Spectral Analysis for Main Annual Streamflow Series

The spectral analysis consists of calculating the energy spectrums of the historical annual mean flow series to identify the dominant frequencies. In the spectral analysis, the mean flow series in the frequency domain are analyzed using the Fast Fourier Transform. Then, the energy spectrum of the fluctuation is calculated using the following equation:

Where:

The following figures (Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 y Figure 15) show the mean annual flow series with their respective dimensionless energy spectrum (using the maximum energy value to nondimensionalize), highlighting the three frequencies considered as dominant in each spectrum (Table 2).

In Figures 6, 11, 12 and 15, the first dominant frequency of the spectrum is marked with a rectangle. These frequencies (greater than the length of the series) were not taken into account because they are the product of trends in the series, or components of the low frequency of the series. This was demonstrated by analyzing synthetic series.

It can be observed in Table 3 that the two dominant periods occurred in the following ranges:

< 3.7 years 2 cases

3.7 - 6.7 years 9 cases

8.5 - 10.7 years 4 cases

10.7 - 18.3 years 2 cases

18.3 - 32 years 11 cases

It is observed that most of the basins (except Colorado, Paraná and Dulce) show fluctuations with dominant periods between 3 and 7 years. Mean flow series of the Colorado, Paraná, Atuel and Dulce rivers have dominant periods between 7 and 11 years. Dominant periods between 13 and 35 years are observed in all river basins (except Atuel, Salado and Pilcomayo basins). Similar results were observed by other authors. For example, ^{Vargas, Minetti and Poblete (2002)} observed quasi-periodic low frequency fluctuations (22 to 26 years) in the mean annual flow series of Paraná River. ^{Compagnucci, Berman, Velasco-Herrera and Silvestri (2014)} observed periodicities of 3.5 in the mean annual flow series of Paraná River corresponding to 9 and 30 years, and for Atuel River periodicities of 4-5 years, corresponding to 7, 11 and 22 years.

In the case of San Juan River, five frequency bands with significant amplitudes were observed, corresponding to periods of approximately 12.3, 7.4, 5.7 and 3.8 years (^{Correa & Guevara, 1992}).

Wavelet coherence analysis

A wavelet coherence analysis was performed for the mean annual flow series, in order to validate the results obtained with the Fourier spectral analysis. In the case of signals that present multiple scales of temporal and spatial variability and frequency, a localized analysis of transformations with wavelet curves can be useful to identify the dominant geometries of the variables analyzed. With regard to dominant frequencies, similar results were obtained with spectral analysis methodologies based on the Fourier transform and wavelet curves. This shows that the signals analyzed do not have a high degree of variability in dominant frequencies, and therefore, satisfactory results would be achieved with the application of the Fourier-based spectral analysis.

The analysis was performed using Morlet wavelet function (ko = 6).

Analysis of the contribution of different processes to the fluctuations observed in the flow series

The analysis of dominant frequencies for mean annual flow series enabled observing the predominant frequency ranges. Then, the variance was calculated (Figure 16). The variance is computed by integrating the energy spectrum in the frequency range analyzed. It allows evaluating the contribution of the fluctuation variances of each frequency range, which is a characteristic of the different processes. Band-pass filters were applied to the energy spectrum between the following frequency ranges:

The band-pass filter used in this paper consisted of a Fourier filter. First, the tool was verified with the analysis of a synthetic series (fluctuations of known periods), making the energy of the fluctuations equal to zero for the frequency ranges not included in the band of the band-pass filter used. Table 4 summarizes the results, indicating the contribution to the total variance of the fluctuations of each frequency range.

The sum is different than 100% because of the correlation between the components.

In N_a, processes less than 2.2 years and the random part of all frequencies are considered. ^{Correa and Guevara (1992)} proposed that residual variance not explained by significant periodicities should be attributed to the stochastic component. This random variance in 86% of the study cases represents between 30% and 50% of the variance of the fluctuations. Thus, the series has a high random component. The rest can be explained by different processes, with characteristic frequencies between 2 and 3 years, 3 and 7 years, 7 and 13 years, 13 and 35 years, and greater than 35 years.

Table 4 shows 9 basins (of 14 studied) where the low frequency processes N_bd dominated (without considering the Na processes), followed in importance by N₅. For two basins, the N_v process was notable, and the N₁₁ process was notable for only one (Paraná). This result shows fluctuations of high, low and medium frequency in the mean annual flow series analyzed, which contribute, with different significance, to the temporal variability of drained flows. This result allows to advance in the study of processes that should be considered in the explanation of hydrological phenomena such as droughts and water excesses, taking into account that the process involves phenomena that have a temporal evolution that includes small, medium and large scales, and random or stochastic processes.

Relation between temporal evolution of hydrological droughts and macro-climatic indices

In this section, macro-climatic and astronomical indices are analyzed using the same spectral analysis of time series, to determine whether there is a relationship between its temporal evolution with the hydrological droughts in the different time scales.

Table 5 describes the macro-climatic indices analyzed.

The selected indices are: ONI, AMM, TSA, AMO, SOI, PDO, Niño 1+2; Niño 3.4 and TNA (see Table 5). Table 5 and Table 6 show the observed dominant periods of the fluctuations and the contributions to the variance (in percentage) of the fluctuations, with the different bandwidths analyzed previously.

Time series of AMO, AMM, ONI, PDO, SOI, Niño 3.4, Niño 1 + 2, TSA and TNA indices were obtained from the NOAA website ^{NOAA (2016)}, and sunspot time series were obtained from SILSO ^{SILSO (2016)}.

In the PDO, TNA, AMM indices, a dominant frequency of 64 years^-1 was observed, and 128 years^-1 for the TSA index. However, they were discarded because of great uncertainty, since the series have a length less than 100 years. This analysis shows a dominant frequency on the order of 4 to 5 years for Niño 1+2, Niño 3.4 and ONI indices. The Sunspots, AMM, SOI, TSA and TNA index have dominant frequencies between 8.5 and 13 years. While the PDO and AMO indices have a dominant multidecadal frequency.

Table 7 shows that the n₁₁ processes contribute 66% to the variance of the energy spectrum, while process n₅ has an important contribution in the ONI, Niño 3.4 and Niño 1+2 indices.

Analysis of correlation between fluctuations of mean annual flow series and macro-climatic indices

Knowing the macro-climatic indices, whose fluctuations can be linked to droughts on different time scales, helps the prediction of these phenomena. In this section, the correlation between fluctuations of mean annual flow series and macro-climatic indices (for different bandwidths) is analyzed. The basins selected for this analysis are Suquía, Dulce, Paraná, San Juan and Atuel rivers, because they are the most extensive and representative time series of each study region. The Fourier band-pass filters used in this analysis are N_bd (between 13 and 35 years), N₁₁ (between 7 and 13 years) and N₅ (between 3 and 7 years). In this analysis, the correlation coefficient between the flow filtered time series of each basin (N_bd, N₁₁, N₅) and the macro-climatic indices filtered series was calculated with the same bandwidth. Then, the series with the best correlation coefficients were graphed. Below are the results for the basins analyzed.