Stochastic modeling of daily summertime rainfall over the

southwestern U.S. Part II: intraseasonal variability

Jingyun Wang, Bruce T. Anderson, and Guido D. Salvucci,

Department of Geography, Boston University

675 Commonwealth Ave.

Boston MA, 02215-1401

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Abstract:

The intraseasonal variability of summertime precipitation over the southwestern U.S.

is examined using stochastic daily occurrence models combined with empirical daily

rainfall distributions to document: 1) the seasonal evolution of the frequency and

intensity of rainfall events across the summertime monsoon season; and 2) the

climatological evolution of wet spells, dry spells and storm events. Study results indicate

that the evolution of the North American Monsoon System (NAMS) is most apparent in

the occurrence of daily rainfall events, which exhibits clear time-dependence across the

summer season over the southwestern U.S. and can be principally portrayed by stochastic

models. In contrast, the seasonal evolution of NAMS is largely absent in the averaged

daily rainfall amount time-series. There is also a significant seasonal evolution in the

length of dry spells. In the central area of the domain (approximately 50 out of 78 stations)

dry spell lengths tend to increase over the course of the summer season, while on the

western fringe (8 out of 78 stations) dry spell lengths tend to decrease. In contrast, wet

spells tend to exhibit constant lengths over the course of the season (48 out of 78 stations).

The seasonal trend for storms indicates that the number of storms and duration of storms

tend to decrease in September, however the storm depths tend to be more intense. Overall

90% of the area-averaged variance for dry spell lengths can be explained by the random

daily evolution of the stochastic model alone. For wet spell lengths, the area-average

variance explained by the stochastic models is 98% and for storm amounts it is 92%.

These results suggest that the characteristics of most intraseasonal events over this region

(i.e. spell lengths and storm amounts) can be captured by the random evolution of daily

rainfall models, even with constant year-to-year statistical parameters, indicating that

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systematic variations in the background climatic conditions from one year to the next

contribute little to the characteristics of these events.

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Introduction

The summertime precipitation over the southwestern U.S. represents the northward

extension of the North American Monsoon System (NAMS), and contributes a large

fraction of the annual total precipitation for the region (e.g. Douglas et al. 1993). It has a

critical influence on local ecological systems. For instance, anomalous events such as

extremely dry/wet summer seasons or the onset of severe storms can adversely affect

ecologic activities over this semi-arid region (Haro and Green, 1996). In order to improve

the predictability for these extreme events, it is necessary to first describe the

characteristics of the daily rainfall series over the southwestern U.S.

The summertime precipitation over this region exhibits complex temporal and spatial

structures. In addition, previous studies indicate the summertime daily precipitation over

the southwestern U.S. is influenced by numerous multi-scalar processes, including: large-

scale atmospheric circulation patterns (Adams and Comrie 1997; Comrie and Glenn 1998;

Ellis and Hawkins 2001; Hawkins et al. 2002); mesoscale convective systems (Carvalho

and Jones, 2001; Nieto-ferreira et al., 2003); mesoscale to synoptic-scale squall lines

(Cohen et al, 1995); the Madden-Julian Oscillation (MJO) (e.g. Higgins et al. 2004);

synoptic-scale meridional transient eddies (e.g. Garreaud 2000); and Gulf of California

surges (Stensrud et al. 1997; Fuller and Stensrud 2000).

Generally, the studies of these phenomena have been process-based and have focused

on particular climate and/or meteorological phenomena that impact the intraseasonal

evolution of rainfall in this region. Other studies that characterize the daily NAMS

precipitation over the southwestern U.S. have been limited to either the use of

observations over small regions (such as over Arizona - e.g. Mullen et al. 1998; Maddox

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et al. 1995) or the use of reanalysis data for large-scale meteorological analyses (such as

NCEP reanalysis data - e.g. Mo 2000; Higgins et al. 2004). For instance, in Arizona it has

been shown that wet spells have statistical means of 2 to 4 days (Mullen et al., 1998) and

spectral peaks in a band of 12 to 18 days (Mullen et al., 1998; Mo, 200). However this

analysis was confined to a localized region in Arizona; different spectral characteristics

have been found over Mexico for instance (Reyes et al., 1994).

Overall, few detailed studies of the statistical characteristics of the daily rainfall

system have been reported for the region as a whole. Hence, this paper aims to better

describe the NAMS precipitation patterns in the southwestern U.S., and in particular the

intraseasonal characteristics of the summertime daily rainfall. To do so, stochastic

precipitation models will be used to analyze the seasonal evolution of monsoon daily

precipitation, wet spells (consecutive days experiencing precipitation), dry spells

(consecutive days without rainfall), and storm amounts (total precipitation over wet

spells). In turn, these studies of intraseasonal variability (e.g. at time scales shorter than

10 days) may provide a new perspective on ways to identify and diagnose the influence

of climatic components, either individually or jointly, upon the summertime precipitation

over this region.

This paper is an accompaniment to a previous study on the interannual variability of

the summertime daily rainfall over the southwestern U.S. (Wang et al. 2005). It is

organized as follows: section 2 describes the observed dataset and stochastic models, and

investigates the characteristics of the model’s parameters; section 3 analyzes the

simulation results of intraseasonal variations including dry spells, wet spells and storms,

and compares them with observations; and section 4 discusses these results.

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2. Data and models

2.1 Data

The data used here are the serially complete daily maximum and minimum

temperatures and precipitation compiled by Eischeid et al. (2000). The latest version of

this dataset comprises daily precipitation observations from at least January 1948 to

August 2003 for 14,317 sites in the U.S., although most stations include observations

prior to 1948. A subsample of summertime daily rainfall at 78 stations from 115W to

102W and 30N to 42N (southwestern U.S. in Figure 1) is extracted. All time series in the

subsample have a sample size longer than 70 years with full observations from July 1st to

September 30th. Years with omitted observations during the summer season were

removed from the dataset.

2.2 Chain dependent models

Chain-dependent stochastic weather models - which are commonly used for studies of

crop development (e.g. Sharpley and Williams, 1990), ecological systems (e.g. Kittel et al

1995), hydrologic systems (e.g. Pickering et al., 1988), and others (e.g., Wilks, 1992,

1999) - treat the occurrence and amount of daily rainfall events separately. The term

“chain-dependence” reflects the statistical structure of the occurrence sequence. For

instance, for a first-order chain dependent process (also termed a Markov Chain process),

the “chain-dependence” means that the state at time t only depends on the state at

time 1!t , and is independent of states at other times. If the state depends on more than

one previous state, the time sequence is said to follow a higher order chain dependent

process and the number of related previous states is termed the chain order.

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2.2.1 Occurrence sub-model

Although the first-order two-state Markov Chain model is the most commonly used for

studying daily precipitation, it is often inadequate for capturing the high frequency

variations in the daily rainfall series and underestimates both the variance and extremes

of the intraseasonal rainfall series (Gates and Tong, 1976; Buishand, 1978; Coe and Stern,

1982; Katz and Parlange; 1998; Madden et al., 1999; Wang et al., 2004).

A natural improvement is to employ higher-order chain dependent models. In studies

of interannual rainfall variability for this region, the second-order chain-dependent model

was found to optimally represent the temporal structure of daily precipitation (Wang et

al., 2005). Hence, we also use the second-order model to investigate the intraseasonal

characteristics of the daily rainfall, and refer to it simply as the “chain dependent model”.

Another alterative to the chain dependent model is to adopt a negative binomial

distribution to represent the joint probability of spells (consecutive dry/wet days - Wilby,

et al., 1998; Wilks, 1999):

{ } ( ) 12

1 1)(,Pr!!+

! !===xkxk

k ppkKxX ; ......3,2,1, =kx (1)

Here X is the length of dry spells, K the length of wet spells, and p the rainfall

probability with 00pp = for dry spells and

11pp = for wet spells. Unlike the chain

dependent model, this model explicitly gives the length of the next dry/wet spell. The

special case 1=K gives the geometric distribution { } ( ) 1

1Pr!

!==x

ppxX , which is

equivalent to the first-order chain dependent model.

As such, two occurrence sub-models – the Chain Dependent model and the Negative

Binomial model – are evaluated in this paper. All model parameters are estimated from

the observed dataset using Maximum Likelihood Estimators (MLE). To begin, the

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temporal homogeneity of these parameters is examined using a 2! test (Anderson and

Goodman, 1957) with degree of freedom )1()1( 21!!=

! Tssdf r ), in which s is the

model’s states, r the model’s order, T the tested sample size, ijkp the transition

probabilities at each test timescale, and 0

ijkp the expectation of transition probabilities.

Test results indicate that no parameters are constant from year-to-year or from day-to-

day. However, no clear pattern is observed in the annual series. In the daily series, two

parameters –1,0

P for the negative binomial model and 1,0,0

p for the chain dependent model

– display clear time-dependent evolution across the summer season (Figure 2). Numerical

tests indicate that at most stations a fitted third-order polynomial curve can significantly

reduce these parameters’ residuals compared to the mean value line (i.e. by more than

15%). In contrast, the fitted curve reduces the residuals for the other parameters by less

than 15%.

As such, to study the intraseasonal characteristics of the daily rainfall system,

seasonally varying occurrence sub-models are adopted. The parameters with a clear

seasonal evolution pattern - 1,0

P for the negative binomial model and 1,0,0

p for the chain

dependent model - are fitted with a third-order polynomial curve, and all other parameters

are set equal to the average values for the full summer season. In addition, since no trend

pattern is observed in the interannual time series, all parameters are assumed to be

constant on a year-to-year basis.

2.3.2 Intensity sub-model

Both theoretical and empirical distributions have been used to simulate the daily

rainfall amounts in stochastic weather models (Swift and Schreuder, 1981, Wilks, 1998,

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1999, Keatinge, 1999). Theoretically, the empirical distribution can reproduce the

observed statistics with only a very small discrepancy due to the binning of observations.

But the empirical distribution may produce a much noisier and more heterogeneous

probability density function (PDF) curve compared to the smooth, function-based,

theoretical distributions, which may introduce errors into the simulation. However,

numerical tests indicate that simulations with the empirical distributions are insensitive to

bin widths from 0.25 to 1.0 mm for this region, and that they can reproduce the daily

variance of observed rainfall better than theoretical distributions, such as the Gamma and

Weibull distributions (Wang et al., 2005). This result suggests the empirical distributions

in this region are smooth enough for our study purpose. Hence, the empirical distribution

is adopted and used in the rest of this paper.

In examining the intraseasonal variability of daily rainfall amounts we find first that

the amount distributions conditioned on storm duration time (one-day or multi-day) or on

previous occurrence state (rain or not) are not significantly different from the

unconditioned (or full) distribution, as determined from the commonly-used

Kolmogorov-Smirnov (K-S) test (Swift and Schreuder, 1981). However, variations in

averaged daily rainfall amounts across the summer season are observed at some stations

(See Section 3). Hence, empirical rainfall distributions are produced separately for each

of the three months (July, August, and September). In addition, the rainfall depth in each

month is assumed to be an independent identically distributed (IID) process for this

region.

3. Results

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All sub-models described in section 2 are run 41092! times, with 92 representing the

length of the daily time series for July-September, and 410 representing the total number

of annual time series. These randomly-generated matrices provide a modeling description

of daily rainfall based upon the statistical frequency and intensity characteristics of the

observations. To characterize and evaluate the intraseasonal features of NAMS daily

rainfall simulations in this region we will calculate the model “overdispersion”

(Reference needed). Here, the overdispersion, representing the underestimate of variance

and extremes compared with observations, can be defined as:

0

0

e

eer

i

i

!= , (2)

Where ir is the overdispersion,

0e the observed variance, and

ie the modeled variance

by the thi model. Simulations of daily rainfall amounts and probabilities, dry spells

(consecutive non-rain days), wet spells (consecutive wet days), and storms (total

precipitation over wet spells) are analyzed in the following sections.

3.1 Daily rainfall amounts and probabilities

The evolution of the average daily rainfall amounts and of the occurrence probabilities

across the summer season are important variables for characterizing the intraseasonal

structure of NAMS precipitation. In general, the observed evolution for daily rainfall

amount, although noisy, are generally constant at about half (40) of the stations (Figure

3b). At the other stations (38) there appears to be a weak trend structure as determined by

a t-test (Figure 3a). Two types of trends, increasing and decreasing, are observed in this

region. At 28 stations (out of 38 in the subset), rainfall amounts and day-to-day variations

are relatively constant in July and August and then increase in September; these stations

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are mainly located in Utah and northwestern Colorado (Figure 4a). At the other 10

stations, which are mainly clustered in eastern Colorado and southern Arizona, relatively

higher rainfall amount are observed in the first two months with decreases in September.

These 38 stations also tend to have seasonal variations in the monthly rainfall

distributions (as identified by a K-S test), with 21 (6) stations displaying heavier (lighter)

than normal distributions in September (Figure 4b).

In contrast, the daily rainfall occurrence probabilities display strong time-dependence

across the summer season (Figure 3). The seasonal evolution of this statistic can be fitted

very well with a third-order polynomial curve. Among 78 stations in total, the observed

residuals at 75 stations are reduced significantly (>15%) by the fitted curves, and at 44

stations the reductions are larger than 50%. The seasonal variation of these daily rainfall

occurrence probabilities can be used to characterize the NAMS evolution: it begins in late

June or early July, reaches its mature phase about one month later, and ends in September.

The two plots in Figure 3 also suggest that the strength and evolution of NAMS can differ

according to location. For instance, over dry regions (e.g. SPANISH FORK PWR

HOUSE) the NAMS precipitation is weaker during the mature period (late July and early

August), with a flatter evolution of rainfall probabilities across the summer season

compared to those regions with more precipitation (e.g. PRESCOTT). This relation

between the evolution of NAMS and the climatological precipitation is generally

observed across this region (not shown).

Both occurrence sub-models can effectively reproduce the seasonal evolution for the

daily rainfall occurrence probabilities although both omit some high frequency variations

found in the observed series (Figures 3c,d). In contrast to the seasonal evolution of the

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daily rainfall probabilities, the daily series of rainfall amounts are mainly related to high-

frequency day-to-day variations (as opposed to low-frequency seasonal evolutions), and

hence are not captured by the seasonally-evolving stochastic rainfall models or the fitted

polynomial curves (Figures 3a,b). These results hold across the domain, in which about

65% of the variance in seasonal rainfall occurrence (calculated as the day-to-day variance

in the climatological rainfall occurrence at each station and then averaged across all

stations) is found in the low-frequency structure as represented by the seasonally-varying

Negative Binomial occurrence model. In contrast, less than one percent of the seasonal

rainfall amount variance is captured by the empirical distribution model (or even the

fitted third-order polynomial curve). As above, these results suggest that the seasonal

evolution of the NAMS precipitation in this region is best represented by the seasonal

evolution in the occurrence of rainfall as opposed to the intensity of rainfall.

3.2 Dry spells, wet spells, and storms

3.2.1 Dry spells

Observed dry spells generally have mean durations of 3 to 7 days with relatively longer

spells (longer than 5 days) clustering over western Utah and western Arizona (Figure 5).

Correspondingly, relatively higher variances are also observed in these regions.

As expected, both occurrence sub-models can reproduce the mean dry durations

exactly, but underestimate the variances (Table 1). Figure 6 shows the simulated variance

for the dry spells returned by the chain dependent model. Also shown on the figure are

the 95% confidence interval (CI) lines and the area-averaged overdispersion. Here, the

confidence interval lines are based upon the observed distribution of dry-spells at each

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station (typically these would be shown as error-bars but for clarity they are presented as

continuous lines). From the figure, more than 90% of the area-averaged variance can be

reproduced by the chain dependent model for the dry spells, and most individual variance

estimates lie between the 95% CI lines. Numerical tests indicate that the unexplained

10% variance is mainly attributable to one or two extremely long dry spells at each

station (not shown). In addition, comparable overdispersions are found at individual

stations regardless of the rainfall variance for that station, indicating that the model’s

approximation capability of the dry spell variance is independent of the variance itself for

this region.

3.2.2 Wet spells

The observed wet spells generally have mean durations between 1 to 3 days in the

southwestern U.S., and display a uniform spatial structure (Figure 7). This can be exactly

reproduced by both occurrence sub-models. In addition, coincident with previous studies

(Katz and Parlange, 1998), nearly 98% of the area-averaged variance in wet-spell length

is explained by the chain dependent model (Table 1). Even better simulations are returned

by the negative binomial sub-model (Table 1) in which all the area-averaged observed

variance is explained. Figure 8 shows simulations by the chain dependent sub-model. All

simulated variances are between the observed 95% CI line and distribute smoothly along

the 1:1 line with only a slight underestimation at the high variance tail. In contrast, the

negative binomial model creates much wider scattering of the estimates around the

observations and tends to overestimate the variance of wet spells both at individual

stations as well as for the region as a whole (not shown).

3.2.3 Storms

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Observed storms have a mean precipitation amount smaller than 25 mm at all stations.

These means display an obvious south-north spatial pattern with typical precipitation

larger than 13 mm in the south, and typical precipitation smaller than 13 mm in the north

(Figure 9). Variances of observed storms display a similar south-north spatial pattern

with higher values in the south.

Statistics of storms are simulated by combining the occurrence sub-models with

empirical rainfall amount distributions. The full model can exactly reproduce the mean

depth of storms, but underestimates the variance (Table 1). The area-average

overdispersion is 8.5% by the chain dependent model and 3.1% by the negative binomial

model. The positive area-averaged overdispersion for storms is larger than that for wet

spells alone. This is attributed to the combination of occurrence and intensity sub-models.

Although individual sub-models can simulate the occurrence or intensity well, the

combination decreases the capabilities of the full model. Figure 10 gives the simulated

variance returned by the chain dependent model as well as the 95% observed CI lines.

From the figure, most simulations are above the lower 95% CI line, although they

systematically underestimate the observed variance (with an area-average overdispersion

of 0.0845). The Negative Binomial model does slightly better but contains more

scattering of the estimates compared with the chain dependent model, which is attributed

to the increased scattering of wet-spell length estimates from the Negative Binomial

model (not shown).

3.3 Intraseasonal evolution of dry spells, wet spells, and storms

In addition to the daily characteristics of precipitation and general features of the

intraseasonal variables, the seasonal evolution of dry spells, wet spells, and storms may

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have a critical influence on local ecologic and hydrologic systems. Here, the intraseasonal

evolution is presented as a time series across the summer season. Each value of the time

series represents the average value (length in days for spells and rainfall amount in mm

for storms) over all historic observations of spells or storms that have the same onset date

at each station.

In general, dry spells exhibit three types of evolution patterns in this region: 1) the

average lengths of dry spells are constant through the first two months, and then increase

in September (termed a Type A dry-spell evolution); 2) the average lengths decrease

across the summer season (termed a Type B dry-spell evolution); and 3) the average

lengths of dry spells are constant across the monsoon season (no evolution pattern). The

evolution pattern at each station can be determined by a t-test ( 1.0=p ) on the monthly

averaged values.

Spatial distributions of the three types of dry-spell evolution are plotted in Figure 11.

Generally the Type A (increasing) patterns are predominant over this region (50 stations

out of 78 in total) and mainly centered in Arizona, central-east Colorado, and

northwestern Texas; the decreasing evolution patterns (Type B) are much fewer (only 10

stations) and mainly found in western Utah and western Arizona. This figure suggests the

evolution of dry periods over the central area of this region differs from the peripheral

monsoon regions, which tend to have shorter dry periods during the mature monsoon

period compared to the rest of the domain.

The observed evolution trend for dry spells can be reproduced by the occurrence sub-

models at most stations. This is mainly attributable to the different seasonally-evolving

transition probabilities at individual stations. At stations with the Type A pattern for dry

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spells, the seasonally-evolving transition probabilities exhibit a stronger decreasing trend

across August and September (see Figure 2 b,d) than those at stations with constant or

Type B evolution (see Figure 2 a,c). However the simulated increase (decrease) for the

Type A pattern (Type B pattern) is much smaller than the observed increase (decrease).

This underestimation of the seasonal cycle is mainly attributable to the use of a low-order

polynomial fitting-curve applied to the seasonal varying transition probabilities, which

omits high-frequency (i.e. sub-monthly) variations in the intraseasonal evolution curves

of the transition probabilities.

In contrast to dry spells, the average lengths for wet spells have a much flatter

evolution through the summer season. For instance at 43 stations (as compared with 18

stations for dry spells) the monthly averaged wet spells do not show a significant

difference as determined by a t-test ( 1.0=p ). Again, though, at the other 35 stations two

types of patterns are observed in the monthly evolutions for wet spells: 1) the average

lengths of wet spells increase slightly into September at 11 stations (termed the Type A*

wet-spell pattern); and 2) the average lengths of wet spells exhibit slightly decreasing

trends through the summer season at 24 stations (termed the Type B* wet-spell pattern).

The spatial distribution of the two types of evolution patterns for the wet spells is plotted

in Figure 12. In comparison to Figure 11, the Type B* wet-spell evolution pattern (which

represents decreasing wet-spell duration) displays a corresponding spatial structure to the

Type A (increasing) evolution pattern for dry spells and is mainly distributed over the

centeral area of the domain. The Type A* wet-spell pattern is observed mainly over

peripheral regions to the west, north, and east of the domain.

Although the overlap between the spatial structure of the dry spell evolution pattern

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and wet spell evolution pattern suggests a compensating balance between the two, fewer

stations show a seasonal evolution for wet spells than for dry spells, suggesting the

evolution of the NAMS system over the southwestern U.S. is more obvious in the

intermission rather than the duration of precipitation. In addition, the Type A* and B*

evolution patterns for wet spells cannot be reproduced by the occurrence sub-models

even though the dry spells can be. Both the chain-dependent and negative binomial sub-

models create flat evolution curves for wet spells at all stations, suggesting that the

inclusion of only one seasonal transition probability in our model influences only the

evolution of dry spells. The intraseasonal evolution for wet spells may be attributed to

other transition probabilities which are assumed constant in this study.

The intraseasonal evolutions for storms are also studied. The time series for the

average rainfall amounts during storms follows the time series of wet spells closely, but

with larger variations (not shown). This suggests a positive influence of the length of wet

spells on the storm amount. However, at many stations the evolution of storms exhibits

an increasing trend across the summer season relative to the seasonal evolution of wet

spells (see below), indicating the average daily rainfall amounts during a given storm

may increase with the evolution of the monsoon system. This coincides with the results in

Section 3.1 that suggest higher probabilities for severe storms in September than during

the first two months at some stations. In addition, the monthly averaged storms also

exhibit two types of seasonal evolution trends (decreasing and increasing) across the

summer season for this region, as determined by a t-test ( 1.0=p ). The spatial

distribution of the two types of evolution patterns for the storm amounts are plotted in

Figure 13, in which the Type A’ (increasing) stations are mainly located over the

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northeastern portion of the domain (Colorado) and eastern Arizona and Type B’

(decreasing) stations are mainly distributed over the western domain including Utah,

western Colorado, and central Arizona.

As discussed above, at many stations the evolution of storms exhibits an increasing

trend across the summer season relative to the seasonal evolution of wet spells. Figure 14

shows stations that have offsets between their seasonal evolutions for storm amounts

compared with wet spell lengths. For instance, if a station shows an increasing

(decreasing) trend in storm amounts but a constant trend in wet spell duration, the offset

would be considered positive (negative). Similarly if a station shows a constant trend in

storm amounts but a decreasing (increasing) trend in wet spell duration, the offset would

again be considered positive (negative). Overall, this figure indicates more stations (27)

show an increasing seasonal evolution in storm amounts relative to wet-spell length,

compared with the number showing negative offsets (9 stations).

Although storm amounts show a relative increase compared to wet spells for this

region, the variation in the evolution patterns for storm amounts cannot be reproduced by

stochastic models either, even when accounting for different monthly rainfall amount

distributions. Similar to wet spells, the stochastic models generate a flat evolution curve

for storms, with similar probability for severe storms over all three months.

4. Conclusions and discussions

The intraseaonal variability of summertime precipitation over the southwestern U.S. is

studied using a daily second-order chain dependent model and a negative binomial model,

both combined with monthly empirical daily rainfall amount distributions.

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Study results indicate that the evolution of NAMS is most apparent in the occurrence

of daily rainfall events, which exhibits clear time-dependence across the summer season

over the southwestern U.S.; this seasonal evolution is largely absent in the series of

averaged daily rainfall amounts. In general, the features of the seasonal evolution of daily

occurrence probabilities can be reproduced at all stations, although some high frequency

variability is omitted. These results indicate that a model incorporating one seasonal

varying transition probability can capture most of the intraseasonal variance in the daily

series of rainfall occurrence probability. In contrast, the evolution for the daily rainfall

intensity depth is much flatter and displays relatively higher mean values and/or larger

variances in September at some stations, suggesting the NAMS may have higher

probabilities for severe storms in September than in the first two months. However, the

seasonal trends contribute little to the variance in daily climatological rainfall amounts

across the season, indicating that the intensity of daily rainfall does not capture the

seasonal evolution of NAMS the way the daily rainfall occurrence values do.

Next, statistics for three intraseasonal characteristics - dry spells, wet spells, and

storms - are studied. In this region, dry spells exhibit an east-west spatial pattern with

longer duration over the western region; wet spells exhibit a uniform spatial distribution;

and storms exhibit a north-south spatial pattern with heavier storms in the south. These

spatial patterns can roughly be reproduced by both models. Respectively, the models can

capture more than 90% of the area-averaged variance for dry spell lengths, 98% for wet

spell lengths, and 90% for storm amounts. The overdispersion for storms is generally

larger than that for the wet spells alone. This is attributable to the fact that the

combination of occurrence and intensity sub-models decreases the model’s capability to

19

simulate storms, although it increases the model’s capability for reproducing seasonal

total precipitation (Wang et al. 2005).

The seasonal evolution of the dry spells, wet spells, and storms are also investigated.

Generally, two types of intraseasonal evolution patterns (increasing and decreasing) are

observed for dry spell lengths in this region. They exhibit clear geographic structure such

that the increasing pattern, which is prominent at two-thirds of the stations (50 out of 78

stations), mainly clusters in the central area of the domain. Only eight stations over the

western fringe of the region display the decreasing dry spell lengths. The spatial structure

of the evolution patterns of dry spells can be reproduced by the occurrence sub-models,

suggesting the seasonal variations of one transition probability in our model can account

for the evolution features of dry spells.

In contrast, wet spells have a much flatter evolution over this region compared to dry

spells; in general most stations (43) exhibit constant wet-spell lengths over the course of

the season. At other stations, the seasonal series for wet spells also display two types of

evolution patterns (increasing and decreasing) over this region. The stations with

decreasing wet spell lengths tend to exhibit a geographic distribution that corresponds to

the increasing patterns for dry spells. However, the seasonal evolution for wet spells

cannot be reproduced by the occurrence sub-models although those for dry spells can be.

This indicates the intraseasonal evolution for wet spells is attributable to variations of

transition probabilities which are assumed to be constant in our study.

The evolution curve for storms follows that for wet spells, but with larger variance.

However, the seasonal trend of storms suggests that although fewer storms (as

represented by the daily rainfall probabilities) and shorter duration of storms (as

20

represented by the wet spells duration) occur in September at many stations, the storm

depths may be more intense than in the first two months. Similar to wet spells, these

evolution characteristics for storms cannot be reproduced by a simple chain-dependent

model, even when incorporating separate empirical rainfall amount distributions for each

month. This further suggests that the weakening of the NAMS over Arizona (and/or

southeastern Colorado) is mainly related to an increase in the amount of dry days as

opposed to a weakening of the monsoon precipitation.

Acknowledgement:

The authors wish to thank Jon Eischeid at NOAA’s Climate Diagnostic Center for

producing and providing the station-based precipitation data products.

This research was funded by a cooperative agreement from NOAA-NA040AR431002.

The views expressed here are those of the authors and do not necessarily reflect the views

of NOAA.

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24

Figures:

Fig. 1 Average observed summertime (July-September) total precipitation (mm) over the

southwest US. The area and shading of the dots are proportional to the amount of

averaged seasonal total precipitation.

Fig. 2. Time-dependence of daily transition probabilities, 1,0,0

p for the chain dependent

model (a & b) and 1,0

p for the Negative Binomial model (c & d), plotted as a function

of Julian day. The triangles are observed transition probabilities. The solid lines are

regression curves returned by a least-square 3rd-order polynomial fit: (a) 1,0

p at

SPANISH FORK PWR HOUSE; (b) 1,0

p at PRESCOTT; (c) 1,0,0

p at SPANISH

FORK PWR HOUSE; and (d) 1,0,0

p at PRESCOTT.

Fig. 3 (a) Average daily rainfall amount at SPANISH FORK PWR HOUSE plotted as a

function of Julian day. The triangles are observed rainfall amounts for that day with

units of mm/day. The solid line is a regression curve returned by a least-square 3rd-

order polynomial fit, and the dashed line is the simulated values returned by the

negative binomial model. (b) Same as (a) except for PRESCOTT; (c) Daily

precipitation probability at SPANISH FORK PWR HOUSE, again plotted as a

function of Julian day. Units are fractions representing the probability of rainfall

occurrence (>0.25 mm) for the given day. (d) Same as (c) except for PRESCOTT.

Fig. 4 Spatial distribution of intraseasonal variations of daily rainfall amounts determined

by applying a t-test on monthly averaged rainfall amounts (a) and by applying a K-S

test on monthly rainfall amount distributions (b). The circles represent a flat evolution

of averaged daily rainfall amounts; the dark triangles represent relatively higher

25

rainfall amounts in September; and the grey squares represent relatively lower rainfall

amounts in September.

Fig. 5 Spatial distribution for the mean and standard deviation of dry spell length

(unit:days). The area and shading of the solid dots are proportional to the values at

each station.

Fig. 6 Relationship between observed and modeled variances for dry spell lengths.

Model results produced using the chain dependent model. Shown in the top left

corner is area-average overdispersion – see text for details.

Fig. 7 Spatial distribution for the mean and standard deviation of wet spell length (unit:

days). The area and shading of the solid dots are proportional to the values at each

station.

Fig. 8 Relationship between observed and modeled variances for wet spell lengths.

Model results produced using the chain dependent model. Area-average

overdispersion shown in top left corner – see text for details.

Fig. 9 Spatial distribution for the mean and standard deviation of storm amount

(unit:mm ). The area and shading of the solid dots are proportional to the values at

each station.

Fig. 10 Relationship between observed and modeled variances for storm amounts. Model

results produced using the chain dependent model combined with the empirical daily

rainfall amount distributions. Area-average overdispersion shown in top left corner –

see text for details.

Fig. 11 Spatial distribution of the three types of varying dry-spell evolution patterns. The

clear circles are stations with a constant evolution, the triangles represent stations

26

with a Type A (increasing) evolution, and the squares represent stations with a Type B

(decreasing) evolution.

Fig. 12 Spatial distribution of the three types of varying wet-spell evolution patterns. The

clear circle are stations with constant evolution, the triangles represent stations with a

Type A* (increasing) evolution, and the squares represent stations with a Type B*

(decreasing) evolution.

Fig. 13 Spatial distribution of the three types of storm-amount evolution patterns. The

clear circles are stations with a constant evolution, the triangles represent stations

with a Type A’ (increasing) evolution, and the squares represent stations with a Type

B’ (decreasing) evolution.

Fig. 14 Spatial distribution for the offsets between the seasonal evolution of wet spells

and storm amounts. The clear circles represent stations with similar seasonal

evolutions for storms and wet spells, the triangles represent stations in which the

seasonal evolution of storm amounts increases relative to the seasonal evolution of

wet-spell lengths, and the squares represent stations in which the seasonal evolution

of storm amounts decreases relative to the seasonal evolution of wet-spell lengths –

see text for details.

27

Fig. 1 Average observed summertime (July-September) total precipitation (mm) over

the southwest US. The area and shading of the dots are proportional to the amount of

averaged seasonal total precipitation.

28

Fig. 2. Time-dependence of daily transition probabilities, 1,0,0

p for the chain dependent

model (a & b) and 1,0

p for the Negative Binomial model (c & d), plotted as a function of

Julian day. The triangles are observed transition probabilities. The solid lines are

regression curves returned by a least-square 3rd-order polynomial fit: (a) 1,0

p at SPANISH

FORK PWR HOUSE; (b) 1,0

p at PRESCOTT; (c) 1,0,0

p at SPANISH FORK PWR

HOUSE; and (d) 1,0,0

p at PRESCOTT.

29

Fig. 3 (a) Average daily rainfall amount at SPANISH FORK PWR HOUSE plotted as

a function of Julian day. The triangles are observed rainfall amounts for that day with

units of mm/day. The solid line is a regression curve returned by a least-square 3rd-order

polynomial fit, and the dashed line is the simulated values returned by the negative

binomial model. (b) Same as (a) except for PRESCOTT; (c) Daily precipitation

probability at SPANISH FORK PWR HOUSE, again plotted as a function of Julian day.

Units are fractions representing the probability of rainfall occurrence (>0.25 mm) for the

given day. (d) Same as (c) except for PRESCOTT.

30

Fig. 4 Spatial distribution of intraseasonal variations of daily rainfall amounts determined

by applying a t-test on monthly averaged rainfall amounts (a) and by applying a K-S test

on monthly rainfall amount distributions (b). The circles represent a flat evolution of

31

averaged daily rainfall amounts; the dark triangles represent relatively higher rainfall

amounts in September; and the grey squares represent relatively lower rainfall amounts in

September.

32

Fig. 5 Spatial distribution for the mean and standard deviation of dry spell length

(unit:days). The area and shading of the solid dots are proportional to the values at each

station.

33

Fig. 6 Relationship between observed and modeled variances for dry spell lengths.

Model results produced using the chain dependent model. Shown in the top left corner is

area-average overdispersion – see text for details.

34

Fig. 7 Spatial distribution for the mean and standard deviation of wet spell length (unit:

days). The area and shading of the solid dots are proportional to the values at each station.

35

Fig. 8 Relationship between observed and modeled variances for wet spell lengths.

Model results produced using the chain dependent model. Area-average overdispersion

shown in top left corner – see text for details.

36

Fig. 9 Spatial distribution for the mean and standard deviation of storm amount

(unit:mm ). The area and shading of the solid dots are proportional to the values at each

station.

37

Fig. 10 Relationship between observed and modeled variances for storm amounts.

Model results produced using the chain dependent model combined with the empirical

daily rainfall amount distributions. Area-average overdispersion shown in top left corner

– see text for details.

38

Fig. 11 Spatial distribution of the three types of varying dry-spell evolution patterns.

The clear circles are stations with a constant evolution, the triangles represent stations

with a Type A (increasing) evolution, and the squares represent stations with a Type B

(decreasing) evolution.

39

Fig. 12 Spatial distribution of the three types of varying wet-spell evolution patterns.

The clear circle are stations with constant evolution, the triangles represent stations with a

Type A* (increasing) evolution, and the squares represent stations with a Type B*

(decreasing) evolution.

40

Fig. 13 Spatial distribution of the three types of storm-amount evolution patterns. The

clear circles are stations with a constant evolution, the triangles represent stations with a

Type A’ (increasing) evolution, and the squares represent stations with a Type B’

(decreasing) evolution.

41

Fig. 14 Spatial distribution for the offsets between the seasonal evolution of wet

spells and storm amounts. The clear circles represent stations with similar seasonal

evolutions for storms and wet spells, the triangles represent stations in which the seasonal

evolution of storm amounts increases relative to the seasonal evolution of wet-spell

lengths, and the squares represent stations in which the seasonal evolution of storm

amounts decreases relative to the seasonal evolution of wet-spell lengths – see text for

details.

42

Table 1. Area-averaged overdispersions, r, returned by a chain-dependent model and a

negative binomial model. .

Dry spells Wet spells storms

Chain-dependent model 0.0936 0.0265 0.0845

Negative binomial model 0.1677 -0.0999 0.0309