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

Bayesian discharge estimation

Equations:

• Manning:

$$ \begin{aligned} \log Q_M &= \frac{5}{3} \log(A_0 + dA) - \log n - \frac{2}{3} \log w + \frac{1}{2} \log S \ &= g_{QM}(A_0, dA, n, w, S) \end{aligned} $$

$$ \begin{aligned} \log (A_0 + dA) &= \frac{3}{5} \log Q_M + \frac{3}{5} \log n + \frac{2}{5} \log w - \frac{3}{10} \log S \ &= g_{AM}(Q_M, n, w, S) \end{aligned} $$

$$ \begin{aligned} \log n &= \frac{5}{3} \log(A_0 + dA) - \log Q_M - \frac{2}{3} \log w + \frac{1}{2} \log S \ &= g_{nM}(Q_M, A_0, dA, w, S) \end{aligned} $$

$$ \begin{aligned} \log w &= \frac{5}{2} \log(A_0 + dA) - \frac{3}{2} \log n - \frac{3}{2} \log Q_M + \frac{3}{4} \log S \ &= g_{wM}(Q_M, A_0, dA, n, S) \end{aligned} $$

$$ \begin{aligned} \log S &= 2 \log Q_M - \frac{10}{3} \log (A_0 + dA) + 2 \log n + \frac{4}{3} \log w \ &= g_{SM}(Q_M, A_0, dA, n, w) \end{aligned} $$

$$ dA(w) = \int_{h_0}^h w(h') dh' $$

$$ 4 \log W - 3 \log S = 10 \log(A_0 + dA) - 6 \log n - 6 \log Q_M $$

Note that this means that SE for the lhs is 6 times SE for Manning's flow estimation.

If A is known (can be used for lower bound on Q):

$$ 4 \log W - 3 \log S - 10 \log A = - 6 \log n - 6 \log Q_M $$

Or (assume depth > 1m)

$$ 4 \log W - 3 \log S - 10 \log (dA + Wo) = - 6 \log n - 6 \log Q_M $$

• AMHG:

$$ \begin{aligned} \log Q_G &= \frac{1}{b} ( \log{w} - \log w_c ) + \log Q_c \ &= g_{QG}(b, w, w_c, Q_c) \end{aligned} $$

$$ \begin{aligned} \log Q_c &= \frac{1}{b} ( \log{w_c} - \log w ) + \log Q_G \ &= g_{Qc}(b, w, w_c, Q) \end{aligned} $$

$$ \begin{aligned} \log w &= b ( \log{Q} - \log Q_c ) + \log w_c \ &= g_{wG}(b, Q, w_c, Q_c) \end{aligned} $$

$$ \begin{aligned} \log w_c &= b ( \log{Q_c} - \log Q ) + \log w \ &= g_{wc}(b, w, Q, Q_c) \end{aligned} $$

$$ \begin{aligned} b &= (\log w_c - \log w) /( \log{Q_c} - \log Q ) \ &= g_{b}(w, w_c, Q, Q_c) \end{aligned} $$

Unknowns:

| Variable | Symbol | number | | -------- | ------ | ------ | | flow | $Q_t$ | $n_t$ | Manning flow | $Q_{M,t}$ | $n_t$ | AMHG flow | $Q_{G,t}$ | $n_t$ | min-flow area | $A_{0,i}$ | $n_x$ | manning coefficient | $n$ | 1 | width AHG exponent | $b_i$ | $n_x$ | AMHG common flow | $Q_c$ | 1 | AMHG common width | $w_c$ | 1

Knowns:

| Variable | Symbol | number | | -------- | ------ | ------ | | area change | $dA_{it}$ | $n_x \cdot n_t$ | slope | $S_{it}$ | $n_x \cdot n_t$ | width | $w_{it}$ | $n_x \cdot n_t$

Probability

• $\log Q_t \sim N(\frac{\sum_i\log Q_{M, it} + \sum_i\log Q_{G, it}}{2 * nx}, \sigma^2_{Q})$
• $\log Q_{it, M} \sim N(g_{QM}(A_{0i}, dA_{it}, n, w_{it}, S_{it}), \sigma^2_{QM})$
• $\log Q_{it, G} \sim N(g_{QG}(b_i, w_{it}, w_c, Q_c), \sigma^2_{QG})$
• $\log (A_{0, i} + dA_{it}) \sim N(g_{AM}(Q_{M, it}, n, w_{it}, S_{it}), \sigma^2_A)$
• $\log n \sim N(g_{nM}(Q, A_0, dA, w, S), \sigma^2_n)$
• $\log S \sim N(g_{SM}(Q, A_0, dA, n, w), \sigma^2_S)$
• $\log w \sim N(g_{wG}(b, Q, w_c, Q_c), \sigma^2_w)$
• $b \sim N(g_{wG}(b, Q, w_c, Q_c), \sigma^2_w)$

Ao prior

Following Durand:

$$ \begin{aligned} \hat{Ao} &= exp(\bar{logQ} + \log n + \frac{2}{3} \bar{logW} - \frac{1}{2} \bar{logS})^{\frac{3}{5}} - \bar{dA} \ &= exp(\frac{3}{5} \bar{logQ} + \frac{3}{5} \log n + \frac{2}{5} \bar{logW} - \frac{3}{10} \bar{logS}) - dA \end{aligned} $$

so variance on $Ao$ is $var(exp(\frac{3}{5} \bar{logQ} + \frac{3}{5} \log n + \frac{2}{5} \bar{logW} - \frac{3}{10} \bar{logS}))$, which is variance of a lognormal.

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