LSF Tides

Least Squares Fit for ROMS Tides

$\phi (t)={\bar {\phi }}+\sum _{k=1}^{N}A_{k}\sin \omega _{k}t+\sum _{k=1}^{N}B_{k}\cos \omega _{k}t$

$\,\!\phi$: state variables

$\,\!\omega _{k}$: tidal frequency

$\,\!A_{k},B_{k}$: amplitude

$\,\!N$: number of harmonics

To minimize cost function $\varepsilon ^{2}$

$\varepsilon ^{2}={\frac {1}{T}}\int _{t_{1}}^{t_{2}}\left[\phi -\left({\bar {\phi }}+\sum _{k=1}^{N}(A_{k}\sin \omega _{k}t)+\sum _{k=1}^{N}(B_{k}\cos \omega _{k}t)\right)\right]^{2}dt$

$\,\!\phi ,A_{k},B_{k}$ 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

${\frac {\partial \varepsilon ^{2}}{\partial {\bar {\phi }}}}=0$

${\frac {\partial \varepsilon ^{2}}{\partial A_{k}}}=0\;\;\;\;\;\;k=1,...,N$

${\frac {\partial \varepsilon ^{2}}{\partial B_{k}}}=0$

${\frac {\partial \varepsilon ^{2}}{\partial {\bar {\phi }}}}=0$

$\sum _{i=1}^{M}\left[-2\phi _{i}+2{\bar {\phi }}+2\sum _{k=1}^{N}(A_{k}\sin \omega _{k}t_{i})+2\sum _{k=1}^{N}(B_{k}\sin \omega _{k}t_{i})\right]=0$

${\frac {\partial \varepsilon ^{2}}{\partial A_{k}}}=0$

${\begin{aligned}\sum _{i=1}^{M}{\Bigg [}&-2\phi _{i}\sin \omega _{k}t_{i}+2{\bar {\phi }}\sin \omega _{k}t_{i}+2\sum _{p=1}^{N}(A_{p}\sin \omega _{p}t_{i}\sin \omega _{k}t_{i})\\&+2\sum _{p=1}^{N}(B_{p}\cos \omega _{p}t_{i}\sin \omega _{k}t_{i}){\Bigg ]}=0\end{aligned}}$

${\frac {\partial \varepsilon ^{2}}{\partial B_{k}}}=0$

${\begin{aligned}\sum _{i=1}^{M}{\Bigg [}&-2\phi _{i}\cos \omega _{k}t_{i}+2{\bar {\phi }}\cos \omega _{k}t_{i}+2\sum _{p=1}^{N}(A_{p}\sin \omega _{p}t_{i}\cos \omega _{k}t_{i})\\&+2\sum _{p=1}^{N}(B_{p}\cos \omega _{p}t_{i}\cos \omega _{k}t_{i}){\Bigg ]}=0\end{aligned}}$

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.

${\begin{array}{cccc}\left[{\begin{array}{cccccc}\\M&\sum \sin \omega _{1}t_{i}&\sum \sin \omega _{2}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}&\cdots \\\\\sum \sin \omega _{1}t_{i}&\sum \sin ^{2}\omega _{1}t_{i}&\sum \sin \omega _{2}t_{i}\sin \omega _{1}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}\sin \omega _{1}t_{i}&\cdots \\\\\sum \sin \omega _{2}t_{i}&\sum \sin \omega _{1}t_{i}\sin \omega _{2}t_{i}&\sum \sin ^{2}\omega _{2}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}\sin \omega _{2}t_{i}&\cdots \\\\\vdots &\vdots &\cdots &\cdots &\cdots &\cdots \\\\\sum \sin \omega _{7}t_{i}&\sum \sin \omega _{1}t_{i}\sin \omega _{7}t_{i}&\sum \sin \omega _{2}t_{i}\sin \omega _{7}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}\sin \omega _{7}t_{i}&\cdots \\\\\sum \sin \omega _{1}t_{i}&\sum \sin \omega _{1}t_{i}\cos \omega _{1}t_{i}&\sum \sin \omega _{2}t_{i}\cos \omega _{1}t_{i}&\cdots &\sum \cos ^{2}\omega _{1}t_{i}&\cdots \\\\\sum \cos \omega _{2}t_{i}&\sum \sin \omega _{1}t_{i}\cos \omega _{2}t_{i}&\sum \sin \omega _{2}t_{i}\cos \omega _{2}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}\cos \omega _{2}t_{i}&\cdots \\\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\\\\sum \cos \omega _{7}t_{i}&\sum \sin \omega _{1}t_{i}\cos \omega _{7}t_{i}&\sum \sin \omega _{2}t_{2}\cos \omega _{7}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}\cos \omega _{7}t_{i}&\cdots \\\\\end{array}}\right]&\left[{\begin{array}{c}\\{\bar {\phi }}\\\\A_{1}\\\\A_{2}\\\\\vdots \\\\A_{7}\\\\B_{1}\\\\B_{2}\\\\\vdots \\\\B_{7}\\\\\end{array}}\right]&=&\left[{\begin{array}{l}\\\sum \phi _{i}\\\\\sum \phi _{i}\sin \omega _{1}ti\\\\\sum \phi _{i}\sin \omega _{2}ti\\\\\vdots \\\\\sum \phi _{i}\sin \omega _{7}ti\\\\\sum \phi _{i}\cos \omega _{1}ti\\\\\sum \phi _{i}\cos \omega _{2}ti\\\\\vdots \\\\\sum \phi _{i}\cos \omega _{7}ti\\\\\end{array}}\right]\\A&x&&b\end{array}}$