The power-law rating curve has been used extensively in hydraulic practice and hydrology. It is given by $Q(h)=a{(h-c)}^b$𝑄(ℎ)=𝑎(ℎ−𝑐)𝑏$Q(h)=a{(h-c)}^b$, where $Q$𝑄$Q$ is discharge, $h$ℎ$h$ is water elevation, $a$𝑎$a$, $b$𝑏$b$, and $c$𝑐$c$ are unknown parameters. We propose a novel extension of the power-law rating curve, referred to as the generalized power-law rating curve. It is constructed by linking the physics of open channel flow to a model of the form $Q(h)=a{(h-c)}^{f(h)}$𝑄(ℎ)=𝑎(ℎ−𝑐)𝑓(ℎ)$Q(h)=a{(h-c)}^{f(h)}$. The function $f(h)$𝑓(ℎ)$f(h)$ is referred to as the power-law exponent and it depends on the water elevation. The proposed model and the power-law model are fitted within the framework of Bayesian hierarchical models. By exploring the properties of the proposed rating curve and its power-law exponent, we find that cross-sectional shapes that are likely to be found in nature are such that the power-law exponent $f(h)$𝑓(ℎ)$f(h)$ will usually be in the interval $[1.0,2.67]$[1.0,2.67]$[1.0,2.67]$. This fact is utilized for the construction of prior densities for the model parameters. An efficient Markov chain Monte Carlo sampling scheme, that utilizes the lognormal distributional assumption at the data level and Gaussian assumption at the latent level, is proposed for the two models. The two statistical models were applied to four datasets. In the case of three datasets the generalized power-law rating curve gave a better fit than the power-law rating curve while in the fourth case the two models fitted equally well and the generalized power-law rating curve mimicked the power-law rating curve. We developed an R package, bdrc, for fitting the power-law and generalized power-law rating curve models. It is available on CRAN.

中文翻译：

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使用流体动力学理论和贝叶斯分层建模的幂律评级曲线的推广

幂律等级曲线已广泛用于水力实践和水文学。它是由$问(H)=\mathrm{一种}{(H-C)}^b$𝑄 ( ℎ ) = 𝑎（ℎ - 𝑐 ）𝑏$Q(h)=a{(h-c)}^b$， 在哪里$问$𝑄$Q$是放电，$H$ℎ$h$是水位高程，$\mathrm{一种}$𝑎$a$,$b$𝑏$b$， 和$C$𝑐$c$是未知参数。我们提出了幂律评级曲线的新扩展，称为广义幂律评级曲线。它是通过将明渠流的物理与以下形式的模型联系起来而构建的$问(H)=\mathrm{一种}{(H-C)}^{F(H)}$𝑄 ( ℎ ) = 𝑎（ℎ - 𝑐 ）𝑓( ℎ )$Q(h)=a{(h-c)}^{f(h)}$. 功能$F(H)$𝑓( ℎ )$f(h)$被称为幂律指数，它取决于水位。所提出的模型和幂律模型适合贝叶斯层次模型的框架。通过探索所提出的评级曲线及其幂律指数的性质，我们发现自然界中可能存在的横截面形状使得幂律指数$F(H)$𝑓( ℎ )$f(h)$通常会在区间$[1.0,2.67]$[ 1 . 0 , 2 . 67 ]$[1.0,2.67]$. 这一事实被用于构建模型参数的先验密度。为这两个模型提出了一种有效的马尔可夫链蒙特卡洛采样方案，该方案在数据层利用对数正态分布假设，在潜在层利用高斯假设。这两个统计模型应用于四个数据集。在三个数据集的情况下，广义幂律评级曲线比幂律评级曲线拟合得更好，而在第四种情况下，两个模型拟合得同样好，广义幂律评级曲线模仿幂律评级曲线. 我们开发了一个 R 包bdrc，用于拟合幂律和广义幂律评级曲线模型。它在 CRAN 上可用。