Supplementary information for the paper "On the long-range dependence properties of annual precipitation using a global network of instrumental measurements"

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Modelling of rainfall maxima at different durations using max-stable processes

<div>The multivariate extreme value distribution (MEVD) has been used to model the dependence of rainfall block maxima at different temporal resolutions, as a means of estimating intensity-duration-frequency (IDF) curves for engineering applications. It is characterized by max-stability, which assumes that under proper renormalization, the rainfall block maxima at different temporal resolutions are extreme value distributed and the degree of their dependence remains invariant to the severity of the event. Due to these properties, and contrary to other commonly used approaches, MEVD allows for more conservative return level estimates at those durations used for model fitting. Max-stable processes are continuous extensions of MEVD, which are more flexible, and allow for extrapolation to temporal resolutions beyond those used for model fitting. Here we: 1) propose using max-stable processes to model rainfall block maxima, 2) apply the Brown-Resnick, Schlather and extremal-t models to hourly rainfall data, and 3) compare the obtained results to traditional approaches for IDF estimation. We discuss advantages and limitations regarding the use of max-stable processes in IDF estimation, and their potential use in hydrologic practice.</div

One-step ahead forecasting of geophysical processes within a purely statistical framework: Supplementary material

Supplementary material for the paper entitled "<b>One-step ahead forecasting of geophysical processes within a purely statistical framework</b>"<div><br></div><div><u>Abstract</u>: The simplest way to forecast geophysical processes, an engineering problem with a widely recognised challenging character, is the so called “univariate time series forecasting” that can be implemented using stochastic or machine learning regression models within a purely statistical framework. Regression models are in general fast-implemented, in contrast to the computationally intensive Global Circulation Models, which constitute the most frequently used alternative for precipitation and temperature forecasting. For their simplicity and easy applicability, the former have been proposed as benchmarks for the latter by forecasting scientists. Herein, we assess the one-step ahead forecasting performance of 20 univariate time series forecasting methods, when applied to a large number of geophysical and simulated time series of 91 values. We use two real-world annual datasets, a dataset composed by 112 time series of precipitation and another composed by 185 time series of temperature, as well as their respective standardized datasets, to conduct several real-world experiments. We further conduct large-scale experiments using 12 simulated datasets. These datasets contain 24 000 time series in total, which are simulated using stochastic models from the families of Autoregressive Moving Average and Autoregressive Fractionally Integrated Moving Average. We use the first 50, 60, 70, 80 and 90 data points for model-fitting and model-validation and make predictions corresponding to the 51st, 61st, 71st, 81st and 91st respectively. The total number of forecasts produced herein is 2 177 520, among which 47 520 are obtained using the real-world datasets. The assessment is based on eight error metrics and accuracy statistics. The simulation experiments reveal the most and least accurate methods for long-term forecasting applications, also suggesting that the simple methods may be competitive in specific cases. Regarding the results of the real-world experiments using the original (standardized) time series, the minimum and maximum medians of the absolute errors are found to be 68 mm (0.55) and 189 mm (1.42) respectively for precipitation, and 0.23 °C (0.33) and 1.10 °C (1.46) respectively for temperature. Since there is an absence of relevant information in the literature, the numerical results obtained using the standardised real-world datasets could be used as rough benchmarks for the one-step ahead predictability of annual precipitation and temperature.<br></div

Univariate time series forecasting properties of random forests

The random forests’ univariate time series forecasting properties have remained unexplored. Here we assess the performance of random forests in one-step forecasting using two large datasets of short time series with the aim to suggest an optimal set of predictor variables. Furthermore, we compare their performance to benchmarking methods. The first dataset consists of 16 000 simulated time series from a variety of Autoregressive Fractionally Integrated Moving Average (ARFIMA) models. The second dataset consists of 135 mean annual temperature time series. The random forests performed better mostly when using a few recent lagged predictor variables. A possible explanation of this result is that increasing the number of lagged variables decreases the length of the training set and simultaneously decreases the information exploited from the original time series during the model fitting phase. Furthermore, the random forests were comparable to the benchmarking methods

Dependence of long-term persistence properties of precipitation on spatial and regional characteristics

<p>The long-term persistence (LTP), else known in hydrological science as the Hurst phenomenon, is a behaviour observed in geophysical processes in which wet years or dry years are clustered to respective long time periods. A common practice for evaluating the presence of the LTP is to model the geophysical time series with the Hurst-Kolmogorov process (HKp) and estimate its Hurst parameter <i>H</i> where high values of <i>H</i> indicate strong LTP.</p> <p>We estimate <i>H</i> of the mean annual precipitation using instrumental data from approximately 1 500 stations which cover a big area of the earth’s surface and span from 1916 to 2015. We regress the <i>H</i> estimates of all stations on their spatial and regional characteristics (i.e. their location, elevation and Köppen-Geiger climate class) using a random forest algorithm. Furthermore, we apply the Mann-Kendall test under the LTP assumption (MKt-LTP) to all time series to assess the significance of observed trends of the mean annual precipitation.</p> <p>To summarize the results, the LTP seems to depend mostly on the location of the stations, while the predictive value of the fitted regression model is good. Thus when investigating for LTP properties we recommend that the local characteristics should be considered. Additionally, the application of the MKt-LTP suggests that no significant monotonic trend can characterize the global precipitation. Dominant positive significant trends are observed mostly in main climate type D (snow), while in the other climate types the percentage of stations with positive significant trends was approximately equal to that of negative significant trends. Furthermore, 50% of all stations do not exhibit significant trends at all.</p