LSF Tides
<wikitex>$$\phi(t) = \bar\phi + \sum_{k=1}^N A_k \sin\omega_k t + \sum_{k=1}^N B_k \cos\omega_k t$$
- $\phi$: state variables
- $\omega_k$: tidal frequency
- $A_k, B_k$: amplitude
- $N$: number of harmonics
To minimize cost function $\varepsilon^2$
- $$\varepsilon^2 = \frac{1}{T} \int_{t_1}^{t_2} \left[\phi - \left(\bar\phi + \sum_{k=1}^N (A_k \sin\omega_k t) + \sum_{k=1}^N (B_k \cos\omega_k t)\right)\right]^2dt$$
- $\phi, A_k, B_k$ are unknowns
In discrete space: $$\varepsilon^2 = \frac{1}{M} \sum_{i=1}^M \left[\phi_i - \left(\bar\phi + \sum_{k=1}^N (A_k \sin\omega_k t_i) + \sum_{k=1}^N (B_k \cos\omega_k t_i)\right)\right]^2$$
at the minimum
$$\frac{\partial \varepsilon^2}{\partial \bar\phi} = 0$$
$$\frac{\partial \varepsilon^2}{\partial A_k} = 0\;\;\;\;\;\;k = 1, ..., N$$
$$\frac{\partial \varepsilon^2}{\partial B_k} = 0$$
$$\frac{\partial \varepsilon^2}{\partial \bar\phi} = 0$$
$$\sum_{i=1}^M \left[-2\phi_i + 2\bar\phi + 2 \sum_{k=1}^N (A_k \sin\omega_k t_i) + 2 \sum_{k=1}^N (B_k \sin\omega_k t_i)\right] = 0$$
$$\frac{\partial \varepsilon^2}{\partial A_k} = 0$$
$$\eqalign{\sum_{i=1}^M \Bigg[ &-2 \phi_i \sin\omega_k t_i + 2 \bar\phi \sin\omega_k t_i + 2\sum_{p=1}^N (A_p \sin\omega_p t_i \sin\omega_k t_i) \cr
&+ 2\sum_{p=1}^N (B_p\cos\omega_p t_i \sin\omega_k t_i)\Bigg] = 0}$$
$$\frac{\partial \varepsilon^2}{\partial B_k} = 0$$
$$\eqalign{\sum_{i=1}^M \Bigg[ &-2 \phi_i \cos\omega_k t_i + 2 \bar\phi \cos\omega_k t_i + 2\sum_{p=1}^N (A_p \sin\omega_p t_i \cos\omega_k t_i) \cr
&+ 2\sum_{p=1}^N (B_p\cos\omega_p t_i \cos\omega_k t_i) \Bigg] = 0}$$
in matrix form (N harmonics). Note: all instances of $\sum$ are actually $\sum_{i=1}^M$ where M is the number of time-steps in the time-averaging window.
$$\matrix{\left[ \matrix{ \cr M & \sum \sin\omega_1 t_i & \sum \sin\omega_2 t_i & \cdots & \sum \cos\omega_1 t_i & \cdots \cr \cr \sum \sin\omega_1 t_i & \sum \sin^2 \omega_1 t_i & \sum \sin\omega_2 t_i \sin\omega_1 t_i & \cdots & \sum \cos\omega_1 t_i \sin\omega_1 t_i & \cdots \cr \cr \sum \sin\omega_2 t_i & \sum \sin\omega_1 t_i \sin\omega_2 t_i & \sum \sin^2 \omega_2 t_i & \cdots & \sum \cos\omega_1 t_i \sin\omega_2 t_i & \cdots \cr \cr \vdots & \vdots & \cdots & \cdots & \cdots & \cdots \cr \cr \sum \sin\omega_7 t_i & \sum \sin\omega_1 t_i \sin\omega_7 t_i & \sum \sin\omega_2 t_i \sin\omega_7 t_i & \cdots & \sum \cos\omega_1 t_i \sin\omega_7 t_i & \cdots \cr \cr \sum \sin\omega_1 t_i & \sum \sin\omega_1 t_i \cos\omega_1 t_i & \sum \sin\omega_2 t_i \cos\omega_1 t_i & \cdots & \sum \cos^2 \omega_1 t_i & \cdots \cr \cr \sum \cos\omega_2 t_i & \sum \sin\omega_1 t_i \cos\omega_2 t_i & \sum \sin\omega_2 t_i \cos\omega_2 t_i & \cdots & \sum \cos\omega_1 t_i \cos\omega_2 t_i & \cdots \cr \cr \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \cr \cr \sum \cos\omega_7 t_i & \sum \sin\omega_1 t_i \cos\omega_7 t_i & \sum \sin\omega_2 t_2 \cos\omega_7 t_i & \cdots & \sum \cos\omega_1 t_i \cos\omega_7 t_i & \cdots \cr \cr } \right] & \left[ \matrix{ \cr \bar\phi \cr \cr A_1 \cr \cr A_2 \cr \cr \vdots \cr \cr A_7 \cr \cr B_1 \cr \cr B_2 \cr \cr \vdots \cr \cr B_7 \cr \cr } \right] & = & \left[ \matrix{ \cr \sum\phi_i \cr \cr \sum\phi_i \sin\omega_1 ti \cr \cr \sum\phi_i \sin\omega_2 ti \cr \cr \vdots \cr \cr \sum\phi_i \sin\omega_7 ti \cr \cr \sum\phi_i \cos\omega_1 ti \cr \cr \sum\phi_i \cos\omega_2 ti \cr \cr \vdots \cr \cr \sum\phi_i \cos\omega_7 ti \cr \cr } \right] \cr A & x & & b}$$
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