LSF Tides

Least Squares Fit for ROMS Tides

A ROMS state variable, $\phi$, can be represented in terms of its time mean, ${\bar {\phi }}$, plus a set of $N$-tidal harmonics of frequency, $\omega _{k}$.

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

The unknown tidal amplitude $A_{k}$, and $B_{k}$ and unknown state $\phi$ coefficients are evaluated by minimizing the least-squares error function defined by:

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

In discrete space:

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

Minimization subject to the additional constraints ${\frac {\partial \varepsilon ^{2}}{\partial {\bar {\phi }}}}=0$, ${\frac {\partial \varepsilon ^{2}}{\partial A_{k}}}=0$, ${\frac {\partial \varepsilon ^{2}}{\partial B_{k}}}=0$ result in a linear set of equations:

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

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

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

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.

$${\begin{matrix}{\begin{bmatrix}\\M&\sum \sin \omega _{1}t_{i}&\sum \sin \omega _{2}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}&\cdots \\\\\sum \sin \omega _{1}t_{i}&\sum \sin ^{2}\omega _{1}t_{i}&\sum \sin \omega _{2}t_{i}\sin \omega _{1}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}\sin \omega _{1}t_{i}&\cdots \\\\\sum \sin \omega _{2}t_{i}&\sum \sin \omega _{1}t_{i}\sin \omega _{2}t_{i}&\sum \sin ^{2}\omega _{2}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}\sin \omega _{2}t_{i}&\cdots \\\\\vdots &\vdots &\cdots &\cdots &\cdots &\cdots \\\\\sum \sin \omega _{7}t_{i}&\sum \sin \omega _{1}t_{i}\sin \omega _{7}t_{i}&\sum \sin \omega _{2}t_{i}\sin \omega _{7}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}\sin \omega _{7}t_{i}&\cdots \\\\\sum \sin \omega _{1}t_{i}&\sum \sin \omega _{1}t_{i}\cos \omega _{1}t_{i}&\sum \sin \omega _{2}t_{i}\cos \omega _{1}t_{i}&\cdots &\sum \cos ^{2}\omega _{1}t_{i}&\cdots \\\\\sum \cos \omega _{2}t_{i}&\sum \sin \omega _{1}t_{i}\cos \omega _{2}t_{i}&\sum \sin \omega _{2}t_{i}\cos \omega _{2}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}\cos \omega _{2}t_{i}&\cdots \\\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\\\\sum \cos \omega _{7}t_{i}&\sum \sin \omega _{1}t_{i}\cos \omega _{7}t_{i}&\sum \sin \omega _{2}t_{2}\cos \omega _{7}t_{i}&\cdots &\sum \cos \omega _{1}t_{i}\cos \omega _{7}t_{i}&\cdots \\\\\end{bmatrix}}\;\;&{\begin{bmatrix}\\{\bar {\phi }}\\\\A_{1}\\\\A_{2}\\\\\vdots \\\\A_{7}\\\\B_{1}\\\\B_{2}\\\\\vdots \\\\B_{7}\\\\\end{bmatrix}}&=&{\begin{bmatrix}\\\sum \phi _{i}\\\\\sum \phi _{i}\sin \omega _{1}ti\\\\\sum \phi _{i}\sin \omega _{2}ti\\\\\vdots \\\\\sum \phi _{i}\sin \omega _{7}ti\\\\\sum \phi _{i}\cos \omega _{1}ti\\\\\sum \phi _{i}\cos \omega _{2}ti\\\\\vdots \\\\\sum \phi _{i}\cos \omega _{7}ti\\\\\end{bmatrix}}\\A&x&&b\end{matrix}}$$