GEOPHYSICAL RESEARCH LETTERS, VOL., 3637 3642, doi:10.2/grl.50728, 13 A probabilistic framework for assessing drought recovery Ming Pan, 1 Xing Yuan, 1 and Eric F. Wood 1 Received 18 June 13; revised 8 July 13; accepted 8 July 13; published 26 July 13. [1] A probabilistic framework is proposed to explore drought recovery and its uncertainty based on any ensemble forecast. First, the joint distribution between precipitation and drought index is established from the forecast ensemble using the copula method, which allows arbitrary marginal distributions and fine-tuned correlation structure. Then, questions like how much precipitation is needed for recovery and its uncertainty? and what is the likelihood that a specified drought index threshold be surpassed given specified cumulative precipitation over a fixed period? are studied. The application investigates how the 12 13 drought over central United States may recover during the forecast period from February to July 13. The ensemble streamflow prediction method is used to create the ensemble forecast, with soil moisture percentile against climatology as the drought index. We find significant uncertainty in drought recovery given the same precipitation, suggesting that a probabilistic analysis will offer important additional information about drought risk. Citation: Pan, M., X. Yuan, and E. F. Wood (13), A probabilistic framework for assessing drought recovery, Geophys. Res. Lett.,, 3637 3642, doi:10.2/grl.50728. 1. Introduction [2] Drought remains a frequent and costly natural disaster [Sheffield et al., 12]. A large part of the central United States has entered an extended dry period in the spring of 12, causing great losses in agriculture. The drought condition still persists as of early 13, as monitored by institutions like Princeton University (PU) (http://hydrology. princeton.edu/forecast/) [Luo et al., 07; Luo and Wood, 08], University of Washington (UW) (http://www.hydro. washington.edu/forecast/monitor/) [Wood, 08], and National Drought Mitigation Center (http://droughtmonitor. unl.edu/), among others. Following the PU drought monitoring and forecast system [Luo et al., 07; Luo and Wood, 08], this study uses the drought index defined as the soil moisture percentile values against climatology (1950 1999) [Sheffield et al., 04] as it measures agricultural drought a primary impact for the central United States. The PU system runs the variable infiltration capacity (VIC) [Liang et al., 1994] land surface model (LSM) continuously forced with the meteorological fields from the near-real-time North American Land Data Assimilation System Phase 2 (NLDAS-2) [Xia et al., 12]. 1 Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey, USA. Corresponding author: M. Pan, Department of Civil and Environmental Engineering, Princeton University, Room E322, E-Quad, Olden St., Princeton, NJ 08544, USA. (mpan@princeton.edu) 13. American Geophysical Union. All Rights Reserved. 0094-8276/13/10.2/grl.50728 3637 [3] Drought management would benefit greatly if more risk-based information is available on how a region in drought may recover [Karl et al., 1987], e.g., the likelihood of recovery under different precipitation scenarios and the related uncertainty. Hydrologically, several factors such as the initial moisture condition, the amount and timing of precipitation, and the temperature will control the recovery process. Also, an LSM-based forecast system would be an ideal tool for resolving the interplays among these factors and predict the moisture states [Wood and Lettenmaier, 06]. The surface meteorological inputs to the system can be either taken from a dynamic forecast model like the Climate Forecast System version 2 [Yuan et al., 11; Mo et al., 12] or randomly selected from historical meteorological records as done by the ensemble streamflow prediction (ESP) method [Day, 1985]. [4] Due to limited forecast skills, climate forecasts are usually made in an ensemble form to help quantify the uncertainty. To better address this uncertainty, we propose a probabilistic framework to assess drought recovery that is based on the joint distribution between the cumulative precipitation the main driver for recovery and the soil moisture based drought index described earlier. With a parametric joint probability distribution established using the copula model [Salvadori and De Michele, 04; Shiau, 06; Wong et al., 10], the related forecast uncertainties can be better quantified under various forecast scenarios, and the drought risks can be evaluated accordingly. In our application, we used the ESP method to create an ensemble of meteorological forecasts as inputs to the LSM, and all recovery analyses are based on these ESP hydrologic forecasts. 2. Methods 2.1. ESP-Based Hydrologic Forecast [5] The ESP method generates meteorological forecasts by random sampling from the historical records instead of running a dynamic model [Day, 1985]. ESP is simple and robust, though unable to differentiate the likelihood of any climate event from its climatological rate of occurrence. Compared to dynamic models, ESP s skill relies mostly on the initial conditions. The 50 year (1950 1999), 0.125 resolution daily meteorological fields compiled at UW [Maurer et al., 02] serve as the historical database. The hydrologic forecast is initialized on 1 February 13, and the VIC model initial condition is taken from the PU drought monitoring system mentioned earlier that runs the same LSM continuously with the NLDAS-2 forcing fields. The forecast length is 6 months (1 February to 31 July 13), and we refer the February/ March/ /July mean forecasts as 0.5/1.5/ /5.5 month lead time forecasts. We sample exhaustively and uniquely from the 50 year historical database, i.e., the February July meteorological fields of every year in 1950 1999 are all sampled exactly once, to create a 50-member ensemble inputs to VIC.
a) b) c) Soil Moisture Percentile 0 d) e) f) 0 0 0 0 Cumulative Precipitation Percentile Figure 1. ESP ensemble of {(p,θ) (i) } i =1,,50 averaged over a 0.5º 0.5º box centered at.25 N, 99.75 W in Nebraska is shown as blue dots. The joint fitted PDF f(p,θ) is shown as gray shading and drought threshold θ drought = 30% as red lines. The red cross is the median cumulative precipitation percentile p median θ = θ drought that is needed to recover to θ drought. The light and dark green lines are the conditional PDF of drought index under two precipitation scenarios f(θ p = 50%)and f(θ p = 70%). The hatched areas under the conditional PDF give the recovery probability (integral values noted on the side). The magenta plus shows the monitoring results. The goodness-of-fit statistic S n is noted together with the type of copula used. The VIC simulation is performed at 0.125 and daily level to create the ESP-based soil moisture forecasts. We calculate the percentile values of the column total soil moisture against the 50 year retrospective VIC simulations (1950 1999) forced with the same 50 year UW database. The resulted column total soil moisture percentile forecasts, noted as θ, are used as the drought index. The percentile values of cumulative precipitation between forecast initialization and verification time, noted as p, are also calculated against the same 50 year database. As the forecast initialization time is fixed at 1 February 13, both θ and p are a function of verification time and location. At every time and location, the ESP-based forecast will provide a 50-member ensemble of p versus θ pairs, {(p,θ) (i) } i =1,,50, and this ensemble will serve as the sample for fitting a joint probability distribution for the subsequent drought recovery analysis. 2.2. Copula-Based Joint Distribution Model for Precipitation Versus Drought Index Relationship [6] For smoothness, we first average the 0.125 results {(p,θ) (i) } i =1,,50 up to 0.5 and then use the copula model to fit a joint distribution between p and θ for every lead time and every 0.5 pixel. The Sklar theorem [Sklar, 1959] states that a multivariate (bivariate here) cumulative distribution function (CDF) F X,Y (x,y) can be written as a function of its marginal CDFs F X (x) and F Y (y)as follows: F X ;Y ðx; yþ ¼ CF ð X ðþ; x F Y ðþ y Þ (1) [7] The function C = C(u,v) is called a copula, and its form reflects the correlation structure of the joint distribution. The Sklar theorem provides a powerful way to construct a joint distribution in two independent steps fitting the marginal distributions and the copula function. Almost any parametric univariate distribution can be used for the marginal, and there exist a large number of copula functions for various correlation structures. Once a joint distribution is constructed as in equation (1), the related conditional distributions like F Y X (y x) can be derived immediately as follows: F YjXðyjxÞ ¼ u CF ð X ðþ; x F Y ðþ y Þ (2) [8] Normally, we use the forms F X (x) and f X (x) for CDF and probability density function (PDF). From here on, we 3638
Figure 2. Maps of median cumulative precipitation percentile p median θ = θ drought required for the drought index to reach the specified threshold θ drought = 30% (Figure 1, red cross). Red colored areas are those unable to recover from drought under any cumulative precipitation scenario, and empty colored areas are those not in drought (θ > θ drought ) as of 1 February 13. (a f) The results for lead times of 0.5 5.5 months. simplify the notation by dropping the subscript so that f X (x)is simplified to f(x) and f Y X (y x) tof(y x). [9] Given the samples of soil moisture and cumulative precipitation percentiles in the ensemble {(p,θ) (i) } i =1,,50,wefit the marginal distributions of p and θ. Since p is already the percentile values calculated against the same 50 year climatology, it follows a uniform distribution (no fitting needed). We choose the Beta distribution for the marginal distribution of θ for its great flexibility [Sheffield et al., 04], and the fitting is done using the maximum likelihood method. The copula function is fitted from three types of single-parameter copulas, namely, Gumbel, Frank, and Clayton [Salvadori and De Michele, 04], separately with the maximum likelihood method, and the one with the best goodness-of-fit statistic S n [Genest et al., 06] is selected as the final choice. S n is asymptotic to the p-value statistic, and smaller S n values mean better fit. [10] The examples of resulted joint PDF f(p,θ) are shown in Figure 1 as gray shading. The input {(p,θ) (i) } i =1,,50 samples (blue dots) come from the ESP ensemble forecast averaged over a 0.5º 0.5º box centered at.25 N, 99.75 W in Nebraska, an area deep in drought as of 1 February. Six lead times from 0.5 (February) to 5.5 months (July) are shown in the panels. Here we observe a positive correlation between p and θ (as expected), and the scattering of samples (and spread of f(p,θ)) increases with longer lead times. Such a growing spread suggests that the temporal variability in the meteorological forcings (rainfall intensity and frequency, temperature, etc.) adds significant uncertainty to the drought forecast given the same initial moisture condition and cumulative precipitation. At 0.5 month lead, θ can never exceed the preset drought index threshold of θ drought = 30% (Figure 1, red lines) given any p, i.e., no historical event of precipitation can recover the drought in the first forecast month. The influence of initial condition is very obvious in the first 3 months (Figures 1a and 1c). Then, the probability of drought recovery increases, and the joint PDF gradually approaches the 1:1 line with longer lead time. The PDF basically follows the 1:1 line at the 5.5 month lead but with a large spread, suggesting a diminished influence of the initial soil moisture condition. The goodness-of-fit statistic S n is generally very good (<0.1), except at 0.5 month lead (S n = 0.191) where the ensemble is highly compact. With such a parametric f(p,θ), we calculate a number of quantities for the drought recovery analysis in Figure 1. [11] 1. The median cumulative precipitation percentile, p median θ = θ drought, needed to recover soil moisture just to threshold θ drought (red cross marks). This quantity gives the most likely precipitation needed for recovery. [12] 2. The conditional PDF of drought index under two cumulative precipitation scenarios: f(θ p = 50%) and f(θ p = 70%) (light and dark green lines). These conditional PDFs quantify the uncertainty in drought index given the specified precipitation scenarios. 3639
PAN ET AL.: PROBABILISTIC DROUGHT RECOVERY Figure 3. Maps of the probability of drought recovery under the median (p = 50%) cumulative precipitation scenario % (Figure 1, θdrought f ðθjp ¼ 50%Þdθ values). Red- and empty-colored areas and panel sequences are the same as in Figure 2. [10] 3. The integration of the above conditional PDF % % above θdrought, i.e., θdrought f ðθjp ¼ 50%Þdθ and θdrought f ðθjp ¼ 70%Þdθ (Figure 1, hatched areas). The integral values (noted next to hatched areas) are the recovery probability under the two precipitation scenarios. [14] The precipitation and soil moisture results from the monitoring effort, i.e., VIC simulation forced with observations, are shown in Figure 1 as magenta plus in months up to May (latest available as of this writing). These above diagnostic quantities are calculated for all 0.5º 0.5º pixels in the contiguous United States (CONUS) area for all lead times. 3. Recovery Analysis [15] Figure 2 shows the maps of the median cumulative precipitation percentile pmedian θ = θdrought for the recovery of soil moisture to the threshold θdrought. This value tells us how much cumulative precipitation is mostly likely needed for recovery. The red color in the figure indicates that even the largest (p = %) historical accumulation is insufficient for recovery. Large areas of such irrecoverable conditions are seen at the 0.5 month lead (Figure 2a) over states like Nebraska, South Dakota, Minnesota, and Oklahoma, but this irrecoverable condition diminishes at the 1.5 and 2.5 months (Figures 2b and 2c). At 0.5 and 1.5 months (Figures 2a and 2b), a very large cumulative precipitation (mostly > 90%) is needed for recovery. The precipitation requirement drops quickly at 2.5 and 3.5 months (Figures 3c and 3d), and the drought can recover under a normal precipitation scenario (p around 50%) at the 5.5 month lead. [16] Figure 3 shows the maps of recovery probability under % the median cumulative precipitation scenario, i.e., θdrought f ðθjp ¼ 50%Þdθ. The smaller this value, the less likely it is to recover and the higher the probability (risk) that the area remains in drought. Figure 3a shows that large parts of CONUS are irrecoverable at 0.5 month (same as in Figure 2a), and the recovery probability is very low. Most areas start to be recoverable from the 1.5 month onward (Figure 3b), but the recovery probability is low (10% %). The recovery probability across CONUS increases at 2.5 and 3.5 months until it reaches the % 90% level at the 5.5 month lead (very likely to recover if median cumulative precipitation is received for 6 months). 4. Validation [17] The monitoring results in Figure 1 provide a partial validation for the recovery analysis, where the monitored precipitation and soil moisture values (magenta plus) fall well within the ESP ensemble and copula-fitted f (p,θ). The recovery analysis is also performed on two historical droughts: the 07 drought in the U.S. Southeast (ESP forecast initialized on 1 May 07) and the 11 drought in the Texas areas (initialized on 1 May 11). The 0.5 scale results are shown in Figure 4 with one pixel in Georgia during the 07 drought (Figures 4a and 4d) and another in Texas during the 11 drought (Figures 4e and 4h). In both 36
a) b) c) d) Soil Moisture Percentile 0 e) f) g) h) 0 0 0 0 0 Cumulative Precipitation Percentile Figure 4. Recovery analysis and validation for two historical droughts (symbols same as in Figure 1). (a d) The results from a 0.5 0.5 box centered at 32.25 N, 81.75 W in Georgia initialized in May 07. (e h) The results from a 0.5 0.5 box centered at 31.25 N, 95.75 W in Texas initialized in May 11. historical droughts, the copula model works very well in PDF fitting with S n < 0.1 all the time. The limitation of the persistence-based ESP method is also seen the recovery probability will always increase in time (which happens in 13 forecast, too). However, in reality, the drought can recover faster than what is expected by ESP, as in the 07 case (Figures 4a and 4d), where the monitored soil moisture (magenta plus) rises above θ drought quite early (1.5 month lead). At the same time, the drought can persist much longer than expected, as in the 11 case (Figures 4e and 4h), where the area stays in drought all throughout the 6 months. Note that the probabilistic analysis here provides additional insights into the uncertainty and risk-given user-selected precipitation scenarios, but it does not modify the forecast skills of the forecast system itself or how accurately the forecast ensemble represents the actual uncertainty. 5. Conclusions [18] A probabilistic drought recovery analysis framework is developed using copula-based joint distribution model for cumulative precipitation and drought index and applied to the recovery analysis for the current central United States drought and two historical droughts in 07 and 11. While the inputs to the recovery analysis may come from any ensemble forecast of precipitation and drought index, this paper uses a 50-member ESP ensemble forecast and the LSM-derived soil moisture percentiles against climatology as the drought index. Significant uncertainty is found to be associated with drought recovery for different locations and lead times, and this uncertainty can be well captured and quantified using conditional probabilities derived from the joint distribution model. The results show that the temporal variability of precipitation and other factors such as temperature over a specified forecast period can result in large uncertainty in drought forecasts. The findings suggest that a deterministic approach to estimating the amount of rainfall required for drought recovery will not provide sufficient information to assess drought risk and that a probabilistic analysis can provide indispensible risk information for drought managers. [19] Acknowledgments. The research was supported by the NOAA Climate Program Office through grants NA17RJ2612 and NA10OAR4310246. [] The Editor thanks Amir AghaKouchak and an anonymous reviewer for their assistance in evaluating this paper. References Day, G. N. (1985), Extended streamflow forecasting using NWSRFS, J. Water Resour. Plann. Manage., 111(2), 157 170. Genest, C., J. Quessy, and B. Remillard (06), Goodness-of-fit procedures for copula models based on the probability integral transformation, Scand. J. Stat., 33(2), 337 366, doi:10.1111/j.1467-9469.06.00470.x. Karl, T., F. Quinlan, and D. S. Ezell (1987), Drought termination and amelioration Its climatological probability, J. Clim. Appl. Meteorol., 26(9), 1198 19, doi:10.1175/15-0450(1987)026<1198:dtaaic>2.0. CO;2. Liang, X., D. P. Lettenmaier, E. F. Wood, and S. J. Burges (1994), A simple hydrologically based model of land-surface water and energy fluxes for general-circulation models, J. Geophys. Res., 99(D7), 14,415 14,428, doi:10.1029/94jd00483. Luo, L., and E. F. Wood (08), Use of Bayesian merging techniques in a multimodel seasonal hydrologic ensemble prediction system for the eastern United States, J. Hydrometeorol., 9(5), 866 884, doi:10.1175/ 08JHM9.1. 3641
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