## 1. Introduction

Central Arizona (Fig. 1a) is a region of the southwestern Unites States that has undergone a dramatic population growth over the last few decades. The largest development has been concentrated in the Phoenix metropolitan area (Fig. 1b), where the population has reached 4.48 million and is expected to increase up to 7 million by 2050, according to projections of the state of Arizona. In this semiarid region, the rainfall regime is characterized by complex features (Sheppard et al. 2002), including (i) low annual rainfall (from 200 to 500 mm depending on elevation); (ii) large interannual variability; (iii) strong seasonality due to the North American monsoon (NAM; Adams and Comrie 1997); and (iv) convective thunderstorms with very high rain intensities, mostly occurring during the NAM. As a result, characterizing the spatial and temporal variability of the rainfall statistical properties in central Arizona is crucial for a number of reasons. First, it is important to support design of water infrastructures (Langousis and Veneziano 2007) and development of sustainable water management strategies, including irrigation schedules (Volo et al. 2015) and rainwater harvesting systems (Wang and Zimmerman 2015). Second, it is useful to quantify risk of floods and flash floods (McCollum et al. 1995), whose potential impacts and costs have increased in time because of urban and infrastructure expansion, and concentration of manufacturing activities in the metropolitan area (Kane et al. 2014). Finally, it is a preliminary step toward the development and calibration of statistical downscaling tools that reproduce the finescale rainfall properties (in time and/or space) from coarse predictions of numerical weather prediction (NWP) models, estimates from satellite products, and simulations of climate models (e.g., Over and Gupta 1994; Deidda et al. 2004; Mascaro et al. 2014; Akrour et al. 2015).

The rainfall statistical variability has been the subject of a large number of studies worldwide, which pursued several goals, including but not limited to estimation and statistical modeling of amount and occurrence of rainfall at different aggregation times (e.g., Mehrotra et al. 2006, and references therein; Deidda 2010) as well as of rainfall extremes (e.g., Papalexiou and Koutsoyiannis 2013; Serinaldi and Kilsby 2014), quantification of interannual and intraseasonal variability (e.g., Krishnamurthy and Shukla 2000; Heinselman and Schultz 2006; Stillman et al. 2013), and study of the spatial correlation structure (e.g., Goovaerts 2000; Krajewski et al. 2003; Ciach and Krajewski 2006). Recently, the increasing availability of long-term records at high temporal resolution has allowed the investigation of rainfall intermittency across a wide range of scales (from tens of seconds to years) using spectral and/or scale invariance analyses (e.g., Badas et al. 2006; Gebremichael et al. 2007; Molini et al. 2009; Verrier et al. 2011; Rysman et al. 2013; Mascaro et al. 2013, 2014; Mandapaka and Qin 2015). These techniques revealed the existence of scaling regimes, that is, time intervals where rainfall properties are linked by power-law relations across scales, which have been related to the dominant meteorological phenomena (Fraedrich and Larnder 1993). Moreover, in a recent study by Mascaro et al. (2013), the availability of dense networks of gauges allowed identifying spatial patterns for the metrics charactering rainfall intermittency that were linked to local terrain features and synoptic circulation patterns.

Most rainfall-related studies in central Arizona focused on understanding the meteorological mechanisms and climatological aspects related to the development, evolution, and interannual variability of the NAM because of its important influence on the rainfall regime (e.g., Carleton 1986; Carleton et al. 1990; Maddox et al. 1995; Adams and Comrie 1997; Heinselman and Schultz 2006; Favors and Abatzoglou 2013). These analyses utilized observations from individual radiosonde sites and satelliteborne sensors, simulations of NWP models, and reanalysis products. Somewhat surprisingly, studies of the rainfall spatial and temporal statistical properties based on gauge observations have been limited, likely because, prior to the early 1980s, rainfall records have only been available at sparse locations. Notable expectations include the works of (i) Balling and Brazel (1987), who analyzed rainfall data from 45 gauges in Arizona with the aim of characterizing the spatial variability of the diurnal cycle of warm season rainfall, and (ii) Svoma and Balling (2009), who expanded this analysis by investigating the influence of monsoonal gulf surges with additional data provided by the Flood Control District of Maricopa County (FCDMC) in the Phoenix area. To date, no study has investigated the statistical properties of rainfall across multiple temporal scales (from minutes to year), their spatial variability in the region using dense networks of gauges, and the relation between these properties and local terrain characteristics.

This paper aims to fill this research gap by analyzing the high-resolution (1 min) rainfall time series collected by the dense network of rain gauges installed and maintained by the FCDMC from the early 1980s in central Arizona (Fig. 1b). The network currently includes 310 gauges, 142 of which have at least 20 years of data. A suite of techniques is applied to investigate the multiscale rainfall statistical properties in space and time. First, analyses are conducted to study the interannual variability of rainfall accumulation at annual and seasonal scales, evaluating the effect of El Niño–Southern Oscillation (ENSO) teleconnections (Simpson and Colodner 1999; Sheppard et al. 2002). Next, metrics are computed at monthly scale to characterize the intraseasonal variability of (i) rainfall accumulation and occurrence; (ii) storm duration; (iii) maximum rainfall intensities at different durations, whose analysis is important given the growing potential impacts of intense storms in the urban area; and (iv) diurnal cycle. The correlation structures of the records sampled at different time resolutions are then analyzed to identify the characteristic spatial scales of the dominant weather phenomena in summer and winter. Finally, the presence of scaling regimes is investigated through the spectral and scale invariance analyses. For the metrics involved in the analyses mentioned above, the spatial variability is quantified along with the effect of local terrain characteristics, which have been demonstrated to exert an important physical control on rainfall properties and regimes in the region (e.g., Carleton 1986; Adams and Comrie 1997).

## 2. Study area and dataset

The climate of central Arizona (Figs. 1a,b) is characterized by dry and warm conditions throughout the year, mainly due to a quasi-permanent subtropical high pressure ridge (Sheppard et al. 2002). According to the 1970–2000 climate normals produced by the National Oceanic and Atmospheric Administration (NOAA)/National Centers for Environmental Information (NCEI), the mean annual temperature and rainfall recorded at Phoenix Sky Harbor International Airport station are 24°C and 204 mm, respectively. Within the year, climate variability is distinguished by marked seasonality. From late fall to early summer, the area is dominated by westerly flow and relatively dry conditions. Occasionally, cold fronts associated with these flows lead to storm systems driven by dynamical lifting with low-to-moderate rainfall intensity, large spatial coverage, and relatively long duration (a few days). Sometimes, convective cells with high rain intensity can also be embedded within these large-scale systems (Sheppard et al. 2002). In early July, a shift in mid- and upper-tropospheric patterns leads to the onset of the NAM that lasts until September. During the NAM, water vapor is advected northward from the eastern tropical Pacific Ocean and Gulf of California and, to a lesser extent, from the Gulf of Mexico (Adams and Comrie 1997). As a result, about one-third of the annual rain falls in central Arizona during this season, mostly in form of high-intensity convective thunderstorms of short lifetime (<1 h) and small spatial coverage (<50 km^{2}). A distinctive feature of summer storms is a strong diurnal variability that depends on local conditions promoting the time of maximum convective activity (Balling and Brazel 1987).

The dataset used for the analyses consists of rainfall time series of the network managed by the FCDMC, a political subdivision of the state of Arizona that has the goal of providing regional flood hazard identification, regulation, remediation, and education to Maricopa County residents. Figures 1a and 1b report the locations of the rain gauges within the state and counties of Arizona and the Phoenix metropolitan area. The rain gauges are mostly installed in Maricopa County, with a larger density in the urban areas. Additional stations are located in the neighboring counties of Pinal, La Paz, and Yavapai to track in real time the movement of regional storm systems. Currently, the network includes a total of 310 gauges that have been gradually installed since the early 1980s, as reported in Fig. 1c. Table 1 summarizes the main characteristics of the FCDMC network in terms of number of stations, years of observations, and density, measured through the mean area per gauge and the intergauge distance. Details are provided for the entire network (all), which covers an area of ~29 600 km^{2}, and separately for the urban Phoenix metropolitan area (~2037 km^{2}) and the nonurban regions mostly occupied by desert, forests, and crops. Table 1 also reports information on the FCDMC network for classes of gauge elevation that ranges from 220 to 2325 m MSL. While most of the gauges (263) are installed at lower elevations up to 800 m MSL, a significant number of stations (46) cover higher altitudes, thus providing a valuable source of information to investigate the orographic effect on rainfall properties in the region.

Characteristics of the rain gauge network of the FCDMC, including number of gauges *N*; number of gauges with more than 10, 20, and 30 years of observation (*N*_{10}, *N*_{20}, and *N*_{30}, respectively); mean area per gauge *A*_{g}; and min, median, and max intergauge distance (*d*_{min}, *d*_{med}, and *d*_{max}, respectively). Data are reported for the entire network (all) and for the portions located in urban and nonurban areas. Variables *N*, *N*_{10}, *N*_{20}, and *N*_{30} are also reported for the gauges divided into elevation classes.

The rain gauges are of the tipping-bucket type, with an accumulated rain depth of 1 mm and a recording apparatus that stores the time of each tip in a digital memory with second precision. The raw rainfall data (i.e., the instants of each tip recorded at the gauges) were obtained from FCDMC and the method of Mascaro et al. (2013, their appendix A) was applied to generate the rainfall signal ^{−1}) at different time resolutions Δ*t* required in our analyses. In brief, the sampling approach for the construction of the rainfall signal assumes that the relation between the cumulative rainfall depth and time is obtained through linear interpolation. Thus, for each time step, the rainfall intensity is derived by (i) computing the difference between the cumulative rainfall at the end and at the beginning of the interval Δ*t* and (ii) dividing this difference by Δ*t*. The dataset covers a period from 1980 to 2013 with a different record length at each station depending on the time of gauge deployment and possible instrument malfunctions. As shown in Figs. 1b and 1c and Table 1, most of the gauges (274) have a record length larger than 10 years, with 142 stations covering a period of observation included between 20 and 34 years. Additional datasets used in this study include (i) the 75-yr-long record of annual rainfall depth observed at the Sky Harbor airport gauge and available from the NCEI database and (ii) the monthly Southern Oscillation index (SOI; Ropelewski and Jones 1987) provided by the NOAA/Climate Prediction Center (CPC) and used to characterize the ENSO influence, as done by Simpson and Colodner (1999) and Pool (2005) in two hydroclimatological studies in Arizona.

## 3. Methods

The techniques and metrics adopted to investigate the spatial and temporal variability of rainfall at multiple resolutions are described in the following subsections. To quantify the effect of elevation, the coefficient of determination *r*^{2} of the linear regression between the metrics and gauge elevation was utilized. In addition, since the rainfall regime in central Arizona is characterized by marked seasonality with presence of distinct storm types in each season (Adams and Comrie 1997; Sheppard et al. 2002; McCabe and Clark 2006), some of the analyses described below were applied separately for the summer monsoon season, including the months from July to September, and the winter period, defined from November to March. In the other months, conditions are extremely dry and storms are associated with mixed weather systems such as dissipating tropical storms that are not representative of the summer and winter seasons (Pool 2005).

### a. Analyses at annual scale

The rainfall interannual variability was analyzed by computing, for each station, the rainfall depth *R* (mm) accumulated over each year and summer and winter seasons. The spatial variability of these metrics was quantified through the spatial coefficient of variation (CV; spatial standard deviation divided by spatial mean) using the stations active in each year. To classify the wetness of each year, we derived the empirical frequency of the annual rainfall depth at the Sky Harbor airport gauge *R*_{SH} based on a record of 75 years. Following the climatological rankings of NOAA, years were classified as below normal, normal, and above normal if the frequency of *R*_{SH} was lower than 0.33, included between 0.33 and 0.66, and higher than 0.66, respectively. Finally, the ENSO teleconnections were explored by computing the Pearson correlation coefficient (CC) between the time series of SOI averaged from June to November, as in Pool (2005) and Simpson and Colodner (1999), and the time series of summer and winter *R*.

### b. Analyses at monthly scale

For each gauge, a set of metrics was calculated at monthly scale, including rainfall depth (i.e., *R*); mean number of consecutive and nonconsecutive rainy days (daily *R* > 0.2 mm day^{−1}); mean storm duration; and maximum rainfall intensities ^{−1}) at different durations *τ* = 5, 15, and 30 min and 1, 3, 6, 12, and 24 h. For the computation of the mean storm duration, the signal was sampled at resolution Δ*t* = 5 min and a single storm was defined as a period of nonzero rainfall with possible intervals of zero rain lasting less than 15 min to account for the relatively high tipping depth of the instrument (1 mm). The monthly climatological means (hereafter called monthly climatologies) of the metrics were computed at each station averaging the values of all available years. These were then used to derive the spatial mean and spatial standard deviation across all stations.

### c. Diurnal cycle

*t*= 1 h and, for each month, the mean frequency of rainfall occurrence

*F*(

*t*) was calculated for each hour of the day (

*t*= 1, 2, …, 24). Next, the harmonic analysis was applied on

*F*(

*t*):

*ω*is the angular frequency equal to 2

*π*/24, with 24 being the number of hours in a day; and

### d. Correlation analysis

*t*ranging from 5 min to 24 h. This was done separately for each season. In each case, the CC was plotted as a function of the separation distance between the pairs of stations

*d*and fitted to the analytical model:

*d*and

*m*and

*n*are parameters. The correlation distance

*m*.

The analysis described above is based on all pairs of gauges and is not able to capture the existence of directions where the correlation is higher because of, for example, preferential directions of storm movement or shape. Given its relatively high density and large spatial coverage, the FCDMC rain gauge network allows for exploring potential anisotropies in the rainfall correlation structures. As a result, the directional spatial correlograms were calculated as described in the following. Let *γ* be the angle measured from the east–west axis and used to identify cardinal and intercardinal directions (*γ* = 0° for the east–west direction, *γ* = 45° for the southwest–northeast direction, etc.). For each gauge *j*, the line with angle *γ* passing through the gauge location was found and the gauges located within a maximum distance *l* (km) from this line were identified. The pairs formed by gauge *j*, and these gauges were then utilized in the construction of the spatial correlogram associated with the direction *γ*. A distance *l* = 10 km was found to be a good compromise between robustness of the results (i.e., the shape of the correlograms becomes stable) and the need to include a significant number of gauges when computing the CC. As done for the isotropic correlogram, the correlation distance was derived by fitting Eq. (2) and estimating the parameter *m*.

### e. Spectral and scale invariance analysis

*f*, the energy

*α*in the different scaling regimes was estimated through linear regression of Eq. (3) in the log–log space. The spectral exponent

*α*quantifies the contribution of the different frequency components of the time series. When

*α*is low, all frequencies have a similar contribution to the overall variability. In contrast, a larger

*α*is associated with a signal where lower frequencies have larger contribution than higher ones. Time series with lower (higher)

*α*have a slower (faster) decay of the autocorrelation with time. Thus, from the physical point of view,

*α*controls the level of organization of weather systems within their respective scaling regime (Mandapaka and Qin 2015, and references therein).

*q*and aggregation scales

*j*= 0, 1, …,

*M*), defined as

*τ*for the

*k*th time step, and

*τ*within the duration

*T*. For a given

*q*, the presence of scale invariance is then investigated by identifying time intervals where the structure function varies with

*τ*according to the power-law relation:

*q*. For rainfall signals with fluctuations,

*q*> 1 becomes larger as the signal intermittency increases. Here, for both summer and winter seasons, the finescale was set to

*M*= 17 (

*T*~91 days) was fixed in summer, thus covering the entire season, while

*M*= 15 (

*T*~23 days) was adopted in winter and several rainfall sequences were extracted within the period from November to March. For each season and gauge, the scale invariance was analyzed on a single set of structure functions computed by averaging

## 4. Results

### a. Rainfall interannual variability and effect of elevation

Figure 2a presents the time series of the spatial mean of annual *R* across all gauges and the frequency of annual *R* at the Sky Harbor airport gauge *R* and *R* is characterized by high interannual variability (Sheppard et al. 2002), with totals that range from 92 to 310 mm and cover almost equally the three climatological rankings of below-normal (9 years), normal (10 years), and above-normal (13 years) conditions. During the study period, the maximum number of consecutive years in below-normal conditions is 2 (1996–97 and 2011–12), while that in above-normal conditions is 5 (1991–95). Figure 2b presents the time series of spatial mean annual and summer *R* along with that of SOI averaged from June to November. On average, the portion of annual *R* provided by summer monsoonal storms is 33% with high year-to-year variability ranging from 13% to 71%, which has been ascribed to the interannual variation of the summer circulation in the region and a land memory effect that was found to be important over multidecadal periods (Carleton et al. 1990; Hu and Feng 2002). As presented in previous studies (e.g., Sheppard et al. 2002; Simpson and Colodner 1999; Pool 2005), teleconnections during ENSO periods lead to higher-than-normal winter *R*, while they do not have a significant influence on summer *R*, as revealed by the high (low) negative correlation between winter (summer) *R* and SOI (CC of −0.51 and −0.16, respectively).

Interannual variability of rainfall. (a) Time series of spatial mean annual *R* and frequency of annual *R* and of the SOI averaged from June to November. (c) Time series of the spatial CV of annual and summer *R*.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Interannual variability of rainfall. (a) Time series of spatial mean annual *R* and frequency of annual *R* and of the SOI averaged from June to November. (c) Time series of the spatial CV of annual and summer *R*.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Interannual variability of rainfall. (a) Time series of spatial mean annual *R* and frequency of annual *R* and of the SOI averaged from June to November. (c) Time series of the spatial CV of annual and summer *R*.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

The spatial variability of annual and summer rainfall is analyzed in Fig. 2c through the spatial CV. The rainfall accumulated during the warm season always has larger spatial variability in the region as compared to the annual totals (the mean plus/minus standard deviation of CV across all years is 0.65 ± 0.14 for summer *R* and 0.46 ± 0.08 for annual *R*). The physical control on the spatial variability is mostly due to elevation, as shown in Fig. 3, where scatterplots are presented between the average across all available years of annual and summer *R* at each gauge and the corresponding elevation. For both time periods, the correlation is high and the relation can be well represented through a linear regression (*r*^{2} ≥ 0.76). Note that the relation between *R* and elevation is robust and does not change in below- or above-normal conditions (not shown). In addition to elevation, the effect of gauge location was also investigated, finding that only latitude has a significant relation with *R*. This is most likely a consequence of the fact that elevation increases from south to north (see Fig. 1; CC between latitude and elevation is 0.70).

Relation between the mean (a) summer and (b) annual *R* observed at each station and the corresponding elevation *z*. The linear regression and *r*^{2} are also plotted for each case.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Relation between the mean (a) summer and (b) annual *R* observed at each station and the corresponding elevation *z*. The linear regression and *r*^{2} are also plotted for each case.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Relation between the mean (a) summer and (b) annual *R* observed at each station and the corresponding elevation *z*. The linear regression and *r*^{2} are also plotted for each case.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

### b. Rainfall intraseasonal variability and spatiotemporal variability of extremes

The spatial mean and standard deviation of the monthly climatologies for a set of metrics are presented in Fig. 4. The monthly rainfall depth (i.e., *R*; Fig. 4a) is characterized by two maxima in summer (July–September) and winter (November–March) separated by two minima in spring (April–June) and fall (October). For reference, the spatial mean *R* for the wettest (1992) and driest (2002) years is also plotted. The spatial variability is larger (smaller) in summer (winter) months, consistent with the more scattered (widespread) storm systems. The mean number of rainy days (Fig. 4b) resembles the pattern of *R*, including the trend of the spatial variability, with more pronounced peaks in summer (seasonal average of 3.38 days) as compared to winter (2.87 days). In contrast, the mean number of consecutive rainy days is fairly constant throughout the year (1.45 days) with negligible spatial variability. Note that the mean number of consecutive rainy days was computed by including only months with observed rainfall, and thus, it can be larger than the mean number of rainy days. Similarly, the mean duration of single storms (Fig. 4c) is rather constant across all months (average across all months of 50.2 min), with relatively large spatial differences (average standard deviation of 35.1 min). To investigate the controls on the spatial variability, Fig. 5 shows the monthly variability of *r*^{2} of the linear regression between each metric and gauge elevation. The monthly *R* and number of rainy days exhibit a significant orographic control in the summer season and, to a lesser extent, in the winter months from November to January. The control on the mean number of consecutive rainy days is moderate from October to April, when large-scale weather systems occur more frequently, and is negligible in summer. Finally, the duration of single storms does not display any significant relation with elevation.

Spatial mean and std dev (bars) of monthly climatologies of (a) *R* (climatology), (b) mean number of rainy days and of consecutive rainy days, and (c) mean duration of rainfall events. In (a) the spatial mean *R* observed during the wettest (1992) and driest (2002) years on record are also plotted.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Spatial mean and std dev (bars) of monthly climatologies of (a) *R* (climatology), (b) mean number of rainy days and of consecutive rainy days, and (c) mean duration of rainfall events. In (a) the spatial mean *R* observed during the wettest (1992) and driest (2002) years on record are also plotted.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Spatial mean and std dev (bars) of monthly climatologies of (a) *R* (climatology), (b) mean number of rainy days and of consecutive rainy days, and (c) mean duration of rainfall events. In (a) the spatial mean *R* observed during the wettest (1992) and driest (2002) years on record are also plotted.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Monthly time series of *r*^{2} of the linear regression between station elevation and climatologies of *R*, mean number of rainy days and consecutive rainy days, and mean storm duration.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Monthly time series of *r*^{2} of the linear regression between station elevation and climatologies of *R*, mean number of rainy days and consecutive rainy days, and mean storm duration.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Monthly time series of *r*^{2} of the linear regression between station elevation and climatologies of *R*, mean number of rainy days and consecutive rainy days, and mean storm duration.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

The spatiotemporal variability of monthly rainfall extremes was explored through the monthly climatologies of maximum rainfall intensities *τ*). Figure 6a presents the monthly time series of spatial mean and standard deviation of *τ* (5 min, 1 h, and 24 h), while Fig. 6b shows the relation between *τ* in three illustrative months (August, January, and April). Results reveal that (i) for all durations, *τ* and become negligible for *τ* = 24 h. These considerations provide insight to support statistical modeling of extreme rainfall, including the potential need to account for seasonal differences at shorter durations. The spatial variability of the extremes is investigated in Fig. 7, which presents the spatial distribution of the average across all years of *τ* = 5 min and *τ* = 24 h. In summer (Figs. 7a,b), the pattern of *τ* = 5 min; Fig. 7c) are randomly distributed in space with no substantial link to elevation. When longer durations are considered (*τ* = 24 h; Fig. 7d), the relation with elevation becomes significant and the largest values of

(a) Spatial mean and std dev of the monthly climatologies of *τ* = 5 min, 1 h, and 24 h. (b) Relation between spatial mean of the monthly climatologies of *τ* in August, January, and April.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

(a) Spatial mean and std dev of the monthly climatologies of *τ* = 5 min, 1 h, and 24 h. (b) Relation between spatial mean of the monthly climatologies of *τ* in August, January, and April.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

(a) Spatial mean and std dev of the monthly climatologies of *τ* = 5 min, 1 h, and 24 h. (b) Relation between spatial mean of the monthly climatologies of *τ* in August, January, and April.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Spatial distribution of mean across all years of *τ* = 5 min and (b),(d) *τ* = 24 h for the summer and winter seasons. The insets show the corresponding relations between *r*^{2}. The legend of the digital elevation model and the urbanized areas shown as background in all maps is reported in (b).

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Spatial distribution of mean across all years of *τ* = 5 min and (b),(d) *τ* = 24 h for the summer and winter seasons. The insets show the corresponding relations between *r*^{2}. The legend of the digital elevation model and the urbanized areas shown as background in all maps is reported in (b).

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Spatial distribution of mean across all years of *τ* = 5 min and (b),(d) *τ* = 24 h for the summer and winter seasons. The insets show the corresponding relations between *r*^{2}. The legend of the digital elevation model and the urbanized areas shown as background in all maps is reported in (b).

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

These findings are further demonstrated and summarized in Fig. 8, where *r*^{2} between mean *τ* for the summer and winter seasons. Extreme rainfall events due to summer thunderstorms are controlled by elevation for all durations, although a slight decrease of *r*^{2} is observed for intermediate *τ*. This suggests that, in central Arizona, elevation enhances the development and strength of convection that originates intense monsoonal storms. This finding supports ground observations evidence of intense convective rainfall events at higher elevation shown by Carleton (1986) in Arizona through satellite observations of clouds and by Rowe et al. (2011) in northern Mexico using observations from weather radars. In contrast, during larger-scale systems occurring in winter, elevation does not control the occurrence of high-intensity storms within the characteristic durations of single cells (*τ* ≤ 1 h) and becomes gradually more significant for larger durations.

Relation between *r*^{2} of the linear regression between *τ* for summer and winter seasons.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Relation between *r*^{2} of the linear regression between *τ* for summer and winter seasons.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Relation between *r*^{2} of the linear regression between *τ* for summer and winter seasons.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

### c. Diurnal cycle of rainfall and its spatial variability

The diurnal cycle of hourly rainfall frequencies in each month averaged across all stations is analyzed in Fig. 9. Specifically, Fig. 9a shows the diurnal cycles (in standardized units) for two representative months of January and August along with the lines fitted through harmonic analysis, while Fig. 9b reports for each month the peak time and amplitude of the first harmonic. Results reveal (i) the presence of a diurnal cycle (standardized amplitude >1) from May through August with a peak around 1900–2200 LT and (ii) the lack of a preferential time of rainfall occurrence in the other months (standardized amplitude <1 and random values of peak time), including at the end of the monsoon season in September. The spatial variability of the diurnal cycle is investigated in Fig. 10. In January (Fig. 10a), the peak times are randomly distributed in the study region. In contrast, in August (Fig. 10b), a pattern emerges along the northwest–southeast direction that is mostly controlled by elevation (Fig. 10c): during early afternoon, summer thunderstorms occur more often at higher elevation, and later on during the day, the occurrence of rainfall gradually shifts toward desert areas at lower elevations, with the nighttime rainfall mostly occurring in the Phoenix metropolitan area. These results are consistent with and further support previous studies in the region (Balling and Brazel 1987; Maddox et al. 1995).

(a) Time series of the diurnal cycle of the mean hourly rain frequency for January and August averaged across all stations, along with the corresponding functions fitted to the data via harmonic analysis. The time of the day is in LST. (b) Monthly time series of the mean across all stations and years of amplitude and peak time of the rainfall diurnal cycle computed from the harmonic analysis.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

(a) Time series of the diurnal cycle of the mean hourly rain frequency for January and August averaged across all stations, along with the corresponding functions fitted to the data via harmonic analysis. The time of the day is in LST. (b) Monthly time series of the mean across all stations and years of amplitude and peak time of the rainfall diurnal cycle computed from the harmonic analysis.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

(a) Time series of the diurnal cycle of the mean hourly rain frequency for January and August averaged across all stations, along with the corresponding functions fitted to the data via harmonic analysis. The time of the day is in LST. (b) Monthly time series of the mean across all stations and years of amplitude and peak time of the rainfall diurnal cycle computed from the harmonic analysis.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Spatial distribution of the mean peak time of the rainfall diurnal cycle in (a) January and (b) August. (c) Relation between gauge elevation and mean peak time (LST) in August.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Spatial distribution of the mean peak time of the rainfall diurnal cycle in (a) January and (b) August. (c) Relation between gauge elevation and mean peak time (LST) in August.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Spatial distribution of the mean peak time of the rainfall diurnal cycle in (a) January and (b) August. (c) Relation between gauge elevation and mean peak time (LST) in August.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

### d. Spatial correlation structure of rainfall

The experimental correlograms based on all gauges are plotted in Figs. 11a and 11b for the aggregation times Δ*t* = 30 min and Δ*t* = 3 h, respectively, chosen to capture the behavior of single and multiple rainfall cells. The fitted analytical model Eq. (2) is also plotted, while the estimated parameters *m* and *n* of Eq. (2) for Δ*t* ranging from 5 min to 24 h are reported in Table 2. The CC is always larger in winter for both aggregation times, providing additional evidence of the differences between the dominant weather systems in the two seasons, with widespread rainfall due to cold fronts in winter and more intermittent and spatially variable storms during the NAM. Interestingly, the correlation structures for Δ*t* = 30 min of the two seasons are similar at short distances, indicating a comparable behavior of single rain cells in the two periods. At larger aggregation times (Δ*t* = 3 h), the CC increases in both seasons, suggesting that rain cells have moved or have been dissipated and created, thus covering larger distances. To summarize the results, Fig. 11c shows the relation between the correlation distance for different Δ*t* and the separation distance. Mascaro et al. (2014) carried out a similar analysis using high-resolution rainfall data from a topographic transect covering a distance of 14 km in the NAM region in northern Mexico. The values obtained here (see Table 2) are very similar to those found by Mascaro et al. (2014) in summer for Δ*t* ≥ 30 min and in winter for 30 min ≤ Δ*t* ≤ 3 h. In addition to possible differences between the rainfall regimes of the two sites, the reasons for the discrepancies in other cases may be also due to (i) diverse tipping depth of the gauges (1 mm here and 0.254 mm in the other study) that may impact results at short aggregation times and (ii) extrapolation beyond a much smaller separation distance carried out by Mascaro et al. (2014) for large Δ*t* in winter.

Analysis of the correlation structure. Spatial correlograms [i.e., relation between CC and *d* (km)] of the rainfall rate at (a) Δ*t* = 30 min and (b) Δ*t* = 3 h resolutions. Pairs of stations have been grouped in bins of equal sample size and the lines are the fitted analytical model Eq. (2) with parameters *m* and *n*. The dashed horizontal lines indicate the value of CC = *e*^{−1}. (c) Correlation distance *t*) of the rainfall signal, as returned by the analytical model. Each panel reports results for the summer and winter seasons.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Analysis of the correlation structure. Spatial correlograms [i.e., relation between CC and *d* (km)] of the rainfall rate at (a) Δ*t* = 30 min and (b) Δ*t* = 3 h resolutions. Pairs of stations have been grouped in bins of equal sample size and the lines are the fitted analytical model Eq. (2) with parameters *m* and *n*. The dashed horizontal lines indicate the value of CC = *e*^{−1}. (c) Correlation distance *t*) of the rainfall signal, as returned by the analytical model. Each panel reports results for the summer and winter seasons.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Analysis of the correlation structure. Spatial correlograms [i.e., relation between CC and *d* (km)] of the rainfall rate at (a) Δ*t* = 30 min and (b) Δ*t* = 3 h resolutions. Pairs of stations have been grouped in bins of equal sample size and the lines are the fitted analytical model Eq. (2) with parameters *m* and *n*. The dashed horizontal lines indicate the value of CC = *e*^{−1}. (c) Correlation distance *t*) of the rainfall signal, as returned by the analytical model. Each panel reports results for the summer and winter seasons.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Parameters *m* and *n* of the analytical model Eq. (2) fitted to the experimental correlogram for different aggregation times Δ*t*. Parameters were estimated through linear regression in the log–log plane. The values of *m* are also equal to the correlation distance in kilometers

The spatial correlation structure was further investigated by calculating the directional correlograms. Figures 12a and 12b show polar diagrams of the correlation distances in the cardinal and intercardinal directions for Δ*t* = 30 min and Δ*t* = 3 h, respectively, and both seasons. Again, the differences between summer and winter are apparent in terms of magnitude of the correlation distance. In addition, results indicate that summer storms have an isotropic spatial structure with no preferential direction of movement or shape of single or multiple rain cells. In contrast, the correlation structure in winter has a preferential direction along the northeast–southwest axis. A possible reason for this finding is the combined effect of circulation patterns associated with westward flow from the Pacific Ocean (Sheppard et al. 2002) and the presence of orographic barriers running along the perpendicular northwest–southeast direction (Fig. 1). However, a thorough investigation of the storm generation mechanisms and circulation patterns that is out of the scope of this paper would be required to explain the underlying physical factors of this outcome.

Analysis of the directional correlation structure. Correlation distance calculated by fitting the analytical model Eq. (2) to the directional correlograms as a function of the direction. Results are shown for the rainfall rate at (a) Δ*t* = 30 min and (b) Δ*t* = 3 h resolutions in the summer and winter seasons.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Analysis of the directional correlation structure. Correlation distance calculated by fitting the analytical model Eq. (2) to the directional correlograms as a function of the direction. Results are shown for the rainfall rate at (a) Δ*t* = 30 min and (b) Δ*t* = 3 h resolutions in the summer and winter seasons.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Analysis of the directional correlation structure. Correlation distance calculated by fitting the analytical model Eq. (2) to the directional correlograms as a function of the direction. Results are shown for the rainfall rate at (a) Δ*t* = 30 min and (b) Δ*t* = 3 h resolutions in the summer and winter seasons.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

### e. Evidence of scaling regimes

The investigation of the scaling law Eq. (3) through the spectral analysis led to the identification of different regimes in summer and winter. Figure 13 shows the spectra obtained in the two seasons for two gauges chosen as examples, along with that obtained by averaging all stations. Note that the spectra were smoothed to reduce noise by dividing the frequencies into bins and averaging *α* estimated in all gauges and each regime are reported in Table 3. The regimes identified through the spectral analysis can be associated with the dominant meteorological phenomena within the respective time intervals (Fraedrich and Larnder 1993). The flat regions in both seasons are transition zones between intraseasonal variability and the next regime. In summer, the regime from 1 day to 2 h represents systems that occupy the area for multiple hours, likely due to transient disturbances propagating through the monsoon region (Adams and Comrie 1997), while the range from 2 h to 5 min is associated with single convective storms. Consistent with this interpretation, the corresponding values of *α* are smaller in the first regime (mean of 0.32) as compared to the second one (mean of 0.83), which is then characterized by a higher level of organization (Purdy et al. 2001). In winter, the regime from 5 days to 2 h is representative of large-scale cyclonic storms lasting multiple days (Sheppard et al. 2002). These systems are more organized than those associated with the range from 1 day to 2 h in summer, as revealed by the higher *α* (mean of 0.62). The two regimes found at shorter time scales characterize the behavior of multiple and single cells: the range from 2 h to 15 min is associated with the alternation of multiple storms less organized in time (low *α* with average of 0.11), while the regime from 15 to 5 min is related to single, more localized rain cells with a shorter lifetime (higher *α* with a mean of 0.62). Finally, the existence in both seasons of a regime from 5 to 2 min characterized by a flat spectrum (*α* close to 0) is likely due to the relatively high rain depth threshold of the tipping-bucket gauges that affects the signal sampled at the highest resolutions. As a result, this regime was not considered as representative of the rainfall physical properties.

Scaling regimes obtained through the spectral analysis for (a) summer and (b) winter rainfall events for two gauges (IDs 4500 and 6955) and averaged across all stations (all). The slopes of the regression lines are estimates of the spectral exponents *α* of Eq. (3). Arbitrary units on the *y* axis are used to display results from different gauges in the same graph.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Scaling regimes obtained through the spectral analysis for (a) summer and (b) winter rainfall events for two gauges (IDs 4500 and 6955) and averaged across all stations (all). The slopes of the regression lines are estimates of the spectral exponents *α* of Eq. (3). Arbitrary units on the *y* axis are used to display results from different gauges in the same graph.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Scaling regimes obtained through the spectral analysis for (a) summer and (b) winter rainfall events for two gauges (IDs 4500 and 6955) and averaged across all stations (all). The slopes of the regression lines are estimates of the spectral exponents *α* of Eq. (3). Arbitrary units on the *y* axis are used to display results from different gauges in the same graph.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Mean plus/minus std dev across the gauges of the spectral exponent *α* in the scaling regimes identified through the spectral analysis in the summer and winter seasons.

The scaling regimes and the values of *α* are consistent with those obtained in other regions of the world, with some differences due to local climatic features. Mascaro et al. (2014) conducted separate spectral analyses in winter and summer in the NAM region in northern Mexico, finding similar regimes in summer but only a single regime in winter (from 2 min to 17.1 h). Numerous studies that analyzed rainfall records in Europe, North America, and Asia (e.g., Fraedrich and Larnder 1993; Molini et al. 2009; Verrier et al. 2011; Rysman et al. 2013; Mascaro et al. 2013; Mandapaka and Qin 2015) found a single set of scaling regimes with no seasonal differences and three breaking points at 1–3 days, 2–3 h, and 2–30 min (the highest resolution of the rainfall signal). Results of previous studies are then comparable with the regimes found here in both seasons, except for the additional breaking point at 15 min identified in winter. This can be due to the presence of a limited number of isolated, high-intensity rain cells embedded within wintertime frontal systems. Finally, the values of *α* estimated in this work and the studies mentioned above vary within fairly consistent ranges in each regime: 0.35–0.60 in the time interval associated with frontal systems, and 0.8–1.5 in the range related to single storms.

The rainfall scaling properties were further investigated through the scale invariance analysis of Eq. (5). Figure 14 shows results in summer and winter for two representative gauges and for the structure functions for *q* = 3 averaged across all stations. The moment *q* = 3 was selected to present results of this analysis because it captures the features of rainfall extremes, while not being significantly biased by few high-intensity events. Rainfall exhibits three scaling regimes in summer—(i) from 91 to 11.4 days, (ii) from 11.4 days to 32 min, and (iii) from 32 to 1 min—and three in winter—(i) from 22 days to 17.1 h, (ii) from 17.1 h to 32 min, and (iii) from 32 to 1 min. As found previously (Verrier et al. 2011; Mascaro et al. 2013, 2014; Mandapaka and Qin 2015), scale invariance and spectral analyses reveal the presence of different regimes, likely due to diverse methods adopted to sample the signals at different resolutions. As for the spectral analysis, the regimes can be related to the dominant meteorological phenomena at the corresponding time scales: (i) a transition zone representative of intraseasonal variability at the larger scales, (ii) frontal systems or multiple storms at intermediate synoptic scales, and (iii) single storms at the smallest time scales. To quantify this analysis, Table 4 reports the mean and standard deviation of the multifractal exponent *K*(3), whose value increases with the signal intermittency. In summer, rainfall is more intermittent in the range from 11.4 days to 32 min because of uneven peaks separated by dry periods that emerge at shorter time scales [larger *K*(3)]. In contrast, within the time resolutions of single rain cells from 32 to 1 min, the statistical proprieties have less variability across scales [smaller *K*(3)]. In winter, the rainfall signal is less intermittent in the regime from 17.1 h (close to the daily resolution) to 32 min as compared to the range of single cells, consistent with the longer lifetime of the systems typical of this season. Results in terms of both scaling regimes and values of *K*(*q*) agree with previous studies that analyzed similar temporal scales (Mascaro et al. 2013, 2014; Mandapaka and Qin 2015).

Scaling regimes obtained through the scale invariance analysis for (a) summer and (b) winter rainfall events, shown as the relation between *y* axis are used to display results from different gauges in the same graph.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Scaling regimes obtained through the scale invariance analysis for (a) summer and (b) winter rainfall events, shown as the relation between *y* axis are used to display results from different gauges in the same graph.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Scaling regimes obtained through the scale invariance analysis for (a) summer and (b) winter rainfall events, shown as the relation between *y* axis are used to display results from different gauges in the same graph.

Citation: Journal of Hydrometeorology 18, 1; 10.1175/JHM-D-16-0167.1

Mean plus/minus std dev across the gauges of the multifractal exponents *K*(3) in the scaling regimes identified through the scale invariance analysis in the summer and winter seasons.

The spatial variability of the metrics *α* and *K*(3) was investigated through the relation with elevation, which was again quantified via *r*^{2} of the linear regression. No significant correlation is obtained between the characteristics of the rainfall spectra and elevation in both seasons (*r*^{2} always lower than 0.08). In contrast, the multifractal exponents in summer tend to decrease with altitude in the regime from 11.4 days to 32 min (*r*^{2} = 0.47), implying that storm systems have higher intermittency in the valley as compared to higher terrain. The trend is instead not significant in the range from 32 to 2 min (*r*^{2} = 0.05), indicating that single rain cells have similar multifractal properties that are not influenced by the local orography. These results further confirm the same type of analyses conducted by Mascaro et al. (2013) in the Mediterranean island of Sardinia and by Mascaro et al. (2014) in northern Mexico. Finally, in winter no significant relation was found between *K*(3) and elevation in all scaling regimes (*r*^{2} < 0.03). This result, along with analyses at annual scale, indicates that the orographic control on winter rainfall mostly affects the local mean, but it does not have a meaningful impact on the scaling properties.

## 5. Summary and conclusions

The statistical properties of rainfall in central Arizona were characterized using a network of high-resolution rain gauges, installed in the early 1980s and currently including 310 stations. A set of techniques was applied to investigate the properties across a wide range of temporal scales from 1 min to years. The availability of a dense network of gauges allowed for identifying the existence of spatial patterns and quantifying the control of terrain characteristics. Results can be summarized as follows.

Rainfall accumulated at annual scale varies significantly from year to year. This variability is partially explained by ENSO teleconnections, which lead to wetter winter seasons (from November to March) during El Niño years. The spatial variability of total annual rainfall is strongly controlled by elevation with values of 200 mm in desert valley locations (<500 m MSL) and more than 400 mm in higher terrain (>1500 m MSL). A marked elevational control is also found for the rainfall accumulated during the summer (from July to September) NAM season, which accounts, on average, for 33% of the annual total.

The rainfall regime is characterized by a clear seasonality with the presence of two distinct maxima in summer and winter of rainfall accumulation and mean number of rainy days. In these two seasons, the orographic control on the spatial variability of these metrics is stronger as compared to the rest of the year. The number of consecutive rainy days and the duration of single storms are instead relatively similar across all months, likely because these metrics capture general features of frontal systems and single storms that are common throughout the year.

The maximum rainfall intensities for different durations are, on average, larger in summer as compared to winter months. Differences are higher for durations up to 1 h and become negligible at daily scale. The spatial variability of the rainfall maxima is controlled by elevation for all durations during the monsoon season, suggesting that higher terrain enhances the strength of thermal convective activity. In winter, the effect of elevation on high-intensity events becomes important for durations larger than 3 h that capture the effect of multiple cells and frontal systems. The occurrence of severe wintertime single storms is instead not affected by terrain characteristics.

A strong diurnal cycle modulates summer thunderstorms with a marked elevational control: convective activity peaks first at higher elevations in the early afternoon, and subsequently, convective maximum is experienced in lower desert regions between late evening and early morning.

The spatial correlation structure of winter and summer rainfall is markedly different, as a result of the dominant weather systems in the two seasons. Specifically, the intergauge correlation of wintertime rainfall is high even at short aggregation times (<1 h) and always larger than the correlation computed in summer because of the widespread nature of wintertime rainfall and the more intermittent occurrence of monsoonal convective thunderstorms. The characteristic spatial scales of single cells (multiple storms or frontal systems) are 10.7 km (43.0 km) in summer and 23.2 km (540.2 km) in winter, as quantified through the correlation distances of the rainfall signal sampled at 30 min (12 h). In addition, the spatial correlation structure of summer rainfall is isotropic, while that of winter rainfall has a preferential direction along the northeast–southwest axis.

The spectral analysis revealed the presence of two significant scaling regimes in summer and three in winter. The scale invariance analysis indicated instead the presence of three regimes in both seasons with a common breaking point at 32 min and a different limit at larger time scales (11.4 days in summer and 17.1 h in winter). For both types of analyses, the regimes could be associated with the dominant meteorological phenomena at the corresponding time scales. Furthermore, while no significant spatial variability was identified through the spectral analysis, the scale invariance analysis revealed that the rainfall signal tends to be more intermittent at lower elevations.

Results of this work support and further expand with analyses of rainfall records from a large number of gauges the conclusions of previous studies focused on the meteorological aspects of the NAM in central Arizona and based on atmospheric soundings, remote sensing data, and numerical models. Specifically, the findings provide empirical evidence for the quantification of the effect of elevation on rainfall diurnal cycle and strength of thermal convection in summer (e.g., Carleton 1986; Balling and Brazel 1987). In addition, this study presents novel contributions aimed at characterizing the rainfall regime in central Arizona, including the systematic quantification of the statistical properties of winter rainfall, the analysis of the spatial structures of different storm systems, and evidence of scaling regimes. These results represent a necessary preliminary effort to inform the development and calibration of downscaling models of coarse rainfall products in time and/or space, including daily outputs of climate models, reanalysis products, and rainfall time series stochastically generated. Downscaling tools would then be highly desirable to improve the simulation of the hydrologic response of localized storms with short duration (<1 h) and high intensity typical of the region. This, in turn, will permit characterizing with higher accuracy occurrence and risk of flooding conditions in residential areas, the impacts of extreme rainfall on transportation and power infrastructures, and the feasibility of rainwater harvesting systems. Based on the evidence of scaling regimes, possible downscaling techniques that could be applied are those that reproduce the scale invariance and multifractal properties of rainfall, which have been successfully tested in the NAM region by Mascaro et al. (2014).

## Acknowledgments

The author thanks two anonymous reviewers for their comments that helped to improve the quality of the manuscript. Stephen D. Waters from the Flood Control District of Maricopa County is also thanked for providing the rainfall data of the network. Prof. Benjamin Ruddell, Prof. Enrique R. Vivoni, and Prof. Roberto Deidda are thanked for their guidance. Suggestions of Prof. Maria Grazia Badas are also highly appreciated. Funding is acknowledged from the National Science Foundation Grant EF1049251, “Assessing Decadal Climate Change Impacts on Urban Populations in the Southwestern United States.”

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