Inverse Problems in the Atmosphere and its Applications

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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_at_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

• Part A Theoretical aspects
• A.1 Whats 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

• 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

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A.1 Whats the variational data assimilation?

• 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

• Stage 1 Objective Analyses
  Interpolating 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

• Adjusting initial field x0 so that x0 is
  compatible with observations y and 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

• 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

• 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 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

• 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
iteration
Approximate solutions
convergence
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Some key difficulties of adjoint method of VAR

• (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
• 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

• 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

• 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

• 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

• 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 J with
iteration number
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B.2 ENSO cycle and parameters inversion

• ENSO
• The acronym of the El Nino -Southern
  Oscillation phenomenon 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
• 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 Nina

• 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

• 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
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 T (solid line) and h (dotted
line)
The phase orbit of T and h (Running clockwise
as the time goes on)
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• Now, we seek optimal parameter and external
  forcing , such that the solution
  satisfies
• the
  terminal control term
• the
  control parameter.

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Blue the observed valuered 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

• 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 dont include the Coriolis
  force . Suppose that over the interval, the
  observational TC track is
  .

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• Now, the goal is to determine the optimal
  initial velocity
• and forces , such that the
  corresponding solution
• makes the functional
• minimal. are referred to as the
  regularization parameters,
• is the restraint parameter at the
  terminal.

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Table1. Main Characteristics of 4 TCs
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Inversion of TC 9804 track
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Retrieved forces for TC 9804
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B.4 Inversion of Radar
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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 radarthe 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 forming process of radial velocity
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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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• Suppose that the observational data
  are known, the aim is to determine 2-D wind
  and . This is a ill-posed problem. We
  introduce the following functional
• where ? ? ? and are weight coefficients.

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true true vortex wind field
retrieved retrieved vortex wind field
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The error between the retrieved
vortex wind field and true wind field
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B.5 Inversion of satellite remote sensing
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
• 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
• Step4. Obtain the gradients of the functional

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Experiments
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• The track of the cost functional descending in
  the process of iteration

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B.7 Variational Adjustment of 3-D Wind Field

• 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
  Sasakis 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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• Suppose that is an observational
  horizontal wind field. Our aim is to seek an
  analytic field satisfying the
  equation of continuity
• and make the functional
• minimal. Here
  and satisfies the boundary
  conditions

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(b)
(a)
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(a)
(b)
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(a)
(b)
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(a)
(b)
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B.8 The model of GPS Dropsonde wind-finding
system
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Introduction to Vaisala Dropsonde RD93
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Now we can get the following adjoint equations
and initial boundary conditions
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And its boundary conditions are
the gradients of J
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(a)x-axis position
Fig. the position of dropsonde
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(b) y-axis position
Fig the position of dropsonde
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(c) z-axis position
Fig. the position of dropsonde
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Fig. the comparison between two kinds of cost
functional decrease
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(a) 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) 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 ,
Fig. the comparison between the true value,
initial value and retrieval value of in the
z-axis
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(b) 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) 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 ,
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 ,
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 ,
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 ,
Fig. the comparison between the true
value,initial value and retrieval value of
updraft flow
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(b) Compared without stabilized fuction ,
Fig. the comparison between the true
value,initial value and retrieval value of
updraft flow
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