Abstract
The Bayesian approach is attractive because it can consider various uncertainties in the inverse process. Although the Bayesian algorithm has strong random ergodicity, it still lacks the ability to perform local optimization. Therefore, an improved single-component adaptive Metropolis (SCAM) algorithm based on Bayesian theory was developed to solve this problem and it was applied to the simultaneous identification of groundwater contaminant sources and simulation model parameters. The nondeterministic simulation model parameters have been introduced into the prior distribution as random variables. However, this will increase the number of random variables in the inverse problem, besides making the solution difficult. To alleviate this difficulty, the SCAM algorithm was applied to groundwater contaminant source identification. The acceptance probability formula was adjusted to enhance the local optimization ability of the SCAM algorithm. This improves the searching efficiency of the algorithm in the second stage, without losing the ergodicity in the first stage. In the inverse process, the simulation model is used multiple times to evaluate the likelihood function. To reduce the computational burden, the likelihood function is calculated by the surrogate model of the simulation model instead of by the simulation model itself, which greatly accelerates the process of Bayesian inversion. The effectiveness of this approach has been demonstrated by a hypothetical case study. Finally, the results of previous and improved algorithms have been compared. The results indicate that the improved SCAM algorithm can identify groundwater contaminant sources and simulation model parameters, simultaneously, with high accuracy and efficiency.
Résumé
L’approche bayésienne est attrayante parce qu’elle peut tenir compte de diverses incertitudes dans le processus inverse. Bien que l’algorithme bayésien ait une forte ergodicité aléatoire, il n’a toujours pas la capacité d’effectuer l’optimisation locale. Par conséquent, un algorithme amélioré d’adaptation à un seul composant Metropolis (SCAM) basé sur la théorie bayésienne a été mis au point pour résoudre ce problème et il a été appliqué à l’identification simultanée des sources de contaminants des eaux souterraines et aux paramètres du modèle de simulation. Les paramètres du modèle de simulation non déterministe ont été introduits dans la distribution antérieure en tant que variables aléatoires. Toutefois, cela augmentera le nombre de variables aléatoires dans le problème inverse, en plus de rendre la solution difficile. Pour atténuer cette difficulté, l’algorithme SCAM a été appliqué à l’identification des sources de contaminants dans les eaux souterraines. La formule de probabilité d’acceptation a été ajustée pour améliorer la capacité d’optimisation locale de l’algorithme SCAM. Cela améliore l’efficacité de recherche de l’algorithme dans la deuxième étape, sans perdre l’ergodicité dans la première étape. Dans le processus inverse, le modèle de simulation est utilisé plusieurs fois pour évaluer la fonction de probabilité. Pour réduire la charge de calcul, la fonction de probabilité est calculée par le modèle de substitution du modèle de simulation plutôt que par le modèle de simulation lui-même, ce qui accélère considérablement le processus d’inversion bayésienne. L’efficacité de cette approche a été démontrée par une étude de cas hypothétique. Enfin, les résultats des algorithmes précédents et améliorés ont été comparés. Les résultats indiquent que l’algorithme amélioré SCAM permet d’identifier simultanément les sources de contaminants des eaux souterraines et les paramètres du modèle de simulation, avec une grande précision et efficacité.
Resumen
El método bayesiano es atractivo porque puede considerar varias incertidumbres en el proceso inverso. Aunque el algoritmo Bayesiano tiene una fuerte ergodicidad aleatoria, todavía carece de la capacidad de realizar una optimización local. Por lo tanto, para resolver este problema se desarrolló un algoritmo de Metrópolis adaptativo y mejorado de un solo componente (SCAM) basado en la teoría bayesiana, que se aplicó a la identificación simultánea de fuentes de contaminantes de aguas subterráneas y a los parámetros del modelo de simulación. Los parámetros del modelo de simulación no determinístico se han introducido en la distribución anterior como variables aleatorias. Sin embargo, esto aumentará el número de variables aleatorias en el problema inverso, además de dificultar la solución. Para aliviar esta dificultad, se aplicó el algoritmo SCAM a la identificación de fuentes de contaminantes de aguas subterráneas. La fórmula de probabilidad de aceptación se ajustó para mejorar la capacidad de optimización local del algoritmo SCAM. Esto mejora la eficiencia de búsqueda del algoritmo en la segunda etapa, sin perder la ergodicidad en la primera etapa. En el proceso inverso, el modelo de simulación se utiliza varias veces para evaluar la función de probabilidad. Para reducir la demanda de cálculo, la función de verosimilitud se calcula mediante el modelo sustitutivo del modelo de simulación en lugar del modelo de simulación propiamente dicho, lo que acelera en gran medida el proceso de inversión bayesiana. La eficacia de este enfoque ha quedado demostrada por un estudio de caso hipotético. Por último, se han comparado los resultados de los algoritmos anteriores y los mejorados. Los resultados indican que el algoritmo SCAM mejorado puede identificar las fuentes de contaminantes de las aguas subterráneas y los parámetros del modelo de simulación, simultáneamente, con alta precisión y eficiencia.
摘要
贝叶斯方法在解逆问题中考虑各种不确定性, 因此备受关注。尽管贝叶斯算法具有很强的随机遍历性, 但仍然难以进行局部优化。因此, 提出了一种基于贝叶斯理论的改进的单组分自适应都会算法(SCAM)来解决该问题, 并将其应用于地下水污染物源的同时识别和确定模拟模型的参数。非确定性仿真模型参数已作为随机变量引入到先验分布中。但是, 这将增加反问题中随机变量的数量, 除了使求解变得困难之外。为了解决这一问题, 将SCAM算法应用于地下水污染物源识别。调整了接受概率公式, 以增强SCAM算法的局部优化能力。这提高了第二阶段算法的搜索效率, 而不会在第一阶段失去遍历性。在解逆过程中, 仿真模型被多次使用以评估似然函数。为了降低计算负担, 似然函数是通过模拟模型的替代模型而不是模拟模型本身来计算的, 这极大地加快了贝叶斯反演的过程。假设的案例研究证明了这种方法的有效性。最后, 比较了先前算法和改进算法的结果。结果表明, 改进的SCAM算法可以同时以高精度和高效率地识别地下水污染物源和模拟模型参数。
Resumo
A abordagem bayesiana é interessante por ser capaz de considerar diversas incertezas no processo inverso. Embora o algoritmo bayesiano tenha uma forte ergodicidade aleatória, ele ainda não tem a capacidade de realizar otimização local. Dessa forma, um algoritmo Metropolis adaptável de componente único aprimorado (SCAM - single-component adaptive Metropolis) baseado na teoria bayesiana foi desenvolvido para resolver este problema, e foi aplicado para a identificação simultânea de fontes de contaminação em águas subterrâneas e de parâmetros do modelo de simulação. Os parâmetros não-determinísticos do modelo de simulação foram introduzidos na distribuição anterior como variáveis aleatórias. Entretanto, isso irá aumentar o número de variáveis aleatórias do problema inverso, além de tornar a solução difícil. Para aliviar essa dificuldade, o algoritmo SCAM foi aplicado para a identificação de fontes de contaminação de águas subterrâneas. A fórmula da probabilidade de aceitação foi ajustada para melhorar a capacidade de otimização local do algoritmo SCAM. Isso aumenta a eficiência de busca do algoritmo no segundo estágio, sem perder a ergodicidade do primeiro estágio. No processo inverso, o modelo de simulação é utilizado diversas vezes para avaliar a função de verossimilhança. Para reduzir o custo computacional, a função de verossimilhança é calculada pelo modelo substituto do modelo de simulação, ao invés do próprio modelo de simulação, o que acelera muito o processo de inversão bayesiana. A eficácia dessa abordagem foi demonstrada por um estudo de caso hipotético. Finalmente, os resultados do algoritmo anterior e do aprimorado foram comparados. Os resultados indicam que o algoritmo SCAM aprimorado é capaz de identificar fontes de contaminação de águas subterrâneas e parâmetros do modelo de simulação, simultaneamente, com elevada precisão e eficiência.
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This research was supported by the National Natural Science Foundation of China (No. 41972252), and the National Key Research and Development Program of China (No. 2018YFC1800403).
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Chang, Z., Lu, W., Wang, H. et al. Simultaneous identification of groundwater contaminant sources and simulation of model parameters based on an improved single-component adaptive Metropolis algorithm. Hydrogeol J 29, 859–873 (2021). https://doi.org/10.1007/s10040-020-02257-0
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DOI: https://doi.org/10.1007/s10040-020-02257-0