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Corruption of accuracy and efficiency of Markov chain Monte Carlo simulation by inaccurate numerical implementation of conceptual hydrologic models

机译：概念性水文模型数值计算不准确导致马尔可夫链蒙特卡罗模拟的准确性和效率下降

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Conceptual rainfall-runoff models have traditionally been applied without paying much attention to numerical errors induced by temporal integration of water balance dynamics. Reliance on first-order, explicit, fixed-step integration methods leads to computationally cheap simulation models that are easy to implement. Computational speed is especially desirable for estimating parameter and predictive uncertainty using Markov chain Monte Carlo (MCMC) methods. Confirming earlier work of Kavetski et al. (2003), we show here that the computational speed of first-order, explicit, fixed-step integration methods comes at a cost: for a case study with a spatially lumped conceptual rainfall-runoff model, it introduces artificial bimodality in the marginal posterior parameter distributions, which is not present in numerically accurate implementations of the same model. The resulting effects on MCMC simulation include (1) inconsistent estimates of posterior parameter and predictive distributions, (2) poor performance and slow convergence of the MCMC algorithm, and (3) unreliable convergence diagnosis using the Gelman-Rubin statistic. We studied several alternative numerical implementations to remedy these problems, including various adaptive-step finite difference schemes and an operator splitting method. Our results show that adaptive-step, second-order methods, based on either explicit finite differencing or operator splitting with analytical integration, provide the best alternative for accurate and efficient MCMC simulation. Fixed-step or adaptive-step implicit methods may also be used for increased accuracy, but they cannot match the efficiency of adaptive-step explicit finite differencing or operator splitting. Of the latter two, explicit finite differencing is more generally applicable and is preferred if the individual hydrologic flux laws cannot be integrated analytically, as the splitting method then loses its advantage.
机译：传统上已应用概念性降雨径流模型，而没有过多关注由水平衡动力学的时间积分引起的数值误差。依靠一阶，显式，固定步骤的积分方法，可以得到易于实现的廉价计算仿真模型。对于使用马尔可夫链蒙特卡洛（MCMC）方法估计参数和预测不确定性而言，计算速度尤其理想。证实了Kavetski等人的早期工作。 （2003年），我们在这里表明，一阶，显式，固定步骤积分方法的计算速度是有代价的：对于具有空间集总概念降雨径流模型的案例研究，它在边际后验中引入了人工双峰。参数分布，在同一模型的数字精确实现中不存在。由此产生的对MCMC模拟的影响包括（1）后参数估计值和预测分布不一致，（2）MCMC算法的性能较差和收敛缓慢，以及（3）使用Gelman-Rubin统计数据进行的收敛诊断不可靠。我们研究了几种替代数值方法来解决这些问题，包括各种自适应步骤有限差分方案和算子拆分方法。我们的结果表明，基于显式有限差分或带有分析积分的算子拆分的自适应步阶二阶方法为准确有效的MCMC仿真提供了最佳选择。固定步长或自适应步长隐式方法也可以用于提高准确性，但是它们不能与自适应步长显式有限差分或算子拆分的效率相匹配。在后两种方法中，显式有限差分法更普遍适用，如果单个的水文通量定律不能进行分析积分，则首选有限差分法，因为分裂方法会失去其优势。

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