2.8 Hybrid models

The time series that characterize the evolution of meteorological events (drought, precipitation) in the temporal domain have localized high- and lowfrequency components with dynamic nonlinearity and non-stationary features. MM models have not always proven to be good at capturing the behavior of the time series. Hybrid models can perform superbly when forecasting hydrological and climatological time series. Different combination techniques have been proposed in order to overcome the deficiencies of single models and improve forecasting performance [47]. Many combined models have been introduced in the literature, for example, ANN-ARIMA [48], SVR-ARIMA [49], etc.

Here we will only focus on WT-ML hybrids, where ML is a machine learning method (e.g., ANN or SVR) and WT is a discrete wavelet transform [50].

#### 2.8.1 Wavelet transform (WT)

WT is a time-dependent spectral analysis that decomposes time series in the time-frequency space and provides a timescale illustration of processes and their relationships. In this method, the data series are broken down by transforming them into "wavelets," which are scaled and shifted versions of a mother wavelet [50]. This allows the use of long time intervals for low-frequency information and shorter intervals for high-frequency information and can reveal aspects of data such as tendencies, breakdown points, and discontinuities that other signal analysis techniques might miss, for example, Fourier transform.

There are two main alternatives for WT: discrete wavelet transform (DWT) and continuous wavelet transform (CWT). For DWT, the WT is applied using a discrete set of the wavelet scaling and shifting, whereas in the case of CWT, this scaling and shifting is continuous—that is, CWT is computationally expensive

3. Meteorological indices

DOI: http://dx.doi.org/10.5772/intechopen.85471

anomalies.

effects [54].

in case studies.

the selected time scale.

and reservoir storage).

11

3.1 Standardized precipitation index (SPI)

scale over the entire length of the record (z-score).

Drought can be defined as a period of unusually arid conditions (usually due to rainfall deficiency) that have lasted long enough to cause non-balance in a region's hydrological situation. Based on its intensity and persistence, drought can be classified into four categories [53]: (1) meteorological drought, which occurs when precipitation is less than usual, is characterized by changes in weather patterns; (2) agricultural (vegetation) drought refers to water deficits in plants; it occurs after meteorological drought and before hydrological drought; (3) hydrological drought ensues when the level of surface water and the groundwater table are less than the long-term average; and finally, (4) socioeconomic drought materializes when water resources required for industrial, agricultural, and household consumption are less than required and thus cause socioeconomic

Satellite Data and Supervised Learning to Prevent Impact of Drought on Crop…

A drought index is an indicator or measure derived from a series of observations that reveals some of the cumulative effects of a prolonged and abnormal water deficit. It integrates pertinent meteorological and/or hydrological parameters (accumulated precipitation, temperature, and evapotranspiration) into a single numerical value or formula and gives a comprehensive picture of the situation [53]. Such an index is more readily usable and comprehensible than the raw data

and, if presented as a numerical value, makes it easier for planners and

policymakers to make decisions. Authorities and public and private committees evaluate the impact of drought using these indices and take measures to prevent its

More than 100 drought indices have so far been proposed, and each one has been formulated for a specific condition [55]. The reclamation drought index (RDI), for example, was developed in the USA to activate drought emergency relief funds associated with public lands affected by drought; the crop moisture index (CMI) was designed to show the effects of water conditions on growing crops in the short term and is not a good instrument for displaying long-term conditions. Here we will only describe the standardized precipitation index, which those indices used

Most of the forecasting works reviewed here are based on SPI [56]. It is perhaps

More specifically, SPI is calculated by building a frequency distribution from historical precipitation data (at least 30 years) at a specific location for the precipitation accumulated during a specified period, for example, 1 month (SPI1), 3 months, (SPI3), 24 months (SPI24), and so on. A theoretical probability density function (usually the gamma distribution) is fitted to the empirical distribution for

SPI1 to SPI6 are considered indices for short-term or seasonal variation (soil moisture), whereas SPI12 is considered a long-term drought index (groundwater

The "drought" part of the SPI range is arbitrarily split into "near normal" (0.99 > SPI > 0.99), "moderately dry" (1.0 > SPI > 1.49), "severely dry"

the most popular index for forecasting meteorological drought and has been recommended by the World Meteorological Organization [57]. It can be defined as the number of standard deviations that the observed cumulative rainfall at a given time scale (1,3,6 month) would deviate from the long-term mean for that same time

Figure 7.

Time series wavelet-ANN conjunction model. (A) Three-level wavelet decomposition tree (DWT). (B) Example of the decomposition of a precipitation signal.

and most researchers use DWT. For more information about CWT, the reader should refer to [51].

DWT operates two sets of functions (scaling and wavelets) viewed as high-pass (HPF) and low-pass (LPF) filters. The signal is convolved with the pair of HPF and LPF followed by subband downsampling producing two components. The first component, which is obtained by passing the signal through the low-pass filter, is called an approximation component (or series), and the other component (fast events) is called a detailed component (Figure 7). This process is iterated n times with successive approximation series being decomposed in turn, so that the original time series is broken down into the minimum number of components needed to reflect the time series according to the mother wavelet.

The filterbank implementation of wavelets can be interpreted as computing the wavelet coefficients of a discrete set of child wavelets for a given mother. This mother wavelet function was defined at scale a and location b as

$$
\psi\_{a,b}(t) = \frac{1}{\sqrt{a}} \psi\left(\frac{t-b}{a}\right) \tag{9}
$$

ψ0,0ð Þt is a mother wavelet prototype and a, b are scaling and shifting parameters, respectively.

Several wavelet families have proven useful for forecasting various hydrological time series. As an example, we can mention Haar, which is also known as daubechies1 or db1 [50]. It is defined as

$$\begin{cases} \quad \mathbf{1} \circ \mathbf{0} \circ t \prec \mathbf{0}. \mathbf{5} \\\\ \quad - \mathbf{1} \circ \mathbf{0}. \mathbf{5} \prec t \prec \mathbf{1} \\\quad \mathbf{0} \quad \text{otherwise} \end{cases} \tag{10}$$

A full description of DWT can be found in [50, 52].

Satellite Data and Supervised Learning to Prevent Impact of Drought on Crop… DOI: http://dx.doi.org/10.5772/intechopen.85471
