LSF Tides - Revision history

Robertson at 17:22, 5 August 2015

2015-08-05T17:22:19Z

Robertson at 10:18, 8 October 2008

2008-10-08T10:18:42Z

← Older revision		Revision as of 10:18, 8 October 2008
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	<div class="title">Least Squares Fit for ROMS Tides</div>		<div class="title">Least Squares Fit for ROMS Tides</div>

	<wikitex>$$\phi~~(t) =~~ \bar\phi ~~+ \sum_{k=1}^N A_k \sin\omega_k t + \sum_{k=1}^N B_k \cos\omega_k t~~$$		<wikitex>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, $\omega_k$.
	~~:$\phi$: state variables~~
	:$~~\omega_k$:~~ tidal frequency
	~~:$A_k~~, ~~B_k~~$~~: amplitude~~
	:$~~N$: number of harmonics~~

	~~To minimize cost function~~ $~~\varepsilon^2~~$		$$\phi(t) = \bar\phi + \sum_{k=1}^N A_k \sin\omega_k t + \sum_{k=1}^N B_k \cos\omega_k t$$
	~~:$$\varepsilon^2 = \frac{1}{T} \int_{t_1}^{t_2} \left[~~\phi ~~- \left~~(\bar\phi + \sum_{k=1}^N (A_k \sin\omega_k t) + \sum_{k=1}^N (B_k \cos\omega_k t~~)\right)\right]^2dt~~$$
	~~:$\phi, A_k, B_k$ are unknowns~~

	~~In discrete space:~~		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}{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~~		$$\varepsilon^2 = \frac{1}{T} \int_{t_1}^{t_2} \left[\phi - \left(\bar\phi + \sum_{k=1}^N (A_k \sin\omega_k t) + \sum_{k=1}^N (B_k \cos\omega_k t)\right)\right]^2dt$$

	~~$$\frac{\partial \varepsilon^2}{\partial \bar\phi} = 0$$~~		In discrete space:

			$$\varepsilon^2 = \frac{1}{M} \sum_{i=1}^M \left[\phi_i - \left(\bar\phi + \sum_{k=1}^N (A_k \sin\omega_k t_i) + \sum_{k=1}^N (B_k \cos\omega_k t_i)\right)\right]^2$$

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


	$$\frac{\partial \varepsilon^2}{\partial ~~B_k~~} = 0$$


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


	$$\sum_{i=1}^M \left[-2\phi_i + 2\bar\phi + 2 \sum_{k=1}^N (A_k \sin\omega_k t_i) + 2 \sum_{k=1}^N (B_k \sin\omega_k t_i)\right] = 0$$		$$\sum_{i=1}^M \left[-2\phi_i + 2\bar\phi + 2 \sum_{k=1}^N (A_k \sin\omega_k t_i) + 2 \sum_{k=1}^N (B_k \sin\omega_k t_i)\right] = 0$$

			$$\sum_{i=1}^M \Bigg[ -2 \phi_i \sin\omega_k t_i + 2 \bar\phi \sin\omega_k t_i + 2\sum_{p=1}^N (A_p \sin\omega_p t_i \sin\omega_k t_i) + 2\sum_{p=1}^N (B_p\cos\omega_p t_i \sin\omega_k t_i)\Bigg] = 0$$

	$$~~\frac{\partial \varepsilon^2}{\partial A_k} = 0$$~~		$$\sum_{i=1}^M \Bigg[ -2 \phi_i \cos\omega_k t_i + 2 \bar\phi \cos\omega_k t_i + 2\sum_{p=1}^N (A_p \sin\omega_p t_i \cos\omega_k t_i) + 2\sum_{p=1}^N (B_p\cos\omega_p t_i \cos\omega_k t_i) \Bigg] = 0$$
	~~$$\eqalign{~~\sum_{i=1}^M \Bigg[ ~~&-2 \phi_i \sin\omega_k t_i + 2 \bar\phi \sin\omega_k t_i + 2\sum_{p=1}^N (A_p \sin\omega_p t_i \sin\omega_k t_i) \cr~~
	~~&+ 2\sum_{p=1}^N (B_p\cos\omega_p t_i \sin\omega_k t_i)\Bigg] = 0}$$~~


	~~$$\frac{\partial \varepsilon^2}{\partial B_k} = 0$$~~
	~~$$\eqalign{\sum_{i=1}^M \Bigg[ &~~-2 \phi_i \cos\omega_k t_i + 2 \bar\phi \cos\omega_k t_i + 2\sum_{p=1}^N (A_p \sin\omega_p t_i \cos\omega_k t_i) ~~\cr~~
	&+ 2\sum_{p=1}^N (B_p\cos\omega_p t_i \cos\omega_k t_i) \Bigg] = 0}$$

			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.
	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.

	$$\matrix{\left[ \matrix{		$$\matrix{\left[ \matrix{
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	} \right] \cr		} \right] \cr
	A & x & & b}$$		A & x & & b}$$

	</wikitex>		</wikitex>

Robertson at 20:13, 29 September 2008

2008-09-29T20:13:46Z

Show changes

Robertson at 20:36, 28 January 2008

2008-01-28T20:36:48Z

← Older revision		Revision as of 20:36, 28 January 2008
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	in matrix form (7 harmonics). '''Note:''' all instances of <math>\sum</math> are actually <math>\sum_{i=1}^M</math>		in matrix form (N harmonics). '''Note:''' all instances of <math>\sum</math> are actually <math>\sum_{i=1}^M</math> where M is the number of time-steps in the time-averaging window.

	:<math>\begin{array}{cccc}		:<math>\begin{array}{cccc}
	\left[ \begin{array} {cccccc}		\left[ \begin{array} {cccccc}
	\\		\\
	M & ~~{\color{Blue}~~\sum \sin\omega_1 t_i} & ~~{\color{Blue}~~\sum \sin\omega_2 t_i}		M & \sum \sin\omega_1 t_i & \sum \sin\omega_2 t_i
	& \cdots & \sum \cos\omega_1 t_i & \cdots \\		& \cdots & \sum \cos\omega_1 t_i & \cdots \\
	\\		\\
	~~{\color{Blue}~~\sum \sin\omega_1 t_i} & ~~{\color{Blue}~~\sum \sin^2 \omega_1 t_i} & \sum \sin\omega_2 t_i \sin\omega_1 t_i		\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 \\		& \cdots & \sum \cos\omega_1 t_i \sin\omega_1 t_i & \cdots \\
	\\		\\
	~~{\color{Blue}~~\sum \sin\omega_2 t_i} & ~~{\color{Blue}~~\sum \sin\omega_1 t_i \sin\omega_2 t_i} & ~~{\color{Blue}~~\sum \sin^2 \omega_2 t_i}		\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 \\		& \cdots & \sum \cos\omega_1 t_i \sin\omega_2 t_i & \cdots \\
	\\		\\
	~~{\color{Blue}~~\vdots} & ~~{\color{Blue}~~\vdots} & \cdots & \cdots & \cdots & \cdots \\		\vdots & \vdots & \cdots & \cdots & \cdots & \cdots \\
	\\		\\
	~~{\color{Blue}~~\sum \sin\omega_7 t_i} & ~~{\color{Blue}~~\sum \sin\omega_1 t_i \sin\omega_7 t_i} & ~~{\color{Blue}~~\sum \sin\omega_2 t_i \sin\omega_7 t_i}		\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 \\		& \cdots & \sum \cos\omega_1 t_i \sin\omega_7 t_i & \cdots \\
	\\		\\
	~~{\color{Blue}~~\sum \sin\omega_1 t_i} & ~~{\color{Blue}~~\sum \sin\omega_1 t_i \cos\omega_1 t_i} & ~~{\color{Blue}~~\sum \sin\omega_2 t_i \cos\omega_1 t_i}		\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 \\		& \cdots & \sum \cos^2 \omega_1 t_i & \cdots \\
	\\		\\
	~~{\color{Blue}~~\sum \cos\omega_2 t_i} & ~~{\color{Blue}~~\sum \sin\omega_1 t_i \cos\omega_2 t_i} & ~~{\color{Blue}~~\sum} \sin\omega_2 t_i \cos\omega_2 t_i		\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 \\		& \cdots & \sum \cos\omega_1 t_i \cos\omega_2 t_i & \cdots \\
	\\		\\
	~~{\color{Blue}~~\vdots} & ~~{\color{Blue}~~\vdots} & \vdots & \vdots & \vdots & \vdots \\		\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
	\\		\\
	~~{\color{Blue}~~\sum \cos\omega_7 t_i} & ~~{\color{Blue}~~\sum \sin\omega_1 t_i \cos\omega_7 t_i} & \sum \sin\omega_2 t_2 \cos\omega_7 t_i		\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 \\ \\		& \cdots & \sum \cos\omega_1 t_i \cos\omega_7 t_i & \cdots \\ \\
	\end{array} \right] &		\end{array} \right] &

Robertson: New page:
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...

2008-01-28T20:06:33Z

New page: <div class="title">Least Squares Fit for ROMS Tides</div> <math>\phi(t) = \bar\phi + \sum_{k=1}^N A_k \sin\omega_k t + \sum_{k=1}^N B_k \cos\omega_k t</math> :<math>\,\!\phi</math>: state...

New page

<div class="title">Least Squares Fit for ROMS Tides</div>

<math>\phi(t) = \bar\phi + \sum_{k=1}^N A_k \sin\omega_k t + \sum_{k=1}^N B_k \cos\omega_k t</math>
:<math>\,\!\phi</math>: state variables
:<math>\,\!\omega_k</math>: tidal frequency
:<math>\,\!A_k, B_k</math>: amplitude
:<math>\,\!N</math>: number of harmonics

To minimize cost function <math>\varepsilon^2</math>
:<math>\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</math>
:<math>\,\!\phi, A_k, B_k</math> are unknowns

In discrete space:
:<math>\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</math>

at the minimum
:<math>\frac{\partial \varepsilon^2}{\partial \bar\phi} = 0</math>

:<math>\frac{\partial \varepsilon^2}{\partial A_k} = 0\;\;\;\;\;\;k = 1, ..., N</math>

:<math>\frac{\partial \varepsilon^2}{\partial B_k} = 0</math>

:<math>\frac{\partial \varepsilon^2}{\partial \bar\phi} = 0</math>
::<math>\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</math>

:<math>\frac{\partial \varepsilon^2}{\partial A_k} = 0</math>
::<math>\begin{align}\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{align}</math>

:<math>\frac{\partial \varepsilon^2}{\partial B_k} = 0</math>
::<math>\begin{align}\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{align}</math>

in matrix form (7 harmonics). '''Note:''' all instances of <math>\sum</math> are actually <math>\sum_{i=1}^M</math>

:<math>\begin{array}{cccc}
\left[ \begin{array} {cccccc}
\\
M & {\color{Blue}\sum \sin\omega_1 t_i} & {\color{Blue}\sum \sin\omega_2 t_i}
& \cdots & \sum \cos\omega_1 t_i & \cdots \\
\\
{\color{Blue}\sum \sin\omega_1 t_i} & {\color{Blue}\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 \\
\\
{\color{Blue}\sum \sin\omega_2 t_i} & {\color{Blue}\sum \sin\omega_1 t_i \sin\omega_2 t_i} & {\color{Blue}\sum \sin^2 \omega_2 t_i}
& \cdots & \sum \cos\omega_1 t_i \sin\omega_2 t_i & \cdots \\
\\
{\color{Blue}\vdots} & {\color{Blue}\vdots} & \cdots & \cdots & \cdots & \cdots \\
\\
{\color{Blue}\sum \sin\omega_7 t_i} & {\color{Blue}\sum \sin\omega_1 t_i \sin\omega_7 t_i} & {\color{Blue}\sum \sin\omega_2 t_i \sin\omega_7 t_i}
& \cdots & \sum \cos\omega_1 t_i \sin\omega_7 t_i & \cdots \\
\\
{\color{Blue}\sum \sin\omega_1 t_i} & {\color{Blue}\sum \sin\omega_1 t_i \cos\omega_1 t_i} & {\color{Blue}\sum \sin\omega_2 t_i \cos\omega_1 t_i}
& \cdots & \sum \cos^2 \omega_1 t_i & \cdots \\
\\
{\color{Blue}\sum \cos\omega_2 t_i} & {\color{Blue}\sum \sin\omega_1 t_i \cos\omega_2 t_i} & {\color{Blue}\sum} \sin\omega_2 t_i \cos\omega_2 t_i
& \cdots & \sum \cos\omega_1 t_i \cos\omega_2 t_i & \cdots \\
\\
{\color{Blue}\vdots} & {\color{Blue}\vdots} & \vdots & \vdots & \vdots & \vdots \\
\\
{\color{Blue}\sum \cos\omega_7 t_i} & {\color{Blue}\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}</math>

LSF Tides - Revision history

Robertson at 17:22, 5 August 2015

Robertson at 10:18, 8 October 2008

Robertson at 20:13, 29 September 2008

Robertson at 20:36, 28 January 2008

Robertson: New page: 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...

Robertson: New page:
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...