One of the world's largest irrigation networks, based on the Indus River system in Pakistan, faces serious scarcity of water in one season and disastrous floods in another. The system is dominated both by monsoon and by snow and glacier dynamics, which confer strong seasonal and inter-annual variability. In this paper two different forecasting methods are utilized to analyse the long-term seasonal behaviour of the Indus River. The study also assesses whether the strong seasonal behaviour is dominated by the presence of low-dimensional nonlinear dynamics, or whether the periodic behaviour is simply immersed in random fluctuations. Forecasts obtained by nonlinear prediction (NLP) and the seasonal autoregressive integrated moving average (SARIMA) methods show that the performance of NLP is relatively better than the SARIMA method. This, along with the low values of the correlation dimension, is indicative of lowdimensional nonlinear behaviour of the hydrological dynamics. A relatively better performance of NLP, using an inverse technique, may also be indicative of the low-dimensional behaviour. Moreover, the embedding dimension of the best NLP forecasts is in good agreement with the estimated correlation dimension. This provides evidence that the nonlinearity inherent in the monthly river flow due to the snowmelt and the monsoon variations dominate over the high-dimensional components and might be exploited for prediction and modelling of the complex hydrological system. Key words forecasting; nonlinear prediction; hydrological time series modelling; identification methods; correlation dimension; seasonality; stochasticity; monsoon; snowmelt; Indus River Analyse non-linéaire de la saisonnalité et de la stochasticité du Fleuve Indus Résumé Un des plus grands réseaux d'irrigation au monde, basé dans le système du Fleuve Indus au Pakistan, fait face à une pénurie d'eau sérieuse pendant une saison et à des inondations catastrophiques pendant une autre. Le système est dominé à la fois par la mousson et par la dynamique de la neige et des glaciers, ce qui lui confère une forte variabilité saisonnière et interannuelle. Dans cet article, le comportement saisonnier de long terme du Fleuve Indus est analysé par deux méthodes de prévision différentes. L'étude évalue également si le fort comportement saisonnier est dominé par la présence de dynamiques non-linéaires de faible dimension, ou si le comportement périodique est simplement noyé dans des fluctuations aléatoires. Les prévisions obtenues par les méthodes de prévision non-linéaire (PNL) et de modèle saisonnier autorégressif à moyenne mobile intégrée (SARIMA) montrent que la performance de PNL est relativement meilleure que celle de SARIMA. Ceci, ainsi que les faibles valeurs de la dimension de corrélation, est indicatif d'un comportement non-linéaire de faible dimension de la dynamique hydrologique. Une performance relativement meilleure de PNL, utilisant une technique inverse, peut aussi être indicative du comportement de faible dimension. De plus, la dimension ...

One of the world's largest irrigation networks of Pakistan is based on Indus River System. These networks face serious scarcity of water in 1 year, and agricultural land destructive disastrous floods in the other. To understand the physical basis of statistical variations in space and time in the river flow, this study assesses the reasons governing the system. Therefore, time series analysis of the mean monthly river flow data is performed. The results show strong linear and seasonal behaviour among the river flow of each station. However, the river is regularly energised by the melting of snow/glaciers, and in the monsoon seasons, it experiences strong control by orographic rainfall. On several occasions, the Indus River shows unpredictable disturbances in its usual natural seasonal characteristic flow and their propagation along the network. Therefore, owing to linear, seasonal and monsoonal behaviours, it is expected some contribution of two major local climatic parameters on the river flow. To examine these relations, this paper analyse sum of monthly precipitation and mean monthly temperature with the mean monthly river flow of each station along the Indus River. The real time impact shows that the temperature is more influential at early stations of the Indus River, whereas the rainfall is more affective in case of middle stations. However, the temperature and rainfall display opposite impacts to the river flow at 1-month delay time lag along the Indus River and it explores that the lower three stations river flows are more influential with precipitation than the temperature.

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This article is a companion to the Lilien (lilien) and modified Lilien commands for computing relative indices in Stata. In this article, we illustrate the main features of the commands with an application to the structural determinants of regional unemployment.

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