Spatio-temporal drought forecasting within Bayesian ...

Journal of Hydrology 512 (2014) 134–146

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Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol

Spatio-temporal drought forecasting within Bayesian networks Shahrbanou Madadgar, Hamid Moradkhani ⇑ Department of Civil and Environmental Engineering, Portland State University, Portland, OR 97201, USA

a r t i c l e

i n f o

Article history: Received 4 September 2013 Received in revised form 6 January 2014 Accepted 15 February 2014 Available online 5 March 2014 This manuscript was handled by Konstantine P. Georgakakos, Editor-in-Chief, with the assistance of Ashish Sharma, Associate Editor Keywords: Drought forecast Standardized Runoff Index Bayesian networks Copula Conditional probability

s u m m a r y Spatial variations of future droughts across the Gunnison River Basin in CO, USA, are evaluated in this study, using a recently developed probabilistic forecast model. The Standardized Runoff Index (SRI) is employed to analyze drought status across the spatial extent of the basin. The runoff generated at each spatial unit of the basin is estimated by a distributed-parameter and physically-based hydrologic model. Using the historical runoff at each spatial unit, a statistical forecast model is developed within Bayesian networks. The forecast model applies a family of multivariate distribution functions to forecast future drought conditions given the drought status in the past. Given the runoff in the past (January–June), the forecast model is applied in estimating the runoff across the basin in the forecast period (July– December). The main advantage of the forecast model is its probabilistic features in analyzing future droughts. It develops conditional probabilities of a given forecast variable, and returns the highest probable forecast along with an assessment of the uncertainty around that value. Bayesian networks can also estimate the probability of future droughts with different severities, given the drought status of the predictor period. Moreover, the model can be used to generate maps showing the runoff variation over the basin with the particular chance of occurrence in the future. Our results indicate that the statistical method applied in this study is a useful procedure in probabilistic forecast of future droughts given the spatio-temporal characteristics of droughts in the past. The techniques presented in this manuscript are suitable for probabilistic drought forecasting and have potential to improve drought characterization in different applications. Ó 2014 Published by Elsevier B.V.

1. Introduction The National Oceanic and Atmospheric Administration’s (NOAA, 2013) National Climate Data Center reported the year 2012 as the warmest year on record for the United States. Over the entire year of 2012, average temperatures of the contiguous United States were 3.2 °F above that of the 20th century. According to the U.S. Drought Monitor, more than 70% of the contiguous United States experienced some level of dry spells being classified from abnormal to exceptional droughts in 2012. The droughts of 2012 extended to the next year and approximately 58% of the contiguous United States was under drought conditions as of January 29, 2013. Many major rivers in the Western U.S., including the Colorado and the Rio Grande, had below average streamflow in the spring and summer of 2013. The ongoing droughts in the North America and many other regions across the globe are referred to the climate change and global warming effects (Trenberth, 2011; ⇑ Corresponding author. Address: 1930 SW 4th Avenue, Suite 200, Portland, OR 97201, USA. Tel.: +1 503 725 2436. E-mail addresses: [email protected] (S. Madadgar), [email protected] (H. Moradkhani). http://dx.doi.org/10.1016/j.jhydrol.2014.02.039 0022-1694/Ó 2014 Published by Elsevier B.V.

Peterson et al., 2012) and the frequency of future droughts is expected to be increasing, rather than decreasing (Sheffield and Wood, 2008; Dai, 2011). Consequently, a reliable hydrologic forecast for a region has a significant role in the efficient planning of available water resources. Droughts have strong impacts on the water supply and quality; society and public health; crop production and agriculture; plants, wild fires, and living environments. A variety of studies in the past decades have examined the different aspects of drought events, such as developing different drought indicators (Niemeyer, 2008; Mishra and Singh, 2010), monitoring and characterizing the droughts (Andreadis and Lettenmaier, 2006; Shukla et al., 2011; Shiau, 2006; Dupuis, 2007), climate change impacts on future droughts (Ghosh and Mujumdar, 2007; Sheffield and Wood, 2008; Burke et al., 2010; Moradkhani et al., 2010; Risley et al., 2011; Madadgar and Moradkhani, 2011), and developing early warning systems to survive in drought conditions (Huang and Chou, 2008). There are also a number of studies focused on drought forecasting and estimating the likely drought conditions in the future. In an earlier study, Karl et al. (1987) evaluated the probability of receiving a sufficient amount of precipitation to recover from an ongoing drought over a particular period of time. They rewrote the

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Palmer Drought Severity Index (PDSI; Palmer, 1965) and utilized the unconditional gamma distribution to obtain the probabilities of future droughts. However, this is disputed for ignoring the dependency and auto-correlation in a precipitation record. Since then, several other methods have been developed and tested in drought forecasting such as Markov Chain model (Lohani and Loganathan, 1997; Steinemann, 2003), stochastic renewal models (Kendall and Dracup, 1992; Loaiciga and Leipnik, 1996), stochastic autoregressive models (Mishra and Desai, 2005), and artificial neural networks (Mishra and Desai, 2006; Barros and Bowden, 2008). Cancelliere et al. (2007) discussed the validity of Markov Chain model for making significant errors in transition probabilities of the Standardized Precipitation Index (SPI; McKee et al., 1993) and then derived the transition probability matrix by an analytical study on the statistics underlying the SPI equations. However, Madadgar and Moradkhani (2013) recently argued, in an analytical framework, that the assumption of independent and normally distributed aggregated precipitation volumes is not always true, especially for other hydrologic variables such as runoff and streamflow. They also discussed that for frequency analysis of different drought states, the intense process of obtaining the transition probability matrix from the index equation could be avoided by using multivariate modeling based on copula functions (Joe, 1997). In several other studies, the climate forecast products of NOAA Climate Prediction Center (CPC) are used for developing probabilistic drought forecasts (Carbone and Dow, 2005; Hwang and Carbone, 2009). However, Steinemann (2006) discussed the poor interpretation of forecast probability and uncertainty information supported by CPC forecast products in real applications. This study extends the application of the recently developed model in drought forecasting by Madadgar and Moradkhani (2013). In the previous application, the forecast model predicted the future droughts of the Gunnison River Basin (GRB) in Colorado, USA, using the flow volume at the basin outlet. The promising results of that study have encouraged the authors to apply their forecast model in estimating the spatial variation of future droughts using the runoff volume at different grid cells across the basin. Copulas (Joe, 1997; Nelsen, 1999), as the main core of the proposed forecast model, are multivariate distribution functions that join the marginal distributions of the variables with some level of dependency and correlation. According to the existent correlations among the hydrologic variables like runoff, streamflow, groundwater level, and many other variables, the copula functions have been examined in different hydrologic applications over the past few years (e.g. Favre et al., 2004; Bárdossy, 2006; Shiau, 2006; Dupuis, 2007; Zhang and Singh, 2007; Salvadori and De Michele, 2010; Kao and Govindaraju, 2008, 2010; Madadgar and Moradkhani, 2011; Madadgar et al., 2012). In drought applications, copulas have been used to characterize the future droughts in terms of estimating the return period of droughts’ severity, intensity, and duration, (Shiau, 2006; Dupuis, 2007; Kao and Govindaraju, 2010; Wong et al., 2010; Madadgar and Moradkhani, 2011). However, in a recent study by Madadgar and Moradkhani (2013), a new application of copula functions in drought forecasting was examined, where instead of analyzing the drought characteristics; they studied the occurrence probability of seasonal droughts regarding streamflow observations. They defined the conditional probability of streamflow via Bayesian networks. According to the correlations among the streamflow of consecutive seasons, the conditional probabilities of seasonal droughts were analyzed using the copula functions adopted in a Bayesian framework. This study aims at extending the application of the proposed copula-based method to estimate the spatial variation of future drought probabilities across the GRB. For this purpose, the runoff generated across the basin is used to evaluate the spatial variation of droughts; while in the previous

study, the streamflow observations at a particular point were used for drought forecasting. The paper is organized as follows. Section 2 explains the drought index to evaluate droughts based on the runoff volume at each spatial unit across the basin. Section 3 describes the hydrologic modeling of GRB and analyzes the historical droughts of the basin. Section 4 elaborates on the probabilistic forecast methodology employed in this study, and is followed by a discussion on the required analyses to apply the forecast model (Section 5). Section 6 demonstrates some forecast products for the study basin and discusses the results. Finally, Section 7 summarizes the major remarks of the study.

2. Standardized Runoff Index (SRI) Standardized Runoff Index (SRI; Shukla and Wood, 2008) is used to evaluate the spatial variation of the hydrologic drought across the study area. As with all Standardized Indices (SI), the SRI of each spatial unit reflects the anomalies of surface runoff from its average value generated in the same unit. Mathematically, SRI is defined as the standardized normal variable of the accumulated surface runoff over a specific time period:

SRIsyr;m;k ¼ /1 ðusyr;m;k Þ usyr;m;k ¼ F X syr;m;k ðX syr;m;k Þ X syr;m;k ¼

ð1Þ

mþk1 X

ysyr;i

i¼m

where usyr;m;k is the probability of the total runoff volume at the spatial unit s in year yr over k months starting from month m; F() is the marginal distribution of accumulated runoff (X syr;m;k ); and ysðÞ is the monthly runoff of the spatial unit s. Therefore, the SRI calculation starts with fitting an appropriate marginal distribution to the total runoff volume over k months and computing the standardized normal variable for each aggregated runoff volume. According to Eq. (1), separate marginal distributions should be fitted to the accumulated runoff beginning from different months to obtain the SRI variation over time for each spatial unit. As explained by Madadgar and Moradkhani (2013), Eq. (1) can preserve the natural periodicity (seasonality) of surface runoff, where the runoff variation among the low-flow and high-flow seasons is appropriately reflected in the definition of SRI. Once the SRI is estimated for each spatial unit, the drought status of each unit can be determined by the U.S. Drought Monitor classification for the standardized drought indices (Table 1). Using this classification, one out of five drought categories can be recognized for a region at any time. The SRI = 0.5 separates the dry periods from the wet periods, while the variation in water availability during a time horizon results in a dynamic transition either between dry and wet spells, or among various drought categories.

Table 1 Drought classification by the U.S. Drought Monitor (http://droughtmonitor.unl.edu/) for the Standardized Indices (SI). Drought category

Drought severity

SI value

D0 D1 D2 D3 D4

Abnormally dry Moderate drought Severe drought Extreme drought Exceptional drought

0.5 0.8 1.3 1.6 2.0

to to to to or

0.7 1.2 1.5 1.9 less

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3. Hydrologic modeling of study area The study basin for the application of drought forecast model is the Gunnison River Basin, one of the headwater sub-basins of the Colorado River Basin (CRB), located in the southwestern United States (DeChant and Moradkhani, 2011a,b). The CRB is divided into upper and lower portions and encompasses parts of seven States: Wyoming, Colorado, Utah, Nevada, California, New Mexico, and Arizona. The GRB consists of seven hydrologic units defined by the U.S. Geological Survey (USGS). Its total drainage area is 2072 mi2 at the conjunction of two upstream reaches: the Tomichi Creek and the Gunnison (Fig. 1). The basin elevation varies between 2348 m and 4221 m; and spring and summer runoff is mostly from the snowmelt from the previous winter. To model the hydrologic response of the study area and evaluate the spatial variation of surface runoff generated across the basin, the region is divided into 37 grid cells with 1/8th degree resolution (12 km), as shown in Fig. 1. A distributed-parameter and physically-based hydrologic model, the USGS Precipitation Runoff Modeling System (PRMS; Leavesley et al., 1983), is used to estimate the runoff volume generated at each grid cell of GRB. PRMS has been widely applied in estimating the hydrologic variables such as runoff and streamflow, and has been recently used in impact analyses of climate change on different regions (e.g. Risley et al., 2011; Najafi et al., 2011; Jung et al., 2011; Madadgar et al., 2012). To apply the PRMS, the basin is partitioned into several Hydrologic Response Units (HRUs), each with various parameters to be calibrated. The predicted runoff in each HRU is the output of a series of conceptual reservoirs including an impervious zone, soil zone, and subsurface and groundwater reservoirs. The final outflow of the basin is the total routed runoffs of all the HRUs that reach the basin outlet at the same time. The HRUs for GRB are set the same as the grid cells shown in Fig. 1.

WA

For the hydrologic modeling of the basin, PRMS requires the daily maximum and minimum temperature, and precipitation for each HRU. The Inverse Distance Squared Weighting (IDSW) method is used to spatially distribute the daily records of a group of SNOTEL and COOP stations among the HRUs. In the IDSW method, the interpolation weights are calculated proportional to the squared inverse distance of the HRUs to the measuring sites. Hence, the measurement sites share more information with the nearby HRUs. Parameters of the hydrologic model are calibrated using the Shuffled Complex Evolution (SCE) global search algorithm (Duan et al., 1994) with the objective function of maximizing the Nash Sutcliffe Efficiency (NSE) over the daily record of the basin outflow:

NSE ¼ 1 

1

r2yobs

"

T 1X 2 ðyt  ytobs Þ T t¼1 sim

# ð2Þ

where ytsim and ytobs are the modeled and observed streamflow at time t, respectively; r2yobs is the variance of observations; and T is the length of the observation record. The GRB is not gauged at its outlet; hence, the unregulated flow of the basin is obtained by adding up the gauge readings of the upstream reaches immediately before the basin outlet at USGS 09119000 (Tomichi Creek River) and USGS 09114500 (Gunnison River). Model parameters are calibrated and validated over the periods of 1979–1989 and 1989–2011, respectively, with the associated NSEs equal to 0.7 and 0.72. According to these measures, the model performance seems reliable, and the runoff for each HRU is assumed to be acceptable. Since the actual runoff is not measurable, this study relies on the simulated runoff by PRMS for the drought assessment. To show the PRMS performance, Fig. 2a plots the modeled against the measured daily outflow of the basin for the entire period of 1979–2011. As seen, the estimated flow is similar to

Gunnison River Basin (GRB)

MT

OR

WY CA

ID NV UT

Upper CRB

Lake Powell

*#

CO

HRU

Lower CRB

Sub-basins of GRB AZ

NM

0

5 10

20

Miles

Fig. 1. Gunnison River Basin, a sub-basin of the Colorado River Basin (CRB), located in southwestern United States.

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12,000 Measured Simulated

Calibration Validation

Basin Outflow [cfs]

10,000

8,000

6,000

4,000

2,000

10/01/1979

10/01/1983

10/01/1987

10/01/1991

10/01/1995

10/01/1999

10/01/2003

10/01/2007

09/30/2011

(a) 4000 Measured 3500

Simulated

Monthly Flow [cfs]

3000

2500

2000

1500

1000

500

0

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

(b) Fig. 2. Comparison of the measured and simulated outflow of the GRB during 1979–2011 in the form of (a) daily timeseries, and (b) interquartile range of monthly mean streamflow.

observed flow specifically in low-flow seasons. The interquartile range of monthly mean streamflow during 1979–2011 is shown in Fig. 2b. Generally, the peak flows occur in June and the low-flow season begins in July when the hydrograph starts descending. Furthermore, the model can capture the low flows better than the high flows. The main reason can be attributed to the elevation of the highest available station whose measurements are used as climate input to PRMS. The highest station is located at the elevation of 3523 m, while the highest elevation of the basin is 4221 m. For a snow-dominated basin like GRB, where the snowmelt plays a

significant role in the basin outflow, missing the climate data of elevated areas can cause under-estimating the high flows (Fig. 2a–b). In this regard, a recent study by Jung et al. (2012) showed the high sensitivity of the hydrologic models’ performance to their parameters in snow-dominated basins. 3.1. Historical droughts in GRB Water supply in the western US is highly dependent on the snowmelt generated from the higher elevation areas and also large

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scale climate predictors (Moradkhani and Meier, 2010; Najafi et al., 2012). Decreased high elevation snowpack has caused droughts of varying intensities in Upper CRB during recent decades, while the most severe one occurred in 2002. Recently, snowpack depletion and early meltout in spring 2012 caused a widespread drought in the states of Colorado, Utah, and Wyoming (reported by Western Water Assessment (WWA) and National Integrated Drought Information System (NIDIS), July 2012). It triggered below-average streamflow, poor pasture and crop conditions, and region-wide wildfires. Despite the low inflow in 2012, the carryover of local reservoirs from the past couple of wet years (2010–2011) could mitigate the drought impact on the water supply throughout the region. Spring of 2012 was the 2nd warmest spring on record in the state of Colorado and the 4th driest spring in Upper CRB since 1900. Continued warm and dry climate increases the probability of contiguous droughts over the region, thus affecting the irrigation and crop production. As reported by NOAA’s National Climate Data Center (NOAA, March 2013), the regions across the Central Plains and Mountain West have already received the below-average precipitation during winter 2013, and are likely to have another dry summer for the second year in a row. While the recent droughts might be the signature of global warming impacts on extreme events across the world, the reliable drought forecast across the CRB seems to be essential for planning and managing the available water in the future. Observed streamflow of the GRB clearly shows the drought of 2002 in Upper CRB (Fig. 3a). The Standardized Streamflow Index (SSI), as plotted in Fig. 3a, shows the drought conditions with respect to the measured streamflow at the basin outlet. Both SSI and SRI are from the SI family of drought indices, where SSI is in respect with the observed streamflow and the SRI is in respect with the predicted runoff. Similar to all other SIs, the line SSI = 0.5 separates the dry and wet periods. As seen in Fig. 3a, the GRB undergoes continuous drought conditions with differing severities from 2000 to 2005. Fig. 3b shows drought condition of the basin given the surface runoff at each HRU in January (on the left) and July (on the right) for the year 2000–2005. Drought categories (D4 to Normal status) in each HRU are determined according to the SRI with 6-month runoff accumulation (Eq. (1)). As seen, the drought of 2002 encompasses the entire basin. Generally, July runoff predictions of 2000–2005 indicate stronger droughts than their respective January runoff predictions, despite the larger monthly mean streamflow in July (Fig. 2b). As discussed in Section 2, drought events over a particular time window are defined relative to the associated average condition. Therefore, drought conditions in high-flow seasons might have totally different characteristics than drought conditions in low-flow seasons. The drought maps in Fig. 3b also indicate that the SRI acquired from estimated runoff by PRMS capture the drought events during 2000–2005.

1.5 1 0.5 0

SSI

138

-0.5 -1 -1.5 -2 -2.5 2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

Year

(a)

2000

2001

2002

2003

D4 D3 D2 D1

2005

2004

D0 Normal / Wet

(b) Fig. 3. Droughts in GRB during the past years, (a) SSI with respect to the observed streamflow at the basin outlet. The line SSI = 0.5 is the threshold to separate the dry and wet periods, (b) spatial variation of drought events throughout the basin. The drought maps are shown for January (on the left) and July (on the right) for each year.

4. Probabilistic drought forecast methodology Drought is an evolving extreme event that occurs over a given period of time. Drought status of a region at any time depends on the water availability (precipitation, soil moisture, runoff, etc.), within the past few months or seasons. In other words, water availability in the past plays a significant role in future drought status. Since drought-related variables (e.g. runoff, streamflow, drought indices, etc.) are statistically dependent on their past status (Madadgar and Moradkhani, 2013), they can be expressed within the Bayesian networks (Pearl, 1985), which are capable of describing the conditional dependencies of a set of random variables with a Directed Acyclic Graph (DAG). A DAG (Thulasiraman and Swamy, 1992) consists of the sequence of events or random variables with direct ordering, such as time evolving events.

Assuming x ¼ fxt1 ; . . . ; xtn g as a time-evolving random variable, the joint probability density function of xt1 ; . . . ; xtn is defined as the product of individual conditional probability density functions within a Bayesian network:

f ðxt1 ; . . . ; xtn Þ ¼

Y

f ðxti jxt1 ; . . . ; xti1 Þ

ð3Þ

ti 2T

where f ðxt1 ; . . . ; xtn Þ is the joint probability density function of dependent random variables and (xt1 ; . . . ; xtn ) denote different random variables with a level of dependency; however, in our application, they refer to the same variable such as runoff that varies in time. The conditional pdf at the right-hand side of Eq. (3) can be found by expanding the equation:

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f ðxt1 ; . . . ; xtn Þ ¼

Y

[0, 1], i.e. C:[0, 1]n ? [0, 1] (Joe, 1997; Nelsen, 1999). Supported by Sklar’s theorem (Sklar, 1959), copula functions can express a multivariate distribution, F(x1, . . ., xn), as follows:

f ðxti jxt1 ; . . . ; xti1 Þ

t i 2ft1 ;...;t n g

f ðxt1 ; . . . ; xtn Þ ¼ f ðxtn jxt1 ; . . . ; xtn1 Þ

Y

f ðxti jxt1 ; . . . ; xti1 Þ ð4Þ

t i 2ft 1 ;...;tn1 g

f ðxt1 ; . . . ; xtn Þ ¼ f ðxtn jxt1 ; . . . ; xtn1 Þf ðxt1 ; . . . ; xtn1 Þ ) f ðxtn jxt1 ; . . . ; xtn1 Þ ¼

Fðx1 ; . . . ; xi ; . . . ; xn Þ ¼ C½F X1 ðx1 Þ; . . . ; F X i ðxi Þ; . . . ; F X n ðxn Þ ¼ Cðu1 ; . . . ; ui ; . . . ; un Þ

f ðxt1 ; . . . ; xtn Þ f ðxt1 ; . . . ; xtn1 Þ

where the product of conditional pdfs in the second line of Eq. (4) is replaced by the associated joint pdf (see Eq. (3)). While direct calculation of conditional pdf from the joint pdfs in Eq. (4) would require intense analytical efforts, Madadgar and Moradkhani (2013) used a family of powerful statistical functions called copula functions to decompose the joint pdfs in the right hand side of Eq. (4) and rewrote the equation as follows (for n = 2):

f ðxt2 jxt1 Þ ¼ cðut2 ; ut1 ÞfX t2 ðxt2 Þ

ð5Þ

where cðut2 ; ut1 Þ is the pdf of copula function which joins the marginal distributions (ut2 ; ut1 ) of the random variables (xt1 ; xt2 ). Copulas are multivariate distribution functions on the n dimensional unit cube with uniform marginals on the interval

ð6Þ

where C refers to the Cumulative Distribution Function (CDF) of copula; and F Xi ðxi Þ is the marginal distribution of the ith variable, which is denoted by variable ui in the cdf of copula. A copula should satisfy the ‘‘boundary’’ and ‘‘increasing’’ conditions defined as follows (for a bivariate copula):

Cðu1 ; 0Þ ¼ Cð0; u2 Þ ¼ 0 Cðu1 ; 1Þ ¼ u1 Cð1; u2 Þ ¼ u2

ð7Þ

Cðu12 ; u22 Þ þ Cðu11 ; u21 Þ P Cðu12 ; u21 Þ þ Cðu11 ; u22 Þ for u11 6 u12

and u21 6 u22

The pdf of copula with an absolutely continuous cdf is defined as follows:

Table 2 Summary of 10, 50, and 90 percentiles of correlation coefficients over all HRUs. In each HRU, the Pearson’s correlation coefficient is estimated for each pair of accumulated runoff (with a particular starting month) having been transformed to the unit interval [0, 1]. The grey cells are associated with the forecast months being six months later than each predictor month in the first column on the left. More details are available in Section 5. Forecast month of the same year 10%

February

March

April

May

June

July

August

September

October

November

December

Predictor month January 1 February 0.24 March 0.26 April 0.26 May 0.21 June 0.18 July 0.18 August 0.10 September 0.07 October 0.12 November 0.21 December 0.41

January

0.90 1 0.25 0.25 0.22 0.18 0.17 0.11 0.07 0.12 0.33 0.14

0.75 0.93 1 0.21 0.21 0.17 0.17 0.10 0.10 0.13 0.39 0.02

0.73 0.85 0.97 1 0.20 0.15 0.16 0.11 0.08 0.15 0.42 0.04

0.64 0.79 0.91 0.90 1 0.17 0.18 0.12 0.08 0.15 0.52 0.13

0.44 0.77 0.84 0.89 0.94 1 0.17 0.12 0.08 0.12 0.50 0.24

0.43 0.58 0.71 0.73 0.82 0.92 1 0.13 0.09 0.20 0.41 0.15

0.29 0.30 0.46 0.50 0.59 0.68 0.81 1 0.11 0.27 0.34 0.08

0.00 0.00 0.17 0.17 0.24 0.28 0.39 0.45 1 0.26 0.30 0.17

0.02 0.03 0.08 0.09 0.08 0.14 0.09 0.17 0.50 1 0.31 0.13

0.13 0.11 0.13 0.12 0.12 0.07 0.08 0.02 0.01 0.06 1 0.12

0.29 0.28 0.29 0.29 0.25 0.21 0.20 0.12 0.06 0.07 0.32 1

50% Predictor month January 1 February 0.00 March 0.02 April 0.05 May 0.03 June 0.04 July 0.04 August 0.05 September 0.12 October 0.08 November 0.44 December 0.95

1.00 1 0.00 0.04 0.04 0.04 0.02 0.03 0.09 0.04 0.40 0.94

0.98 1.00 1 0.02 0.04 0.05 0.04 0.05 0.05 0.02 0.37 0.89

0.97 0.99 1.00 1 0.01 0.05 0.03 0.05 0.05 0.01 0.36 0.87

0.91 0.95 0.97 0.97 1 0.08 0.04 0.07 0.05 0.05 0.27 0.82

0.90 0.93 0.94 0.95 0.98 1 0.05 0.09 0.08 0.01 0.37 0.80

0.87 0.91 0.92 0.93 0.94 0.97 1 0.11 0.07 0.02 0.29 0.64

0.80 0.85 0.87 0.86 0.87 0.89 0.93 1 0.08 0.02 0.15 0.54

0.62 0.67 0.68 0.68 0.69 0.74 0.69 0.79 1 0.01 0.03 0.33

0.39 0.45 0.45 0.43 0.44 0.40 0.46 0.50 0.73 1 0.02 0.24

0.11 0.10 0.09 0.10 0.10 0.14 0.13 0.15 0.20 0.32 1 0.11

0.04 0.05 0.03 0.02 0.02 0.00 0.01 0.03 0.12 0.09 0.55 1

1.00 1.00 1.00 1 0.15 0.18 0.20 0.19 0.24 0.30 0.88 0.99

0.96 0.97 0.98 0.99 1 0.21 0.21 0.23 0.26 0.24 0.83 0.94

0.95 0.96 0.97 0.98 0.99 1 0.25 0.25 0.26 0.26 0.82 0.93

0.94 0.95 0.96 0.96 0.98 0.99 1 0.25 0.27 0.19 0.73 0.92

0.90 0.92 0.92 0.94 0.96 0.97 0.99 1 0.21 0.15 0.54 0.88

0.82 0.85 0.87 0.87 0.91 0.92 0.95 0.98 1 0.20 0.25 0.73

0.67 0.76 0.78 0.79 0.82 0.83 0.88 0.94 0.98 1 0.19 0.45

90% Predictor month January 1 1.00 1.00 February 0.25 1 1.00 March 0.24 0.23 1 April 0.25 0.23 0.20 May 0.17 0.18 0.16 June 0.20 0.19 0.20 July 0.20 0.19 0.19 August 0.21 0.20 0.21 September 0.30 0.27 0.23 October 0.48 0.36 0.32 November 0.98 0.96 0.87 December 1.00 1.00 0.99 Forecast month of the next year

0.37 0.41 0.41 0.42 0.42 0.41 0.46 0.51 0.53 0.70 1 0.27

0.27 0.27 0.24 0.24 0.24 0.18 0.21 0.25 0.35 0.43 0.92 1

140

cðu1 ; . . . ; un Þ ¼

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@ n Cðu1 ; . . . ; un Þ @u1 ; . . . ; @un

ð8Þ

The joint probability density function of (x1, x2, . . ., xn) can be expressed as the product of the pdf of copula (Eq. (8)) and the marginal density function of each variable:

f ðx1 ; . . . ; xn Þ ¼ cðu1 ; . . . ; un Þ

n Y

fX i ðxi Þ

ð9Þ

i¼1

Replacing the joint density functions in Eq. (4) with the righthand side of Eq. (9), the conditional pdf of Eq. (4) is simplified to Eq. (5) in a bivariate case. The advantage of Eq. (5) over Eq. (4) is that using the copula function avoids the excessive effort to find the joint density function of the random variables. It might be analytically difficult to directly model the joint behavior of hydrologic variables, whereas fitting a copula function to the marginal distributions takes less effort. Using Eq. (5) and the same methodology introduced by Madadgar and Moradkhani (2013), the probabilistic forecast of drought status in time t2 given the drought condition at time t1 is examined for the runoff volume of each HRU. Since this study utilizes the 6 month SRI to estimate the drought status of the basin, the runoff volume should be accumulated over 6 months for drought forecasting. Therefore, xt1 and xt2 in Eq. (5) are defined as the runoff volume accumulated over 6 months, beginning from the predictor month t1 and the forecast month t2, respectively. The predictor month is set to January and the forecast month is set to July, with a 6-month accumulation window (i.e. January–June, and July– December, respectively). A lapse of 6 months is fitted between the predictor and forecast months to avoid an overlap of the accumulation periods. In this study, the drought status at the forecast month is assumed to be only dependent on the drought status of one timestep behind, although the drought status may show more persistence in time. Exploring the dependency of drought conditions within several consecutive time steps may inspire future studies. However, the auto-correlation of hydrologic variables fades as the time span grows; and recent study by Madadgar and Moradkhani (2013) showed that the highest auto-correlation in the observed flow of GRB is between two consecutive time steps. After fitting a copula function to the marginal distributions for a particular HRU, the only knowledge required to estimate the drought status of the forecast month (t2) via Eq. (5) is the runoff volume of the predictor month (t1). Hence, there is no need for an initial guess of the forecast variable by the hydrologic model. Using Eq. (5) relaxes the need for a primary estimate of the accumulated runoff over the forecast period (July–December); and this is an advantage of using the proposed technique over other hydrologic forecast methods. The proposed technique only requires the knowledge about the predictor and forecast variables to establish the forecast model communicating between them. Although it is a purely statistical forecast model, the parameter estimation of the copula function and marginal distributions (Eq. (5)) is totally based on the joint behavior of the predictor and forecast variables in the past. The forecast model is setup with runoff estimation by PRMS during a historical time period, where the physical processes within the basin are appropriately considered by the hydrologic model, and consequently the inherent dependencies among the predictor and forecast variables are preserved.

5. Correlation analysis and copula fitting Drought forecasts in the GRB, based on the runoff volume produced in each of the 37 HRUs of the basin (Fig. 1), are examined through the copula-based technique described in the previous

section. The runoff of each HRU is accumulated over six months, beginning from different months (January–December) during 32 years from 1979 to 2011. To find the predictor and forecast months with reasonable dependency in associated accumulated runoffs, a correlation analysis is examined for different months (Table 2). For each HRU, the accumulated runoff over six months are transformed to the unit interval [0, 1] by the associated marginal distributions, and then the Pearson correlation coefficient is obtained for any possible pair of transformed variable:

qX;Y ¼

Cov ðX; YÞ

rX rY

ð10Þ

where X and Y are the transformed accumulated runoff over six months beginning from months m1 and m2. The transformed variables, X and Y, are identical to us:;m1 ;k and us:;m2 ;k in Eq. (1) for the spatial unit s. A correlation matrix is then obtained for each spatial unit (HRU). Table 2 summarizes the 10, 50, and 90 percentiles of correlation coefficients over all HRUs. For instance, the correlation coefficient of 0.44 in row January and column June in 10% matrix means that the correlation coefficient of 10% of the HRUs is less than 0.44 for the transformed accumulated runoff beginning from months January and June. The correlation coefficient for the same periods increase to 0.9 and 0.95 for 50 and 90 percent of the HRUs, respectively, which indicates a rather high correlation for these particular periods. According to the window size of six months for accumulated runoff, the beginning of the forecast period should be located later than six months from the beginning of the predictor period. Otherwise, the forecast period would have some overlap with the predictor period. For January to June (as the beginning of the predictor period), the forecast period with 6-month lag is issued at some time in the same year (upper triangle of Table 2). However, for July to December, the forecast period locates in the next coming year (lower triangle). As seen, the correlation matrix of Table 2 is not symmetric; e.g. correlation coefficients of January/July and July/January are not equal. In the January/July case, January is the beginning of the predictor period and the correlation coefficient with the forecast period beginning from July would be 0.87 in the 50% table. Otherwise, if July is chosen as the beginning of the predictor period (July/January case), the correlation coefficient significantly decreases to 0.04. In general, the numbers in lower triangle of Table 2 are smaller than those in upper triangle. The reason should be explored in the coherence of monthly runoff for consecutive months. As shown in Fig. 2b, the outflow of the basin is much larger during April–July than the rest of the year. Therefore, if some highflow months fit in a 6-month accumulation period, the aggregated runoff would suddenly increase, and the corresponding marginal probability would move towards the tail of the distribution. According to the influence of high-flows in increasing the accumulated runoff, a high correlation is guaranteed if the high flows in the accumulation window for predictor month are followed by the high flows in accumulation window for the forecast month. In other words, the high flows should occur in consecutive months when the predictor and forecast periods are connected adjacently. Looking at Fig. 2b, the predictor period beginning from January, which includes April–May–June in its accumulation window, is followed by the high flows of July and August (of the same year). Hence, the high flows should be from consecutive months to expect a high correlation. This is the reason that the upper triangle of the correlation matrix has greater values than the lower triangle (Table 2). In July/January case, the high flows in predictor period (July and August) are not followed by the high flows in the forecast period and a low correlation is thus expected for that case. Summaries of 10, 50, and 90 percent of correlation coefficients over all HRUs indicate that the most correlated predictor and forecast periods are January–June and July–December, respectively. The correlation coefficient decays for later predictor periods, and in general, the

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S. Madadgar, H. Moradkhani / Journal of Hydrology 512 (2014) 134–146 Table 3 Summary of bivariate copula functions in Archimedean and Elliptical families. Copula

Function

Gaussian Cðu1 ; u2 Þ ¼

Support Z

U1 ðu2 Þ

Z

U1 ðu1 Þ



1 1

1

1

2pð1  q2 Þ2

exp 

x21

x22



þ  2qx1 x2 dx1 dx2 2ð1  q2 Þ

x1, x2 e R

u1 = U(x1), u2 = U(x2)

q: Linear correlation coefficient U: Standard normal cumulative distribution function t Cðu1 ; u2 Þ ¼

Gumbel Clayton Frank

Z

t 1 v ðu2 Þ

Z

t 1 v ðu1 Þ

1 1

 ðv þ2Þ=2 x2 þ x22  2qx1 x2 exp 1 þ 1 dx1 dx2 v ð1  q2 Þ

1 1 2pð1  q u1 = tv(x1), u2 = tv(x2) tm: Cumulative distribution function of t distribution with m degree of freedom C(u1, u2) = exp{[(ln u1)h + (ln u2)h]1/h} h: Measure of dependency between u1 and u2 (estimated by correlation coefficients)  1=h h Cðu1 ; u2 Þ ¼ uh 1 þ u2  1  hu    e 1  1 ehu2  1 1 Cðu1 ; u2 Þ ¼  ln 1 þ h h e 1 2 Þ2

predictor periods beginning from the winter months (January, February, March) show high correlation with the forecast periods beginning from the summer months (July, August, September). Thus, January–June is selected as the predictor period and July– December is set to be the forecast period to evaluate our forecast technique in the remainder of this paper. The primary assignment to apply Eq. (5) in drought forecasting is to find a copula function to appropriately join the marginal distributions of correlated and dependent variables. Table 3 summarizes the copula functions that are tested in this study. Gumbel, Clayton, and Frank copulas are known as the Archimedean copulas; and the rest (t and Gaussian copulas) come from the Elliptical family of copulas (Embrechts et al., 2003; Nelsen, 1999). The method of Inference Function for Margins (IFM; Joe, 1997) is used to estimate the parameters of copula functions. Among several goodness-of-fit (GOF) tests for selecting the best alternative in a group of copulas (Genest et al., 2009), we employ the method of parametric bootstrapping introduced by Genest and Rémillard (2008). This GOF test returns the Cramér-von Mises statistic (S) as a measure of distance between the empirical and parametric copulas (CEmp and Ch, respectively), fitted to n data points:

Z

DCðuÞ2 dCðuÞ pffiffiffi DC ¼ nðC Emp  C h Þ S¼

u

ð11Þ

The p-value of the test is found by bootstrap sampling (set for 1000 replications, here) via the Monte Carlo approach (Genest et al., 2009). The null hypothesis of the test is the acceptance of parametric copula (H0:CEmp e Ch), which is evaluated by the p-value. For a particular copula, the p-value is sufficient to determine the acceptance/rejection of the null hypothesis with the significance level of a; whereas, in a group of different copulas, the best alternative is the one with the smallest S and the greatest p-value. For each HRU, a separate copula is required to join the marginal distributions of the accumulated runoff during the predictor and forecast periods (January–June and July–December). Thus, each HRU is assigned a particular copula function, set as the best fit from those listed in Table 3. Before finding the best choice of copula function, several marginal distributions are tested to fit the accumulated runoff volume generated at each HRU. Gamma, Generalized Extreme Value, Lognormal, Gaussian, Weibull, Gumbel, and Exponential distributions are the alternatives. The method of Maximum Likelihood Estimation (MLE) is used for parameter estimation of each distribution, and the best marginal distribution is then found by the Kolmogorov–Smirnov (K–S) and the Akaike Information Criterion (AIC; Akaike, 1974) tests. The K–S test returns the p-value to evaluate

x1, x2 e R

h e [1, 1) h e (0, h) heR

the acceptance/rejection of each parametric distribution given a significance level, and the AIC test compares the goodness-of-fit of different distributions. A distribution should pass the K–S test first, and then its superiority to other competing distributions is evaluated by the AIC test, where the one with smallest AIC is designated as the best fit. 6. Drought forecasting products Future droughts can be analyzed from different perspectives and various products can be generated using the methodology explained in this study. One useful analysis is the forecast of droughtrelated variables such as runoff, soil moisture, and snow pack across the study area. It would be even more useful to estimate the uncertainty range of forecast variables as well. An advantage of the forecast model proposed in this study is the ability to approximate the forecast uncertainty, via the estimated conditional pdfs of the forecast variable (Eq. (5)). The forecast variable can be shown by a particular uncertainty bound around the pdf mean (median), rather than a single deterministic value. Fig. 4 shows forecast uncertainty for the runoff produced in a few HRUs across the basin during the hind-cast period from 1980 to 2010. Note that the forecast variable is called as the hindcast variable during a historical time period. The predictor month is January followed by the forecast month, July. Observed runoff (solid black dots) during the hindcast period is estimated by the deterministic PRMS and the forecast runoff is shown within the 5–95% uncertainty bound around the pdf median (dash line). As seen, the uncertainty bound fairly encompasses the observed runoff of associated HRUs and the median of forecast pdf (dash line) generally passes through the observations. The uncertainty bound is found to be rather large for high flows and small for low flows, which is quite reasonable due to the heteroscedastic nature of streamflow. Probabilistic estimation of future droughts is the primary feature of the presented technique. Through probabilistic prediction, the probability of a particular future drought condition can be estimated conditional on the past drought status. In other words, the question is to find the probability of a specific drought at time t + 1, given the status at time t. Another feature of probabilistic prediction is the identification of the future drought state associated with a particular probability. This alternative asks for the future drought at time t + 1 associated with a specific probability. Using the copula-based forecasting method, Madadgar and Moradkhani (2013) developed the conditional pdf and cdf of future droughts, given the drought status of an earlier time, based on the basin outflow in the seasonal drought forecasting of GRB. Fig. 5 simply

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Runoff [KAF]

15

10

5

Runoff [KAF]

45 40 0 1980 35

1985

1990

1995

2000

2005

1985

1990

1995

2000

2005

30 25 20 15 10

30

5 25 1980

Runoff [KAF]

0

20 15 10

80 5

70 60 1985

1980

Runoff [KAF]

0

1990

1995

2000

2005

50 40 30 20 10 0

1980

1985

1990

1995

2000

2005

shows the two possible cases described above; where X2 and X1 are the drought-related variables in the future and in the past, respectively. The probability of future drought given a particular drought in an earlier time (X1 = x1) is equal to the area under the conditional pdf, f(X2|X1 = x1). Given X1, the former case (explained earlier) asks for the probability P(X2 < x2|X1 = x1) of a particular x2, while the latter asks for the X2 associated with a particular P(X2|X1 = x1). Thus, a probability map of future droughts using the runoff variable at each cell (Fig. 6) can be produced, as well as the runoff map with particular chance of occurrence in the forecast period (Fig. 7). Fig. 6 displays the matrix of probability maps in the forecast month. It shows the probability of drought status being equally wet or wetter than a particular drought status in the forecast month (July), given the drought status in the predictor month (January). These probability maps are identical to the first case explained earlier, where the conditional probability is estimated for a particular future drought, given the condition in the past. The estimated probability at each HRU is equal to the area under

f (X2 | X1)

Fig. 4. Runoff volume accumulated over July–December estimated by the forecast model for a few HRUs across GRB in the hindcast period (1980–2010). The hindcast is shown within the 5–95% uncertainty bound along with the corresponding observations (black dots); the dash line is showing the median of the conditional pdf (Eq. (5)).

P(X28 [KAF]

Fig. 7. Maps of runoff volume in the forecast month (July) associated with different non-exceedance probabilities. Row labels show the drought status of the predicting month (January).

7. Summary and conclusion Since frequent droughts have recently affected the southwestern U.S. with different water issues, reliable forecast of future droughts are essential for this region of the United States. The historical records across the Colorado River Basin denote the water year 2012 as the 4th driest year of the region in the past century (since 1904), with consequences like insufficient water supply, poor pasture and crop conditions, and region-wide wildfires. Whereas the recent droughts of CRB might be referred to the worldwide impacts of global warming on extreme events, accurate estimation of ongoing droughts across the region is crucial for future planning and managements of water resources in the area. This study developed a forecast model to predict the spatial variation of future droughts across the Gunnison River Basin, one of the headwater sub-basins of the Upper Colorado River Basin. Using the runoff volume generated at each spatial unit of the basin, the SRI drought index was obtained across the region for the historical time period. The USGS Precipitation Runoff Modeling System

(PRMS) was utilized to estimate the hydrological response of each grid cell including the runoff. Using the estimated runoff in a primary analysis of historical droughts verified the appropriate performance of PRMS in estimating the runoff and capturing the historical droughts, e.g. the severe drought in 2002. Given the drought status in the past, a forecast model was then established between the runoff volumes to predict the future droughts in each spatial unit. To attain reliable forecasts, the most correlated runoffs over six-month time periods were chosen as the predictor and forecast variables. The auto-correlation analysis of the historical time period indicated that the highest correlation of six-month accumulated runoff is between the runoffs over (January–June) and (July–December) periods. Hence, January–June and July– December were taken as the predictor and forecast periods, respectively. Bayesian networks were employed to model the conditional probabilities of the forecast variable given the predictor variable; however, the intense analytical effort of estimating the conditional pdf via the Bayesian network was then simplified with the help of copula functions.

S. Madadgar, H. Moradkhani / Journal of Hydrology 512 (2014) 134–146

The main advantage of the proposed forecast methodology is its probabilistic features. It develops the conditional probabilities of the forecast variable returning the most likely forecast within an uncertainty bound. The model could estimate the 5–95% uncertainty bound of the accumulated runoff at each grid cell with sufficient width to encompass the observed runoff over July– December during a hindcast period. The width of estimated uncertainty bound was larger for the high flows, which would reflect the greater uncertainty in high flows than the low flows. As the next output, the forecast model can estimate the chance of a particular drought in the forecast period, given the drought status of an earlier time. In this application, the forecast model produced the probability maps of the drought status in July–December, given the observations in January–June. According to the results, the more intense droughts are expected in July–December as the January– June period gets drier. In other words, the probability of dry status in the forecast period increases as the predictor period undergoes intensive dry conditions. These probabilistic maps are also useful to approximate the chance of drought recovery (normal/wet condition) in the forecast period, given the drought status observed in the predictor period. Another product of the forecast model was the runoff estimation at each grid cell with a particular chance of occurrence. The variation of runoff across the basin with particular probabilities (0.25, 0.5, and 0.75) in the forecast period was evaluated given different drought statuses in the predictor period. The forecast methodology applied in this study shows promise in developing various products using its probabilistic features. With the application of copula functions, the proposed methodology can generate useful products in estimating the spatial variation of future droughts. Similar analyses within this forecast methodology would help the water managers and decision makers to regulate their policies according to the uncertainties in the future droughts. To account for other sources of uncertainty, blending of such a statistical approach with a dynamic modeling based on ensemble data assimilation (Moradkhani, 2008; Moradkhani et al., 2012) is suggested and our study is underway to be reported in our future publications. Acknowledgments Partial financial support for this study was provided by NOAAMAPP Grant NA110AR4310140. References Akaike, H., 1974. A new look at the statistical model identification. IEEE Trans. Autom. Control 19 (6), 716–723. Andreadis, K.M., Lettenmaier, D.P., 2006. Trends in 20th century over the continental United States. Geophys. Res. Lett. 33, L10403. http://dx.doi.org/ 10.1029/2006GL025711. Bárdossy, A., 2006. Copula-based geostatistical models for groundwater quality parameters. Water Resour. Res. 42, W11416. http://dx.doi.org/10.1029/ 2005WR004754. Barros, A.P., Bowden, G.J., 2008. Toward long-lead operational forecasts of drought: an experimental study in the Murray-Darling River Basin. J. Hydrol. 357, 349– 367. Burke, E.J., Perry, R.H.J., Brown, S.J., 2010. An extreme value analysis of UK drought and projections of change in the future. J. Hydrol. 388, 131–143. Cancelliere, A., Mauro, G.D., Bonaccorso, B., Rossi, G., 2007. Drought forecasting using the Standardized Precipitation Index. Water Resour. Manage. 21, 801– 819. Carbone, G.J., Dow, K., 2005. Water resource management and drought forecasts in South Carolina. J. Am. Water Resour. As. (JAWRA) 41 (1), 145–155. Dai, A., 2011. Drought under global warming: a review. WIREs Clim. Change 2, 45– 65. http://dx.doi.org/10.1002/wcc.81. DeChant, C., Moradkhani, H., 2011a. Radiance data assimilation for operational snow and streamflow forecasting. Adv. Water Resour. 34, 351–364. DeChant, C., Moradkhani, H., 2011b. Improving the characterization of initial condition for ensemble streamflow prediction using data assimilation. Hydrol. Earth Syst. Sci. 15, 3399–3410. http://dx.doi.org/10.5194/hess-15-3399. Duan, Q., Sorooshian, S., Gupta, V.K., 1994. Optimal use of the SCE-UA global optimization method for calibrating watershed models. J. Hydrol. 158, 265–284.

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