International Letters of Natural Sciences | 2021
Seasonal and Annual Probabilistic Forecasting of Water Levels in Large Lakes (Case Study of the Ladoga Lake)
Abstract
The production functions of water-dependent sectors of the economy can include the water level in the lake as a natural resource. This characteristic must be able to reliably predict for the effective functioning of sectors of the economy. In the article the main attention is paid to the methods of forecasting based on the extrapolation of natural variations of the large lakes water level. As an example, Lake Ladoga is considered. In this paper, it is assumed that the level varies accordingly to a stochastic multi-cycle process with principal energy-containing zones in frequency bands associated with seasonal and multi-annual variations. Hence, the multi-year monthly and yearly averaged time series are represented by the ARIMA (auto-regression integrated moving average) processes. Forecasts are generated by using of the seasonal ARIMA-models, which take into account not only the seasonal but also the evolution non-stationarity. To compare the forecasts and the actual values, the relative errors are computed. It is shown that implementation of the models mainly allows receiving good and excellent forecasts. Subject Classification Numbers: UDC 556.555.2.06(4) Introduction Many sectors of the economy, whose activities are closely related to the use of water resources, require hydrological forecasts. A reliable forecast allows you to optimize economic activities taking into account the needs of production (see, eg [1, 2]). A human has many consumer interests in water, and this interest may be generalized in the language of the economy. If water is regarded as a natural resource, then hydrological variables in economic models may act either as parameters (then hydrology is an external reality for the economy) or on the contrary they can be influenced by economic variables (then we are dealing with an expanded subject area combining hydrology and economy). Fig. 1 shows production and technological interpretation of the economy. The production characterizes the following aspects: labor (L); instrument of labor or fixed capital stock (K); subject of labor ( W W W S + = ~ ), comprising natural resources WS and subject of labor returned to the production as a part of gross product X. International Letters of Natural Sciences Submitted: 2020-08-04 ISSN: 2300-9675, Vol. 82, pp 13-19 Revised: 2021-01-14 doi:10.18052/www.scipress.com/ILNS.82.13 Accepted: 2021-01-27 CC BY 4.0. Published by SciPress Ltd, Switzerland, 2021 Online: 2021-04-28 This paper is an open access paper published under the terms and conditions of the Creative Commons Attribution license (CC BY) (https://creativecommons.org/licenses/by/4.0) Figure 1. Interrelation of the production elements. In this situation the condition of economy is determined by the capital-labor ratio k (K/L), and its management – by labor performance x (X/L) and a share of non-production consumption u (C/Y). With the use of theory of sufficient conditions of optimality, we may show that optimum labor performance x is ensured at x = f (k, t). The production function f (K, L, t) reflects a relation between production factors and its result. Parameter t may be used to account for the influence of external factors, including scientific-technical progress and variability of the natural resources, on the economy (its model). Further we show how hydrological consequences of the climate change in economic calculations may be taken into account (see, eg [3]). The purpose of the study is to consider methods and mathematical models of probabilistic forecasting, taking into account possible climatic changes, the water level in large lakes in Europe, using the example of Lake Ladoga, as a natural factor included in production functions. Level regime of lakes is formed under impact of active and adaptive factors, which, in their turn, are influenced by modern climate and anthropogenic pressure. Climatic factors are usually considered as active, while factors of the underlying surface are adaptive [4]. Various combinations of the climate signal and the underlying surface response, in particular, define certain differences in patterns of seasonal [5, 6] and multi-years fluctuations of the lakes water level. In this paper, the Ladoga Lake is considered as an example of a large European lake [7]. The seasonal cycle of its water level is quite smooth, while its multi-years fluctuations are basically represented by the overlapped low-frequency oscillation and slightly adjusted characteristic elements of the seasonal cycle. Previously performed researches [8, 9] showed that contribution of the seasonal fluctuations dispersion into the total dispersion is significantly less than that of the lowfrequency oscillation. Materials and Methods Application of the autoregressive-integrated-moving-average (ARIMA) methods [10] for the observation time series analysis opens wide perspectives for statistical forecasting of the large lakes water stage. In our research, the Ladoga Lake was considered. Monthly averages of the water level observed at the gauge in Syas’skie Ryadki in the period from 1881 to 2004 were used to simulate their seasonal fluctuations in the typical (in terms of water content) years (1923–1925, 1986, 1990, 1993, 1995, 1998, 1999 и 2003). Quantile analysis of the yearly averages has shown that, in 1923– 1925, 1986, 1990, 1993 and 1995, the Ladoga Lake stages were above the multi-years median. Meanwhile in 1924, a historical maximum (609 cm) was recorded. In 1998, 1999 and 2003, average 14 ILNS Volume 82