Many drought indices have been developed for the purpose of drought monitoring, based on meteorological and hydrological variables which show the size, duration, severity and spatial extent of droughts. The PDSI is such an example. It was developed by  \citet{Palmer1965} and it is one of the most well-known and widely used drought indices in the U.S. (e.g.  \citealt*{Karl1982}\citealt*{Byun_1999}) and beyond (\citealt{Dai1998,Dai2004,Shabbar2004,Tatli2015}Skinner, 2004van der Schrier et al., 2006). PDSI values are computed along the soil moisture balance that requires time series of temperature, precipitation, ground moisture content (or available water-holding capacity) and potential evapotranspiration. The calculation algorithm of PDSI - either in its original version by  \citet{Palmer1965} or in modified ones (e.g. \citealt{Wells2004}) - is thus a reflection of how much soil moisture is currently available compared to that for normal or average conditions. The PDSI incorporates both precipitation and temperature data in a simplified, though reasonably realistic, water balance model that accounts for both supply (rain or snowfall water equivalent) and demand (temperature, transformed into units of water lost through evapotranspiration), which affect the content of a two-layer soil moisture reservoir model (a runoff term is also activated when the reservoir is full). Not explicitly bounded, the PDSI typically falls in the range from -4 (extreme drought) to +4 (extremely wet). This dimensionless quantity allows the PDSI to be compared between regions with radically different precipitation regimes.
Land-atmosphere interactions can introduce persistence into droughts because reduced precipitation lowers soil moisture, reduces surface evapotranspiration and, with less vapor in the atmosphere, further reduces precipitation. In this sequence, soil moisture adjustment occurs with a length of time which introduces a lag and a memory. Depending on situations, there might be a strong coupling between soil moisture and precipitation, and land surface processes can lead to persistence \citep{Koster2004}. The calculation of PDSI is intended to model persistence in the soil moisture balance. The combination of past wet/dry conditions with past PDSI data means that the PDSI for a given time step (generally one month) can be seen as a weighted function of current moisture conditions and a contribution of PDSI over previous times \citep{Cook2007}. In the light of this persistence structure, PDSI chronologies can be used to reconstruct drought conditions, but persistence can also be a criterion to be used as a measure of predictability \citep{Tatli2015}.
This paper deals with time series analysis (TSA) related to PDSI dynamics. Several statistical TSA approaches have been applied to predict climate variables, including their extremes (e.g. Han et al., 2012; Huang et al., 2014). The potential ability of these modeling approaches to forecast drought has been demonstrated by Mossad and Alazba (2015). However, drought forecasts are often performed at monthly time-scale (e.g., Mishra and Desai, 2005; Fernández et al., 2009). Here, we target annual to decadal time scales. We investigate to what extent TSA model simulations may give accurate forecasts of future hydrological changes. Although research on meteorological drought is particularly difficult because of the complex and heterogeneous character of drought processes, their temporal trends respond to climate fluctuations (e.g. large-scale atmospheric circulations). Specifically, the work explores a homogenized long series of annual PDSI data (1801-2014) as derived for California by  \citet*{Griffin2014}. Then the study assesses the response of an Exponential Smoothing (ES) model, using an Ensemble Prediction approach. ES \citep{Holt2004}(Gardner, 2006) and autoregressive integrated moving average (ARIMA) models (Box and Jenkins, 1970) are the most representative methods in TSA. In this study, ES was used because it is known to be optimal for a broader class of state-space models than ARIMA models (Gardner, 2006). ES responds easily to changes in the pattern of time series \citep{McClain1974} and is often referred to as a reference model for time-pattern propagation into the future  \citep{Taylor2003,Hyndman2008}. It is also less complex in its formulation and, as such, it was expected to be easier in identifying the causes of unexpected results. The ensemble approach has been adopted as a way to consider uncertainty in hydrological forecasting, and thus enhance accuracy by combining forecasts made at different lead times, as in  \cite{Armstrong_2001} and in previous authors’ papers (e.g.  \citealt{Diodato2014,Diodato2017} and \citealt*{Diodato2014a} ). A lengthy PDSI series offers a unique opportunity to explore past interannual-to-interdecadal climate variability, and to use its internal dependence structure to (try to) replicate future PDSI ramifications. This approach is compared with the more traditional TSA approach using transfer function models, introduced by \citet*{m1976} and re-visited by \citet*{Shumway2017}. In this case input time series of El Niño Southern Oscillation (ENSO) and the the Pacific Decadal Oscillation (PDO) are considered as impulse inputs to the output PDSI time series. An approximate ensemble approach is also developed under the transfer function models framework for comparison purposes with the ES results.
The paper is organized along the following lines. Data resources and modeling approaches are described in Section \ref{853859}. The empirical results on data analysis and ensemble forecasts are described in Section \ref{995174}. A comparison of the ES model with the transfer function modeling approach is presented in Section \ref{631165}. The results are discussed and put in perspective with previous studies in Section \ref{348939}, which closes the paper with some concluding remarks.