1. Introduction
Earth experiences numerous extreme weather-induced disasters: droughts, floods, hurricanes, heat waves, bushfires, insect infestations and many others. Among them, drought is one of the most damaging environmental phenomena [
1]. Timely information about the onset of droughts, their extent, intensity, duration and impact is useful for alleviating the losses of life, human suffering and decreasing damages to economy and environment. In recent years, droughts have been occurring frequently, and their impacts are being aggravated by the rise in water demand and the variability in hydro-meteorological variables due to climate change [
2]. Because of this, droughts have been receiving much attention. The causes for the occurrence of droughts are complex, because they depend not only on atmospheric phenomena but also on hydrologic processes, which makes it difficult to forecast [
3].
Dozens of commonly used drought indices are proposed in recent decades. In general, a drought index is a prime variable for assessing the effects of a drought and defining different drought parameters, which include intensity, duration, severity and spatial extent. A drought variable should be able to quantify the drought at different time scales for which a long time series is essential. The time series of a drought index provides a framework for evaluating drought parameters of interest. A number of different indices are developed to quantify the drought, such as Standardize Precipitation Index (SPI) [
4] and Palmer Drought Severity Index (PDSI) [
5]. Traditionally, these indices are based on data collected from the conventional ground measurements.
Satellite remote sensing provides a synoptic view of the land and a spatial context for estimating drought impacts over large areas timely and spatially [
6]. Many studies show that Normalized Difference Vegetation Index (NDVI) can be a useful index for studying vegetation and ecosystems in semi-arid environments where vegetation cover is less than 30% [
7,
8,
9]. Significant relationships between time series of NDVI and various vegetation indicators including green leaf area index, green biomass production, as well as rainfall or soil moisture in semi-arid environments have been reported [
10,
11,
12,
13,
14,
15]. Consequently, Advanced Very High Resolution Radiometer (AVHRR) instruments on board National Oceanic and Atmospheric Administration (NOAA) derived NDVI and other related indices have been successfully used to identify and monitor areas affected by droughts at regional and local scales [
16,
17,
18,
19,
20]. Tucker and Choudhury [
17] found that NDVI could be used as a response variable to identify and quantify drought disturbance in semi-arid and arid lands since its low values correspond to stressed vegetation. Furthermore, Ji and Peters [
20] found that NDVI is an effective indicator of vegetation response to drought in the Great Plains of the USA. Since NDVI has been proved to represent vegetation responses in a timely manner to climate variability as a normalized ratio, the Vegetation Condition Index (VCI), which is NDVI normalization for each pixel based on minimum and maximum NDVI values over time, was developed by Kogan [
21] in order to relatively assess changes in the NDVI signal through time by reducing the influence of local ecosystem variables. In addition, the probability of the standardized NDVI anomaly, the Standardized Vegetation Index (SVI), has been used to monitor areas affected by droughts and vegetation conditions in terms of relative greenness at pixel level over time periods [
22]. Gutman [
23] showed that the thermal data from polar orbiters might be useful for detecting the inter-annual changes in surface moisture, when the change in the vegetation index fails to produce a significant signal. However, NDVI has two main limitations for drought monitoring, first the apparent time lag between rainfall and the NDVI response and second, little influence of significant precipitation events later in the growing season (plant seed production period) on NDVI. Several authors used thermal (e.g., land surface temperature (LST)) products of the NOAA/AVHRR and MODIS to provide a more ecological and physical interpretation of remotely sensed data for examining drought conditions [
24,
25].
The combined application of the NDVI and LST for mapping and monitoring land surface provides a more mechanistic interpretation of the remotely sensed data. Goetz [
26] reported that the negative correlation between LST and NDVI observed at several scales (25 m
2 to 1.2 km
2), was largely due to changes in vegetation cover and soil moisture, and indicted that the surface temperature can rise rapidly with water stress. Nemani et al [
27] found the slope of LST versus NDVI to be negatively correlated to a crop-moisture index. Therefore, the ratio of LST/NDVI increases during times of drought. There are two methods currently being put forward. The first is a progression from the slop of LST versus NDVI approach, which describes the data as falling into a triangle [
28,
29,
30,
31]. The second, the Vegetation Index/Temperature Trapezoid (VITT) [
32,
33], is an evolution of the Crop Water Stress Index (CWSI) [
34], which promotes the ideas of data falling into a trapezoid. Based on AVHRR brightness temperature, temperature condition index (TCI) have been developed and used for monitoring drought [
35]. On the basis of the triangular space of LST and NDVI, the vegetation temperature condition index (VTCI) approach was proposed for monitoring drought occurrence at a regional level [
36], and the temperature vegetation dryness index (TVDI) was developed for assessing soil surface moisture status [
37]. VTCI is time-dependent and usually region-specific, and is better used during plant growing seasons. It was determined that the VTCI can be used to monitor droughts in a near real time at regional level, to better indicate the severity of droughts in the Guanzhong Plain of China and the Great Plains of the USA [
24,
25]. Lin et al. [
38] validated that VTCI drought monitoring method at scale of 10 days is more suitable. Zhang et al. [
39] classified and validated drought categories for the VTCI. Wang et al. [
40] estimated soil water in northern China based on VTCI and determined that VTCI method has good precision and can monitor the drought conditions in northern China more accurately.
Drought forecasting is very important for effectively assisting local authorities to mitigate the drought impacts, and to reasonably use water resources. Nowadays, the time series forecasting has been widely applied and has become an important approach of drought forecasting. In a time series model, the past observations are analyzed to formulate a model describing the inner correlation among them. Then, the time series is extrapolated into the future according to the model. The Markov chain approach was used by Lohani and Loganathan [
41] to develop an early warning tool. A non-homogeneous Markov chain formulation was adopted to derive drought characteristics and assess dry spells from long-term records of the PDSI in two climatic areas of Virginia, USA. Rao and Padmanahan [
42] investigated the stochastic characteristics of yearly and monthly Palmer’s Drought Index (PDI) to characterize those using valid stochastic models to forecast and simulate PDI series. Sen [
43] predicted the possible critical drought durations that may result from any hydrologic phenomena during any future periods using the second order Markov chain. Kim and Valdes [
44] used the PDSI as drought parameter to forecast droughts in the Conchos River Basin of Mexico.
Another most widely used time series model is the AutoRegressive Integrated Moving Average (ARIMA) model [
45]. However, it is basically a linear model assuming that time series data are stationary, and have a limited ability to capture non-stationarities and nonlinearities in the data. The main advantage of ARIMA model forecasting is that it only requires the time series data. Thus, in time series analyses of transportation freight and transportation demand, ARIMA models are mostly used [
46]. ARIMA modeling has been also implemented in runoff and inflow [
47,
48], electric grid [
49].
ARIMA models effectively consider serial linear correlation among observations, whereas Seasonal AutoRegressive Integrated Moving Average (SARIMA) models can satisfactorily describe time series that exhibits non-stationarity both within and across seasons. The SARIMA models describe the seasonality and autocorrelation structure of the time series of VTCI more complexly and appear to be more suitable for assessing the relations within the series. In the AR(1) models’ development [
50], there is only one parameter to be estimated. If the estimated parameter is larger than the one, the forecasting results will be larger than the monitoring ones, and with the increase of the forecasting steps, the forecasting values are gradually increased. The SARIMA modeling approach has been promoted as the statistical method of choice in the case of data arising from observations collected over long periods of time. The main objective of the present study is to model time series of VTCI for drought forecasting by applying SARIMA models in the Guanzhong Plain of China.
3. Results
The acquired time series is from the first ten days of March to the last ten days of May during 2000–2009. The dataset from the first ten days of March of the year 2000 to the last ten days of March of the year 2009 was used for model development and for obtaining the fitted model for each weather station. First, start with model and analyze the time series of VTCI in each weather station, and take Tongchuan station as an example to present the subsequent modeling process. The graph and ACF of the time series VTCI in Tongchuan station are shown in
Figure 2.
As can be seen in
Figure 2, the series contains non-stationary characteristics, since it represents “uncontrolled” behaviors. The dataset is formed by ten years with nine data in each year, which indicates the periodicity, as
equals to nine, as shown in
Figure 2a. The estimated ACF of the series indicates that at least one differencing is necessary in
Figure 2b. Therefore, considering first simple differencing (
) and first seasonal differencing (
) of
, the ACF and PACF of
are plotted in
Figure 3. The ACF of
are small after the second lag, which shows the series
is stationary, and the seasonal and trend components within the time series of VTCI are eliminated. The next step is to identify the values of
,
,
and
.
The ACF and PACF both tail off exponentially in
Figure 3, which indicates a combined autoregressive moving average (ARIMA) process that contains a
order autoregressive component and a
order moving average component.
is smaller than 3 (
Figure 3b),
is smaller than 2 (
Figure 3a), and
and
are both equal to zero. In order to get the fitting model among the different combination models, the AIC values of each model for Tongchuan weather station are calculated using Equation (6), as shown in
Table 1.
In
Table 1, the values of AIC reach their minimum when
and
, and therefore, ARIMA(1, 1, 1)(0, 1, 0)
9 is considered the suitable model for Tongchuan weather station. After the estimation steps, the diagnostic check is necessary. The autocorrelations of the ARIMA(1, 1, 1)(0, 1, 0)
9 model residuals are shown in
Table 2.
The diagnostic check is provided by the quantity test (Equation (7)), where , , which is approximately distributed as with 25 degrees of freedom. The observed value of is smaller than , and the residual is white noise. The described diagnostic check shows that the model is adequate.
The same process is applied to determine the parameters of VTCI dataset in the other 16 weather stations. The ACF of the datasets does not die out rapidly, suggesting that the time series of VTCIs of the dataset are non-stationary. Therefore, the first simple differencing and first seasonal differencing are used to process the datasets to stationary as
. From the ACFs and PACFs of the 16 time series of VTCIs, the values of
and
are both equal to zero, while the values of
and
of the models vary, as seen in
Table 3. Most values of
are equal to one or two, and values of
are all equal to one except for one model in which
equals to zero.
Although only the model structures of Baoji (ARIMA(2, 1, 0)(0, 1, 0)
9) and Liquan (ARIMA(0, 1, 1)(0, 1, 0)
9) areas in particular are mentioned in
Table 3, observe that the models of the 17 parts can be mainly divided into two model types: ARIMA(1, 1, 1)(0, 1, 0)
9 and ARIMA(2, 1, 1)(0, 1, 0)
9. Integration of ground observed data in the entire area as well as in the specified areas shows that most of the parts identified in ARIMA(2, 1, 1)(0, 1, 0)
9 models are irrigated farmlands, while the ARIMA(1, 1, 1)(0, 1, 0)
9 types are rainfed farmlands. These results show that the irrigated farmlands of the current ten-day conditions are associated with the past first and second ten-day intervals, while in the rainfed farmlands, the current ten-day conditions are only relative with the previous ten-day interval. We can see the whole area is relatively uniform. Furthermore, the ARIMA(2, 1, 1)(0, 1, 0)
9 models are gathered in the middle part of the Guanzhong Plain (
Table 3 and
Figure 1), and ARIMA(1, 1, 1)(0, 1, 0)
9 models are in the east part of the Plain (
Table 3 and
Figure 1). Baishui is the only mountain county in Weinan City, differing from its surrounding counties, which leads to the different model structure. In this respect, the identified models are reasonable for the Plain.
Regarding the values of
and
identified in each part of
Table 3, the SARIMA approach has been applied to forecast droughts in the whole area of the Guanzhong Plain. Our former developed AR(1) models were also used for the forecasting [
50]. Employing the results shown in
Figure 4, the forecasting and monitoring results for the ten days periods of the SARIMA and the AR(1) models are compared in order to determine the better performing model. The forecasting results of the SARIMA models show that the east of the Guanzhong Plain has drought conditions in the second and last ten days of April 2009, and the VTCI’s values in the west of the study area are higher than in the east, which shows that the drought conditions are not severe in the west of the Plain. The AR(1) models results show that the values of VTCI are relatively high and uniform compared to the results of the SARIMA models in the whole study area, and from the
Figure 4 (
Figure 4g–i) we cannot find obvious drought conditions from the ten days of April, 2009. In general, the monitoring results (
Figure 4) show that there is drought occurrence in the northeast of the Plain in the middle and last ten days of April 2009. The forecasting results of the SARIMA models are in good agreement with the monitoring ones up to a certain extent, while those of the AR(1) models approaches are not in good agreement with the actual drought conditions.
5. Conclusions
The Guanzhong Plain is one of the areas subjected to droughts. In this study, the seasonal ARIMA models are developed to forecast droughts using the quantitative drought monitoring index called VTCI, and compared with Han’s AR(1) models for better performance of the models. The structures of the models are identified in 17 pixels of the selected area where the weather stations are located, and the results of model identification show that the whole area can be divided into two types of model structures, which are related to irrigated farmlands and rainfed farmlands. Moreover, the absolute errors of the AR(1) models are less than those of the SARIMA models, while the forecasted drought conditions of the AR(1) models are not in good agreement with the monitoring ones. Based on the categorized drought forecasting results, the AR(1) models can predict no drought grades very well, and the SARIMA models can not only predict the no drought grades but also the drought grades, which indicates that the performance of the SARIMA models is better than that of the AR(1) models. Result shows that the selected SARIMA models are appropriate for drought forecasting.
Further, the SARIMA models describe the seasonality and autocorrelation structure of the time series of VTCI more complexly and appear to be more suitable for assessing the relations within the series. The SARIMA modeling approach has been promoted as the statistical method of choice in the case of data arising from observations collected over long periods of time. Compared with the AR(1) modeling approach, the SARIMA modeling approach has several advantages, in particular, its forecasting capability and its richer information on time-related changes. The VTCI time series of this study are constructed by 10 years, and has seasonal variations and fluctuations, Overall, the SARIMA models are better than the AR(1) models. The forecasting abilities of the SARIMA models decrease with the increase of the forecasting steps.
This research work contributes in developing a feasible approach for drought forecasting in the Guanzhong Plain. Results demonstrate that the SARIMA models can be used for drought forecasting; however, the absolute errors of the SARIMA models are still large and challenging. Further studies to improve the precision of the models by employing other methods to identify seductive models are being considered.