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misc [2015/10/17 18:48] – external edit 127.0.0.1misc [2023/03/28 09:14] (current) – external edit 127.0.0.1
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   where you can quickly type some ideas in LaTeX and the other   where you can quickly type some ideas in LaTeX and the other
   person with internet connection can read it ...    person with internet connection can read it ... 
 +
 +===== Da Redesign 1 =====
 +
 +$a$: all information on bonds, spots or collections or records of observations
 +
 +$dim_a$: dimension of bonds
 +
 +$y$: Observation vector, List of observation values
 +
 +$dim_y$: dimenension of obs
 +
 +$is$: Ispace, Interpolation space, list of interpolation points and variables
 +
 +$dim_i$: dimension of interpolation space
 +
 +$y-i-list$: values needed for y from $is$, this is a vector of some dimension $dim_{yi}$, referenced by vectors yii and yil of integers, where yii(jy) is the starting index and yil(jy) is the number of points needed for the observation with index jy.
 +===== Da Redesign 2 =====
 +==== Maths Notation ====
 +
 +Linearisierung
 +$$
 +f(x) = f(x_0) + df*(x-x_0), \hspace{1cm}
 +q_i(t,q) = q_i(t_0,q_0) + \frac{dq_i}{dt} (t-t_0) + \frac{dq_i}{dq} (q-q_0)
 +$$
 +
 +$$
 +A = \left( \begin{array}{cc}
 +1 & 0 \\
 +0 & 1 \\
 +v_1 & v_2 
 +\end{array} \right), \hspace{1cm}
 +A^{T} = \left( \begin{array}{cc}
 +1 & 0 & v_1\\
 +0 & 1 & v_2 
 +\end{array} \right)
 +$$
 +
 +$$
 +x = \left( \begin{array}{c} t \\ q \end{array}\right), \hspace{1cm} 
 +A * x = \left( \begin{array}{c} t \\ q \\ q_i \end{array} \right)
 +$$
 +
 +$z$ be the reflectance using $H_{s}$ MFASIS, then: 
 +$$
 +H_{s} = (\alpha_1, \alpha_2, \alpha_3), 
 +\hspace{1cm} \left( \begin{array}{c} t \\ q \\ q_i \end{array} \right) = 
 +\left( \begin{array}{c} \alpha_1 z \\ \alpha_2 z \\ \alpha_3 z \end{array} \right) = H_s^T z
 +$$
 +
 +$$
 +A^T H^T z = A^T \left( \begin{array}{c} \alpha_1 z \\ \alpha_2 z \\ \alpha_3 z \end{array} \right) 
 += \left( \begin{array}{c} (\alpha_1  + v_1 \alpha_3 ) z \\
 +(\alpha_2  + v_2 \alpha_3 ) z 
 +\end{array} \right) \hspace{1cm}
 +H_s * A = \left( \begin{array}{cc} (\alpha_1  + v_1 \alpha_3 ) & (\alpha_2  + v_2 \alpha_3 ) 
 +\end{array} \right) 
 +$$
 +
 +
 +
 +
 +==== Maths Notation ====
 +
 +$$
 +\frac{dx'}{dt}(\rho,t)\Big|_{t=0} = 0   \hspace{3cm} (3.4)
 +$$
  
 ==== Talk Data Assimilation ==== ==== Talk Data Assimilation ====
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 [[misc2]][[feglt]] [[misc2]][[feglt]]
    
 +===== another statement =====
 +
 +\begin{eqnarray}
 +\tilde{Y}^{T} \tilde{R}^{-1} \tilde{Y} & = &  Y^{T} A^{T} (A R A^{T})^{-1} A Y \nonumber \\
 +& = & Y^{T} A^{T} (A^{T})^{-1} R^{-1} A^{-1} A Y \nonumber \\
 +& = & Y^{T} R^{-1} Y
 +\end{eqnarray}
 +
 +Which metric to use?
 +
 +$$
 +D{y} := \frac{y_{2}-y_{1}}{h}
 +$$
 +
 +in other words
 +
 +$$
 +{\bf y} := \left( \begin{array}{c}
 +y_{1} \\ D{y}
 +\end{array} \right)
 +$$
 +and 
 +$$
 +\| {\bf y} - {\bf y^{(o)}} \|^2 := \| y_{1} - y_{1}^{(o)} \|^2 + \| D{y} - D{y}^{(o)} \|^2 
 +$$
 +or the original one
 +\begin{eqnarray}
 +\| {\bf y} - {\bf y^{(o)}} \|^2 & := & \| {\bf y} - {\bf y^{(o)}} \|_{\tilde{R}}^2 \nonumber \\
 +& = & 
 +({\bf y} - {\bf y^{(o)}})^{T} (A^{T})^{-1} R^{-1} A^{-1} ({\bf y} - {\bf y^{(o)}})
 +\end{eqnarray}
 +
 +$$
 +A = \left( \begin{array}{cc}
 +1 & 0 \\
 +-\frac{1}{h} & \frac{1}{h}
 +\end{array}\right)
 +$$
 +
misc.1445100535.txt.gz · Last modified: 2023/03/28 09:14 (external edit)