2.1 GRACE

The GRACE mission operated from April 2002 to June 2017, with a primary goal to track water redistribution on Earth and to improve our understanding of the global (Eicker et al., 2016; Fasullo et al., 2016) and regional water cycle (Singh et al., 2018; Springer et al., 2017). The GRACE-based TWSA includes integrated water mass changes in a vertical column, which may consist of rivers, lakes, snow, ice, glaciers, soil moisture, permafrost, swamps, groundwater, etc. We downloaded the GRACE mascon (RL06) solutions from the Jet Propulsion Laboratory (JPL) website https://grace.jpl.nasa.gov (last access: 3 March 2019; Wiese et al., 2018). The gravity field signals of GRACE are preprocessed to monthly gridded equivalent water height (EWH) variations by JPL (Watkins et al., 2015; Wiese et al., 2016). The mascons are estimated as being 3^∘ spherical caps, where 3^∘ indicates the radius of the spherical cap. The 3^∘ spherical cap mascon estimates are then represented on a $\mathrm{0.5}{}^{\circ }\times \mathrm{0.5}{}^{\circ }$ grid. The shape and size of the mascon caps vary with latitude. Therefore, the gridded mascon solutions are multiplied by a scaling factor grid (https://grace.jpl.nasa.gov/data/get-data/jpl_global_mascons/, last access: 21 March 2019) to improve the interpretation of signals at the submascon resolution. Since 2011, the GRACE data set has data gaps of 1–2 months in every 5–6 months due to the aging batteries of the satellites. However, to compare precipitation and storage variability, a continuous monthly TWSA time series is required. Therefore, the data gaps in the time series are filled by cubic convolution interpolation (Keys, 1981). Comparison between different GRACE solutions are discussed in the Supplement.

2.2 Global Precipitation Climatology Project (GPCP)

The Global Precipitation Climatology Project (GPCP) is widely used global precipitation data. Most of the other observational products do not produce precipitation estimates beyond 60^∘ S/N for a longer historical period (1979–present). Besides, GPCP applies gauge under catch correction to in situ precipitation measurements, which has been found important for improving snowfall measurements (Behrangi et al., 2018). The latest global monthly precipitation data are obtained from the GPCP V2.3 from their website https://www.esrl.noaa.gov/psd/ (last access: 10 May 2019; Adler et al., 2003) for 1979–2017. It is a combined satellite-based product adjusted by rain gauge analysis. The downloaded 2.5^∘ resolution data are regridded to 0.5^∘ by using bilinear interpolation to harmonize the data with the GRACE grid. The spatial resolution of the original GRACE solution (3^∘ mascon) and GPCP (2.5^∘) are comparable. However, as mascon size varies with latitude, both data sets are adjusted to the 0.5^∘ grid to improve the interpretation.

3.1 Storage deficit

It is useful to know the total amount of missing water from an ecosystem in order to characterize a drought so that an explicit target can be assumed that defines a drought recovery. Currently, global gridded total water storage variations can only be obtained from GRACE TWSA. The TWSA is first smoothed by a 3-month moving average filter, followed by the removal of a linear trend to reduce the impact of long-term signals on the storage. A linear trend in the storage variability can be caused by continuous/long-term processes other than just precipitation, like upstream water abstraction, groundwater pumping, increase/decrease in snowmelt, etc. We acknowledge the caveat of the possibility of pseudo-trends due to the unusual signal at the beginning or end of the record in some regions. The reduced TWSA is termed the deviation of storage (dTWSA). The dTWSA from its normal water storage cycle (i.e., its historical climatology) can give an idea of the severity of drought phenomena. Here, we define recovery as a return to the climatological storage state for a given month. The climatology of the time series is estimated over the 15-year GRACE record (April 2002–March 2017) by averaging values from the same months of each year (i.e., all Januaries, all Februaries and so on). The negative residuals of the dTWSA from its climatology are considered as being a water storage deficit in a grid cell (Thomas et al., 2014). If the duration of negative residuals is longer than 3 months, we designated it as a drought event. If recurring drought happens within a 1 month gap (i.e., recovery shorter than a 1-month duration), we considered it a continuation of the same drought. The green plot in Fig. 1 shows the duration and severity of recurring drought in an example location in Australia (centered on 133.75^∘ E, 16.75^∘ S). Using this approach, we produce a global gridded drought characteristics record, which includes the frequency, intensity and duration of drought, for the 2002–2017 period. For any instance and location, the state of drought and its length can be identified by quantifying the water storage deficit from the dTWSA. Eventually, the recovery duration for each drought can also be observed, i.e., how long negative residuals from climatology continued. For instance, Fig. 1 shows three major droughts and their respective recovery periods (of nearly 4, 3, and 1 years) for a sample location in Australia.

Figure 1Water storage deficit from GRACE. The smoothed and detrended TWSA (dTWSA – red plot) is reduced by its climatology (black plot) to estimate the deviation from the climatology. The negative residuals from the climatology are plotted on the upper axis as a green shaded area and scaled on the right side. The gray shading indicates ±1 SD (standard deviation) of the climatology.

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3.2 Estimation of the required precipitation for storage deficit

The water balance equation, based on hydrological fluxes (Eq. 1), shows that the change in terrestrial water storage (dS) in a region for a given month (dt) depends on is the monthly precipitation (P; millimeters per month), evapotranspiration (ET; millimeters per month) and the streamflow (R, which includes both surface water and subsurface water; Swenson and Wahr, 2006). Assuming the relationship between precipitation and ET+R remains constant for a region, the variability in precipitation gives an idea of the possible variation in the storage. The amount of required precipitation needed to overcome a deficit can be estimated using the association between precipitation and the water storage anomaly (TWSA).

Monthly GPCP observations are first reduced by their mean for the April 2002–March 2017 period (i.e., the 15-year GRACE data record) to obtain the precipitation anomaly. Then, the relationship between precipitation and storage anomalies is derived. For this, first, both variables are smoothed by a 3-month moving average low-pass filter to remove high-frequency noise. Then, their linear trends are removed to reduce the impact of other processes, like groundwater, upstream abstraction, glacier melts, etc. (as discussed above), and to focus our analysis on subdecadal drought events within the GRACE period. The smoothed and detrended precipitation anomaly is then integrated in time to obtain the storage anomaly, which is termed the cumulative detrended smoothed precipitation anomaly (cdPA). Finally, the cdPA is compared with the smoothed and detrended storage anomaly (dTWSA).

An ecosystem may behave differently under stress (a deficit period) than under an excess water situation. In this study, the linear relationship between storage (dTWSA) and precipitation (cdPA) has been analyzed only during historical deficit periods as the system behaves differently under stress (Famiglietti et al., 1998; Vereecken et al., 2007). Several researchers used rainfall–runoff curves, like the Soil Conservation Service curve number (SCS-CN) for the computation of surface runoff, based on precipitation, with an assumption of a stable relation between rainfall and abstraction (Mishra et al., 2006; Singh et al., 2015; Verma et al., 2017). This study also assumes that the precipitation intensity for a region does not change significantly over time; consequently, the relationship between precipitation and storage variability can be considered stable.

Figure 2 shows the strength of this relationship with the correlation coefficients (Fig. 2a) and linear regression coefficients (Fig. 2b). Based on the linear relationship between dTWSA and cdPA, the required precipitation has been estimated. Regression coefficients greater than 1 means the required precipitation is more than the amount of missing water. This is because precipitation lost in other hydrological processes, like evapotranspiration and runoff (Eq. 1), is not observed by storage variability. A coefficient equal to 1 means the amount of required precipitation is the same as that storage loss, which means there is no other dominant process in the region. Coefficients less than 1 are the regions of weak precipitation–storage coupling, which can be due to other physical processes, like melting of snow/frozen surfaces, groundwater extraction, irrigation, etc. (non-red regions in Fig. 2a). Therefore, for most of the regions, the required precipitation is more than the amount of missing water (i.e., regression coefficients greater than 1), except for the regions with weak precipitation–storage coupling. For example, in higher latitudes, mass loss observed by GRACE during spring snowmelt is not directly linked to precipitation. Additionally, highly arid regions also have weak precipitation and storage signals. Therefore, the proposed method is not suitable for regions with weak precipitation–storage coupling. These regions of the weak association are identified based on regression coefficients below 1 (Fig. 2b), as less than 1, or a negative relationship between storage variability and precipitation, may describe a case in which storage variability is not linked to a direct precipitation effect. Also, locations with less than 5 months of drought in 15 years are considered as regions of the weak association because we do not have enough drought samples to derive their association. The regions of weak association (with regression coefficients less than 1) are considered as being unsuitable for the GRACE-based recovery analysis and have been masked out in this study.

Figure 2(a) Correlation coefficients and (b) regression coefficients between cumulative detrended precipitation anomalies (cdPA) and the detrended terrestrial water storage anomaly (dTWSA).

Based on the derived linear relationship between cdPA and dTWSA (Fig. 2b), a required precipitation is estimated for each regional drought period. The method for the estimation of required precipitation is shown in Fig. 3 at an example location (133.75^∘ E, 16.75^∘ S) in Australia. Figure 3a shows an agreement between cdPA (black plot) and dTWSA (red plot). In Fig. 3b, an absolute required precipitation (blue plot) is calculated, by adding precipitation climatology to the estimated surplus required precipitation (magenta plot), to fill the storage deficit (green plot). Analogous to an accounting methodology, this approach applies the assumption that generally more precipitation than usual (climatology) is required to replenish the losses incurred during drought. The example location has a strong annual signal (5–150 mm, with predominantly winter rain), which led to a relatively high ratio of required precipitation to the amount of missing water.

Figure 3Estimation of the required precipitation at an example location. (a) Cumulative detrended precipitation anomaly (cdPA) compared with the detrended storage anomaly (dTWSA). (b) Surplus required precipitation is estimated (magenta plot), from the linear relationship between dTWSA and cdPA, to fill the storage deficit (green plot). Then, precipitation climatology is added to obtain the absolute required precipitation (blue plot).

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3.3 Historical precipitation analysis

Historical precipitation data from GPCP (1979 to 2017) are statistically analyzed, using signal decomposition, in order to create a simplistic precipitation forecast. Note that the motivation for providing a precipitation forecast here is not to present a state-of-the-art precipitation prediction but to demonstrate the potential utility of the terrestrial water storage deficit in determining required precipitation and estimating a likely time for recovery. This methodology could be augmented with any type of more complex precipitation forecasting approach.

3.3.1 Precipitation signal decomposition

Historical precipitation data are decomposed into a linear trend, inter-annual signal, annual/climatological cycle and the sub-seasonal components in order to explore temporal variability. First, a linear trend and an annual signal (mean of each month; e.g., all of January, February, etc.) are extracted from the original signal. Then, the residual signal is filtered by a 12-month low-pass window to split it into a smooth inter-annual signal and a high-frequency sub-seasonal signal. Together, the linear trend and inter-annual signal are considered to contribute to long-term variability. The individual variance of the annual, long-term and sub-seasonal signals is normalized by their sum in order to obtain their fractional contribution to local variability (Fig. 4). This provides an overview of the relative importance and spatial distribution of these components in global temporal variability. Figure 4 shows the fractional variance of the decomposed signal. For most regions, the annual signal dominates in precipitation (Fig. 4a). However, regions for which the wet season is not explicit in their climatology, a high-frequency signal plays a major role, for example in central Europe, eastern Siberia, western North America, southern Australia, etc. (Fig. 4c). Contrarily, the long-term signal obtained by combining linear trend and the inter-annual signal has the least variability globally (Fig. 4b). These smooth signals are driven by climate indices like the El Niño–Southern Oscillation (ENSO), Pacific Decadal Oscillation (PDO), and North Pacific Mode (NPM), etc. (Özger et al., 2009). The annual and long-term signals are directly applied for the signal reconstruction, with the assumption that a similar trend will continue.

Figure 4Fractional variance in the decomposed signal to the full signal. (a) Annual signal, (b) long-term signal and (c) sub-seasonal high-frequency signal.

3.3.2 Signal reconstruction and forecasting skill

Based on the above findings, we formulate a statistical model for hindcasting precipitation. The annual signal and the linear trend extracted by signal decomposition (Sect. 3.3.1) are directly used for the precipitation reconstruction, with the assumption of the continuation of the similar variability. Furthermore, inter-annual variability in the precipitation data is added by autoregression for 10–14 months, depending on the duration of significant autocorrelation. Finally, the sub-seasonal signal is added, which is obtained from the residual of the inter-annual signal. This high-frequency signal has only 0–3 months of temporal autocorrelation; accordingly, we have limited skill in synthesizing the sub-seasonal signal.

Figure 5 shows the precipitation hindcast for January 2016–December 2017 at an example location (56.25^∘ W, 27.75^∘ S) in the La Plata basin. Figure 5a shows that the reconstructed precipitation (red plot), compared to its climatology (blue plot) and GPCP observations (black plot) for the same duration. Figure 5b shows the reconstructed inter-annual precipitation by autoregression. The figure shows that inter-annual autoregression (blue plot) signals have a good association with the observed inter-annual signal (black plot) for the first 11 months. The sub-seasonal autoregression is significant only for 2 months in the example location. The final hindcast is an integration of a linear trend, climatology, sub-seasonal and inter-annual autoregression.

Figure 5Reconstruction of precipitation signal for 2016–2017. (a) The reconstructed signal compared with the GPCP observations and climatology. (b) The reconstruction of a long-term secular signal from the linear trend and inter-annual and sub-seasonal autoregression compared to GPCP inter-annual signal.

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The precipitation reconstruction skill is used for a simplistic normal forecast. Furthermore, two additional precipitation scenarios are simulated by adding, respectively, 1 and 2 standard deviations (SDs) of precipitation to the normal forecast, which is used in probability recovery analysis.

3.4 Probabilistic recovery

Precipitation is the major control on drought dynamics. Knowing the amount of precipitation required to overcome a drought (at any instance and any location globally) presents the opportunity for the estimation of a likely drought recovery period. We can apply a probabilistic approach by using the historical precipitation forecast model to simulate different precipitation scenarios based on the historical distribution of precipitation for each region. Here, we propose three precipitation scenarios, namely (1) normal precipitation (as described in Sect. 3.3.2), (2) 1 standard deviation wetter than normal precipitation (assumed as being a wet month) and (3) 3 standard deviations wetter than normal precipitation (assumed as being an exceptionally wet month). The latter two scenarios are based on 1 standard deviation from the local precipitation climatology to simulate average rainy and extremely rainy months, respectively. Again, we assume that, in order to overcome a deficit due to drought, the ecosystem needs to receive a surplus of water that surpasses the climatological average. It follows that if drier than normal conditions were to persist indefinitely, then a drought could theoretically go on forever. The climatological average is integrated with the estimated surplus required precipitation (Fig. 3b; magenta plot) to obtain the absolute required precipitation (Fig. 3b; blue plot). Whenever precipitation is more than the absolute required precipitation, the system advances in recovery to its predrought state. Based on this hypothesis, we simulated the three scenarios for how long any instance of drought will continue, given the expected three precipitation cases. Note that the scenarios suggest the needed recovery time for normal, wet and exceptionally wet years, hence providing a minimum baseline for the duration of drought recovery.

4.1 Observed recovery time based on GRACE and GPCP observation

In this study, drought is defined by the negative deviation of TWSA from its record-length climatology. The observed recovery duration is measured directly from the storage deficit, as described previously (Fig. 1,; Thomas et al., 2014). For our approach, we need to know when the observed precipitation is more than the absolute required precipitation (Sect. 3.2). Figure 6 shows the recovery estimation of all the droughts that occurred during 2002–2017 at four random example locations, namely northwestern tropical Australia (123.25^∘ E, 17.75^∘ S), northeastern Argentina in the La Plata basin (56.25^∘ W, 27.75^∘ S), northern India in the Ganges Basin (78.75^∘ E, 27.75^∘ N) and northern Brazil in the Amazon basin (57.25^∘ W, 2.25^∘ S). Whenever the observed precipitation (Fig. 6; red plot; i.e., GPCP) is larger than the required precipitation (blue plot) for its respective month, the drought should end. Ideally, GRACE should also observe it simultaneously.

Figure 6Validation of the required precipitation estimate by drought recovery estimates at example locations. The different instances of drought show that drought ends (from the perspective of TWSA) whenever the observed precipitation (red plot) exceeds the required precipitation (blue plot).

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The figure shows that the precipitation during a drought typically stays below its monthly required precipitation until the end of the drought. In most cases, precipitation crossed the required precipitation limit in precisely the same month when GRACE observed the end of the storage deficit. Even for the case of recurring droughts with 2 or more months' gap, both methods observed the end of the drought in approximately the same month. To examine our method in detail, we randomly selected a drought month and validated our approach and estimated the recovery time based on different precipitation scenarios in the following section.

4.2 Example of storage deficit and required precipitation

In this section, we discuss drought in an example month of January 2016. During the study period (2002–2017), the year 2015–2016 was the strongest El Niño on record, and many regions experienced drought. Nevertheless, this is done to demonstrate the recovery analysis and can be applied to any other time window. Figure 7 shows the regions under drought in January 2016 (Fig. 7a) and the estimated required precipitation to overcome the drought (Fig. 7b).

Figure 7(a) Storage deficit in an example month (January 2016). (b) The amount of required precipitation to fill the deficit.

Here, the severity of a drought is defined by the amount of water shortage in 1 month. All colors, other than white, in the figure are the drought-affected regions in January 2016, within the region of strong precipitation–storage relations (discussed in Sect. 3.2). The color bar demonstrates the severity of the drought, i.e., the amount of missing water (Fig. 7a) and the respective amount of required precipitation (Fig. 7b). Figure 7a shows that eastern Amazon, southern Australia, southeastern Africa and northern India were under severe drought in 2016 winter. For most of the region in the Southern Hemisphere, the amount of required precipitation is double the storage deficit because January is a summer month and water demand is higher.

Figure 8Validation of the estimated required precipitation by the recovery duration from the January 2016 drought observed from (a) GRACE and (b) estimated, by the discussed method, using GRACE and GPCP observations. (c) Consistency in the observed recovery duration by GRACE and GPCP (1=1–2 months' difference, 2=3–4 months' difference, 3=5–8 months' difference and $\mathrm{4}=\mathrm{9}+$ months' difference).

The expected 1-month recovery state

Spatiotemporal patterns of drought at the global scale are largely uncharacterized. Often, 1 month of surplus precipitation is not enough to fill the entire deficit. However, if rain is significantly above average immediately after/during the drought, the recovery time decreases dramatically. We simulated a 1-month (February 2016) recovery percentage for the January 2016 drought, given the three different precipitation scenarios (discussed in Sect. 3.4). The surplus precipitation within 1 month (February) is divided by the required reconstructed precipitation to calculate the percentage recovery. In most of the drought-affected regions, the recovery percentage of our forecasted normal precipitation (Sect. 3.3.2) for February 2016 is more than the recovery percentage of observed GPCP precipitation (Fig. 9d). This indicates that February 2016 was drier than our estimated normal. Most of the region recovered in an extremely wet scenario (Fig. 9c) within 1 month, except for regions dominated by summer monsoons (Fig. 9c; orange/yellow area), which had less than 30 % recovery, as February is not a rainy season for this region. This shows a case in which regions with high-amplitude seasonal cycles in precipitation mostly recover during their rainy season, which varies globally.

Figure 9Expected percent recovery in 1 month, given the three different precipitation scenarios and the observed GPCP precipitation.

Best estimated time for recovery

Recovery time varies from immediate (i.e., 1 month) to several years across different climate zones and depends on the severity of the drought. Figure 10 shows the predicted recovery duration of the January 2016 drought state, which ranges from 1 month (yellow) to not recoverable within the study period of 15 months (black). Figure 10d shows the recovery duration observed by GRACE, which is considered as being the truth. Figure 10a and b show that most of the region under severe drought in 2016 did not recover with even 1 standard deviation of wetter than normal precipitation, and the drought in this region continued beyond a year. In the extremely wetter (3 standard deviations) than normal situation (Fig. 10c), most of the regions recovered within 4–5 months, except for regions of the most severe drought, such as the southeastern Amazon and southern Africa. Even in the extremely wet scenario, the monsoon regions (Fig. 10c) recovered only during their rainy season (6–7 months after January 2016). This demonstrates that information on the state of precipitation, compared to its usual levels, can provide an idea of the expected drought recovery duration, provided we know the amount of precipitation required.

Figure 10Duration of drought recovery from January 2016, given the three different precipitation scenarios, as observed by GRACE.