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APPENDIX C. Exercises
The number of stars indicate roughly the degree of difficulty.
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(i) Prove equation (A4) giving  if  is optimal. |
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(ii) Prove directly the equations
given in the section on the scalar case. |
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(iii) Prove the theorem on preconditioning,
including the case where the square root of  is used. Does the condition number depend on the choice of square
root matrix? |
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(iv) Compare the BLUE equations
with the linear regression equations between the model and observation
values. |
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(v) Write and comment on the BLUE
analysis in a one-dimensional model, with one and then with 2 observations. |
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(vi) rewrite the KF equations
in the scalar case and examine its convergence in time if the model
is the identity and if  and  are constant. |
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(vii) Calculate the product of
a vector with the Hessian using the simulator operator only. |
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(viii) The SWM (Sherley-Woodbury-Morrisson)
approximation of a positive definite matrix is  ( are positive scalars,  are vectors). Prove that it is positive definite, and derive
its inverse and a symmetric square root. |
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(ix) * Calculate the normalization
factors to define properly the Gaussian pdfs for the background and
the analysis states. |
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(x) Write the algorithm to implement
a Cressman analysis. What happens if the observing network is very
dense? |
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(xi) a primitive analysis technique
is to fit a set of polynomials to the observations. Derive the algorithm
in a one-dimensional framework. |
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(xii) * Generalize the polynomial
fit technique to give different weights to different observations. |
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(xiii) Prove that  . Is it a sufficient
condition for the covariance matrix to be positive definite? |
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(xiv) Prove that a covariance
matrix can be factorized in the form  and describe
some numerical methods to do it. |
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(xv) * Give examples in which
the adjoint is not the inverse, and examples in which it is. |
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(xvi) * Derive in the scalar case
what is the analysis error if the weight is calculated using an assumed
 that is not the genuine background standard error. |
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(xvii) * Prove that the background
error covariance matrix can be factorized as  where  is a diagonal matrix and  is the correlation matrix. What is the physical meaning of  ? |
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(xviii) * Rewrite the 4D-Var algorithm
using the inverse of the model (assuming it exists), putting the analysis
time at the end of the time interval. |
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(xix) * (physics regularization)
In the scalar case, considering the observation operator  , design a continuously differentiable observation operator
 with a tunable "regularization" parameter
so that  can be
as small as required and  outside
a small interval around zero. |
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(xx) ** Design a scalar example
using the previous observation operator, in which the cost-function
has one or two minima, depending on the value of the regularization
parameter. |
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(xxi) * Prove that the scalar
KF, with the model equal to the identity and constant error statistics,
is equivalent to a running average that is defined, in the limit of
a continuous time variable, by an exponential weighting function.
How does the e-folding time depend on the error statistics? |
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(xxii) * (adaptive filter) Rewrite
the KF equation as an adaptive statistical adaptation scheme:  , where the model state is the two scalars  and  is the scalar observation,  is an externally defined function of time. The forecast model
is assumed to be the identity. |
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(xxiii) ** Generalize the Cressman
algorithm in order to retain some background information at the analysis
points, as in the least-squares analysis. |
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(xxiv) ** (retrieval and super-obing)
Modify the BLUE equation for when the observations are replaced by
a linear combination of them through a retrieval algorithm  , i.e.  . |
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(xxv) ** Precondition the PSAS
cost function with the symmetric square root of  and prove that
the condition number is then the same as 3D-Var preconditioned by
the symmetric square root of  . |
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(xxvi) *** (control variable remapping)
In a continuous one-dimensional model, derive the adjoint of the "remapping"
operator  where  is the space coordinate and  is an
invertible, continuously differentiable function. Does this make sense
in a discrete model? |
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(xxvii) *** Derive the 4D-Var
equations by expressing the minimization problem constrained by the
model equations with its Lagrangian, and comment on the physical meaning
of the Lagrange multiplier at the analysis point. |
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(xxviii) ** (flow-dependency in
4D-Var) Derive the Hessian of a 4D-Var in which there is one single
observation at the end of the analysis interval. How does the analysis
increment compare with the singular vectors of the model? (see the
training course on predictability) |
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(xxix) *** The NMC method assumes
that the covariances of forecast error differences (differences between
two forecasts starting from 2 consecutive analyses and valid at the
same time) are similar to the forecast error covariances. Formulate
this using the KF notation and discuss the validity of the assumption. |
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(xxx) *** (lagged innovation covariances)
Assuming that the observing network is always the same in the KF,
prove that if the analysis weight is optimal, then the innovation
departures are not correlated in time. |
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(xxxi) *** (fixed-lag Kalman smoother)
Derive the equations for the 1-lag Kalman smoother, i.e. a generalization
of the KF equations in which the observations at both times of the
current analysis and of the next one are used at each analysis step.
Tip: extend the KF control variable to include the model state at
both analysis times. |
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