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).
The full log-likelihood, \(\ell\), for the parameters given the data is as follows:
\[ \ell(A_{0,s}, q_t, n, \sigma^2 | \delta A_{st}, x_{st}) = - \frac{NT}{2\sigma^2} - \sum_{s = 1}^N\sum_{t = 1}^T \Big[\frac{3}{5}\log(A_{0,s} + \delta A_{st}) + \frac{1}{2 \sigma^2} \big(\log({A_{0,s} + \delta A_{st}}) + x_{st} - \log(q_tn)\big)^2 \Big] , \\ A_{0,s} > -\min_t \delta A_{st}; q_t > 0; n > 0; \sigma > 0 \]
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.
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.
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).