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Theoretical Analyses and Numerical Tests of Variational Data Assimilation with Regularization Methods. Huang Sixun P.O.Box 003, Nanjing 211101,P.R.China Email: huangsxp@yahoo.com.cn. Canada-China Workshop on Industrial Mathematics HongKong Baptist University, 2005.

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Theoretical Analyses and Numerical Tests of Variational Data Assimilation with Regularization Methods

Huang Sixun

P.O.Box 003, Nanjing 211101,P.R.China

Email: huangsxp@yahoo.com.cn

Canada-China Workshop on Industrial Mathematics

HongKong Baptist University, 2005


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It is well known that numerical prediction of atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc. VAR is using all the available information (e.g., observational data from satellites, radars, and GPS, etc.) to determine as accurately as possible the state of the atmospheric or oceanic flow.


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Contents atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

Part ATheoretical aspects

  • A.1 What’s the variational data assimilation?

  • A.2 Idea of adjoint method of VAR

  • A.3 3D-VAR

  • A.4 4-D VAR


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Part B Applications atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

  • B.1 variational assimilation for one-dimensional ocean temperature model

  • B.2 ENSO cycle and parameters inversion

  • B.3 Assimilation of tropical cyclone(TC) tracks

  • B.4 Inversion of radar

  • B.5 Inversion of satellite remote sensing data and its numerical calculation

  • B.6 Generalized variational data assimilation with non- differential term

  • B.7 Variational adjustment of 3-D wind field

  • B.8 The model of GPS dropsonde wind-finding system


A 1 what s the variational data assimilation l.jpg
A.1 What’s the variational data assimilation? atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

Talagrand 1995

Assimilation: using all the available information, determine as accurately as possible the state of the atmospheric or oceanic flow

Variational Data Assimilation: study assimilation through variational analytical method(adjoint method)


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Data assimilation undergoes the following stages atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

Stage 1 Objective AnalysesInterpolating observational data at irregular observational points to regular grid points by statistical methods, which would be taken as initial fields

Stage 2 Initialization

Filtering high frequency components in initial fields so as to reduce prediction errors


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Stage 3 3D –VAR atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

Adjusting initial field x0so that x0is compatible with observations yand background xb , i.e. to make the following cost function minimum

H----observation operator( nonlinear operator)

y---observational field

xb- --- background field

B---covariance matrix of background

O---covariance matrix of observation


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Stage 4 4D-VAR atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

  • Case 1

    State equations

    F is the classical PDO

    Observation Xobs [0,T]

    Cost functional

    C---linear operator

    It means that gives the “true value of the field at the point (in space and /or in time) of observation

    This is optimal control of PDEs


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Case 2 atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

  • Model

    w(t) is assumed to have 0 mean and covariance matrix error Q(t)

  • information

    background fields xb

    covariance matrix of

    background error


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observational data atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc. y

e(t) is assumed to have 0 mean and covariance matrix O(t). e(t) is white process, and also assumed to be uncorrelated with the model error w(t).

  • cost functional


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A.2 Idea of adjoint method of VAR atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

  • As an example, we consider the inversion of IBVC for the following problem

    ----- observational data

    the cost functional is


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Idea: solving an optimization problem by descent algorithm atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

iteration

Approximate solutions

convergence


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Some key difficulties of adjoint method of VAR atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

(1) Ill-posedness

During iteration, the cost functional oscillates, and decreases slowly so as to lead too low accuracy. The reason: ill-posedness

(2) Error of BVC


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(3) Local observations atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

In some cases, especially in the oceans, observations are not incomplete, e.g., observations are obtained from ships, sounding balloons, which will lead to calculation unstable, and therefore is worth studying further.

(4) Variational data assimilation with non-differentiable term (on-off problem)

The adjoint method holds only with differentiable term; for systems containing non-differentiable physical processes( called as “on-off” ) , a new method must be developed.


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A3 3D -VAR atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

If H is linear operator , we obtain the optimal estimate

And the error estimate matrix is


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Some key difficulties in 3D-VAR atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

  • H is an on observational operator

    Prob.1: How to find H ?

    Prob.2: H is not a surjection. How to deal with it ?

  • B is non-positive

  • O is non-positive

  • The hypothesis of unbiased errors is a difficult one in practice, because there often

    as significent biases in the background fields(caused by biases in the forecast

    model) and in the observations ( or in the observational operators)

  • The hypothesis of uncorrelated errors

  • H is a nonlinear operator, which leads to J = min! is not unique, i.e. ill-posedness


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A4 4D-VAR atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc.

  • Model


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If we suppose ,then the direct equations and adjoint equations are not coupled, except at the initial time t0


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Part B Applications adjoint equations are not coupled, except at the initial time t

  • B.1 variational assimilation for one-dimensional ocean temperature model

  • B.2 ENSO cycle and parameters inversion

  • B.3 Assimilation of tropical cyclone(TC) tracks

  • B.4 Inversion of Radar

  • B.5 Inversion of satellite remote sensing data and its numerical calculation

  • B.6 Generalized Variational Data Assimilation for Non- Differential System

  • B.7 Variational Adjustment of 3-D Wind Field

  • B.8 The model of GPS Dropsonde wind-finding system


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B.1 variational assimilation for one-dimensional ocean temperature model

  • The one-dimensional heat-diffusion model for describing the vertical distribution of sea temperature over time is,

  • Here is sea temperature, is the vertical eddy diffusion coefficient,

    is the sea water density, is the sea water specific heat capacity, is the light diffusion coefficient, is the depth of ocean upper layer, is the transmission component of solar radiation at sea surface, is the net heat flux at sea surface. It is known that there exists the unique solution of the model if the initial boundary condition and the model parameters are known and smooth.


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  • Assume , are known constants, the initial boundary conditions , and model parameters , are not known exactly, e.g., they have unknown errors and need to be determined by data assimilation. Now a set of observations of sea temperature is given on the whole domain. A convenient cost functional formulation is thus defined as

    Where is a stable functional and is a regularization parameter. The problem is: Find the optimal initial boundary conditions and model parameters

    , such that J is minimal.


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Decreasing of the cost functional initial boundary conditions , and model parameters , are not known exactly, e.g., they have unknown errors and need to be determined by data assimilation. Now a set of observations of sea temperature is given on the whole domain. A convenient cost functional formulation is thus defined asJ with iteration number


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B.2 ENSO cycle and parameters inversion initial boundary conditions , and model parameters , are not known exactly, e.g., they have unknown errors and need to be determined by data assimilation. Now a set of observations of sea temperature is given on the whole domain. A convenient cost functional formulation is thus defined as

  • ENSO:

    The acronym of theEl Nino -Southern Oscillationphenomenon which is the most prominent international oscillation of the tropical climate system.


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  • The phase of the Southern Oscillation on El Nino initial boundary conditions , and model parameters , are not known exactly, e.g., they have unknown errors and need to be determined by data assimilation. Now a set of observations of sea temperature is given on the whole domain. A convenient cost functional formulation is thus defined as

    • High temperature over eastern Pacific; High surface pressure over the western and low surface pressure over the south-eastern tropical Pacific coincide with heavy rainfall, unusually warm surface waters, and relaxed trade winds in the central and eastern tropical pacific


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La initial boundary conditions , and model parameters , are not known exactly, e.g., they have unknown errors and need to be determined by data assimilation. Now a set of observations of sea temperature is given on the whole domain. A convenient cost functional formulation is thus defined asNina

  • The phase of the Southern Oscillation on La Nina

    • Surface pressure is high over the eastern but low over the western tropical Pacific, while trades are intense and the sea surface temperature and rainfall are low in the central and eastern tropical Pacific


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A nonlinear dynamical system for ENSO: initial boundary conditions , and model parameters , are not known exactly, e.g., they have unknown errors and need to be determined by data assimilation. Now a set of observations of sea temperature is given on the whole domain. A convenient cost functional formulation is thus defined as

: Sea Surface Temperature Anomaly (SSTA)

: thermocline depth anomaly

: a monotone function of the air-sea coupling coefficient

: external forcing

: constants .


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Observation: initial boundary conditions , and model parameters , are not known exactly, e.g., they have unknown errors and need to be determined by data assimilation. Now a set of observations of sea temperature is given on the whole domain. A convenient cost functional formulation is thus defined as

Obtain the time series of T and h(denoted by and from the observational data set TAO (Tropical Atmosphere and Oceans)


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The time series of initial boundary conditions , and model parameters , are not known exactly, e.g., they have unknown errors and need to be determined by data assimilation. Now a set of observations of sea temperature is given on the whole domain. A convenient cost functional formulation is thus defined asT (solid line) and h (dotted line);

The phase orbit of T and h (Running clockwise as the time goes on)


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Blue : the observed value , such that the solution satisfies red : the value predicted by the original modelblack: the value predicted by the improved model whengreen: the value predicted by the improved model when


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B. 3 Assimilation of tropical cyclone(TC) tracks , such that the solution satisfies

  • A TC is regarded as a point vortex, whose motion satisfies

  • Here , , are the velocity and coordinates of TC center respectively, and is the force exerted on TC, but don’t include the Coriolis force . Suppose that over the interval, the observational TC track is .


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The Definition of Radar

Radar is an acronym for “Radio Detecting And Ranging”.

Radar systems are widely used in air-traffic control, aircraft navigation, marine navigation and weather forecasting.


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The Definition of Doppler Radar

Doppler radar:the radar can detect both reflectivity intensity and radial velocity of the moving objects with the “Doppler effect”.

The right graphic show:

The forming process of reflectivity



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  • The 2-D horizontal wind is governed by the following conservation of reflectivity factor of Radar and of mass in the polar coordinates

  • where are time , redial distance and azimuth respectively, is the reflectivity factor of Radar, are redial and azimuthal velocity respectively.

    is eddy diffusion coefficient. is given by diagnosis. The inversion domain is


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true the aim is to determine 2-D wind and . This is a ill-posed problem. We introduce the following functional

true vortex wind field

retrieved

retrieved vortex wind field


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The error between the retrieved vortex wind field the aim is to determine 2-D wind and . This is a ill-posed problem. We introduce the following functional and true wind field


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B.5 Inversion of satellite remote sensing the aim is to determine 2-D wind and . This is a ill-posed problem. We introduce the following functional

data and its numerical calculation


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  • With the use of techniques in nonlinear problems, the IDP (improved discrepancy principle) method has been proposed to the optimal smooth factor (parameter ) in the inversion process of atmosphere profiles from satellite observation. This method has also been used to inverse atmospheric parameters from the observation of new generation geostationary operational environmental satellite(GOES-8). Results show that this method is more accurate than that in use.


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If the atmosphere scatter effect is ignored, then the infrared radiance of the earth atmosphere system that goes to satellite sensor is

R---- the spectral radiance of a channel(given)

B ---- Plank function

---- the total atmosphere transmittance above the

pressure level

---- surface emissivity

---- reflected radiation of the sun

---- surface value of physical quantities


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B.6 Generalized Variational Data Assimilation with Non-Differential Term

The simple ordinary differential equation with non-differential term:(Zou X.,1993):

Here is Heaviside function.

Problem:

Supposing the equation has a unique solution and the observation is known, our goal is to find the initial value and critical value that can make functional


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Step1. Introduce a weak form: Non-Differential Term

Step2. The weak form is disturbed as the following :

Here is the time at that time


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Step3. Introduce the adjoint system: Non-Differential Term

Step4. Obtain the gradients of the functional:


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Experiments: Non-Differential Term



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B.7 Variational Adjustment of 3-D Wind Field of iteration

The vertical velocity of an air parcel is a very important quantity in atmospheric sciences. However, its magnitude is so small that it can not be measured accurately by meteorological apparatus, but rather inferred from the fields measured directly, such as the horizontal velocity, temperature , pressure, and so on.

Three commonly used methods for inferring the vertical velocity are the kinematical method, the adiabatic method, and the variational analysis method (VAM) suggested by Sasaki(1969,1970). However, It turns out that Sasaki’s VAM can not adjust 3-D wind field well for observational wind containing high frequency components, even if filtering is applied. Here we combine VAM with the regularization method and filtering to deal with this problem (GVAM).


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(b) field. Our aim is to seek an analytic field satisfying the equation of continuity

(a)


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(a) field. Our aim is to seek an analytic field satisfying the equation of continuity

(b)


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(a) field. Our aim is to seek an analytic field satisfying the equation of continuity

(b)


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(a) field. Our aim is to seek an analytic field satisfying the equation of continuity

(b)


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B.8 The model of field. Our aim is to seek an analytic field satisfying the equation of continuity GPS Dropsonde wind-finding system


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Introduction to Vaisala Dropsonde RD93 field. Our aim is to seek an analytic field satisfying the equation of continuity



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And it boundary conditions’s boundary conditions are::

the gradients of J:


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boundary conditionsa)x-axis position

Fig. the position of dropsonde


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boundary conditionsb) y-axis position

Fig the position of dropsonde


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boundary conditionsc) z-axis position

Fig. the position of dropsonde



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(a) decrease Compared without stabilized fuction ,

Fig. the comparison between the true value, initial value and retrieval value of in the x-axis


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(b) decrease Compared with stabilized fuction ,

Fig. the comparison between the true value, initial value and retrieval value of in the x-axis


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(a) Compared without stabilized function decrease ,

Fig. the comparison between the true value, initial value and retrieval value of in the z-axis


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(b) decrease Compared with stabilized fuction ,

Fig. the comparison between the true value,initial value and retrieval value of in the z-axis


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(a) decrease Compared without stabilized fuction ,

Fig. the comparison between the true value,initial value and retrieval value of wind (x direction)


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(a) Compared with stabilized fuction decrease ,

Fig. the comparison between the true value,initial value and retrieval value of wind (x direction)


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(a) Compared without stabilized fuction decrease ,

Fig. the comparison between the true value,initial value and retrieval value of wind(y direction)


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(b) Compared without stabilized fuction decrease ,

Fig. the comparison between the true value,initial value and retrieval value of wind(y direction)


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(a) Compared without stabilized fuction decrease ,

Fig. the comparison between the true value,initial value and retrieval value of updraft flow


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(b) Compared without stabilized fuction decrease ,

Fig. the comparison between the true value,initial value and retrieval value of updraft flow


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