Abstract
Identification of groundwater contamination sources, as a solution to an inverse problem, is mainly based on simulation optimization methods. However, the establishment and solution of an optimization model are extremely complicated for the inverse problem, where many decision variables need to be identified. Combining simulation optimization methods with other mathematical methods can effectively avoid this shortcoming. In order to identify features of contamination sources, the Kalman filter is combined with a mixed-integer nonlinear programming optimization model (MINLP). Firstly, the Kalman filter is improved by U-D decomposition (a matrix decomposition method) to identify the number and approximate locations of real contamination sources among potential contamination sources. A 0–1MINLP solution is then employed to identify the accurate location and release history of the contamination source. Kriging is used to produce a surrogate model for the numerical simulation model. The surrogate model approximates the numerical simulation model, which can be embedded in the 0–1MINLP instead of the numerical simulation model and it can be called directly during the iteration of the optimization model, thereby avoiding manually coupling the numerical simulation model with the optimization model during the solution process. Results demonstrate that the improved Kalman filter exhibits better numerical stability, where it can effectively avoid the filter divergence problem as well as solve the number and approximate locations of contamination sources in the inverse identification problem. The accurate location and release history of the contamination source can be effectively identified by using the 0–1MINLP.
Résumé
L’identification des sources de contamination des eaux souterraines, comme solution à un problème inverse, est principalement basée sur des méthodes d’optimisation de la simulation. Toutefois, l’établissement et la résolution d’un modèle d’optimisation sont extrêmement compliqués pour le problème inverse, dans lequel beaucoup de variables décisionnelles doivent être répertoriées. Combiner des méthodes d’optimisation de la simulation et d’autres méthodes mathématiques peut réellement éviter cet écueil. Afin de définir les caractéristiques des sources de contamination, le filtre de Kalman est combiné avec un modèle d’optimisation par programmation non -linéaire de nombres entiers mélangés (MOPNL). Tout d’abord, le filtre de Kalman est amélioré par une décomposiation U-D (une méthode de décompostion matricielle) afin d’identifier le nombre et la localisation approximative des sources de contamination réelles parmi les sources de contamination potentielles. Une solution 0–1 MOPNL est ensuite utilisée pour décrire la localisation exacte et l’historique des émissions de la source de contamination. Un krigeage est mis en oeuvre pour créer un modèle de substitution au modèle de simulation numérique. Le modèle de substitution, qui se rapproche du modèle de simulation numérique, peut être implanté à sa place dans le 0–1 MOPNL et peut être appelé directement durant l’itération du modèle d’optimisation, évitant ainsi le couplage manuel du modèle de simulation numérique et du modèle d’optimisation pendant le processus de résolution. Les résultats démontrent que le filtre de Kalman amélioré fait preuve d’une meilleure stabilité numérique, dans la mesure où il peut réellement éviter le problème de divergence du filtre aussi bien qu’éclairer le nombre et la localisation approximative des sources de contamination dans le problème inverse d’identification. La localisation exacte et l’historique des émissions de la source de contamination peuvent être effectivement définis en utilisant 0–1 MOPNL.
Resumen
La identificación de fuentes de contaminación de aguas subterráneas, como solución a un problema inverso, se basa principalmente en métodos de optimización de simulación. Sin embargo, el establecimiento y la solución de un modelo de optimización son extremadamente complicados para el problema inverso, donde es necesario identificar muchas variables de decisión. La combinación de métodos de optimización de simulación con otros métodos matemáticos puede evitar eficazmente esta deficiencia. Para identificar las características de las fuentes de contaminación, el filtro Kalman se combina con un modelo de optimización de programación no lineal de números enteros mixtos (MINLP). En primer lugar, el filtro Kalman se mejora con la descomposición U-D (un método de descomposición matricial) para identificar el número y la ubicación aproximada de las fuentes reales de contaminación entre las posibles fuentes de contaminación. Luego se emplea una solución 0-1MINLP para identificar la ubicación exacta y el historial de liberación de la fuente de contaminación. Kriging se utiliza para producir un modelo sustituto para el modelo de simulación numérica. El modelo sustituto se aproxima al modelo de simulación numérica, que puede ser embebido en el 0-1MINLP en lugar del modelo de simulación numérica y puede ser llamado directamente durante la iteración del modelo de optimización, evitando así el acoplamiento manual del modelo de simulación numérica con el modelo de optimización durante el proceso de solución. Los resultados demuestran que el filtro Kalman mejorado exhibe una mejor estabilidad numérica, donde puede evitar eficazmente el problema de divergencia del filtro, así como resolver el número y la ubicación aproximada de las fuentes de contaminación en el problema de identificación inversa. La ubicación exacta y el historial de liberación de la fuente de contaminación pueden ser identificados efectivamente usando el 0-1MINLP.
摘要
模拟优化方法是解决地下水污染源识别这类反问题的主要方法之一。然而,对于具有多决策变量待于识别的反问题,优化模型的建立和求解都十分复杂。将模拟优化方法和其它数学方法相结合,可以有效地避免这一缺点。为了对地下水污染源的特征进行识别,将卡尔曼滤波器与混合整数非线性规划结合运用。首先,利用U-D分解(矩阵分解法)对卡尔曼滤波器进行改进,以识别潜在污染源中真实污染源的个数和大致位置。然后,应用0–1混合整数非线性优化模型识别污染源的准确位置和释放历史。运用克里金方法建立数值模拟模型的替代模型。替代模型在功能上逼近数值模拟模型,可以代替模拟模型嵌入于0–1混合整数非线性优化模型,在优化模型求解的迭代过程中替代模型可以直接被调用,从而避免在求解过程中,人工重复将模拟模型与优化模型耦合的操作。结果表明改进的卡尔曼滤波器具有更好的数值稳定性,可有效避免滤波发散的缺点,解决污染源反演识别研究中个数和大致位置识别问题。应用0–1混合整数非线性优化模型,可以有效识别出污染源的准确位置和释放历史。
Resumo
A identificação de fontes de contaminação de águas subterrâneas, a partir de uma solução de problema inverso, baseia-se principalmente em métodos de otimização de simulação. No entanto, estabelecer e solucionar um modelo de otimização são extremamente complicados para o problema inverso, já que há muitas variáveis de decisão para ser identificadas. Combinar métodos de otimização de simulação com outros métodos matemáticos pode efetivamente evitar esse problema. Para identificar características de fontes de contaminação, o filtro de Kalman é combinado com um modelo de otimização de programação não-linear inteira mista (MINLP). Em primeiro lugar, o filtro de Kalman é melhorado pela decomposição U-D (um método de decomposição de matriz) para identificar o número e a localização aproximada de fontes reais de contaminação entre fontes potenciais de contaminação. Uma solução 0–1MINLP é então empregada para identificar a localização precisa e o histórico de liberação da fonte de contaminação. Krigagem é usada para produzir um metamodelo para o modelo de simulação numérica. O metamodelo aproxima o modelo de simulação numérica, que pode ser incorporado ao modelo 0–1MINLP ao invés do modelo numérico de simulação, e pode ser chamado diretamente durante a iteração do modelo de otimização, evitando assim o acoplamento manual entre o modelo de simulação numérica e o modelo de otimização durante o processo de solução. Os resultados demonstram que o filtro de Kalman melhorado apresenta melhor estabilidade numérica, onde pode efetivamente evitar o problema de divergência do filtro, bem como resolver o número e localização aproximada das fontes de contaminação no problema de identificação inversa. A localização precisa e o histórico de versões da fonte de contaminação podem ser efetivamente identificados usando o 0–1MINLP.










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Special thanks are given to the journal editors and anonymous reviewers for their valuable comments and suggested revisions.
Funding
This study was supported by the National Nature Science Foundation of China (No. 41672232) and Jilin Province Science and Technology Development Project (No. 20170101066JC).
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Li, J., Lu, W., Wang, H. et al. Identification of groundwater contamination sources using a statistical algorithm based on an improved Kalman filter and simulation optimization. Hydrogeol J 27, 2919–2931 (2019). https://doi.org/10.1007/s10040-019-02030-y
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DOI: https://doi.org/10.1007/s10040-019-02030-y