In markwh/bamr: Bayesian AMHG and Manning discharge estimation using SWOT-like river observations

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Definitions

SWOT observes the dataset ${x_{st}, \delta A_{st}}$, where $x_{st} = -\frac{2}{3} \log W_{st} + \frac{1}{2} \log S_{st}$. $s = 1, \dots N$ indexes space and $t = 1 \dots T$ indexes time. Based on Mass-conserved Manning's equation, these data are assumed to be generated according to the following pdf:

$$ f(x_{st}, \delta A_{st}) = \frac{1}{(A_{0,s} + \delta A_{st}) \sqrt{2 \pi\sigma^2}} \exp\Big(-\frac{1}{2\sigma^2}\big(\frac{3}{5}\log({A_{0,s} + \delta A_{st}}) + x_{st} - \log(q_tn)\big)^2\Big) $$

where $\sigma$ is the standard deviation of log-transformed errors (errors are assumed to be lognormal).

Full likelihood - Manning's equation

The full log-likelihood, $\ell$, for the parameters given the data is as follows:

$$ \ell(A_{0,s}, q_t, n, \sigma^2 | \delta A_{st}, x_{st}) = - \frac{NT}{2\sigma^2} - \sum_{s = 1}^N\sum_{t = 1}^T \Big[\frac{3}{5}\log(A_{0,s} + \delta A_{st}) + \frac{1}{2 \sigma^2} \big(\log({A_{0,s} + \delta A_{st}}) + x_{st} - \log(q_tn)\big)^2 \Big] , \ A_{0,s} > -\min_t \delta A_{st}; q_t > 0; n > 0; \sigma > 0 $$

Conditional log-likelihood

For sampling purposes (e.g. Gibbs sampler) it is often useful to express the likelihood of a single parameter conditional on all other parameters and the data.

Conditional likelihood for $\log(q_tn)$

Note that in the above, the same likelihood value is obtained from any pair of $q_t, n$ values such that $q_tn = c$ for any constant $c$. In other words, the model is not fully identifiable with respect to the parameters $q_t$ and $n$; these parameters cannot be determined from the likelihood alone. BAM addresses this inference problem by using fairly informative priors on $q_t$ and $n$. However, the model is identifiable with respect to the product $q_tn$, and thus parameterized the conditional log-likelihood is as follows:

$$ \ell(q_tn | \delta A_{st}, x_{st}, A_{0,s}, \sigma^2) = - \sum_{s = 1}^N \Big[\log(A_{0,s} + \delta A_{st}) + \frac{1}{2 \sigma^2} \big(\frac{3}{5} \log({A_{0,s} + \delta A_{st}}) + x_{st} - \log(q_tn)\big)^2 \Big] , \ q_t > 0; n > 0 $$

Note that this is exactly a lognormal likelihood, meaning that the posterior for $q_tn$ could be sampled directly using a Gibbs sampler, provided that the prior distributions for $q_t$ and $n$ are also lognormal.

Conditional likelihood for $A_0$

The conditoinal log likelihood for $A_0$ is given by:

$$ \ell(A_{0,s} | \delta A_{st}, x_{st}, q_tn, \sigma^2) = - \frac{T}{2\sigma^2} - \sum_{t = 1}^T \Big[\log(A_{0,s} + \delta A_{st}) + \frac{1}{2 \sigma^2} \big(\log({A_{0,s} + \delta A_{st}}) + \frac{5}{3} x_{st} - \frac{5}{3} \log(q_tn)\big)^2 \Big] , \ A_{0,s} > -\min_t \delta A_{st} $$

If $A_{0,s}$ has a prior distribution with pdf $\pi(A_{0,s})$, then the conditional posterior for $A_{0,s}$ is given by adding $\log \pi(A_{0,s})$ to the above:

$$ p(A_{0,s} | \delta A_{st}, x_{st}, q_tn, \sigma^2) = - \frac{T}{2\sigma^2} - \sum_{t = 1}^T \Big[\log(A_{0,s} + \delta A_{st}) + \frac{1}{2 \sigma^2} \big(\frac{3}{5} \log({A_{0,s} + \delta A_{st}}) + x_{st} - \log(q_tn)\big)^2 \Big] + \log\pi(A_{0,s}) , \ A_{0,s} > -\min_t \delta A_{st} $$

Unlike $q_tn$, this posterior cannot be sampled from directly (no matter what the choice of prior distribution), and therefore other sampling methods must be used (e.g. Metropolis).

markwh/bamr documentation built on Aug. 7, 2020, 11:52 p.m.