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--- misc [2014/11/11 22:35] – [Talk Data Assimilation] potthast
+++ misc [2023/03/28 09:14] (current) – external edit 127.0.0.1
@@ Line 10: / Line 10: @@
   person with internet connection can read it ...
-==== Talk Data Assimilation ====
+===== Da Redesign 1 =====
-A general introductory talk into data assimilation for numerical weather
+$a$: all information on bonds, spots or collections or records of observations
-prediction: [[talk_potthast_reading_2014_11.pdf]]
+$dim_a$: 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
+$$
+f(x) = f(x_0) + df*(x-x_0), \hspace{1cm}
+q_i(t,q) = q_i(t_0,q_0) + \frac{dq_i}{dt} (t-t_0) + \frac{dq_i}{dq} (q-q_0)
+$$
+$$
+A = \left( \begin{array}{cc}
+& 0 \\
+& 1 \\
+v_1 & v_2
+\end{array} \right), \hspace{1cm}
+A^{T} = \left( \begin{array}{cc}
+& 0 & v_1\\
+& 1 & v_2
+\end{array} \right)
+$$
+$$
+x = \left( \begin{array}{c} t \\ q \end{array}\right), \hspace{1cm}
+A * x = \left( \begin{array}{c} t \\ q \\ q_i \end{array} \right)
+$$
+$z$ 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 ====
+A general introductory talk into data assimilation for numerical weather
+prediction: [[http://scienceatlas.com/potthast/_media/talk_potthast_reading_2014_11.pdf]]
+and [[http://scienceatlas.com/potthast/_media/potthast_icon_bacy_tour.pdf]]
 ===== KENDA Concept =====
@@ Line 293: / Line 358: @@
 [[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}
+$$
+A = \left( \begin{array}{cc}
+& 0 \\
+-\frac{1}{h} & \frac{1}{h}
+\end{array}\right)
+$$