# Decadal Oscillation in the Predictability of Palmer Drought Severity Index in California

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Environmental Setting and Data

#### 2.2. Exponential Smoothing

_{t}is the smoothed PDSI at decadal scale centered on time-year t (Equation (2)); α is the smoothing parameter for the data; I

_{t−p}is the smoothed cycle index at the end of period t, its number being defined by the periods p in the seasonal cycle (Equation (3))

#### 2.3. Transfer Function Models

_{1}(t), X

_{2}(t), …, X

_{k}(t) are the input time series to be considered as explanatory variables contributing to the temporal dynamics of the output series Y(t) and η(t) is a stationary random process. The terms α

_{1}(B), α

_{2}(B), …, α

_{k}(B) are fractional polynomials in the back-shift operator B (such that B

^{S}(X(t) = X(t − s)) of the form:

#### 2.4. Model Validation Methods

## 3. Results and Discussion

#### 3.1. Data Analysis

#### 3.2. Validation Results and PDSI Time Series Predictability

#### 3.3. Simulation Experiment

#### 3.4. Comparison with the Transfer Function Modelling Approach

_{1}presents the sample cross-correlation function (CCF) between El Niño 3.4 series (X) and the PDSI series (Y), and the sample CCF between the pre-whitened X series (residuals after fitting an ARIMA model) and the filtered Y series (after applying the AR(2) filter) in Figure 10b

_{1}. Figure 9b1,b2 show a significant spike at lag = −1 for the CCF between Niño3.4, and PDSI series and PDO and PDSI series. After filtering the input and output time series to discard the autocorrelation effects, Figure 9a1,a2 show the persistent significant leading impact of the Niño3.4 and the PDO series on the PDSI series one year in advance (lag = −1). From the sample CCF functions and according to Box and Jenkins [42], a transfer function model of order $\left({r}_{1},{s}_{1},{d}_{1}\right)=\left(1,1,1\right)$ for input series ${X}_{1}\left(t\right)$ = Niño3.4, and order $\left({r}_{2},{s}_{2},{d}_{2}\right)=\left(1,0,1\right)$ for input series ${X}_{2}\left(t\right)=PDO$, was proposed for this data set. In this case ${\alpha}_{1}\left(B\right)=\left({\delta}_{01}+{\delta}_{11}B\right){B}^{1}/\left(1-{\omega}_{11}{B}^{1}\right)$ and ${\alpha}_{2}\left(B\right)=\left({\delta}_{02}\right){B}^{1}/\left(1-{\omega}_{12}{B}^{1}\right)$.

#### 3.5. Ensemble Forecast with the Transfer Function Model

#### 3.6. Limitations and Perspectives

## 4. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Mapped patterns of reconstructed PDSI for some intervals in 19th century across the U.S. (modified from Cole et al. [5]).

**Figure 2.**(

**a**) Rainfall monthly regime with relative bioclimatic patterns for California; (

**b**) mean annual smoothed 20-km spatial precipitation over the period 1961–1990; (

**c**) the corresponding annual reference evapotranspiration (arranged via LocClim FAO software, http://www.fao.org/land-water/land/land-governance/land-resources-planning-toolbox/category/details/en/c/1032167).

**Figure 3.**(

**a**) Observed Palmer Drought Severity Index time-series (blue curve 1801–2014) with training and validation periods; (

**b**) for the validation period, the simulated series (plume, light grey) with both the ensemble mean (red curve) and the observed Gaussian Filter with 11-year smoothing (bold grey curve).

**Figure 4.**Smoothed periodogram of the PDSI time series (bandwidth is a measure of the width of the frequency interval used in the smoothing procedure).

**Figure 5.**(

**a**,

**a**) Residual histogram and normal Q-Q plot between PDSI forecasted and observed in validation period for the official run; (

_{1}**b**,

**b**) Residual histogram and normal Q-Q plot between PDSI forecasted and observed in validation period for the ensemble mean.

_{1}**Figure 6.**Evolution of observed annual PDSI (black curve) with its smoothed 11-year Gaussian filter in bold blue curve (1942–2014), and exponential smoothing forecasts (2015–2054) with plume prediction (light gray) and the ensemble mean value (bold black line). PDSI classes are also reported.

**Figure 7.**Time series of PDO, ENSO indices (source: https://www.esrl.noaa.gov/psd/data/climateindices/list) and PDSI. MEI-1871 and MEI-1950 are the Multivariate ENSO Index series starting in 1871 and 1950 simultaneously; indices Niño1+2, Niño3, Niño34 and Niño4 are the mean Sea Surface Temperature anomalies in the Pacific Ocean regions: 0–10 S, 90 W–80 W; 5 N–5 S, 150 W–90 W; 5 N–5 S, 170–120 W; 5 N–5 S, 160 E–150 W, respectively; ONI is the Oceanic Niño Index; PDO is the Pacific Decadal Oscillation; SOI is the Southern Oscillation Index.

**Figure 8.**Sample cross-correlation functions between the PDSI series and all ENSO indices shown in Figure 7. Numbers indicate the associated lag at the peak value.

**Figure 9.**(

**a1**) Sample cross-correlation function (CCF) between the pre-whitened Pacific Decadal Oscillation (PDO) series denoted as X and the filtered PDSI series denoted as Y for the training period; (

**a2**) the sample CCF between the PDO series and the PDSI series; (

**b1**) CCF pre-whitened Niño3.4 and the filtered PDSI; (

**b2**) the sample CCF between the Niño3.4 series and the PDSI series.

**Figure 10.**(

**a**) Cross-correlation and (

**b**) partial autocorrelation functions (ACF and PACF, respectively) of the estimated residuals (${\widehat{\eta}}_{t}$) for the fitted model.

**Figure 11.**Observed PDSI time series (black line) with the training dataset to build the model (blue line) for the period 1900–1953, including the filtered observed values (navy blue). Also comparison between the observed values (black line) and predicted values with the TF model for the validation period (1954–2014) (red line), including the corresponding 95% confidence intervals (red dash).

**Figure 12.**Residual histogram and normal Q-Q plot between the PDSI predicted values and observed PDSI time series for the validation period (1954–2014).

**Figure 13.**Observed PDSI time series (black line) with the simulation plume (grey lines) for the period 2015–2054, including the filtered observed series (navy-blue line), the median of the simulated values (thick grey line) and the 2.5% (bottom dashed line) and 97.5% quantile (top dashed line) of the simulated values.

**Figure 14.**Comparison between estimates from the exponential smoothing model (ESM) and the Transfer Function model (TFM) for both the validation period (1954–2014) and the forecast period (2015–2054).

**Table 1.**Estimated values of the Hurst (H) exponent (with two methods) for the PDSI annual series as a whole and for a reduced number of years.

Hurst (H) Exponent/Estimation Method | Whole Series (1801–2014) | Reduced Series (1901–2014) |
---|---|---|

Rescaled range (R/S) | 0.611 | 0.743 |

Ratio variance of residuals | 0.611 | 0.550 |

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**MDPI and ACS Style**

Diodato, N.; De Guenni, L.B.; Garcia, M.; Bellocchi, G.
Decadal Oscillation in the Predictability of Palmer Drought Severity Index in California. *Climate* **2019**, *7*, 6.
https://doi.org/10.3390/cli7010006

**AMA Style**

Diodato N, De Guenni LB, Garcia M, Bellocchi G.
Decadal Oscillation in the Predictability of Palmer Drought Severity Index in California. *Climate*. 2019; 7(1):6.
https://doi.org/10.3390/cli7010006

**Chicago/Turabian Style**

Diodato, Nazzareno, Lelys Bravo De Guenni, Mariangel Garcia, and Gianni Bellocchi.
2019. "Decadal Oscillation in the Predictability of Palmer Drought Severity Index in California" *Climate* 7, no. 1: 6.
https://doi.org/10.3390/cli7010006