability of precipitation over the southwestern United States. Section 2 describes the simulated and observed datasets. Section 3 outlines two separate techniques for characterizing and evaluating large-scale summertime precipitation patterns within the observed and simulated systems; the spatiotemporal regionalization of intraseasonal precipitation over the southwestern United States is then described. Section 4 extends these evaluation techniques by analyzing the large-scale dynamic fields that correspond to the observed/simulated precipitation regimes. Section 5 summarizes the results.

2. Datasets

a. Regional model description

The simulation of the summertime monsoon is performed using NCEP's regional spectral modeling system (Juang and Kanamitsu 1994; Juang et al. 1997). The NCEP global to regional modeling system used in this study contains two components—a low-resolution global spectral model (GSM) and a regional spectral model with a single nest (RSM). The nesting method is a one-way, noninteractive procedure that is designed to calculate regional responses (or adjustments) of the RSM to the large-scale background fields provided by the coarser-resolution GSM. This nesting procedure is performed through the entire domain, not only at the lateral boundary zones and is therefore referred to as a perturbation method (see Juang and Kanamitsu 1994; Juang et al. 1997 for details). Details of the full modeling system are found in appendix A.

The nesting allows the model system to capture the finescale features in the RSM, mainly due to enhanced orography of the study region, while still incorporating the development of large-scale synoptic features in the GSM. The resolution of the GSM is triangular-62 (approximately 200 km); the resolution for the RSM is 25 km × 25 km (Fig. 1). Vertically, the prognostic equations are solved on 18 levels represented in sigma coordinates (where σ = p/ps). The RSM domain extends from northern California to central Mexico and includes the Sierra Nevada and Rocky Mountain ranges in the United States, the Baja and Sierra Madre mountain ranges in Mexico, along with the Gulf of California. Also shown in Fig. 1 are selected radiosonde observing sites and binned precipitation locations (see next section) used for evaluation of the simulated dynamic and hydrodynamic fields. For this research, a “continuous” 92-day simulation is performed (see appendix A), starting at 0000 UTC 1 July 1999. Prognostic and diagnostic output is archived every 6 model hours. The 6-h integrating intervals for all diagnostic terms (including precipitation) are from 0000–0600, 0600–1200, 1200–1800, and 1800–0000 UTC; for this study, daily integrated values will be estimated from these four-times daily values.

b. Surface observations

To evaluate the regional model simulations, daily surface observations from all stations in Arizona, New Mexico, Utah, and Colorado are taken from the National Climatic Data Center's (NCDC's) daily surface data archive for 1 July–30 September 1999. This station data is then averaged into 25 km × 25 km cells centered on the RSM grid points; most grid points do not have a surface station associated with them. The majority of those that do have only one station located within their domain; the maximum number of stations per grid cell is three. This averaging ignores subgrid-scale effects associated with changes in elevation and surface characteristics (such as rain shadows); however, at the spatial scale of 25 km these effects are expected to be minimal given the spatial scale of the features of interest (see later). In addition, it ignores precipitation over western Mexico in the core monsoon region. Therefore we decided to focus on evaluating and analyzing the summertime precipitation only over the southwestern United States where high-resolution precipitation data is more readily available; an additional paper on the core monsoon region will be developed later. For this study, missing data are treated as rain-free days; overall, missing data only constitute 1% of the total number of binned observations, hence this approximation does not affect the results presented here.

Using the location of the surface stations, it is also possible to produce a truncated version of the model simulation data by only analyzing those grid points that correspond to a station location. This parsed simulation dataset will be referred to as the gridded simulations.

3. Characterization and evaluation of the simulated precipitation

Previously, the NCEP Regional Spectral Model has been used to identify, describe, and diagnose the dynamics of low-level summertime winds over the Gulf of California and the southwestern United States (Anderson et al. 2000a, 2001). Here, this research is extended to investigate the intraseasonal variability of precipitation in the same region. Figure 2 presents the 3-month mean of daily precipitation as seen in the full RSM simulation, gridded simulation, and surface observations; also shown are the seasonal mean differences between the simulations and observations. In general, the two products show good agreement in magnitude and geographic structure. Within the United States both data products indicate a precipitation maximum in the elevated regions over northwestern New Mexico extending into eastern Colorado; the absolute maximum in the RSM product is found along the western side of the Sierra Madre Occidental. The simulation indicates slightly too much precipitation over western Colorado and fails to capture the local maximums found along the Arizona–Mexico border. However, it does capture the local maximums situated along the Mogollon rim in central Arizona, albeit with slightly smaller magnitudes.

To examine temporal agreement between the RSM and observations, equitable threat scores (Colle et al. 1999) are computed using 24-h accumulations from collocated observation and simulation grid points. These scores represent a ratio between the number of simultaneous rainfall events (above a given threshold) at collocated grid points in both datasets to the total number of events involving precipitation (at the same grid point) in either of the datasets. Numerically,
i1520-0442-15-23-3321-eq1
where ET(x) is the equitable threat score at grid point x, H is the number of simultaneous precipitation events in both the observations and simulation, O is the number of precipitation events in the observations, S is the number of precipitation events in the simulations, and R is the simultaneous occurrence based on random chance: R = SO/N (where N is the number of verified observations).

Figure 3a shows the domain-averaged equitable threat scores for a range of thresholds from 1 to 20 mm day−1. The average scores drop dramatically for larger precipitating events but remain above 0.1 through a threshold of 6 mm day−1. Figures 3b,c,d show the gridpoint equitable threat scores for small (>2 mm day−1), medium (>8 mm day−1), and large (>16 mm day−1) precipitation thresholds. For small precipitation events, the model shows good skill, particularly over most of New Mexico and along the Mogollon rim, but also over the elevated portions of Colorado and in Utah. For the slightly higher threshold, threat scores drop but still show some skill over New Mexico and also eastern Colorado. For larger events, threat scores drop still further, but again show some skill over eastern Colorado and the Mogollon rim. In general, these equitable threat scores are of the same size as the typical Eta and medium-range forecast/aviation (MRF/AVN) forecast values (between 0.15 and 0.25, see Staudenmaier 1996; Chen et al. 1999; Hamill 1999), suggesting the model, although not perfect, is performing reasonably.

The other typical metric for precipitation is bias scores, defined as B = S/O. These scores relate the ratio of simulated events to observed events; scores of unity are considered good while scores less than 1 indicate a model deficit and scores above 1 indicate a model surplus. Figure 4a shows the domain-averaged bias scores for various thresholds, again ranging from 1 to 20 mm day−1. It appears that the model underestimates smaller precipitation events while overestimating the larger events (>15 mm day−1). Gridpoint bias scores for small (>2 mm day−1), medium (>8 mm day−1), and large (>16 mm day−1) precipitation thresholds are shown in Figs. 4b,c,d. As suggested by the seasonal mean difference maps, for the smallest precipitation threshold the model shows a positive bias over eastern New Mexico, which probably contributes to the high equitable threat scores in this same region (Hamill 1999); negative biases are found over southern Arizona as well as western Colorado. Bias scores appear reasonable in eastern Colorado. For medium precipitation, bias scores actually improve over southern Arizona. Eastern New Mexico still shows positive bias scores, as does eastern Colorado and Utah; bias characteristics are similar for larger precipitation events as well.

Equitable threat and bias scores are typically presented for precipitation because of the discontinuous and sporadic nature of the field, particularly in a semiarid region such as the desert Southwest. Unfortunately, this same temporal discontinuity results in fields that are non-Gaussian and hence greatly affects traditional statistical measures such as covariance and correlation. Previous authors have found that the non-Gaussian nature of precipitation can be removed to some extent by raising the absolute fields (as opposed to the anomalies) by some fractional power (Comrie and Glenn 1998). For all three data products, the precipitation fields can be transformed and “normalized” in this manner by simply raising them to the one-fifth power (0.2). Figure 5 shows the probability distribution plot for precipitation from the full RSM and observed data sources compared with that for precipitation to the 0.2 power. As can be seen, the latter fields show a much more linear profile, indicating a more Gaussian, or normal, distribution.

Using the transformed datasets, it is possible to produce maps of temporal correlations between the transformed simulation and observation precipitation anomalies at collocated grid points (Fig. 6). (For these figures, and throughout the rest of the paper, we present minimum correlation values of ±0.4; technically, given a normal distribution of 90 independent events, correlations of approximately 0.26 and 0.20 are statistically significant at the 99% and 95% level, respectively. It is recognized that precipitation events have a temporal decorrelation length that reduces the effective degrees of freedom. However, correlation values of ±0.4 are above the 95% significance level even for sample sizes of approximately 25, suggesting that the values shown are indeed significant.) In agreement with the equitable threat scores, these figures indicate statistical correlation between the simulated and observed fields over the eastern portion of New Mexico and to a lesser extent over the Mogollon rim and central Utah. Interestingly, these correlations are not extremely large, suggesting that the model may be doing a poor job of reproducing the observed surface fields. However, simple discrete statistics do not account for spatial and temporal structures in the fields themselves, which may be well represented by the model even if the individual gridpoint values are not. For instance, it has been shown that gridpoint estimation of precipitation based upon large-scale atmospheric circulation patterns results in low skill; however, there is a much stronger relationship between these same atmospheric circulation patterns and the geographical distribution of rainfall (Mamassis and Koutsoyiannis 1996).

To evaluate the large-scale temporal and spatial precipitation patterns within the simulations and observations, as well as to detect and isolate the major modes of variability for this region, we compute the empirical orthogonal functions (EOFs) for each transformed dataset. To further distinguish the spatial and temporal patterns within each dataset, a varimax rotation of the spatial EOFs for each is performed (Richman 1986). Varimax rotation entails using an iterative procedure to rotate a subset of the weighted spatial (or principal) components in eigenspace such that overlapping spatial patterns are incorporated into the same rotated component, thereby regionalizing the spatial structures. It is important to note here that no information from any of the three fields is used to reproduce the rotated spatiotemporal components in the other two fields; that is, this analysis isolates only those modes of variability intrinsically found in the associated dataset.

For this analysis we use the first three principal components as input to the varimax algorithm; these components capture 40%, 50%, and 30% of the variance of the full simulated, gridded simulated, and observed precipitation, respectively. Only three principal components are used because inclusion of more than that results in rotated patterns that have significantly correlated temporal loadings (with correlation coefficients greater than 0.4), indicating that the rotated patterns may not in fact represent independent modes. In addition, the inclusion of four or more components produces pairwise plots of the rotated components that violate the concept of simple structure as discussed in Richman (1986). Additional analysis using promax rotation techniques did not isolate any additional simple structures beyond those discussed here; in addition, it produced spatial patterns whose temporal loadings were more significantly correlated than those derived from the varimax rotation. Hence, it appears that the varimax algorithm is the most appropriate one to use in this case and that a maximum of three simple structures can be isolated. More discussion of the spatial and temporal independence of the various patterns follows.

To derive the temporal loadings associated with each rotated component, the original transformed anomaly field is spatially correlated with the corresponding rotated spatial pattern at each point in time. The resulting time series are normalized such that their inner-products are unity (this makes for easy comparison with other time series as well as for easy correlation with other anomaly fields). The rotated spatial patterns, which actually represent weighted components, are then divided by the gridpoint standard deviations of the corresponding transformed field to arrive at correlation coefficient values. See appendix B for details. Spatial correlation coefficients are plotted because the weighted components derived from the transformed fields do not have any physical units. In addition, these weighted components cannot be converted back to real precipitation values because of the nonlinearity involved in transforming using the 0.2 power. However, the spatial maps of correlation values for the transformed datasets are very similar to those derived using the same procedure with the original precipitation anomalies (not shown). The difference in the two methods lies in the temporal loadings associated with each map; for the untransformed data, the corresponding time series show unrealistic negative spikes that the EOF algorithm introduces in order to balance large (and non-Gaussian) precipitating events.

It should be noted that for all subsequent figures, temporal patterns are normalized in the manner described above (such that their inner-product is unity) and all spatial patterns are divided by the gridpoint standard deviations of the associated transformed field, such that correlation maps are produced. In these cases, both spatial and temporal patterns are referred to as components or factors although neither actually represents a weighted field per se. In addition, only correlation coefficients greater than ±0.4 are shown; as before, given a normal distribution of approximately 90 independent events, correlations of approximately 0.26 and 0.20 are technically statistically significant at the 99% and 95% level, respectively. To test this assumption for the analysis presented here, for each transformed dataset (i.e., the full simulation, binned simulation, and observations) the order of the time series of precipitation at each grid point is randomized independent of all other grid points. This procedure maintains the amount of variance of precipitation at each grid point but makes the evolution of the gridpoint precipitation temporally independent from that at the other gridpoints (other than from what arises by chance). The EOF and rotating procedure is then repeated for 100 resamplings. None of the resulting spatial fields show correlations above 0.15 for any of the datasets, suggesting as before that the values presented in the figures are statistically significant.

Figure 7 shows the correlation maps for the first rotated principal component (PC) of precipitation from the gridded simulation and observation fields along with the second principal component from the full simulated field. Also shown are the time series for these three patterns. (The fact that the principal component number is different is a reflection of the fact that the two gridded datasets simply reverse the ordering of the first two PCs compared with the full simulated dataset. We have grouped them here according to spatial and temporal characteristics and not principal component number.) The spatial patterns all show strong similarities. This pattern, termed the western plateau precipitation regime, is characterized by coherent precipitation anomalies over the Sierra Madre Occidental extending into the western and northern portions of Arizona. Both the RSM and gridded simulations reproduce the general spatial structure of the surface observations, although the precipitation in the observed field is slightly less well correlated, especially over the southern portion of Utah extending into western Colorado. As can be seen, there is a strong correlation between all three time series (r = 0.66, 0.65 for the full-RSM/observed and gridded-RSM/observed time series, respectively). The time series indicate two large events at the beginning of July with numerous intermittent events extending from August through to the end of the simulation.

As mentioned earlier, no information from any of the three fields is used to reproduce the other two patterns; that is, this spatiotemporal mode of variability is intrinsically found in all three datasets. It has been shown that EOF and rotated EOF patterns can have preferred structures depending upon domain size, shape, and resolution (Richman 1986), which might suggest that the similarity in the three spatial patterns seen here is a statistical artifact. However, the fact that the three independently derived time series also have strong correlation indicates that the model and observations are both producing similar temporal evolution of this precipitation regime (see later).

The spatial and temporal patterns for the second rotated PCs from the gridded simulation and surface-observed fields, along with the first rotated PC from the full RSM precipitation field, are presented in Fig. 8. The spatial patterns, termed the eastern plateau precipitation regime, are characterized by a precipitation maximum centered over southern and eastern New Mexico extending into southern Colorado. As before, the correlated precipitation in the regional model is slightly stronger than that seen in surface observations, particularly over eastern New Mexico. The temporal correlation between the two simulated fields and the observed fields are not as good as in the previous case, although they are still significant (r = 0.58 and 0.61 for the full-RSM/observed and gridded-RSM/observed time series, respectively). The time series indicate that these events have a timescale slightly longer than the synoptic (approximately 5–6 days) with four major events occurring over the 3-month time span.

Finally, Fig. 9 shows the spatial and temporal patterns for the third rotated PC for all fields. As before, the spatial patterns all have a strong similarity between one another with slightly higher correlations in the RSM- and gridded-simulation fields. This pattern is characterized by a large precipitation maximum situated over eastern Colorado. The time series indicate that these events are shorter lived and more frequent than the two described previously. Some of these events resulted in extreme precipitation in this region with observed rainfall of 20 cm at Colorado Springs between 31 July and 4 August. The correlations between the three time series are not as large as for the first two patterns, although both are statistically significant (r = 0.43 and 0.58 for the full-RSM/observed and gridded-RSM/observed time series, respectively). In particular the moderate correlation between the gridded-RSM and observed patterns suggests that the model is capturing this mode of observed variability as well.

If the pairwise plots of the spatial PCs (e.g., Richman 1986) are examined for the three datasets, the western plateau pattern taken from all three contains a unique simple structure compared with either the eastern plateau or elevated plains precipitation patterns (not shown). However, in all three datsets, there is only a weak pairwise separation between the eastern plateau and elevated plains precipitation regimes, suggesting that indeed this latter pattern may be a statistical artifact produced by the EOF and rotating algorithms. This apparent redundancy will be discussed later.

In addition, it should be noted that additional preliminary studies using complex PC analysis for the simulated and observed data indicate that the two main modes of variability identified here (the western and eastern plateau regimes) do in fact have weak propagating signals over other regions of the domain. However, they still remain independent modes of variability, both in the time and frequency domain. Hence, the analysis presented here is not merely isolating two related patterns that are offset in time, but it is in fact isolating stationary modes of variability. Possible reasons for this stationarity will be discussed later when the large-scale forcings are examined in more detail.

It is interesting to note that these three rotated patterns, derived from intraseasonal variability in observations and model simulations, have a qualitative resemblance to the three monsoon subregions identified by Comrie and Glenn (1998) using observed interannual precipitation anomalies for the southwestern United States and northwestern Mexico (see Comrie and Glenn 1998, their Fig. 9). In that paper, these three regions were termed the monsoon west (here, the western plateau precipitation region), the monsoon east (here, the eastern plateau precipitation region) and the monsoon north (here, the elevated plains precipitation). The spatial correspondence between these patterns strongly suggests a link between synoptic-scale precipitation events and anomalous summertime rainfall in the various regions. However, it is still unclear whether this link is established via the number of synoptic events that occur during a given season or via the amount of rainfall per event. Multiyear records of daily rainfall anomalies for the southwestern United States are presently being used to investigate this issue.

In general, the temporal correlations of the individual rotated patterns shown earlier indicate that the model has skill in simulating spatial and temporal characteristics of precipitation for this region. Another way to examine the statistical robustness of the patterns presented here is to do a simple canonical correlation analysis (CCA) between the transformed observed field and the transformed, full simulated field (results are qualitatively similar if the gridded simulated field is used instead; here, we present the full simulated field to extend coverage over the entire domain).

Details for this procedure are found in appendix B. Briefly, this analysis entails minimizing the variance between a subset of the observed EOF time series (i.e., the temporal principal components) and a subset of the full simulated EOF time series to produce a set of canonical factor time series sorted by descending covariance explained (as opposed to variance explained as in the case of EOFs). In order to examine only the largest modes of variability, the subset of EOFs used for the CCA is limited to the first five for both the simulated and observed precipitation fields. The next step in performing the CCA is to calculate the cross-covariance matrix between the predictor, in this case the time series of simulated EOFs, and the predictand, that is, the time series of observed EOFs (the same procedure can be performed with the designations reversed and there is no loss of generality). The autocorrelation matrix for each of the fields is also required; because the time series of the EOFs are used, this autocorrelation matrix is simply a diagonal matrix that normalizes the values in the cross-covariance matrix. A single-value decomposition is then performed on a modified version of the covariance matrix, which produces a diagonal matrix of eigenvalues (equal to the squared correlation of the reconstructed time series) and a transformation matrix. The transformation matrix is used to reconstruct the time series of the canonical factors (CFs) from the original temporal principal components. The corresponding spatial patterns are derived by equating the reconstructed CF-based anomaly field with the anomaly field represented by the original EOFs and then solving for the spatial loadings. As with the rotated EOF analysis, the resultant time series are normalized such that their inner-product is unity. The spatial patterns are divided by the collocated standard deviations in order to arrive at spatial maps of correlation coefficients. By construction, the canonical factor time series for a given field are orthogonal to one another, although the spatial patterns are not necessarily so.

Whereas the previous figures, derived from the rotated EOF analysis, isolated spatial patterns and compared the resulting time series to evaluate various regimes, this procedure will isolate time series that show maximum correlation between the simulation and the observed fields; the resulting spatial patterns can then be compared to once again evaluate the two products. The difference between the two methodologies lies in the quantity the respective algorithms are attempting to maximize. For the rotated EOF fields, the algorithm is selecting those spatial features that explain maximum variance within a given field (hence it is an exploratory exercise). For the CCA fields, the algorithm is selecting those temporal features in the model that are best correlated with temporal features in the observations; the spatial patterns that are produced represent those precipitation patterns that the model is best able to simulate (hence it is an evaluative exercise). By only doing a rotated EOF analysis, it is only possible to capture the intrinsic modes of variability within the various datasets. Although the time series for these rotated EOF modes can (and are) compared, this evaluation does not give a quantitative sense of the performance of the model (especially since the time series are well, but not highly, correlated). By doing an independent analysis using CCA, it is possible to confirm the qualitative conclusions drawn from the rotated EOF analysis, namely that the model simulates the large-scale patterns of variability in the observed precipitation fields. In contrast, if only the CCA computations are performed it indicates that the model is performing well in capturing certain modes of intraseasonal variability; however, there would be no method of determining whether those modes are of significant importance to the summertime precipitation over the southwestern United States (other than that they are contained within the first five unrotated EOF modes for each of the transformed datasets). The use of both algorithms in combination allows us to fully characterize (via the rotated EOF fields) and evaluate (via the CCA fields) the simulated and observed precipitation.

Figure 10 shows the correlation maps of the first canonical factor of simulated and observed precipitation as derived via canonical correlation, along with the corresponding time series. As suggested earlier, the time series for the two fields are well correlated (r = 0.78) although this result is to be expected. They are also similar to the western plateau precipitation time series identified via EOF rotation (r = 0.77 and 0.89 for the simulated and observed PC/CF time series correlations, respectively). More interestingly, there is strong similarity in the two CF spatial patterns, with collocated precipitation correlation patterns over northern Arizona, eastern Utah, and western Colorado. As with the time series, this pattern qualitatively resembles the previously identified western plateau precipitation regime isolated using the independent rotated EOF fields. It should be noted, however, that the spatial patterns derived here and those derived via EOF rotation are not identical, particularly over the southeastern corner of Arizona. Hence, these results suggest that the model best simulates precipitation from the Mogollon rim into Utah (as seen in the CF patterns), although there is significant self-correlated precipitation in southeastern Arizona associated with the western plateau precipitation regime within each dataset (as isolated via the rotated EOF algorithms) that is not captured by the model. Importantly though, the fact that there is qualitative agreement between the spatial (and temporal) features isolated via the CCA algorithm and those isolated via the rotated EOF algorithm indicates that the model is indeed successively simulating one of the predominant modes of variability in the observed system.

Figure 11 shows the second CF maps of the simulated and observed precipitation as derived via canonical correlation, along with the corresponding time series. Again, the time series are well correlated (r = 0.75); they indicate strong precipitation events in the middle of July and at the beginning of August, with smaller events seen at the beginning of July and at the beginning and middle of September, similar to that seen in the rotated EOF eastern plateau time series (r = 0.64 and 0.80 for the simulated and observed PC/CF time series correlations, respectively). Importantly, there is also strong similarity between the two CF spatial patterns. These are characterized by maximum precipitation correlations over eastern New Mexico and eastern Colorado, similar to that seen for the eastern plateau regime. As before, the spatial (and temporal) patterns do not exactly match those isolated via EOF rotation, although the results from the CCA algorithm again suggest that this mode of variability is effectively captured by the modeling system.

In addition, these results, combined with those from the first CF, also strongly suggest that the spatial patterns represented by the independently derived rotated EOFs do in fact depict distinct precipitation regimes. Unlike the rotated EOF algorithm, there is no orthogonality condition for the spatial CF patterns, yet the CCA algorithm delineates and identifies these two patterns as temporally independent modes of variability. Similarly, there is no orthogonality condition for the rotated EOF time series, yet the correlation between the first and second temporal factors within any given dataset is weak and most likely insignificant (r = 0.19 and 0.25 for the simulated and observed PC(1)/PC(2) time series, respectively), again suggesting that these two regimes are indeed independent.

It should be noted that, as mentioned in the discussion of the rotated EOF patterns, it is unclear whether the eastern plateau regime and the elevated plains regime actually represent unique simple structures. From the CCA results, it appears that the eastern plateau CF discussed earlier is actually an amalgamation of the eastern plateau and elevated plains precipitation patterns isolated via the rotated EOF analysis; this finding strengthens the suggestion that, indeed, the two patterns should not necessarily be thought of as independent modes, at least given this 3-month dataset. Longer datasets from additional summers may effectively separate the modes discussed here.

4. Discussion

In the preceding section, we presented two separate techniques for isolating and evaluating large-scale summertime precipitation patterns within the observed and simulated systems. Together, the spatial and temporal EOF and CCA patterns indicate that there is a strong regionalization of intraseasonal precipitation over the desert Southwest with distinct spatial and temporal patterns. To show how these evaluation techniques can be extended to help analyze the observed system, we begin to investigate the large-scale dynamic fields that correspond to these precipitation regimes and to suggest a physical interpretation of the results presented so far. Further analysis of the simulated hydrologic cycle associated with these patterns is also feasible and is presented in another paper (Anderson 2002). To perform this analysis of the combined observed/simulated system, the first two CF time series of simulated precipitation, as derived via CCA with the observed fields, are correlated with low-level and midlevel dynamic fields taken from the regional model simulation. The CF time series are used because they contain information from both the observed and simulated precipitation fields, allowing for more robust analysis of the associated dynamic fields; results are qualitatively similar if the independent rotated EOF time series from any of the three datasets are used.

To perform this analysis, positive (precipitating) and negative (nonprecipitating) composite fields are first created. This procedure entails taking only the positive (or negative) values of the normalized time series for each canonical factor and using them to weight the gridpoint RSM geopotential height and wind fields for each day. At each gridpoint, the weighted values are then summed over time and divided by the sum of the time series values themselves (again only using the positive or negative values). Hence, one arrives at a “composite map” in which the gridpoint values represent a composite weighted by the magnitude of the events as determined by the time series of the canonical factor. An anomaly field can then be computed by taking the difference between the two maps (this anomaly field is equivalent to twice the weighted covariance map derived by taking the correlation with the full time series). Figure 12 shows the 500-mb anomaly winds and geopotential heights taken from the RSM, along with 500-mb wind vector anomalies taken from daily radiosonde observations, for the western plateau precipitation regime. Also shown in Fig. 12 are the full composite fields for precipitating (positive) and nonprecipitating (negative) periods. From the anomaly field, it can be seen that precipitation over the western portion of the plateau is associated with an anomalous upper-level trough over the midlatitude eastern Pacific and an anomalous ridge over the central United States. Examining the full field patterns for the two composites, it becomes apparent that the anomalous midtroposphere low represents a southward and eastward displacement of the monsoon ridge, in agreement with previous findings (e.g., Carleton 1986).

A similar figure can be produced for the eastern plateau precipitation regime (Fig. 13). Interestingly, the anomaly and composite fields for the upper-air geopotential heights and winds are very similar to those seen for precipitation over the western portion of the plateau. There is anomalous troughing over the eastern midlatitude Pacific. However, in the case of the eastern plateau precipitation, there is a high pressure anomaly over the south-central United States (outside the domain), resulting in anomalous southerly flow over the region of interest. Examining the full field composite for precipitating events again suggests that the anomalous low over the Pacific produces a southward and eastward migration of the midtropospheric ridge, resulting in southwesterly mean flow over the Rocky Mountain plateau. Absent the high pressure anomaly over the central United States (as seen in the western plateau precipitation composite), it is feasible that the southwesterly flow can protrude further inland, resulting in an eastward shift in the precipitation patterns (note, e.g., the directional change in anomalous wind vectors at Amarillo during the two separate regimes). More quantitative distinction between the two patterns is discussed later.

Although these results are in contrast to previous findings linking severe thunderstorm activity in this region to the northward and westward extension of the subtropical ridge (Maddox et al. 1995), it should be noted that here we are referring to large-scale, synoptically driven precipitating events; more localized convective precipitation may have very different dynamic characteristics from that presented here. Instead, these results more closely fit those of Carleton (1986) who suggested that monsoon bursts in this region are related to the intrusion of the midtroposphere low and, via the associated cyclonic flow, to the advection of cold air over the southwestern United States, which destabilizes the atmosphere and results in more intense convective activity.

Our results, however, suggest a different dynamic mechanism. Again, looking at the full field composite during precipitating events (Figs. 12, 13b), it appears that the southward and eastward displacement of the monsoon ridge results in southwesterly flow over the precipitating regions. From quasigeostrophic theory, this southwesterly flow, positioned over the semipermanent monsoon surface low centered on the Rocky Mountain plateau, will produce large-scale vertical motion in this region via the differential advection of relative vorticity associated with the pressure centers. During nonprecipitating events (Figs. 12, 13c), the monsoon ridge expands north and west, resulting in westerly and northwesterly flow over the precipitating regions; again from quasigeostrophic theory, this flow, positioned over the monsoon surface low, will produce no, or weak negative, large-scale vertical velocities.

To further test this hypothesis, the large-scale 500-mb omega values are taken from the archived daily averaged NCEP–National Center for Atmospheric Research (NCAR) reanalysis (see acknowledgements) and correlated with the first and second canonical factor time series (Fig. 14). The omega values, which are equal to the vertical motion in pressure coordinates at a given level, are a diagnostic proxy for vertical velocities in which negative values indicate upward motion. As can be seen from Fig. 14, for both regimes there are associated anomalous large-scale vertical velocities centered over the precipitating regions, in agreement with the presumed motions associated with the large-scale 500-mb geopotential height fields. The 500–1000-mb thickness anomalies associated with the two canonical factor time series (not shown) indicate that, for the western plateau regime, the vertical velocity anomalies over Arizona and New Mexico are related to changes in 500-mb heights associated with the intrusion of the midlatitude trough aloft. In contrast, the eastern plateau precipitation regime shows negative thickness anomalies over both the southwestern and central United States. The thickness anomalies over the western portion of the United States are due again to the intrusion of the midlatitude trough at the 500-mb level; however, over the central United States, the thickness anomalies are more attributable to a ridge in the 1000-mb heights. The extension of these negative thickness anomalies results in an eastward extension of positive anomalous vertical velocities and hence an intensification of the vertical motions over the eastward plateau and a shift in the precipitation patterns.

These results are in agreement with previous studies utilizing the regional model's diagnostic moisture tendency equations (Anderson 2002). In general, over both the western and eastern portion of the Rocky Mountain plateau, precipitation appears to be predicated upon anomalous midtroposphere large-scale moisture convergence associated with anomalous vertical advection of moisture by the transient midtroposphere, midlatitude synoptic patterns described here. This anomalous large-scale vertical convergence of moisture augments the seasonal mean convective moisture convergence, producing rainfall in the two regions.

It should be noted here that the variability seen in this CCA-based analysis is related to midlatitude synoptic forcing and is “nonmonsoonal” in as much as it is not related to dynamics in the core monsoon region centered on western Mexico. However, it is important to note that although it is nonmonsoonal, these large-scale patterns capture 20%–50% of summertime observed rainfall variability in the given regions. Hence, the analysis is capturing, and explaining, an important component of intraseasonal summertime rainfall variability over the southwestern United States, although it may not be explicitly related to variability in the monsoon itself.

5. Summary

Using three months of simulated and observed daily precipitation data for the summer of 1999, it was possible to evaluate and characterize spatiotemporal patterns of precipitation over the southwestern United States. In general, the mean values, equitable threat maps, and covariance plots indicated some model skill in simulating the observed gridpoint precipitation for this region.

In order to more fully evaluate the large-scale temporal and spatial precipitation patterns within the simulations and observations, as well as to detect and isolate the major modes of variability for this region, two eigenvalue-based approaches were also employed. Spatial and temporal patterns derived through these methods—namely rotated principal component and canonical correlation analysis—indicated that the model does well at representing the larger-scale precipitation fields associated with major modes of observed variability. In addition, these two analysis techniques were able to identify and describe two robust spatiotemporal precipitation regimes for this region. The first precipitation regime is associated with rainfall anomalies over the Sierra Madre Occidental and eastern Arizona, with additional anomalies extending well into Utah and western Colorado. The second precipitation regime is characterized by rainfall anomalies over the eastern portion of New Mexico and southern Colorado. Interestingly, these two regimes have also been identified as regions of significant seasonal variability using observed interannual precipitation anomalies for the southwestern United States and northwestern Mexico (Comrie and Glenn 1998), suggesting a possible geographic link between synoptic-scale precipitation events and regional summertime rainfall anomalies.

The results from these analysis algorithms were also used to examine the large-scale environmental conditions associated with each regime. It appears that both patterns are related to the synoptic modification of the midlevel monsoon ridge by midlatitude low pressure anomalies. The interaction of the resulting midtropospheric pressure patterns with the quasi-stationary monsoon surface patterns results in large-scale vertical motions that can lead to rainfall during the two regimes. For the western plateau precipitation, the intrusion of the midtroposphere low results in significant large-scale vertical velocity anomalies predominantly over Arizona and Utah. For the eastern plateau regime, the large-scale anomalous vertical velocities over the precipitating region appear to result both from the presence of the 500-mb trough as well as an anomalous 1000-mb ridge centered over the central United States. In combination, these two shift the vertical velocities farther to the east, resulting in an eastward shift in the location of the rainfall.

It is recognized that these patterns represent only those regimes that were prevalent during the 1999 summertime season and that additional regimes may be more pronounced during other years. Preliminary analysis of 20 years of observed data for this region suggests that, indeed, there are additional modes of variability that are not captured in this 1-yr subset; however, the two that are presented here are robust from year to year and do represent a predominant component of intraseasonal variability in this region. Characterization and analysis of additional regimes will be part of future research incorporating multiyear observations as well as simulations.

For this study, however, we chose to focus on evaluating and characterizing the regional model simulations because we were additionally interested in investigating the RSM-generated diagnostic terms; as mentioned, these diagnostic terms, particularly those related to the water vapor tendency equation, are analyzed in a separate paper (Anderson 2002). Overall, these model-produced fields provide value-added knowledge of the dynamic and hydrodynamic processes related to simulated precipitation events (and by inference, observed events) that would be difficult to analyze from observations alone. In addition, we chose to use the RSM data because preliminary analysis of the GSM and reanalysis output indicates that the RSM, through its finer-scale resolution and orography, can better represent the spatial and temporal distribution of these events. Unfortunately, because of the high computational cost of performing these high-resolution simulations, it is extremely difficult to perform long-term integrations of the model system. Therefore, in order to study interannual variability in seasonal rainfall anomalies, it may be preferable to use daily surface observations of precipitation instead. For instance, using observational data alone it is possible to quantitatively investigate how the characteristics (i.e., frequency and intensity) of intraseasonal precipitation events (as discussed here) can influence anomalous summertime rainfall in various regions of the southwestern United States, a study that would be much more time-consuming to perform using just regional simulation data.

Acknowledgments

This research was funded by cooperative agreements NOAA—NA77RJ0453, NA16GP1622, and NA17R1231. The views expressed herein are those of the authors and do not necessarily reflect the views of NOAA. The authors thank Jack Ritchie for running and archiving the RSM data from daily Experimental Climate Prediction Center simulations and forecasts for the southwestern United States. Thanks are also extended to the reviewers as well as numerous other readers for all their insightful and constructive comments. NCEP-NCAR reanalysis data provided by the NOAA–CIRES Climate Diagnostics Center, Boulder, CO, from their Web site at: http://www.cdc.noaa.gov.

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APPENDIX A

Regional Spectral Modeling System

The NCEP global to regional modeling system used in this study contains two components—a low-resolution global spectral model (GSM) and a regional spectral model with a single nest (RSM). The nesting method is a one-way, noninteractive procedure that is designed to calculate regional responses (or adjustments) of the RSM to the large-scale background fields provided by the coarser-resolution GSM. This nesting procedure is performed through the entire domain, not only at the lateral boundary zones and is therefore referred to as a perturbation method (see Juang and Kanamitsu 1994; Juang et al. 1997 for details).

The GSM and RSM use the same primitive hydrostatic sigma-coordinate equations expressed in slightly different forms (Juang and Kanamitsu 1994; hereafter JK). The GSM equations consist of the vorticity, divergence, virtual temperature, and conservation equation for water vapor on 18 sigma-layer coordinates and a mass continuity equation for surface pressure (Kanamitsu 1989). The regional model differs slightly in that it utilizes the momentum equations instead of the vorticity/divergence equations (JK). The model physics were described in Kalnay et al. (1996). Listed below are some of the important parameterizations:

Deep convection

Deep convection is modeled using a simplified Arakawa–Schubert parameterization (Kalnay et al. 1996). This scheme assumes a quasi-equilibrium available buoyant energy; changes in buoyant energy associated with convective activity are provided by large-scale changes associated with the environmental stability of the atmosphere. The instability is removed by relaxing the temperature and moisture profiles towards equilibrium values using a prescribed time interval (Grell 1993). In the simplified version, cloud size is effectively prescribed as opposed to being determined from a parameterization-dependent spectrum. In addition Hong and Pan (1996) detail modifications to this scheme. In particular, in the subcloud layers, the level of maximum moist static energy represents the level of origination for updraft air; however, if the depth between this level and the level of free convection is greater than a certain threshold, convection is suppressed. Convection can also be produced in regions with large convective available potential energy (Hong and Pan 1996). Importantly, the scheme accounts for downdrafts associated with convection; in previous modeling attempts over the southwest United States; it was shown that the parameterization scheme needs to consider evaporation of precipitation below the cloud base in order to create outflow and increased convection in the southwestern United States (Dunn and Horel 1994b).

Large-scale precipitation

A vertical iteration starting from the top sigma level checks for supersaturation at each level. Supersaturated layers are set to a near-saturated state using an approximate wet-bulb process; all excess liquid water goes into the form of precipitation. As the precipitation moves downward through the column, it is either augmented (by passage through additional supersaturated layers) or reduced (by passage through unsaturated layers). If it is reduced due to passage through unsaturated layers, evaporation in that layer occurs where the rate of evaporation is dependent upon the drop-size distribution (NOAA/NMC Development Division 1988).

Boundary layer processes

Boundary layer processes (Hong and Pan 1996) are based upon a nonlocal boundary layer vertical diffusion in which the surface layer and boundary layer are coupled using a prescribed profile with similarity-based scale parameters. Included in this scheme are also countergradient diffusion processes, which are shown to be important for mixing low-level moisture upward more efficiently (Hong and Pan 1996). Importantly, the scheme strongly couples the boundary layer physics and the convective processes described earlier, improving the precipitation forecasts for heavy rain events (Hong and Pan 1996). Boundary layer height is computed iteratively by first computing the boundary height without accounting for virtual temperature instability near the surface; from this estimate, calculations of vertical velocities can be obtained, which then allow estimation of the virtual temperature instability and subsequently modified PBL heights (Hong and Pan 1996). The diffusion parameters for momentum and mass can then be determined from the boundary layer height using the prescribed profile shape. Above the boundary layer, vertical eddy transfer uses a more traditional Richardson number–dependent diffusion process.

Soil model processes

The soil model in the RSM is based upon the Oregon State University scheme as described in Chen et al. (1996). The scheme includes an explicit vegetation canopy, soil hydrology, and soil thermodynamics. For the soil scheme, it uses a two-level (0.1 and 1.9 m thick) prognostic soil model for both moisture and temperature, with parameterized diffusion between the two levels and a spatial distribution of soil type and soil properties as described in Betts et al. (1997). Also included in the prognostic scheme are equations for moisture contained in vegetation and snow (Chen et al. 1996). Evaluation and analysis of the soil model performance has been done both within the NCEP Eta (Chen et al. 1996; Betts et al. 1997) and RSM (Roads and Chen 2000) modeling systems.

For the horizontal spectral representation, the GSM uses a triangular-62 (T62) truncation of the spherical harmonics as the basis functions. The regional model expresses the prognostic model variables in terms of the specified large-scale fields and their perturbations on sigma surfaces. The regional model perturbations use trigonometric functions as their basis functions; in particular cosine–cosine basis functions are used for all perturbations except the zonal momentum perturbation, which uses sine–cosine functions, and the meridional momentum perturbation, which uses cosine–sine functions (see JK for details). All nonlinear dynamic and physical terms are computed in gridpoint space for both. The RSM perturbation tendencies are forced to zero at the regional lateral-boundaries by a relaxation towards the external boundary. Semi-implicit time integration is subsequently performed to suppress the higher-frequency waves and allow the use of a longer time step.

The nesting method itself is a one-way, noninteractive procedure designed to calculate regional perturbations (or adjustments) of the RSM to the large-scale background fields provided by the coarser-resolution GSM. For this research, a 92-day simulation is performed, started at 0000 UTC 1 July 1999. The nesting method begins by using daily 0000 UTC NCEP–NCAR global reanalysis data as initial conditions for a 24-h GSM integration; model output is archived every 6 h. The linearly interpolated GSM field taken between two consecutive output times is used as the base field for the RSM forecast. Interpolation of the GSM data from the global to regional grid is done directly using a spectral-to-gridpoint conversion of the global fields. As pointed out by Juang et al. (1997), it is important to include the GSM integration in order to avoid large-scale perturbation growth that results when using just the reanalysis base fields. Since all surface initial conditions for the RSM except for orography are provided by the coarser GSM, the sea surface temperature in the Gulf of California, which is absent from the global data, is assigned a constant value equal to the temperature at the mouth of the Gulf (∼29°C). This assignment is based upon results of previous modeling studies indicating the importance of proper SST fields in the Gulf (Dunn and Horel 1994b); sensitivity studies indicate that finer-scale SST resolution does not quantitatively affect the dynamic fields in the region (Anderson 1998).

In order to produce a “continuous” 92-day simulation of the RSM (as opposed to a set of ninety-two 24-h weather forecasts), the coarse-scale GSM and finescale RSM short-term forecast fields at 2400 UTC (and not the reinitialized 0000 UTC reanalysis field) are used as the initial conditions for the next RSM nesting period. Numerically, this technique allows the RSM forecast, which inherently contains corrections for anomalies associated with enhanced orography and resolution, to carry these corrections forward from one nesting period to the next, thereby reducing spinup effects. Regional surface prognostic fields, however, are not reinitialized and represent a continuous forecast simulation.

APPENDIX B

Varimax Rotation and Canonical Correlation Analysis

The varimax rotation performed here is based on the algorithm as coded for MATLAB proprietary software in Reyment and Jöreskog (1993). An excellent resource for varimax rotation is Richman (1986); this work also provides a comparison between various rotation techniques and their use in the atmospheric sciences.

In general, varimax rotation entails using an iterative procedure to isolate regional structures from a subset of orthogonal spatial principal components derived from the single-value decomposition of a spatial–temporal anomaly field. For this study, we select the first three spatial principal components, weighted by their eigenvalues in order to maintain variance associated with each spatial pattern. This set of PC spatial patterns for a given precipitation transformed dataset is designated 𝗣(x, m), in which the rows represent the position in geographic space and the columns represent the position in EOF space. The varimax algorithm then produces a rotated set of spatial patterns, 𝗣r(x, n), in which the rows represent the position in geographic space and the columns represent the position in the rotated EOF space. In order to calculate the temporal loadings, the spatial pattern is normalized such that its inner-product is unity for each component. Then the normalized spatial pattern, 𝗽r(x, n), is spatially correlated with the original transformed anomaly field, 𝗙(x, t), to compute the temporal component, 𝗧r(t, n), such that 𝗧r(t, n) = 𝗙(x, t)′ · 𝗽r(x, n). The temporal field is then normalized such that its inner-product is unity for each component, producing 𝘁r(t, n). The gridpoint values from the weighted rotated spatial pattern, 𝗣r(x, n), are divided by the collocated temporal standard deviations as calculated from the original transformed anomaly field to produce spatial maps of correlation coefficients for each rotated component, 𝗖r(x, n), where
i1520-0442-15-23-3321-eqb1

It should be noted that rotation of the principal components as done here preserves orthogonality of the components in space but does not necessarily preserve orthogonality of the associated time series.

The canonical correlation analysis (CCA) performed here is similar in nature to a simple EOF analysis, although the field on which the single-value decomposition is performed differs in that it contains a cross-correlation matrix relating a predictand field to a predictor field. An excellent resource for CCA is Bretherton et al. (1992); this work also provides a comparison between CCA and other multivariate analysis techniques. Other descriptions, in various forms, can be found in Barnett and Preisendorfer (1987), Graham et al. (1987), and Cherry (1996).

For this study, we select the first five weighted temporal EOFs (or principal components) of simulated and observed precipitation anomalies. Given this set of PCs for the observed precipitation 𝗢(t, m) and the simulated precipitation 𝗦(t, m), in which the rows represent the position in time and the columns represent the position in EOF space, a cross-correlation matrix 〈𝗢𝗦〉 can be calculated as the inner-product 𝗦′(t, m) · 𝗢(t, m) and autocorrelation matrices 〈𝗢𝗢〉 and 〈𝗦𝗦〉 calculated as 𝗢′(t, m) · 𝗢(t, m) and 𝗦′(t, m) · 𝗦(t, m), respectively where brackets 〈 〉 represent the expectation operator. The matrix equation that needs to be solved is then
i1520-0442-15-23-3321-eqb2
where λ represents the diagonal matrix of eigenvalues (equal to the squared correlation of the reconstructed time series) and 𝗔(m, n) represents a transformation matrix that is used to reconstruct the time series of the canonical factors from the time series of the PCs. The elements of 𝗔(m, n), whose rows represent the position in EOF space and whose columns represent position in CF space, give the weighting of each of the original predictand PCs that comprise the CF time series. Hence, to calculate the CF time series of the predictand, in this case the observed precipitation temporal patterns 𝗢CF(t, n), the inner-product between the original weighted time series and the transformation matrix is computed, that is, 𝗢CF(t, n) = 𝗢(t, m) · 𝗔(m, n). To calculate the corresponding spatial patterns, it is assumed that the reconstructed anomaly field should be equal to the anomaly field represented by the original EOFs, that is, 𝗫CF(x, n) · 𝗢CF(t, n)′ = 𝗫(x, m) · 𝗢(t, m)′ where 𝗫CF(x, n) and 𝗫(x, m) represent the spatial CF and PC loading patterns for the observed precipitation anomalies respectively. Hence,
i1520-0442-15-23-3321-eqb3
To reconstruct the predictor fields, that is, the simulated precipitation fields, a similar procedure is used but the matrix equation that needs to be solved is now
i1520-0442-15-23-3321-eqb4

Note that the eigenvalues, λ, are the same for each equation.

As with the rotated EOF analysis, the resultant time series are normalized such that their inner-product is unity. The spatial patterns are also divided by the collocated standard deviations in order to arrive at spatial maps of correlation coefficients.

Fig. 1.
Fig. 1.

RSM [ ·  (25-km resolution), as seen over Pacific Ocean] and GSM [+ (T62)] grid points and RSM orography contours. Orography contour interval is 250 m; shading interval is 750 m with max shaded elev of 3000 m. Location of important geographic features also indicated. Selected upper-air observing stations (○) are also shown: Tucson (TUS); Albequerque (ALB); Amarillo (AMA); Denver (DEN); Salt Lake City (SLC); Miramer AFB, San Diego (NKX). Grid points for the binned observed precipitation sites are represented by filled circles (•). Lat–long lines are shown every 5°

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 2.
Fig. 2.

(a) Mean daily summertime (JAS) precipitation from the RSM. Shading interval is 2 mm day−1; min shading is 2 mm day−1. Also shown are the RSM orography contours. The contour interval is 500 m; the min contour is 250 m. (b) Mean daily precipitation from the binned observation sites. Hatching interval is 2 mm day−1; all values shown. Area of dots is proportional to rainfall amount. (c) Same as (b) except for the mean daily precipitation from the RSM at only the binned observation sites. (d) Same as (b) except for the mean difference between binned RSM and observed precipitation. Dark shading represents positive differences; light shading represents negative differences. Area of dots is proportional to gridpoint difference

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 3.
Fig. 3.

(a) Domain-averaged equitable threat scores (ETS) between the binned RSM and observed precipitation for various precipitation thresholds. (b) ETS between the binned RSM and observed precipitation for a threshold of 2 mm day−1. Shading interval is 0.1; min shading is 0.001. Area of dots is proportional to the ETS. Also shown are the RSM orography contours. The contour interval is 500 m; the min contour is 250 m. (c) Same as (b) except for a threshold of 8 mm day−1. (d) Same as (b) except for a threshold of 16 mm day−1

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 4.
Fig. 4.

(a) Domain-averaged bias scores between the binned RSM and observed precipitation for various precipitation thresholds. (b) Bias scores between the binned RSM and observed precipitation for a threshold of 2 mm day−1. Shading interval corresponds to interval indicated. Area of dots is the same for all grid points. Also shown are the RSM orography contours. The contour interval is 500 m; the min contour is 250 m. (c) Same as (b) except for a threshold of 8 mm day−1. (d) Same as (b) except for a threshold of 16 mm day−1

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 5.
Fig. 5.

(a) Probability distribution function of daily RSM summertime precipitation at all grid points and all days. Here, x axis is linear; y axis is based on the normal probability scale. (b) Probability distribution function of daily RSM summertime precipitation raised to the 0.2 power for all grid points and all days. (c) Same as (a) except for daily binned observed precipitation. (d) Same as (b) except for daily binned observed precipitation to the 0.2 power

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 6.
Fig. 6.

Gridpoint correlations for the daily binned RSM and observed precipitation, both raised to the 0.2 power. Shading interval is 0.2; min shading is 0.4. Area of dots is proportional to correlation

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 7.
Fig. 7.

Geographic plots of the gridpoint correlations for the rotated principal components representing the western plateau regime. (a) Second rotated principal component of transformed RSM precipitation. Shading interval is 0.2; min shading is 0.4. Negative values are not shown. (b) First rotated principal component of transformed binned observed precipitation. Shading interval is 0.2; min shading is 0.4. Positive values are shaded; negative values are black. Area of dots is proportional to correlation. (c) Same as (b) except for the first rotated principal component of transformed RSM precipitation at only the binned observation sites. (d) Normalized time series of the first principal component of observed precipitation (solid line), first principal component of gridded RSM precipitation (dashed line), and second principal component of full RSM precipitation (dash–dot line)

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 8.
Fig. 8.

Geographic plots of the gridpoint correlations for the rotated principal components representing the eastern plateau regime. (a) First rotated principal component of transformed RSM precipitation. Shading interval is 0.2; min shading is 0.4. Negative values are not shown. (b) Second rotated principal component of transformed binned observed precipitation. Shading interval is 0.2; min shading is 0.4. Positive values are shaded; negative values are black. Area of dots is proportional to correlation. (c) Same as (b) except for the second rotated principal component of transformed RSM precipitation at only the binned observation sites. (d) Normalized time series of the second principal component of observed precipitation (solid line), second principal component of gridded RSM precipitation (dashed line), and first principal component of full RSM precipitation (dash–dot line)

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 9.
Fig. 9.

Geographic plots of the gridpoint correlations for the rotated principal components representing the elevated plains regime. (a) Third rotated principal component of transformed RMS precipitation. Shading interval is 0.2; min shading is 0.4. Negative values are not shown. (b) Third rotated principal component of transformed binned observed precipitation. Shading interval is 0.2; min shading is 0.4. Positive values are shaded; negative values are black. Area of dots is proportional to correlation. (c) Same as (b) except for the third rotated principal component of transformed RSM precipitation at only the binned observation sites. (d) Normalized time series of the third principal component of observed precipitation (solid line), third principal component of gridded RSM precipitation (dashed line), and third principal component of full RSM precipitation (dash–dot line)

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 10.
Fig. 10.

(a) Normalized time series of the first canonical factor of transformed observed precipitation (solid line) and full RSM precipitation (dashed line), representing the western plateau regime. (b) Geographic plot of the gridpoint correlations for the first canonical factor of transformed binned observed precipitation. Shading interval is 0.2; min shading is 0.4. Positive values are shaded; negative values are black. Area of dots is proportional to correlation. (c) Geographic plot of the gridpoint correlations for the first canonical factor of transformed RSM precipitation. Shading interval is 0.2; min shading is 0.4. Negative values are not shown

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 11.
Fig. 11.

(a) Normalized time series of the second canonical factor of transformed observed precipitation (solid line) and full RSM precipitation (dashed line), representing the eastern plateau regime. (b) Geographic plot of the gridpoint correlations for the second canonical factor of transformed binned observed precipitation. Shading interval is 0.2; min shading is 0.4. Positive values are shaded; negative values are black. Area of dots is proportional to correlation. (c) Geographic plot of the gridpoint correlations for the second canonical factor of transformed RSM precipitation. Shading interval is 0.2; min shading is 0.4. Negative values are not shown

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 12.
Fig. 12.

(a) Simulated 500-mb geopotential height and wind vector anomalies, along with observed 500-mb wind vector anomalies from selected upper-air observing stations, for western plateau precipitation calculated from the difference between the composites for precipitating and nonprecipitating periods associated with the first canonical factor (CF) time series. Also shown are the gridpoint correlations of transformed simulated precipitation for the first CF. Contour interval for the geopotential height anomalies in 10 m. Unit wind vector anomalies shown in the right-hand corner are 20 m s−1. Shading interval for precipitation correlation map is 0.2; min shading is 0.4. (b) Composite plot of simulated 500-mb geopotential heights and wind vectors, along with observed 500-mb wind vectors, for precipitating periods associated with the first CF time series. Also shown are the gridpoint correlations of transformed simulated precipitation. Contour intervals for the geopotential heights are every 25 m. Unit wind vectors shown in the right-hand corner are 50 m s−1. Shading interval for precipitation correlation map is 0.2; min shading is 0.4. (c) Same as (b) except for nonprecipitating periods associated with the first CF time series

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 13.
Fig. 13.

(a) Simulated 500-mb geopotential height and wind vector anomalies, along with observed 500-mb wind vector anomalies from selected upper-air observing stations, for eastern plateau precipitation calculated from the difference between the composites for precipitating and nonprecipitating periods associated with the second CF time series. Also shown are the gridpoint correlations of transformed simulated precipitation for the second CF. Contour interval for the geopotential height anomalies is 10 m. Unit wind vector anomalies shown in the right-hand corner are 20 m s−1. Shading interval for precipitation correlation map is 0.2; min shading is 0.4. (b) Composite plot of simulated 500-mb geopotential heights and wind vectors, along with observed 500-mb wind vectors, for precipitating periods associated with the second CF time series. Also shown are the gridpoint correlations of transformed simulated precipitation. Contour intervals for the geopotential heights are every 25 m. Unit wind vectors shown in the right-hand corner are 50 m s−1. Shading interval for precipitation correlation map is 0.2; min shading is 0.4. (c) Same as (b) except for nonprecipitating periods associated with the second CF time series

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

Fig. 14.
Fig. 14.

(a) The 500-mb omega anomalies correlated with the first CF time series. Data are taken from the archived 2.5° daily averaged NCEP–NCAR reanalysis. The omega values are equal to the vertical motion in pressure coordinates at a given level such that negative values indicate upward motion. Positive values are shaded; negative values are contoured. Shading interval is 0.01 Pa s−1. (b) Same as (a) except for second CF time series

Citation: Journal of Climate 15, 23; 10.1175/1520-0442(2002)015<3321:RSOSPO>2.0.CO;2

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  • Fig. 1.

    RSM [ ·  (25-km resolution), as seen over Pacific Ocean] and GSM [+ (T62)] grid points and RSM orography contours. Orography contour interval is 250 m; shading interval is 750 m with max shaded elev of 3000 m. Location of important geographic features also indicated. Selected upper-air observing stations (○) are also shown: Tucson (TUS); Albequerque (ALB); Amarillo (AMA); Denver (DEN); Salt Lake City (SLC); Miramer AFB, San Diego (NKX). Grid points for the binned observed precipitation sites are represented by filled circles (•). Lat–long lines are shown every 5°

  • Fig. 2.

    (a) Mean daily summertime (JAS) precipitation from the RSM. Shading interval is 2 mm day−1; min shading is 2 mm day−1. Also shown are the RSM orography contours. The contour interval is 500 m; the min contour is 250 m. (b) Mean daily precipitation from the binned observation sites. Hatching interval is 2 mm day−1; all values shown. Area of dots is proportional to rainfall amount. (c) Same as (b) except for the mean daily precipitation from the RSM at only the binned observation sites. (d) Same as (b) except for the mean difference between binned RSM and observed precipitation. Dark shading represents positive differences; light shading represents negative differences. Area of dots is proportional to gridpoint difference

  • Fig. 3.

    (a) Domain-averaged equitable threat scores (ETS) between the binned RSM and observed precipitation for various precipitation thresholds. (b) ETS between the binned RSM and observed precipitation for a threshold of 2 mm day−1. Shading interval is 0.1; min shading is 0.001. Area of dots is proportional to the ETS. Also shown are the RSM orography contours. The contour interval is 500 m; the min contour is 250 m. (c) Same as (b) except for a threshold of 8 mm day−1. (d) Same as (b) except for a threshold of 16 mm day−1

  • Fig. 4.

    (a) Domain-averaged bias scores between the binned RSM and observed precipitation for various precipitation thresholds. (b) Bias scores between the binned RSM and observed precipitation for a threshold of 2 mm day−1. Shading interval corresponds to interval indicated. Area of dots is the same for all grid points. Also shown are the RSM orography contours. The contour interval is 500 m; the min contour is 250 m. (c) Same as (b) except for a threshold of 8 mm day−1. (d) Same as (b) except for a threshold of 16 mm day−1

  • Fig. 5.

    (a) Probability distribution function of daily RSM summertime precipitation at all grid points and all days. Here, x axis is linear; y axis is based on the normal probability scale. (b) Probability distribution function of daily RSM summertime precipitation raised to the 0.2 power for all grid points and all days. (c) Same as (a) except for daily binned observed precipitation. (d) Same as (b) except for daily binned observed precipitation to the 0.2 power

  • Fig. 6.

    Gridpoint correlations for the daily binned RSM and observed precipitation, both raised to the 0.2 power. Shading interval is 0.2; min shading is 0.4. Area of dots is proportional to correlation

  • Fig. 7.

    Geographic plots of the gridpoint correlations for the rotated principal components representing the western plateau regime. (a) Second rotated principal component of transformed RSM precipitation. Shading interval is 0.2; min shading is 0.4. Negative values are not shown. (b) First rotated principal component of transformed binned observed precipitation. Shading interval is 0.2; min shading is 0.4. Positive values are shaded; negative values are black. Area of dots is proportional to correlation. (c) Same as (b) except for the first rotated principal component of transformed RSM precipitation at only the binned observation sites. (d) Normalized time series of the first principal component of observed precipitation (solid line), first principal component of gridded RSM precipitation (dashed line), and second principal component of full RSM precipitation (dash–dot line)

  • Fig. 8.

    Geographic plots of the gridpoint correlations for the rotated principal components representing the eastern plateau regime. (a) First rotated principal component of transformed RSM precipitation. Shading interval is 0.2; min shading is 0.4. Negative values are not shown. (b) Second rotated principal component of transformed binned observed precipitation. Shading interval is 0.2; min shading is 0.4. Positive values are shaded; negative values are black. Area of dots is proportional to correlation. (c) Same as (b) except for the second rotated principal component of transformed RSM precipitation at only the binned observation sites. (d) Normalized time series of the second principal component of observed precipitation (solid line), second principal component of gridded RSM precipitation (dashed line), and first principal component of full RSM precipitation (dash–dot line)

  • Fig. 9.

    Geographic plots of the gridpoint correlations for the rotated principal components representing the elevated plains regime. (a) Third rotated principal component of transformed RMS precipitation. Shading interval is 0.2; min shading is 0.4. Negative values are not shown. (b) Third rotated principal component of transformed binned observed precipitation. Shading interval is 0.2; min shading is 0.4. Positive values are shaded; negative values are black. Area of dots is proportional to correlation. (c) Same as (b) except for the third rotated principal component of transformed RSM precipitation at only the binned observation sites. (d) Normalized time series of the third principal component of observed precipitation (solid line), third principal component of gridded RSM precipitation (dashed line), and third principal component of full RSM precipitation (dash–dot line)

  • Fig. 10.

    (a) Normalized time series of the first canonical factor of transformed observed precipitation (solid line) and full RSM precipitation (dashed line), representing the western plateau regime. (b) Geographic plot of the gridpoint correlations for the first canonical factor of transformed binned observed precipitation. Shading interval is 0.2; min shading is 0.4. Positive values are shaded; negative values are black. Area of dots is proportional to correlation. (c) Geographic plot of the gridpoint correlations for the first canonical factor of transformed RSM precipitation. Shading interval is 0.2; min shading is 0.4. Negative values are not shown

  • Fig. 11.

    (a) Normalized time series of the second canonical factor of transformed observed precipitation (solid line) and full RSM precipitation (dashed line), representing the eastern plateau regime. (b) Geographic plot of the gridpoint correlations for the second canonical factor of transformed binned observed precipitation. Shading interval is 0.2; min shading is 0.4. Positive values are shaded; negative values are black. Area of dots is proportional to correlation. (c) Geographic plot of the gridpoint correlations for the second canonical factor of transformed RSM precipitation. Shading interval is 0.2; min shading is 0.4. Negative values are not shown

  • Fig. 12.

    (a) Simulated 500-mb geopotential height and wind vector anomalies, along with observed 500-mb wind vector anomalies from selected upper-air observing stations, for western plateau precipitation calculated from the difference between the composites for precipitating and nonprecipitating periods associated with the first canonical factor (CF) time series. Also shown are the gridpoint correlations of transformed simulated precipitation for the first CF. Contour interval for the geopotential height anomalies in 10 m. Unit wind vector anomalies shown in the right-hand corner are 20 m s−1. Shading interval for precipitation correlation map is 0.2; min shading is 0.4. (b) Composite plot of simulated 500-mb geopotential heights and wind vectors, along with observed 500-mb wind vectors, for precipitating periods associated with the first CF time series. Also shown are the gridpoint correlations of transformed simulated precipitation. Contour intervals for the geopotential heights are every 25 m. Unit wind vectors shown in the right-hand corner are 50 m s−1. Shading interval for precipitation correlation map is 0.2; min shading is 0.4. (c) Same as (b) except for nonprecipitating periods associated with the first CF time series

  • Fig. 13.

    (a) Simulated 500-mb geopotential height and wind vector anomalies, along with observed 500-mb wind vector anomalies from selected upper-air observing stations, for eastern plateau precipitation calculated from the difference between the composites for precipitating and nonprecipitating periods associated with the second CF time series. Also shown are the gridpoint correlations of transformed simulated precipitation for the second CF. Contour interval for the geopotential height anomalies is 10 m. Unit wind vector anomalies shown in the right-hand corner are 20 m s−1. Shading interval for precipitation correlation map is 0.2; min shading is 0.4. (b) Composite plot of simulated 500-mb geopotential heights and wind vectors, along with observed 500-mb wind vectors, for precipitating periods associated with the second CF time series. Also shown are the gridpoint correlations of transformed simulated precipitation. Contour intervals for the geopotential heights are every 25 m. Unit wind vectors shown in the right-hand corner are 50 m s−1. Shading interval for precipitation correlation map is 0.2; min shading is 0.4. (c) Same as (b) except for nonprecipitating periods associated with the second CF time series

  • Fig. 14.

    (a) The 500-mb omega anomalies correlated with the first CF time series. Data are taken from the archived 2.5° daily averaged NCEP–NCAR reanalysis. The omega values are equal to the vertical motion in pressure coordinates at a given level such that negative values indicate upward motion. Positive values are shaded; negative values are contoured. Shading interval is 0.01 Pa s−1. (b) Same as (a) except for second CF time series