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Chaos theory applied to the modelling of karst springs: first results from univariate time series

La théorie du chaos appliquée à la modélisation des sources karstiques: premiers résultats à partir de séries temporelles univariées

La teoría del caos aplicada a la modelización de surgencias kársticas: primeros resultados basados en series temporales univariantes

混沌理论在岩溶泉模拟中的应用:从单变量时间序列得到的首次结果

Teoria do caos aplicada à modelagem de nascentes cársticas: primeiros resultados de séries temporais univariadas

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Abstract

The hydrological dynamics of karstic systems are generally highly nonlinear and weakly predictable. This paper introduces, for the first time, a hydrological modelling approach based on chaos theory. Although this modelling approach may be extended to multi-variables, as a first step in exploring its applicability, the focus is on the simple case of single-variable modelling for karst springs, which in practice corresponds to basins where rainfall is ungauged or poorly constrained. Chaos modelling is applied to the discharge of two karstic springs, the Doubs and the Lez springs in France, selected because they represent very different geological settings, climatic conditions, anthropogenic forcings and discharge dynamics. A deterministic model of autonomous ordinary differential equations is obtained for each spring. The models have chaotic behavior in both cases. The forecasting skills of these chaotic models are assessed. Forecasting performance estimates suggest that, under real conditions, forecasting could be performed for time horizons of ~16 h for Doubs and ~19 h for Lez (±1,000 L/s, 95% of confidence). This analysis offers new evidence for chaos in hydrogeology: the dynamic of discharge of karst springs is both deterministic and highly sensitive to the initial conditions, and it can be approximated by low-dimensional models.

Résumé

Les dynamiques hydrologiques des systèmes karstiques sont généralement fortement non linéaires et faiblement prévisibles. Cet article présente, pour la première fois, une approche de modélisation hydrologique basée sur la théorie du chaos. Bien que cette approche de modélisation puisse être étendue à des variables multiples, la présente étude, destinée en première phase à explorer l’applicabilité de l’approche, se concentre sur la modélisation d’une unique variable de débit, qui correspond en pratique à des bassins insuffisamment jaugés ou entièrement non jaugés. La modélisation chaotique est appliquée au débit de deux sources karstiques, les sources du Doubs et du Lez en France, choisies en raison de configurations géologiques, de conditions climatiques, d’influences anthropogéniques et de dynamiques de débit, très différents. Un modèle déterministe aux équations aux dérivées ordinaires autonomes est obtenu pour chaque source. Les modèles présentent un comportement chaotique dans chacun des deux cas. Les performances prédictives des deux modèles sont évaluées. Les performances prévisionnelles suggèrent qu’en condition réelles des prévisions pourraient être effectuées à des horizons temporels de ~16 heures pour le Doubs et ~19 heures pour le Lez (±1,000 L/s, 95% de confidence). Cette analyse présente une preuve nouvelle de chaos en hydrogéologie : la dynamique du débit des sources karstiques est à la fois déterministes et hautement sensible aux conditions initiales, et elle peut être approchée par des modèles de petites dimension.

Resumen

La dinámica hidrológica de los sistemas kársticos es, por lo general, marcadamente no lineal y débilmente predecible. Este artículo introduce, por primera vez, un método de modelización hidrológica basado en la teoría del caos. Aunque este método de modelización podría extenderse a múltiples variables, como primer paso en la exploración de su aplicabilidad, se centra en el caso más simple de modelización univariante de surgencias kársticas, que en la práctica corresponde a cuencas no aforadas o poco restringidas. La modelización del caos se aplica a la descarga de dos surgencias, la del Doubs y la del Lez en Francia, elegidas porque representan marcos geológicos, condiciones climáticas, forzamientos antropogénicos y dinámicas de descarga muy diferentes. Para cada surgencia, se ha obtenido un modelo determinístico de ecuaciones diferenciales ordinarias autónomas. Los modelos tienen comportamiento caótico en los dos casos. Se ha evaluado la capacidad de predicción de estos modelos caóticos. Las estimaciones indican que, en condiciones reales, se pueden realizar previsiones para horizontes de tiempo de ~16 horas para Doubs y ~19 horas para Lez (±1,000 L/s, 95% de confianza). De este análisis se desprenden nuevas evidencias para el caos en hidrogeología: la dinámica de descarga en fuentes kársticas es simultáneamente determinística y muy sensible a las condiciones iniciales, y puede aproximarse mediante modelos de baja dimensionalidad.

摘要

岩溶系统的水文动态一般具有很高的非线性,很弱的预测性。本论文第一次介绍了一个基于混沌理论的水文模拟方法。尽管这个模拟方法可能扩展到多变量,但为了探索其初步的应用性,我们把焦点放在岩溶泉的单变量模拟状况,在实际中这种模拟状况与无法测量或难于追踪降雨量的流域相关联。混沌模拟被应用于两个岩溶泉的流量: 法国的杜河泉 (Doubs spring) 和雷河泉 (Lez spring),选择这两个泉是基于它们代表非常不同的地质环境,气候条件,人类活动影响和流量动态。对于每一个泉我们都获得了具有自治常微分方程的确定性模型。各个泉的模型都有混沌行为。我们对这些混沌模型的预测能力进行了评价。对预测性能的估算显示,在实际情况下,预测可实施的时间范围是:对杜河 (Doubs) 为16小时,对雷河 (Lez) 为19小时 (±1,000 升/秒,置信区间为95%)。 该分析为水文地质学的混沌提供了新的证据:岩溶泉的流量动态对初始条件既有确定性又有高敏感性,而且它可以通过低维模型来近似。

Resumo

As dinâmicas hidrológicas de sistemas cársticos são altamente não-lineares e pouco previsíveis. Este artigo introduz, pela primeira vez, uma abordagem de modelagem hidrológica baseada na teoria do caos. Embora esta abordagem de modelagem possa ser estendida a multivariáveis, como um primeiro passo para explorar sua aplicabilidade, o foco é no simples caso de modelagem de variável única para nascentes cársticas que, na prática, corresponde a bacias onde a precipitação é insuficiente ou pouco restrita. A modelagem de caos é aplicada à descarga de duas nascentes cársticas, as nascentes Doubs e Lez, na França, selecionadas porque representam ambientes geológicos, condições climáticas, alterações antropogênicas e dinâmicas de descarga muito diferentes. Um modelo determinístico de equações diferenciais ordinárias autônomas é obtido para cada nascente. Os modelos possuem comportamento caótico em ambos os casos. As habilidades de previsão desses modelos caóticos são avaliadas. Estimativas de desempenho de previsão sugerem que, sob condições reais, a previsão pode ser realizada para horizontes de tempo de ~16 horas para Doubs e ~19 horas para Lez (±1,000 L/s, 95% de confiança). Esta análise oferece novas evidências para o caos na hidrogeologia: a dinâmica de descarga de nascentes cársticas é tanto determinante quanto altamente sensível às condições iniciais, e pode ser aproximada por modelos de baixa dimensão.

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Acknowledgements

The authors thank the associate editor Michael Sukop and four anonymous reviewers for their constructive comments which enabled significant improvements to the manuscript. Data sources: Banque HYDRO - MEEDDAT/DGPR/SRNH.

Funding

This work was supported by the Défi InFiNiTi program of the CNRS (Musc and SlowFast project), and the earlier developments of the algorithms (GPoM) used to model chaos were supported by the Les Enveloppes Fluides et l’Environnement (LEFE) program of INSU (MoMu project of MANU section).

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Authors and Affiliations

  1. Centre d’Études Spatiales de la Biosphère, UPS-CNRS-CNES-IRD-INRA, Observatoire Midi-Pyrénées, 18 avenue Édouard Belin, 31401, Toulouse, France

    S. Mangiarotti

  2. Hydrogeology Laboratory, UMR EMMAH, Université Avignon-INRA, Avignon, France

    Y. Zhang & M. Leblanc

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  1. S. Mangiarotti
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  2. Y. Zhang
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  3. M. Leblanc
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Correspondence to S. Mangiarotti.

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Mangiarotti, S., Zhang, Y. & Leblanc, M. Chaos theory applied to the modelling of karst springs: first results from univariate time series. Hydrogeol J 27, 2027–2043 (2019). https://doi.org/10.1007/s10040-019-01971-8

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