I4DVAR ANA SENSITIVITY - Revision history

Robertson: Text replacement - "IS4DVAR" to "I4DVAR"

2020-07-21T18:06:51Z

Text replacement - "IS4DVAR" to "I4DVAR"

Robertson at 19:10, 20 July 2020

2020-07-20T19:10:00Z

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Robertson: Robertson moved page I4DVAR ANA SENSITIVTY to I4DVAR ANA SENSITIVITY without leaving a redirect: Name title typo

2020-07-20T19:08:09Z

Robertson moved page I4DVAR ANA SENSITIVTY to I4DVAR ANA SENSITIVITY without leaving a redirect: Name title typo

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Robertson at 19:02, 20 July 2020

2020-07-20T19:02:16Z

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Robertson: Robertson moved page IS4DVAR SENSITIVITY to I4DVAR ANA SENSITIVTY: Option renamed in SVN revision 1022

2020-07-20T18:57:07Z

Robertson moved page IS4DVAR SENSITIVITY to I4DVAR ANA SENSITIVTY: Option renamed in SVN revision 1022

← Older revision	Revision as of 18:57, 20 July 2020
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Robertson at 01:35, 18 May 2016

2016-05-18T01:35:28Z

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Robertson at 19:29, 8 January 2010

2010-01-08T19:29:45Z

Robertson: moved OBS SENSITIVITY to IS4DVAR SENSITIVITY: CPP Option was renamed in revision 431.

2010-01-08T19:25:31Z

moved OBS SENSITIVITY to IS4DVAR SENSITIVITY: CPP Option was renamed in revision 431.

← Older revision	Revision as of 19:25, 8 January 2010
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Robertson: should only be used INSIDE sections so that the table of contents will work

2008-07-15T15:34:10Z

<wikitex> should only be used INSIDE sections so that the table of contents will work

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	==Mathematical Formulation==		==Mathematical Formulation==
			<wikitex>The mathematical development presented here closely parallels that of [[Bibliography#ZhuY_2008a \| Zhu and Gelaro (2008)]]. The sensitivity of any functional, $\cal J$, of the analysis $\bf\Psi_a$ or forecast $\bf\Psi_f$ can be efficiently computed using the adjoint model which yields information about the gradients of ${\cal J}({\bf\Psi_a})$ and ${\cal J}({\bf\Psi_f})$. We can extent the concept of the adjoint sensitivity to compute the sensitivity of the [[IS4DVAR]] cost function, $J$, in (1) and any other function $\cal J$ of the forecast $\bf\Psi_f$ to the observations, $\bf y$.
	The mathematical development presented here closely parallels that of [[Bibliography#ZhuY_2008a \| Zhu and Gelaro (2008)]]. The sensitivity of any functional, $\cal J$, of the analysis $\bf\Psi_a$ or forecast $\bf\Psi_f$ can be efficiently computed using the adjoint model which yields information about the gradients of ${\cal J}({\bf\Psi_a})$ and ${\cal J}({\bf\Psi_f})$. We can extent the concept of the adjoint sensitivity to compute the sensitivity of the [[IS4DVAR]] cost function, $J$, in (1) and any other function $\cal J$ of the forecast $\bf\Psi_f$ to the observations, $\bf y$.

	In [[IS4DVAR]], we define a quadratic cost function:		In [[IS4DVAR]], we define a quadratic cost function:
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	which again can be readily evaluated using the adjoint model denoted by ${\bf R}^{T}(t,t_0+\tau)$ and the adjoint of [[IS4DVAR]], ${\bf K}^{T}$.		which again can be readily evaluated using the adjoint model denoted by ${\bf R}^{T}(t,t_0+\tau)$ and the adjoint of [[IS4DVAR]], ${\bf K}^{T}$.
			</wikitex>
	==Technical Description==		==Technical Description==
			<wikitex>This ROMS driver can be used to evaluate the impact of each observation on the [[IS4DVAR]] data assimilation algorithm by measuring their sensitivity over a specified circulation functional index, ${\cal J}$. This algorithm is an extension of the adjoint sensitivity driver, [[AD_SENSITIVITY]]. As mentioned above, this algorithm is equivalent to taking the adjoint of the [[IS4DVAR]] algorithm.
	This ROMS driver can be used to evaluate the impact of each observation on the [[IS4DVAR]] data assimilation algorithm by measuring their sensitivity over a specified circulation functional index, ${\cal J}$. This algorithm is an extension of the adjoint sensitivity driver, [[AD_SENSITIVITY]]. As mentioned above, this algorithm is equivalent to taking the adjoint of the [[IS4DVAR]] algorithm.

	The following steps are carried out in [[obs_sen_ocean.h]]:		The following steps are carried out in [[obs_sen_ocean.h]]:

Arango at 18:25, 11 July 2008

2008-07-11T18:25:35Z

← Older revision		Revision as of 18:25, 11 July 2008
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	{{#lst:Options\|OBS_SENSITIVITY}}		{{#lst:Options\|OBS_SENSITIVITY}}
			<wikitex>
	__TOC__		__TOC__
	~~<wikitex>~~
	==Mathematical Formulation==		==Mathematical Formulation==

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	==Mathematical Formulation==		==Mathematical Formulation==
	The mathematical development presented here closely parallels that of [[Bibliography#ZhuY_2008a \| Zhu and Gelaro (2008)]]. The sensitivity of any functional, <math>\cal J</math>, of the analysis <math>\bf\Psi_a</math> or forecast <math>\bf\Psi_f</math> can be efficiently computed using the adjoint model which yields information about the gradients of <math>{\cal J}({\bf\Psi_a})</math> and <math>{\cal J}({\bf\Psi_f})</math>. We can extent the concept of the adjoint sensitivity to compute the sensitivity of the [[~~IS4DVAR~~]] cost function, <math>J</math>, in (1) and any other function <math>\cal J</math> of the forecast <math>\bf\Psi_f</math> to the observations, <math>\bf y</math>.		The mathematical development presented here closely parallels that of [[Bibliography#ZhuY_2008a \| Zhu and Gelaro (2008)]]. The sensitivity of any functional, <math>\cal J</math>, of the analysis <math>\bf\Psi_a</math> or forecast <math>\bf\Psi_f</math> can be efficiently computed using the adjoint model which yields information about the gradients of <math>{\cal J}({\bf\Psi_a})</math> and <math>{\cal J}({\bf\Psi_f})</math>. We can extent the concept of the adjoint sensitivity to compute the sensitivity of the [[I4DVAR]] cost function, <math>J</math>, in (1) and any other function <math>\cal J</math> of the forecast <math>\bf\Psi_f</math> to the observations, <math>\bf y</math>.

	In [[~~IS4DVAR~~]], we define a quadratic cost function:		In [[I4DVAR]], we define a quadratic cost function:

	{\| class="eqno"		{\| class="eqno"
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	where <math>\psi_a</math> is referred to as the analysis increment, the desired solution of the incremental data assimilation procedure.		where <math>\psi_a</math> is referred to as the analysis increment, the desired solution of the incremental data assimilation procedure.

	The [[~~IS4DVAR~~]] analysis can be written in the more traditional form as <math>{\bf\Psi_a} = {\bf\Psi_b} + {\bf K} {\bf d}</math> ([[Bibliography#DaleyR_1991a \| Daley, 1991]]), where <math>\psi_a = {\bf K} {\bf d}</math>, the Kalman gain matrix <math>{\bf K} = {\cal H}^{-1} {\bf G}^T {\bf O}^{-1}</math>, and <math>{\cal H} = {\partial^{2}J}/{\partial\psi^2} = {\bf B}^{-1} {\bf G}^T {\bf O}^{-1} {\bf G}</math> is the Hessian matrix. The entire [[~~IS4DVAR~~]] procedure is therefore neatly embodied in <math>\bf K</math>. At the cost function minimum, (1) can be written as <math>J = \frac{1}{2} {\bf d}^T ({\bf I} - {\bf K}^T {\bf G}^T) {\bf O}^{-1} {\bf d}</math> which yields the sensitivity of the [[~~IS4DVAR~~]] cost function to the observations.		The [[I4DVAR]] analysis can be written in the more traditional form as <math>{\bf\Psi_a} = {\bf\Psi_b} + {\bf K} {\bf d}</math> ([[Bibliography#DaleyR_1991a \| Daley, 1991]]), where <math>\psi_a = {\bf K} {\bf d}</math>, the Kalman gain matrix <math>{\bf K} = {\cal H}^{-1} {\bf G}^T {\bf O}^{-1}</math>, and <math>{\cal H} = {\partial^{2}J}/{\partial\psi^2} = {\bf B}^{-1} {\bf G}^T {\bf O}^{-1} {\bf G}</math> is the Hessian matrix. The entire [[I4DVAR]] procedure is therefore neatly embodied in <math>\bf K</math>. At the cost function minimum, (1) can be written as <math>J = \frac{1}{2} {\bf d}^T ({\bf I} - {\bf K}^T {\bf G}^T) {\bf O}^{-1} {\bf d}</math> which yields the sensitivity of the [[I4DVAR]] cost function to the observations.

	In the current [[~~IS4DVAR~~]]/[[LANCZOS]] data assimilation algorithm, the cost function minimum of (1) is identified using the Lanczos method [[Bibliography#GolubG_1989a \| (Golub and Van Loan, 1989)]], in which case:		In the current [[I4DVAR]]/[[LANCZOS]] data assimilation algorithm, the cost function minimum of (1) is identified using the Lanczos method [[Bibliography#GolubG_1989a \| (Golub and Van Loan, 1989)]], in which case:

	{\| class="eqno"		{\| class="eqno"
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	\|}		\|}

	where <math>{\bf Q}_k = ({\bf q}_i)</math> is the matrix of <math>k</math> orthogonal Lanczos vectors, and <math>{\bf T}_k</math> is a known tridiagonal matrix. Each of the <math>k-</math>iterations of [[~~IS4DVAR~~]] employed in finding the minimum of <math>J</math> yields one column <math>{\bf q}_i</math> of <math>{\bf Q}_k</math>. From (3) we can identify the Kalman gain matrix as <math>{\bf K} = - {\bf Q}_k {\bf T}_{k}^{-1} {\bf Q}_{k}^{T} {\bf G}^T {\bf O}^{-1}</math> in which case <math>{\bf K}^T = - {\bf O}^{-1} {\bf G} {\bf Q}_k {\bf T}_{k}^{-1} {\bf Q}_{k}^{T}</math> represents the adjoint of the entire [[~~IS4DVAR~~]] system. The action of <math>{\bf K}^T</math> on the vector <math>{\bf G}^T {\bf O}^{-1} {\bf d}</math> in (3) can be readily computed since the Lanczos vectors <math>{\bf q}_i</math> and the matrix <math>{\bf T}_k</math> are available at the end of each [[~~IS4DVAR~~]] assimilation cycle.		where <math>{\bf Q}_k = ({\bf q}_i)</math> is the matrix of <math>k</math> orthogonal Lanczos vectors, and <math>{\bf T}_k</math> is a known tridiagonal matrix. Each of the <math>k-</math>iterations of [[I4DVAR]] employed in finding the minimum of <math>J</math> yields one column <math>{\bf q}_i</math> of <math>{\bf Q}_k</math>. From (3) we can identify the Kalman gain matrix as <math>{\bf K} = - {\bf Q}_k {\bf T}_{k}^{-1} {\bf Q}_{k}^{T} {\bf G}^T {\bf O}^{-1}</math> in which case <math>{\bf K}^T = - {\bf O}^{-1} {\bf G} {\bf Q}_k {\bf T}_{k}^{-1} {\bf Q}_{k}^{T}</math> represents the adjoint of the entire [[I4DVAR]] system. The action of <math>{\bf K}^T</math> on the vector <math>{\bf G}^T {\bf O}^{-1} {\bf d}</math> in (3) can be readily computed since the Lanczos vectors <math>{\bf q}_i</math> and the matrix <math>{\bf T}_k</math> are available at the end of each [[I4DVAR]] assimilation cycle.

	Therefore, the basic observation sensitivity algorithm is as follows:		Therefore, the basic observation sensitivity algorithm is as follows:
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	\|}		\|}

	which again can be readily evaluated using the adjoint model denoted by <math>{\bf R}^{T}(t,t_0+\tau)</math> and the adjoint of [[~~IS4DVAR~~]], <math>{\bf K}^{T}</math>.		which again can be readily evaluated using the adjoint model denoted by <math>{\bf R}^{T}(t,t_0+\tau)</math> and the adjoint of [[I4DVAR]], <math>{\bf K}^{T}</math>.

	==Technical Description==		==Technical Description==
	This ROMS driver can be used to evaluate the impact of each observation on the [[~~IS4DVAR~~]] data assimilation algorithm by measuring their sensitivity over a specified circulation functional index, <math>{\cal J}</math>. This algorithm is an extension of the adjoint sensitivity driver, [[AD_SENSITIVITY]]. As mentioned above, this algorithm is equivalent to taking the adjoint of the [[~~IS4DVAR~~]] algorithm.		This ROMS driver can be used to evaluate the impact of each observation on the [[I4DVAR]] data assimilation algorithm by measuring their sensitivity over a specified circulation functional index, <math>{\cal J}</math>. This algorithm is an extension of the adjoint sensitivity driver, [[AD_SENSITIVITY]]. As mentioned above, this algorithm is equivalent to taking the adjoint of the [[I4DVAR]] algorithm.

	The following steps are carried out in [[obs_sen_is4dvar.h]]:		The following steps are carried out in [[obs_sen_is4dvar.h]]:
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	<ol style="list-style-type:lower-roman">		<ol style="list-style-type:lower-roman">
	<li>We begin by running the incremental strong constraint 4DVar data assimilation algorithm ([[~~IS4DVAR~~]]) with the Lanczos conjugate gradient minimization algorithm ([[LANCZOS]]) using <math>k</math> inner-loops (<math>\text{Ninner}=k</math>) and a single outer-loop (<math>\text{Nouter}=1</math>) for the period <math>t=[t_0,t_1]</math>. We will denote by <math>{\bf\Psi_b}(t_0)</math> the background initial condition, and the observations vector by <math>\bf y</math>. The resulting Lanczos vectors that we save in the adjoint NetCDF file will be denoted by <math>{\bf q}_i</math>, where <math>i=1,2,\ldots,k</math>.</li>		<li>We begin by running the incremental strong constraint 4DVar data assimilation algorithm ([[I4DVAR]]) with the Lanczos conjugate gradient minimization algorithm ([[LANCZOS]]) using <math>k</math> inner-loops (<math>\text{Ninner}=k</math>) and a single outer-loop (<math>\text{Nouter}=1</math>) for the period <math>t=[t_0,t_1]</math>. We will denote by <math>{\bf\Psi_b}(t_0)</math> the background initial condition, and the observations vector by <math>\bf y</math>. The resulting Lanczos vectors that we save in the adjoint NetCDF file will be denoted by <math>{\bf q}_i</math>, where <math>i=1,2,\ldots,k</math>.</li>
	<li>Next we run the nonlinear model for the combined assimilation plus forecast period <math>t=[t_0,t_2]</math>, where <math>t_2\geq t_1</math>. This represents the final sweep of the nonlinear model for the period <math>t=[t_0,t_1]</math> after exiting the inner-loop in the [[~~IS4DVAR~~]] plus the forecast period <math>t=[t_1,t-2]</math>. The initial condition for the nonlinear mode at <math>t=t_0</math> is <math>{\bf\Psi_b}(t_0)</math> and not the new estimated initial conditions, <math>{\bf\Psi}(t_0) = {\bf\Psi_b}(t_0) + \psi(t_0)</math>. We save the basic state trajectory, <math>{\bf\Psi_b}(t)</math>, of this nonlinear model run for use in the adjoint sensitivity calculation next, and for use in the tangent linear model run later in step (vii). Depending on time for which the sensitivity functional <math>{\cal J}(t)</math> is defined, this will dictate <math>t_2</math>. For example, if <math>{\cal J}(t)</math> is a functional defined during the forecast interval <math>t_1<t<t_2</math>, then <math>t_2>t_1</math> for this run of the nonlinear model. However, if <math>{\cal J}(t)</math> is defined during the assimilation interval <math>t_0<t<t_1</math>, then <math>t_2=t_1</math>. That is, the definition of <math>t_2</math> should be flexible depending on the choice of <math>\cal J</math>.</li>		<li>Next we run the nonlinear model for the combined assimilation plus forecast period <math>t=[t_0,t_2]</math>, where <math>t_2\geq t_1</math>. This represents the final sweep of the nonlinear model for the period <math>t=[t_0,t_1]</math> after exiting the inner-loop in the [[I4DVAR]] plus the forecast period <math>t=[t_1,t-2]</math>. The initial condition for the nonlinear mode at <math>t=t_0</math> is <math>{\bf\Psi_b}(t_0)</math> and not the new estimated initial conditions, <math>{\bf\Psi}(t_0) = {\bf\Psi_b}(t_0) + \psi(t_0)</math>. We save the basic state trajectory, <math>{\bf\Psi_b}(t)</math>, of this nonlinear model run for use in the adjoint sensitivity calculation next, and for use in the tangent linear model run later in step (vii). Depending on time for which the sensitivity functional <math>{\cal J}(t)</math> is defined, this will dictate <math>t_2</math>. For example, if <math>{\cal J}(t)</math> is a functional defined during the forecast interval <math>t_1<t<t_2</math>, then <math>t_2>t_1</math> for this run of the nonlinear model. However, if <math>{\cal J}(t)</math> is defined during the assimilation interval <math>t_0<t<t_1</math>, then <math>t_2=t_1</math>. That is, the definition of <math>t_2</math> should be flexible depending on the choice of <math>\cal J</math>.</li>
	<li>The next step involves an adjoint sensitivity calculation for the combined assimilation plus forecast period <math>t=[t_2,t_0]</math>. The basic state trajectory for this calculation will be that from the nonlinear model run in step (ii).</li>		<li>The next step involves an adjoint sensitivity calculation for the combined assimilation plus forecast period <math>t=[t_2,t_0]</math>. The basic state trajectory for this calculation will be that from the nonlinear model run in step (ii).</li>
	<li>After running the regular adjoint sensitivity calculation in (iii), we will have a full 3D-adjoint state vector at time <math>t=t_0</math>. Let's call this vector <math>\psi^\dagger(t_0)</math>. The next thing we want to do is to compute the dot-product of <math>\psi^\dagger(t_0)</math> with each of the Lanczos vectors from the previous [[~~IS4DVAR~~]] run. So if we ran [[~~IS4DVAR~~]] with <math>k</math> inner-loops, we will have <math>k-</math>Lanczos vectors which we denote as <math>{\bf q}_i</math> where <math>i=1,2,\ldots,k</math>. So we will compute <math>a_i = \psi^T {\bf q}_i</math> where <math>\psi^T</math> is the transpose of the vector <math>\psi^\dagger(t_0)</math>, and <math>a_i</math> for <math>i=1,2,\ldots,k</math> are scalars, so there will be <math>k</math> of them.</li>		<li>After running the regular adjoint sensitivity calculation in (iii), we will have a full 3D-adjoint state vector at time <math>t=t_0</math>. Let's call this vector <math>\psi^\dagger(t_0)</math>. The next thing we want to do is to compute the dot-product of <math>\psi^\dagger(t_0)</math> with each of the Lanczos vectors from the previous [[I4DVAR]] run. So if we ran [[I4DVAR]] with <math>k</math> inner-loops, we will have <math>k-</math>Lanczos vectors which we denote as <math>{\bf q}_i</math> where <math>i=1,2,\ldots,k</math>. So we will compute <math>a_i = \psi^T {\bf q}_i</math> where <math>\psi^T</math> is the transpose of the vector <math>\psi^\dagger(t_0)</math>, and <math>a_i</math> for <math>i=1,2,\ldots,k</math> are scalars, so there will be <math>k</math> of them.</li>
	<li>The next step is to invert the tridiagonal matrix associated with the Lanczos vectors. Let's denote this matrix as <math>\bf T</math>. So what we want to solve <math>{\bf T}{\bf b}={\bf a}</math>, where <math>\bf a</math> is the <math>k\times 1</math> vector of scalars <math>a_i</math> from step (iv), and <math>\bf b</math> is the <math>k\times 1</math> vector that we want to find. So we solve for <math>b</math> by using a tridiagonal solver.</li>		<li>The next step is to invert the tridiagonal matrix associated with the Lanczos vectors. Let's denote this matrix as <math>\bf T</math>. So what we want to solve <math>{\bf T}{\bf b}={\bf a}</math>, where <math>\bf a</math> is the <math>k\times 1</math> vector of scalars <math>a_i</math> from step (iv), and <math>\bf b</math> is the <math>k\times 1</math> vector that we want to find. So we solve for <math>b</math> by using a tridiagonal solver.</li>
	<li>The next step is to compute a weighted sum of the Lanczos vectors. Let's call this <math>\bf z</math>, where <math>{\bf z} = \sum_i (b_i {\bf q}_i)</math>		<li>The next step is to compute a weighted sum of the Lanczos vectors. Let's call this <math>\bf z</math>, where <math>{\bf z} = \sum_i (b_i {\bf q}_i)</math>
	for <math>i=1,2,\ldots,k</math>. The <math>b_i</math> are obtained from solving the tridiagonal equation in (v), and the <math>{\bf q}_i</math> are the Lanczos vectors. The vector <math>z</math> is a full-state vector and be used as an initial condition for the tangent linear model in step (vii).</li>		for <math>i=1,2,\ldots,k</math>. The <math>b_i</math> are obtained from solving the tridiagonal equation in (v), and the <math>{\bf q}_i</math> are the Lanczos vectors. The vector <math>z</math> is a full-state vector and be used as an initial condition for the tangent linear model in step (vii).</li>
	<li>Finally, we run the tangent linear model from <math>t=[t_0,t_1]</math> using <math>\bf z</math> from (vi) as the initial conditions. During this run of the tangent linear model, we need to read and process the observations <math>\bf y</math> that we used in the [[~~IS4DVAR~~]] of step 1 and write the solution at the observation points and times to the [[MODname]] NetCDF file. The values that we write into this [[MODname]] are actually the values multiplied by error covariance <math>-{\bf O}^{-1}</math> assigned to each observation during the ~~IS4DVAR~~ in step (i).</li>		<li>Finally, we run the tangent linear model from <math>t=[t_0,t_1]</math> using <math>\bf z</math> from (vi) as the initial conditions. During this run of the tangent linear model, we need to read and process the observations <math>\bf y</math> that we used in the [[I4DVAR]] of step 1 and write the solution at the observation points and times to the [[MODname]] NetCDF file. The values that we write into this [[MODname]] are actually the values multiplied by error covariance <math>-{\bf O}^{-1}</math> assigned to each observation during the I4DVAR in step (i).</li>
	</ol>		</ol>

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	<wikitex>This ROMS driver can be used to evaluate the impact of each observation on the [[IS4DVAR]] data assimilation algorithm by measuring their sensitivity over a specified circulation functional index, ${\cal J}$. This algorithm is an extension of the adjoint sensitivity driver, [[AD_SENSITIVITY]]. As mentioned above, this algorithm is equivalent to taking the adjoint of the [[IS4DVAR]] algorithm.		<wikitex>This ROMS driver can be used to evaluate the impact of each observation on the [[IS4DVAR]] data assimilation algorithm by measuring their sensitivity over a specified circulation functional index, ${\cal J}$. This algorithm is an extension of the adjoint sensitivity driver, [[AD_SENSITIVITY]]. As mentioned above, this algorithm is equivalent to taking the adjoint of the [[IS4DVAR]] algorithm.

	The following steps are carried out in [[~~obs_sen_ocean~~.h]]:		The following steps are carried out in [[obs_sen_is4dvar.h]]:

	:[[Image:obs_sens_time_diagram.png]]		:[[Image:obs_sens_time_diagram.png]]

	<ol style="list-style-type:lower-roman">		<ol style="list-style-type:lower-roman">
	<li>We begin by running the incremental strong constraint 4DVar data assimilation algorithm ([[~~IS4dVAR~~]]) with the Lanczos conjugate gradient minimization algorithm ([[LANCZOS]]) using $k$ inner-loops ($\hbox{Ninner}=k$) and a single outer-loop ($\hbox{Nouter}=1$) for the period $t=[t_0,t_1]$. We will denote by ${\bf\Psi_b}(t_0)$ the background initial condition, and the observations vector by $\bf y$. The resulting Lanczos vectors that we save in the adjoint NetCDF file will be denoted by ${\bf q}_i$, where $i=1,2,\ldots,k$.</li>		<li>We begin by running the incremental strong constraint 4DVar data assimilation algorithm ([[IS4DVAR]]) with the Lanczos conjugate gradient minimization algorithm ([[LANCZOS]]) using $k$ inner-loops ($\hbox{Ninner}=k$) and a single outer-loop ($\hbox{Nouter}=1$) for the period $t=[t_0,t_1]$. We will denote by ${\bf\Psi_b}(t_0)$ the background initial condition, and the observations vector by $\bf y$. The resulting Lanczos vectors that we save in the adjoint NetCDF file will be denoted by ${\bf q}_i$, where $i=1,2,\ldots,k$.</li>
	<li>Next we run the nonlinear model for the combined assimilation plus forecast period $t=[t_0,t_2]$, where $t_2\geq t_1$. This represents the final sweep of the nonlinear model for the period $t=[t_0,t_1]$ after exiting the inner-loop in the [[IS4DVAR]] plus the forecast period $t=[t_1,t-2]$. The initial condition for the nonlinear mode at $t=t_0$ is ${\bf\Psi_b}(t_0)$ and not the new estimated initial conditions, ${\bf\Psi}(t_0) = {\bf\Psi_b}(t_0) + \psi(t_0)$. We save the basic state trajectory, ${\bf\Psi_b}(t)$, of this nonlinear model run for use in the adjoint sensitivity calculation next, and for use in the tangent linear model run later in step (vii). Depending on time for which the sensitivity functional ${\cal J}(t)$ is defined, this will dictate $t_2$. For example, if ${\cal J}(t)$ is a functional defined during the forecast interval $t_1<t<t_2$, then $t_2>t_1$ for this run of the nonlinear model. However, if ${\cal J}(t)$ is defined during the assimilation interval $t_0<t<t_1$, then $t_2=t_1$. That is, the definition of $t_2$ should be flexible depending on the choice of $\cal J$.</li>		<li>Next we run the nonlinear model for the combined assimilation plus forecast period $t=[t_0,t_2]$, where $t_2\geq t_1$. This represents the final sweep of the nonlinear model for the period $t=[t_0,t_1]$ after exiting the inner-loop in the [[IS4DVAR]] plus the forecast period $t=[t_1,t-2]$. The initial condition for the nonlinear mode at $t=t_0$ is ${\bf\Psi_b}(t_0)$ and not the new estimated initial conditions, ${\bf\Psi}(t_0) = {\bf\Psi_b}(t_0) + \psi(t_0)$. We save the basic state trajectory, ${\bf\Psi_b}(t)$, of this nonlinear model run for use in the adjoint sensitivity calculation next, and for use in the tangent linear model run later in step (vii). Depending on time for which the sensitivity functional ${\cal J}(t)$ is defined, this will dictate $t_2$. For example, if ${\cal J}(t)$ is a functional defined during the forecast interval $t_1<t<t_2$, then $t_2>t_1$ for this run of the nonlinear model. However, if ${\cal J}(t)$ is defined during the assimilation interval $t_0<t<t_1$, then $t_2=t_1$. That is, the definition of $t_2$ should be flexible depending on the choice of $\cal J$.</li>
	<li>The next step involves an adjoint sensitivity calculation for the combined assimilation plus forecast period $t=[t_2,t_0]$. The basic state trajectory for this calculation will be that from the nonlinear model run in step (ii).</li>		<li>The next step involves an adjoint sensitivity calculation for the combined assimilation plus forecast period $t=[t_2,t_0]$. The basic state trajectory for this calculation will be that from the nonlinear model run in step (ii).</li>
	<li>After running the regular adjoint sensitivity calculation in (iii), we will have a full 3D-adjoint state vector at time $t=t_0$. Let's call this vector $\psi^\dagger(t_0)$. The next thing we want to do is to compute the dot-product of $\psi^\dagger(t_0)$ with each of the Lanczos vectors from the previous [[IS4DVAR]] run. So if we ran [[IS4DVAR]] with $k$ inner-loops, we will have $k\hyphen$Lanczos vectors which we denote as ${\bf q}_i$ where $i=1,2,\ldots,k$. So we will compute $a_i = \psi^T {\bf q}_i$ where $\psi^T$ is the transpose of the vector $\psi^\dagger(t_0)$, and $a_i$ for $i=1,2,\ldots,k$ are scalars, so there will be $k$ of them.</li>		<li>After running the regular adjoint sensitivity calculation in (iii), we will have a full 3D-adjoint state vector at time $t=t_0$. Let's call this vector $\psi^\dagger(t_0)$. The next thing we want to do is to compute the dot-product of $\psi^\dagger(t_0)$ with each of the Lanczos vectors from the previous [[IS4DVAR]] run. So if we ran [[IS4DVAR]] with $k$ inner-loops, we will have $k\hyphen$Lanczos vectors which we denote as ${\bf q}_i$ where $i=1,2,\ldots,k$. So we will compute $a_i = \psi^T {\bf q}_i$ where $\psi^T$ is the transpose of the vector $\psi^\dagger(t_0)$, and $a_i$ for $i=1,2,\ldots,k$ are scalars, so there will be $k$ of them.</li>
	<li>The next step is to invert the tridiagonal matrix associated with the Lanczos vectors. Let's denote this matrix as $\bf T$. So what we want to solve ${\bf T}{\bf b}={\bf a}$, where $\bf a$ is the $k\times 1$ vector of scalars $a_i$ from step (iv), and $\bf b$ is the $k\times 1$ vector that we want to find. So we solve for $b$ by using a tridiagonal solver.</li>		<li>The next step is to invert the tridiagonal matrix associated with the Lanczos vectors. Let's denote this matrix as $\bf T$. So what we want to solve ${\bf T}{\bf b}={\bf a}$, where $\bf a$ is the $k\times 1$ vector of scalars $a_i$ from step (iv), and $\bf b$ is the $k\times 1$ vector that we want to find. So we solve for $b$ by using a tridiagonal solver.</li>
	<li>The next step is to compute a weighted sum of the Lanczos vectors. Let's call this $\bf z$, where ${\bf z} = \sum_i (b_i {\bf q}_i)$ for $i=1,2,\ldots,k$. The $b_i$ are obtained from solving the tridiagonal equation in (v), and the ${\bf q}_i$ are the Lanczos vectors. The vector $z$ is a full-state vector and be used as an initial condition for the tangent linear model in step (vii).</li>		<li>The next step is to compute a weighted sum of the Lanczos vectors. Let's call this $\bf z$, where ${\bf z} = \sum_i (b_i {\bf q}_i)$ for $i=1,2,\ldots,k$. The $b_i$ are obtained from solving the tridiagonal equation in (v), and the ${\bf q}_i$ are the Lanczos vectors. The vector $z$ is a full-state vector and be used as an initial condition for the tangent linear model in step (vii).</li>
	<li>Finally, we run the tangent linear model from $t=[t_0,t_1]$ using $\bf z$ from (vi) as the initial conditions. During this run of the tangent linear model, we need to read and process the observations $\bf y$ that we used in the [[IS4DVAR]] of step 1 and write the solution at the observation points and times to the [[MODname]] NetCDF file. The values that we write into this [[MODname]] are actually the values multiplied by error covariance $-{\bf O}^{-1}$ assigned to each observation during the IS4DVAR in step (i).</li>		<li>Finally, we run the tangent linear model from $t=[t_0,t_1]$ using $\bf z$ from (vi) as the initial conditions. During this run of the tangent linear model, we need to read and process the observations $\bf y$ that we used in the [[IS4DVAR]] of step 1 and write the solution at the observation points and times to the [[MODname]] NetCDF file. The values that we write into this [[MODname]] are actually the values multiplied by error covariance $-{\bf O}^{-1}$ assigned to each observation during the IS4DVAR in step (i).</li>
	</ol>		</ol>
	</wikitex>		</wikitex>