Assessment of hydrological drought based on nonstationary runoff data


 A nonstationary standardized runoff index (NSRI) is proposed by using the GAMLSS framework to assess the hydrological drought under nonstationary conditions. The definition of the NSRI is similar to that of SRI, but using a nonstationary Gamma distribution by incorporating meteorological variables and antecedent runoff as covariates to describe the characteristics of runoff series. The new drought index is then applied to the upper reach of the Heihe River basin. Four models are developed, in which one is stationary, and the other three are nonstationary with one, two and three covariates, respectively. Results show that, for the nonstationary runoff series, the nonstationary models are more robust and reliable than the stationary one. Among these models, the model with two covariates performs the best. For the model with one covariate, the precipitation shows better in the fitting as a covariate in rainy seasons, and the antecedent runoff shows better in dry seasons. The NSRI identifies more drought events than SRI does, and the drought conditions in our case are mainly affected by precipitation. It is proved that the proposed new drought index is a more effective method for drought assessments under nonstationary conditions.

found that the runoff series in the continental United States have significant change points during the 20th century. These mean that many runoff series are not stationary any more, and the SRI with the assumpation of the runoff data to be analyzed keeping stationay is no longer suitable for fitting such nonstationaty data (Wang et al. b). Therefore, a modified SRI, namely, a nonstationary standardized runoff index (NSRI), will be proposed in this study with incorporating the nonstationary characteristic of runoff series.
NSRI is defined similarly with the SRI, but using a nonstationary Gamma distribution by incorporating covariates.
Seeking appropriate covariates is the first, also one of the vital steps in constructing the nonstationary drought index (Vu & Mishra ). A single covariate is usually considered in previous literatures. For example, time or climatic variables are usually used to describe the precipitation changes in developing the nonstationary meteorological drought index (Russo et al. ; Li et al. ; Wang et al. a; Javad & Somayeh ). However, as known, the runoff changes are always affected by the recharge sources.
The recharge sources for the upper reach of the Heihe River basin include precipitation, lacial and snow melting, frozen soil thawing and groundwater (Ding et al. ; Wang et al. ). These sources are affected by meteorological elements like temperature, relative humidity and wind speed. Therefore, these meteorological elements are considered as covariates herein. In addition, evapotranspiration is an important process of water transfer in the hydrosphere and atmosphere, playing a vital role in the hydrological cycle and the runoff processes (Han et al. b; Aa et al. ). The antecedent runoff (runoff in the previous month) contributes to the runoff changes at that month especially in non-flood seasons (Ding et al. ). Consequently, the evapotranspiration and the antecedent runoff are also taken into account as alternative covariates. As the actual evapotranspiration data are not available and not easy to calculate, potential evapotranspiration is used here. In summary, the six variables, precipitation, temperature, relative humidity, wind speed, potential evapo-

Study area
The Heihe River originates from Qilian County in the northeastern part of Qinghai Province. The river basin, between 97 37 0 E-102 06 0 E and 37 44 0 N-42 40 0 N, is located in the middle of Hexi Corridor in northwest of China, with an area of 13,40,000 km 2 . The circulation of the westerly belt in the middle and high latitudes and the polar air mass make the climate across the basin dry, strong wind and sunshine, large temperature differences, less precipitation and great evaporation, resulting in the scarcity of water resources in this basin (Gao & Zhao ).
The basin is divided into three reaches, the upper reach, the middle reach and the lower reach. The annual precipitation for the upper reach is about 350 mm, with 70% occurring from June to October. The water resources over the basin are mainly generated from the upper reach and consumed in the middle and lower reaches. The water resources over the basin is 2.62 billion m 3 , supporting for 1.5 million people and 384,000 ha of farmland irrigation (Ling ). The amounts and variations of water resources in the upper reach have great effects on agriculture, economic development and ecological environment for the middle and lower reaches. Thus, it is particularly necessary to carry on the researches about hydrological droughts in the upper reach.

Data
The monthly runoff data at three hydrological stations (Qilian, Zhamushike and Yingluoxia) and the meteorologi-  Figure 1 and the detailed information are displayed in Table 1. These datasets were obtained from the China Meteorological Data Network (http://data.cma.cn/) and the Hydrology Yearbook. (1) Let x(n) represents the runoff data at the nth month. 54 years  of runoff data are avaibale here, thus n ranges from 1 to 648. For a given time scale of k months (a time scale of 12-month is considered here), the cumulative runoff x k (n) is calculated as:

Standardized runoff index
(2) Fitting a two-parameter Gamma distribution, denoted as where μ and σ are the location and scale parameters in Gamma, μ > 0 and σ > 0. Γ( Á ) is the mathematical Gamma function.
(3) The cumulative probability for a given time scale can be calculated as: (4) Converting cumulative probability to a standard normal distribution function: where c 0 ¼ 2.515517, c 1 ¼ 0.802853, c 2 ¼ 0.010328, d 1 ¼   or random effects. In GAMLSS, the cumulative runoff It is assumed that the changes of runoff with covariates obey the following distribution parameters: with the worm plot, is used to test its goodness of fit.
Since the NSRI proposed here is normalized in the same way as the traditional SRI, the same drought-level standards for both indices are suggested (Guttman ). As shown in

Stationary test
Since it is the cumulative runoff series fitted by a distribution, the stationarity of cumulative runoff series is to be  As shown in Table 3, the Dickey-Fuller statistics at the QL station are less than À3.5, and the corresponding P-values are less than 0.05, which means that, at the 0.05 significance level, the cumulative runoff series at the QL station exhibit a stationary state. While at ZMSK and YLX stations, the results are opposite. Therefore, we conclude that the cumulative runoff at these two stations can be considered as nonstationary series. To model such series, the NSRI needs to be developed.

Correlation test
The selection of a suitable covariate is of great importance in developing a NSRI. To test whether the runoff and the alternative variables over the study area are related to  <0.05, there is a significant correlation between the two variables, and the higher the absolute value of r, the stronger the correlation. Among the six variables, five of them (precipitation, temperature, relative humidity, potential evapotranspiration and antecedent runoff) are strongly correlated with runoff with correlation coefficients greater than 0.5, and wind speed shows weak correlations. Therefore, except for wind speed, all variables are selected as covariates to develop the NSRI.  Table 5     Note: P, precipitation; T, temperature; H, relative humidity; E, potential evapotranspiration; R, antecedent runoff. The smallest AIC value in each type of model is denoted in bold font.

Drought characteristics
The occurrence frequency of different drought grades is calculated from both the SRI and the NSRI. As shown in It also can be found that, in Figure

).
A threshold of À1.0 was set to identify the occurrence of drought events in this study. Drought peak, duration and severity were then extracted from the drought events according to the run-length theory. As shown in Table 6, the QL station identifies 11 drought events, of which the drought peak reaches À2.92, the longest drought duration lasts for 25 months and the maximum drought severity is À59.83.
The ZMSK station identifies more drought events (18 events) based on the NSRI, with higher drought peak (À2.42) and higher drought severity (À33.62), compared with the results of SRI (12 events, À2.29, À26.69). The YLX station also identifies more drought events (19 events) based on the NSRI, while the drought peak (À2.47) and the maximum drought severity (À37.66) are lower than those from the SRI (À2.63 and À43.73). The longest drought durations are quite close from both the NSRI and the SRI for this station. Here, six variables are considered, and three nonstationary models are developed. When only one variable is considered, M1, in which the variable of precipitation is taking as a covariate, shows the best for 9 of 24 series, and the series are concentrated basically in April, May, August, September and October (Table 5). It indicates that precipitation is the main factor affecting runoff changes during these months. It is understandable since these months are usually with more precipitation in the study area. M15, taking the variable of antecedent runoff as a covariate, shows the best for 13 of 24 series, and most of them are concentrated from November to about next April. This finding keeps consistent with the river recharge resources in different seasons. In rainy seasons (e.g., summer), precipitation contributes the runoff most, while in dry seasons (e.g., winter and spring), the river is mostly recharged by glacial and snow melting water, permafrost melting water and  Note that many studies also included the time as a covariate in a nonstationary analysis (Russo et al. ; Wang et al. a; Javad & Somayeh ), although some studies have proved that time is not suitable in their cases due to its linear monotonic trends (Li et al. ). It is considered as an additional covariate for comparison to discuss the nonstationary runoff process in this study. As illustrated in Table 7, the corresponding AIC value is moderate, slightly higher than those from the selected models (M21 for the ZMSK station and M22 for the YLX station) and lower than those from the stationary models (M0). Besides, since using time as an explanatory variable has no physical meaning, it can only describe the general trend of a hydrological sequence over time (Vu & Mishra ). Thus, the time is not considered as a covariate in our nonstationary analysis.

Effects of covariates to drought index
To discuss the effects of different covariates to the drought index, we compare the drought index calculated from three types and 16 nonstationary models (five models in M1, seven in M2 and four in M3). It is observed that in For the ZMSK station, the period of October 1990 to January 1991 is detected as normal state from the SRI, while moderate drought from the NSRI. It is found that the precipitation sliding forward for 4 months (from June to September in 1990) was 328.7 mm, lower than the multi-year mean value of 333.7 mm for the same period.
Less precipitation is more likely to cause drought, and thus the conclusion drawn from the NSRI (drought) seems to be more reliable. For the period of June to July 2011, it is found that the observed precipitation from February to March 2011 (6.4 mm) is less than the multi-year mean value (11.0 mm), and the mean temperature (9.04 C) shows higher than the multi-year mean value (8.28 C).
Figure 6 | Differences in the NSRI with different covariates in the study area (the models in the legend are the same as those in Table 5).
Owing to the less precipitation and the higher temperature during this period, it is concluded that drought is more likely to occur, and thus the detection from the NSRI (drought) is more reasonable than that from the SRI (wet). We also calculated the NSRI with time as a covariate (denoted as the TSRI here) for comparison (Figure 7). It is observed that TSRI differs a lot with SRI especially for the first and last several years. This is because that the location parameter is constant and close to its mean value in the SRI, while it is time-dependent in the TSRI. This condition leads to more differences between TSRI and SRI for the early and last years of the record period and less differences for the middle part of the period (Javad & Somayeh ). Furthermore, the linear monotonic trend of time also makes it not as appropriate as other meteorological variables in characterizing the nonstationary processes of runoff series (Li et al. ).
These results support that, compared with the traditional SRI, the new developed NSRI, with considering meteorological variables as covariates, is proved to be more robust and more applicable in this drought assessment.

CONCLUSIONS
Appropriate drought indices are often important tools for regional drought assessments. In the context of changing environment, the traditional SRI which is defined based on a stationary Gamma distribution is no longer suitable due to the nonstationary characteristics of runoff series (Villarini et al. ; Russo et al. ). Thus, the NSRI is developed in this study within the GAMLSS framework.
The proposed new drought index considers climate change and incorporates six variables into calculation, accounting for the nonstationary features of runoff data. As a quantitative indicator, just like the SRI, the NSRI can assess drought events at different time scales (e.g., 3, 6, 9 and 12 months) and allow comparison of climatic conditions at different locations. The 12-month time-scale NSRI is applied to the upper reach of the Heihe River basin.
The conclusions can be summarized as follows. For the study area, (1) the fitting results of the nonstationary models to those nonstationary runoff data (ZMSK and YLX stations) are much better than the stationary models according to AIC values; (2) for the nonstationary models with one covariate, precipitation shows better in the fitting as a covariate in rainy seasons, and antecedent runoff shows better in dry seasons, which is highly related with the recharge sources in different seasons; the nonstationary models with two covariates perform better than those with one or three covariates; (3) the NSRI identifies more drought events than the SRI, and precipitation is the major factor affecting the hydrological drought conditions over the study area.
In general, the current study sloves the problems of hydrological drought assessment based on nonstationary runoff data. It is proved that the proposed new drought index, NSRI, is capable of providing more reliable results for drought assessment over the study area. Revelant findings are also very helpful in developing strategise for coping with local droughts and water resource risk management. Noted that, although the findings here are in the context of NSRI at 12-month scale and limited stations, the framework and the methods in this study could be