# Robust Decadal Hydroclimate Predictions for Northern Italy Based on a Twofold Statistical Approach

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## Abstract

**:**

## 1. Introduction

^{2}(Figure 1a). The course of the Po and its tributaries have been subject to major structural changes since the mid-20th century, mainly due to flood mitigation measures [13]. Nevertheless, several assessments based on observed and reconstructed precipitation and river discharge data converge in indicating that North Italian hydroclimate evolution is dominated by prominent interannual variability as well as persistent near-decadal variability [6]. This coherent evolution of river discharges and precipitation indicates that Po River discharges can be considered as a reliable descriptor of rainfall variability over Northern Italy [5,6,14], with only negligible influence from direct human alterations of runoff processes.

## 2. Data and Methods

#### 2.1. Data

#### 2.2. Prediction Strategy

^{18}O time series [28,29]. The idea is that a reliable prediction of the process underlying a time series can be obtained if the forecasting methods are applied not on the original record, but separately on its statistically significant variability modes. Such modes consist of the deterministic, and therefore predictable, part of the signal and can be detected using reliable spectral analysis methods. The advantages of this procedure are based on (i) the zeroing of the noise level (i.e., the removal of random and unpredictable components of the record) and (ii) the possibility to adjust the prediction algorithms to best fit the specific time scale of each considered variability mode [30]. Predictions are performed with two different algorithms, namely autoregressive models and feed-forward neural networks. Both methods rely on parameters evaluated through a training procedure, applied to the longest portion of the time series called the learning section (time interval 1807–1992). The performance of both methods is quantified comparing predictions with the observed data over the last 25 years, representing the test section. The choice of fixing the test section over the last portion of the time series relies on the fact that the significant components detected in the record could change their amplitude and phase over the time interval. In fact, the singular spectrum analysis (SSA) method we adopt is particularly efficient in detecting changes in the behavior of the oscillatory components inside a climate record, usually due to long-term trends in the climate system. In order to verify if our training algorithms would be able to capture properly such variability, we decide to focus the quantification of the prediction skill over the last portion of the time interval. Finally, we select the best models to forecast Po discharges for 25 years in the future, namely the period 2018–2042 (forecast section). The schematic diagram of this procedure is shown in Figure 2a,b. Both AR and NN algorithms are applied to each SSA–reconstructed component (RC) and the final forecast is obtained as the sum of the individual predictions.

#### 2.2.1. Detection of Deterministic Components of the Time Series

#### 2.2.2. AR Method

_{AR}previous values, where M

_{AR}is the model order. We chose the order a posteriori using goodness-of-fit criteria to allow for a selection of the simplest possible model, i.e., the model with the fewest parameters that adequately describes the observations [35]. We use the final prediction error (FPE; [36]) and the Akaike information criterion (AIC; [37]) as estimators: the best order is the one that minimizes the values of both FPE and AIC. According to this procedure, the obtained results are: M

_{AR,12yr}= 16, M

_{AR,8yr}= 15 and M

_{AR,3yr}= 31. The AR model coefficients were computed with the Burg’s algorithm [38] over the learning section (Figure 2a) and the predictions are obtained by applying an iterative one-step-ahead procedure in the test and forecast sections. In order to quantify the prediction errors, we consider a section including the last 75 points of the time series (so-called cross-validation subsection, Figure 2a) corresponding to three times the value of the maximum lead time L

_{MAX}= 25 (25 years). The same procedure for the prediction error estimate is applied for both the hindcasts in the test section and the forecast in the forecast section. More details can be found in Appendix B.

#### 2.2.3. NN Method

_{1}and H

_{2}of neurons in the hidden layers, are specifically evaluated for each component. A neural network with only one hidden layer was selected for the decadal component, while the other variability modes were predicted using two hidden layers. This difference is due to the fact that a very simple architecture is needed to obtain reliable and stable predictions for the 12-year component, which is the component with the longest period. The input vector length varies between 6 and 24 datasets, while the hidden layers contain 3–5 neurons. The NN predictions are obtained using the MATLAB Neural Network Toolbox. Further details are provided in Appendix C.

#### 2.3. Metrics for Forecast Skill and Robustness

_{i}is the value of the SSA component at the time i and p

_{i}is the corresponding predicted value. We compare RMSE values obtained from our AR and NN forecasts with those obtained from a persistence model. The persistence forecast assumes that future conditions will be the same as past conditions and is commonly used as a benchmark for decadal climate forecasts [41]. Specifically, we assume as a null hypothesis that in the next 25 years the discharge values (the denoised component) would remain constant and equal to the average value over the entire period covered by the record.

_{M}is the average value of the denoised discharge record. When CE ~ 1 it means that the predictions match quite perfectly the observed data, while CE ~ 0 implies that predictions are as accurate as the mean of the observed data and, finally, CE < 0 is obtained when the residual variance (described by the numerator), is larger than the data variance (the denominator).

_{i}≤ 1, namely when the difference is compatible with 0 in 1σ range.

_{i,AR}and f

_{i,nn}are the forecasted values at the time step i for the AR and NN methods, respectively, and σ

_{i,AR}and σ

_{i,NN}the associated uncertainties.

#### 2.4. Drought Severity Quantification

## 3. Results

#### 3.1. Po River Spectral Analysis

#### 3.2. Hindcasts for the Last 25 Years

#### 3.3. Forecast for the Next 25 Years

_{i}) between the forecasts obtained with AR and NN methods for each time step i. When D

_{I}≤ 1, the twofold forecasts are defined to be robust. Figure 4e (right column) illustrates the result of the compatibility test. The total forecast shows overall robustness between both methods (except for two prediction years), thus confirming the strength of this twofold methodology. Considering the single components, the decadal and eight-year components result to be robust only for a few years. However, the decadal component predictions are compatible in the 2σ range (D

_{I}≤ 2) over almost the entire forecasted period. On the contrary, the eight-year predictions are scarcely compatible, because of the phase lag affecting the AR method.

^{3}/s between 2027 and 2030 is quite abrupt. Moreover, the forecasted drought in the late 2020s/early 2030s is expected to be of the same amplitude, or even more dramatic, than the droughts observed starting from 2003, a year characterized by the reduction of water flows of about 50–75% due to scarce precipitation in spring and high temperatures, over the seasonal average [48]. Since the statistical methods are applied to the denoised part of the discharge signal, they could explain the future variability of only ~40% of the total record. In order to provide a variability range effective for the total annual discharge, we add to the uncertainties associated with AR and NN predictions also the contribution of residuals, namely the random and unpredictable part of the discharge record. In Figure 5a, the total prediction bands are represented by the dotted red and blue curves. The contribution of residuals was evaluated as 95% c.l. of their cumulative distribution. Therefore, in this case, we can compare the raw discharge data for the period 2018–2019 with our predictions (green dots).

#### 3.4. Evaluation of Drought Severity

_{5}, Figure 5b) and 10 years (SPI

_{10}, Figure 5c). Similar results are obtained with the NN method (not shown).

_{5}reveals that the Po River experienced an increasing frequency of multiannual droughts in the course of the last centuries: only a moderate drought episode is detected during the 19th century (in the 1830s), whereas the 20th century is characterized by several moderate events (in the 1910s, 1940s, 1980s). The drought event in the first decade of this millennium (around 2003) is the only minimum classified as extreme according to the SPI

_{5}. The drought predicted in the late 2020s/early 2030s is extreme as well, following the historical tendency of increasing frequency and magnitude of multiannual droughts.

_{10}indicates neither moderate nor extreme decadal droughts during the 19th century. Two events are classified as extreme during the 20th century (in the 1940s and 1960s) and one as moderate (in the 1980s). The minimum around 2003 ranks as extreme, whereas the forecasted drought in the late 2020s/early 2030s ranks as moderate.

_{5}and SPI

_{10}reveal that the intensity of the drought event in 2020 may be neither moderate nor extreme. Our forecasts thus support the intermittent occurrence of moderate/extreme decadal droughts over Northern Italy paced at intervals of about 20 years as observed in the past 60 years.

## 4. Discussion

#### 4.1. Rainfall vs. Runoff Processes

^{−5}) except for the first decade of this millennium, when strong annual precipitation contrasts with low discharges. This discrepancy seems to trace back to the quality of the HISTALP data in this period, as the E-OBS dataset provides precipitation values in the same period that are consistent with the discharge record. Even if discharges are determined both by direct precipitation and snow melting—besides evapotranspiration and runoff processes—no apparent correlation is detectable between annual discharge and near-surface air temperature recorded over neither the whole river basin nor the Alpine region (not shown). Therefore, if we consider discharge measurements close to the river mouth, annual precipitation variability dominates over temperature changes.

#### 4.2. Methodological Aspects

#### 4.3. Drought Severity and Attribution

#### 4.4. Broader Climatic and Socioeconomic Implications

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Singular Spectrum Analysis (SSA)

_{D}of the data (we used the approach by Vautard and Ghil [58]); and (c) diagonalizing C

_{D}, thus evaluating Λ

_{D}= E

_{D}

^{T}C

_{D}E

_{D}, where Λ

_{D}= diag(λ

_{1},λ

_{2},λ

_{3},…λ

_{M}), with λ

_{1}> λ

_{2}> λ

_{3}> … > λ

_{M}> 0, and E

_{D}is the M x M matrix having the corresponding eigenvectors E

_{k}, k = 1,M as its columns. For each E

_{k}we construct the time series, of length N – M + 1, called the k-th principal component (PC); this PC represents the projection of the original time series on the eigenvector E

_{k}(also called empirical orthogonal function (EOF)). Each eigenvalue λ

_{k}gives the variance of the corresponding PC; its square root is called singular value. Given a subset of eigenvalues, it is possible to extract time series of length N by combining the corresponding PCs; these time series are called reconstructed components (RCs) and capture the variability associated with the eigenvalues of interest. In order to reliably identify the trend and oscillations in a series, the Monte Carlo method (MC-SSA) is used [33]. In this approach, we assume a model for the analyzed time series (null hypothesis) and we determine the parameters using a maximum-likelihood criterion. Then a Monte Carlo ensemble of surrogate time series (size 10,000) is generated from the model and SSA is applied to data and surrogates (EOFs of the null hypothesis basis are used), in order to test whether it is possible to distinguish the series from the ensemble. Since a large class of geophysical processes generates series with larger power at lower frequencies, we assume autoregressive lag-1 noise in evaluating evidence for trend and oscillations. This is done to avoid overestimating the system predictability, by underestimating the amplitude of the stochastic component of the time series. SSA is particularly useful for climatic time series, which are most often short and noisy. For the calculations, we used the freeware SSA-MTM Toolkit [59,60].

## Appendix B

#### Autoregressive Model Method

- Selection of the best order of the AR method. The AR method turns out to be most reliable and robust when the order M
_{AR}of the autoregressive model is not too large with respect to the length N of the time series, since the variance of the AR-coefficient estimates increases with the order. For this analysis, the choice of a suitable AR order is done a posteriori using goodness-of-fit criteria [35], namely the final prediction error (FPE; [36]) and the Akaike information criterion (AIC; [37]). We perform the predictions over the test section using a wide range of values of the AR model (between 1 and 60) and calculate both AIC and FPE values. The value of M_{AR}which minimizes these indices is selected to perform the forecasts. - Evaluation of the AR model coefficients. The values of the M
_{AR}coefficients of model are evaluated with Burg’s algorithm [38] applied to the learning section. - Quantification of the prediction error. In order to quantify the uncertainty associated with this method, we perform 25-year predictions over different portions of this time interval (namely the cross-validation section, Figure 2a). More specifically, we repeat the procedure of the previous point varying the length of the learning section. In this way we obtain an ensemble of 25-year-predictions translated in time. By evaluating the root-mean-square-error between the predicted and original data as a function of the lead time (RMSE(l)) we obtain the uncertainty associated with the predictions. A useful scheme describing this procedure can be found in Alessio et al. [10].

## Appendix C

#### Neural Network Method

- Partition of the learning section. While in the AR method all the data in the learning section are used to evaluate the coefficients of the model, in this case this, section is divided into two subsections: training (82% of the data in the learning section) and validation (18%). The partition can consist of continuous blocks, as shown in Figure 2b, or also in a random division of the data into the two subsets.
- Training of the network. The training set is used to compute the error gradient and update the weights and biases according to the Levenberg–Marquardt algorithm [63,64]. The validation set serves to assess the predictive skills of the NN being trained. More specifically, the error on the validation set—namely the mean squared error between predicted and observed data—is monitored during the training process and normally decreases during the initial phase of training, as does the training set error. When the network begins to overfit the data, the error on the validation set typically begins to rise. Thus, the final network weights and biases are those yielding the minimum error on the validation set. The parameter ranges of the NN architecture, namely the length I of the input vector and the numbers H
_{1}and H_{2}of neurons in the hidden layers, are specifically evaluated for each component. The transfer functions used to evaluate the neuron scalar output are the sigmoid hyperbolic-tangent function for the hidden layers and the linear function for the output one. - Prediction and quantification of the error. Each trained network is used to forecast the component in the test section, and the corresponding RMSE is calculated between predicted and observed data. Then, among all the values of I, H
_{1}and H_{2}, the network architecture that best reproduces the samples in the test section is chosen. The trained network is finally used for the predictions over the forecast section. Since the training process depends upon the random choice of the initial guesses for weights and biases, the procedure described above is repeated 100 times for each component, thus yielding 100 predictions. Then, the average of all the predictions in both the test and the forecasting section is evaluated, after discarding potential anomalous scenarios. Error bands associated with the predictions correspond to one standard deviation, while the standard error of the mean is used as the uncertainty associated with the average prediction.

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**Figure 1.**(

**a**) Po River basin and location of the Pontelagoscuro gauge station (44°53’19.34’’ N, 11°36’29.60’’ E; red diamond). This map was obtained with MATLAB using the m_map package [19] and the GSHHG dataset for coastlines [20,21,22] and the ETOPO one [23] for elevation and bathymetry. The black rectangle represents the area over which total annual precipitations were evaluated over the Po plain (44–46° N, 7–12° E). (

**b**) Comparison between annual Po discharges (black curve) and total annual precipitation (blue curve) over the Po basin using data from the HISTALP dataset. From the year 1950 onward, precipitation data from the E-OBS dataset (green curve in (

**b**)) are also considered.

**Figure 2.**(

**a**,

**b**) Schematic diagram of the sections in which the input data series were divided according to the prediction method selected. For both autoregressive (AR) and neural network (NN) models, the length of the learning, test and forecast sections are the same. Nevertheless, while the AR algorithm uses all the available data to evaluate the coefficients of the autoregressive model (

**a**), the NN method requires that the learning section be further divided into training and validation subsections (

**b**), as described in the text. Both methods were applied to evaluate the hindcasts over the test section in order to compare the predictions to the data included in the last portion of the time series. Finally, the forecasts for the next 25 years are evaluated for the forecast section. Uncertainties associated with AR forecasts were obtained performing 25-year predictions over different portions of this time interval (cross-validation subsection) and then evaluating the RMSE between the predicted and original data as a function of the lead time. (

**c**–

**e**) Significant oscillatory components in the Po River discharge series, obtained with singular spectrum analysis (SSA): reconstructed components (RCs) 1–2,5 (~12-year period; (

**c**)), 8–10 (~8-year period; (

**d**)) and 3–4,6–7,11–14 (~3-year period; (

**e**)).

**Figure 3.**Spectral properties of the Po River discharge series. (

**a**) Monte Carlo-SSA test performed using EOFs 1–14 and AR(1) as the null hypothesis model (window width W = 80 y and Monte Carlo ensemble size 10,000). The red dots mark the eigenvalues corresponding to the empirical orthogonal functions (EOFs) included in the null hypothesis. No excursions occur outside the 99% limits, indicating that the series is well explained by this model. The periods associated with the reconstructed components (RCs) are: ~12 y (RCs 1−2,5), ~8 y (RCs 8−10) and ~3 y (3−4, 6−7, 11−14). (

**b**) Power spectrum obtained performing the spectral analysis of the sum of all the significant components of the Po discharge series, performed using the maximum entropy method (order 20, see [35] and references therein).

**Figure 4.**(

**a**–

**d**) Prediction (left column) and forecast (right column) results for the three components of the Po River discharges over the test (1993–2017) and the forecast (2018–2042) sections. In black, the RCs reconstructed by SSA are plotted, while in red the predictions obtained by the AR method and in blue those obtained by the NN method are shown with the associated error bands. All components are represented as zero-mean. Black diamonds in (

**d**) represent the values of the SSA components obtained analyzing the record including the 2018–2019 period. This SSA analysis was applied using a window width w = 80 y and the significant components extracted are RCs 1−13,15−16. (

**e**) Left column: progressive RMSE as a function of lead time evaluated for both the AR and NN methods (dotted red and blue curves, respectively) and for the persistence hypothesis (dotted black curve). The RMSE is evaluated between the denoised discharge time series (black curve in (

**d**)) and the corresponding predictions. Right column: results of the compatibility test performed between the AR and NN forecasts for each time step. Black dots indicate when the predictions are robust, namely when their difference is compatible with 0 (in the 1σ range). Empty green dots indicate when the difference is null in the 2σ range.

**Figure 5.**Standardized precipitation index (SPI) time series calculated for Po River discharge denoised series and the forecasted time period 2018–2042. (

**a**) Po River discharges (dotted black curve) and corresponding denoised series (black curve) with the associated forecast obtained using the AR model (red curve) and the feed-forward neural network (FFNN) method (blue curve). Dotted blue and red curves represent the total uncertainty associated with the forecasts, considering also the contribution of the residuals. Green dots represent raw annual average discharge data for the period 2018–2019. (

**b**–

**c**) Five-year and ten-year SPI time series. The horizontal lines individuate the thresholds for the categories of severity by [38], namely moderate (SPI = −1, green line) and extreme (SPI = −2, red line) droughts. The colored areas indicate periods when the Po River basin experienced moderate (green areas) and extreme (red areas) droughts.

**Table 1.**Performance indices evaluated between the AR and NN total predictions and the sum of significant SSA components. The definition of the indices is reported in Section 2.3.

r (p-Value) | CE | PI | |
---|---|---|---|

AR | 0.95 (<10^{−4}) | 0.84 | 0.79 |

NN | 0.96 (<10^{−3}) | 0.89 | 0.86 |

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**MDPI and ACS Style**

Rubinetti, S.; Taricco, C.; Alessio, S.; Rubino, A.; Bizzarri, I.; Zanchettin, D.
Robust Decadal Hydroclimate Predictions for Northern Italy Based on a Twofold Statistical Approach. *Atmosphere* **2020**, *11*, 671.
https://doi.org/10.3390/atmos11060671

**AMA Style**

Rubinetti S, Taricco C, Alessio S, Rubino A, Bizzarri I, Zanchettin D.
Robust Decadal Hydroclimate Predictions for Northern Italy Based on a Twofold Statistical Approach. *Atmosphere*. 2020; 11(6):671.
https://doi.org/10.3390/atmos11060671

**Chicago/Turabian Style**

Rubinetti, Sara, Carla Taricco, Silvia Alessio, Angelo Rubino, Ilaria Bizzarri, and Davide Zanchettin.
2020. "Robust Decadal Hydroclimate Predictions for Northern Italy Based on a Twofold Statistical Approach" *Atmosphere* 11, no. 6: 671.
https://doi.org/10.3390/atmos11060671