Roberto Buizza

European Centre for Medium-Range Weather Forecasts

(Extract from Buizza 2001: Chaos and weather prediction, Il Nuovo Cimento C, 24, 273-302)

• Singular vector definition
• Projection operators
• References and related publications

Singular vector definition

Farrell (1982), studying the growth of perturbations in baroclinic flows, showed that, although the long time asymptotic behavior is dominated by discrete exponentially growing normal modes when they exist, physically realistic perturbations could present, for some finite time intervals, amplification rates greater than the most unstable normal mode amplification rate. Subsequently, Farrell (1988, 1989) showed that perturbations with the fastest growth over a finite time interval could be identified solving the eigenvalue problem of the product of the tangent forward and adjoint model propagators. His results supported earlier conclusions by Lorenz (1965) that perturbation growth in realistic models is related to the eigenvalues of the operator product.

Kontarev (1980) and Hall and Cacuci (1983) first used the adjoint of a dynamical model for sensitivity studies. Later on, Le Dimet & Talagrand (1986) proposed an algorithm, based on an appropriate use of an adjoint dynamical equation, for solving constraint minimization problems in the context of analysis and assimilation of meteorological observations. More recently, Lacarra & Talagrand (1988) applied the adjoint technique to determine optimal perturbations using a simple numerical model. Following Urban (1985) they used a Lanczos algorithm (Strang, 1986) in order to solve the related eigenvalue problem. For a bibliography in chronological order of published works in meteorology dealing with adjoints up to the end of 1992, the reader is referred to Courtier et al. (1993).

After Farrell and Lorenz, calculations of perturbations growing over finite-time intervals were performed, for example, by Borges & Hartmann (1992) using a barotropic model, and by Molteni & Palmer (1993) using a barotropic and a 3-level quasi-geostrophic model at spectral triangular truncation T21. Buizza (1992) and Buizza et al. (1993) first identified singular vectors in a primitive equation model with a large number of degrees of freedom.

Let c(t) be the state vector of a generic autonomous system, whose evolution equations can be formally written as

(Eq. 1)

Denote by c(t) an integration of Eq. 1 from t₀ to t which generates a trajectory from an initial point c₀ to c₁=c(t). The time evolution of a small perturbation x around the time evolving trajectory c(t) can be described, in a first approximation, by the linearized model equations

(Eq. 2)

where A_l is the tangent operator computed at the trajectory point c(t).

Let L(t,t₀) be the integral forward propagator of the dynamical Eq. 2 linearized about a non-linear trajectory c(t)

(Eq. 3)

that maps a perturbation x at initial time t₀ to the optimization time t. The tangent forward operator L maps the tangent space P₀, the linear vector space of perturbations at c₀, to P₀, the linear vector space at c₁.

Consider two perturbations x and y, e.g. at c₀, a positive definite Hermitian matrix E, and define the inner product (..;..)_E as

(Eq. 4)

on the tangent space P₀ in this case, where á ..;..ñ identifies the canonical Euclidean scalar product,

(Eq. 5)

Let ||..||_E be the norm associated with the inner product (..;..)_E ,

(Eq. 6)

Let L^*E be the adjoint of L with respect to the inner product (..;..)_E ,

(Eq. 7)

The adjoint of L with respect to the inner product defined by E can be written in terms of the adjoint L^* defined with respect to the canonical Euclidean scalar product,

(Eq. 8)

From Eqs. 8 and 3 it follows that the squared norm of a perturbation x at time t is given by

(Eq. 9)

Equation 9 shows that the problem of finding the phase space directions x for which ||x(t)||²/||x(t)||² is maximum can be reduced to the search of the eigenvectors v_I(t₀)

(Eq. 10)

with the largest eigenvalues s_i².

The square roots of the eigenvalues, s_i, are called the singular values and the eigenvectors v_i(t₀) the (right) singular vectors of L with respect to the inner product E (see, e.g., Noble & Daniel, 1977). The singular vectors with largest singular values identify the directions characterized by maximum growth. The time interval t-t₀ is called optimization time interval.

Unlike L itself, the operator L^*EL is normal. Hence, its eigenvectors v_I(t₀) can be chosen to form a complete orthonormal basis in the N-th dimensional tangent space of the perturbations at c ₀. Moreover, the eigenvalues are real, s_i^2³ 0.

At optimization time t, the singular vectors evolve to

(Eq. 11)

which in turn satisfy the eigenvector equation

(Eq. 12)

From Eqs. 9 and 12 it follows that

(Eq. 13)

Since any perturbation x(t)/||x(t₀)||_E can be written as a linear combination of the singular vectors v_I(t), it follows that

(Eq. 14)

Thus, maximum growth as measured by the norm ||..||_E is associated with the dominant singular vector v₁ .

Given the tangent forward propagator L, it is evident from Eq. 10 that singular vectors' characteristics depend strongly on the inner product definition and to the specification of the optimization time interval.

The problem can be generalized by selecting a different inner product at initial and optimization time. Consider two inner products defined by the (positive definite Hermitian) matrices E₀ and E, and re-state the problem as finding the phase space directions x for which

(Eq. 15)

is maximum. Equation 15 can be transformed into

(Eq. 16)

Since

(Eq. 17)

the phase space directions which maximize the ratio in Eq. 16 are the singular vectors of the operator E^-1/2LE₀^-1/2 with respect to the canonical Euclidean inner product. With this definition, the dependence of the singular vectors' characteristics on the inner products is made explicit.

At ECMWF, due to the very large dimension of the system, the eigenvalue problem that defines the singular vectors is solved by applying a Lanczos code (Golub & Van Loan 1983).

The set of differential equations that defines the system evolution can be solved numerically with different methods. For example, they can be solved with spectral methods, by expanding a state vector onto a suitable basis of functions, or with finite-difference methods in which the derivatives in the differential equation of motions are replaced by finite difference approximations at a discrete set of grid points in space. The ECMWF primitive equation model solves the system evolution equations partly in spectral space, and partly in grid point space.

Denote by x_g the grid point representation of the state vector x, by S the spectral-to-grid point transformation operator, x_g=Sx, and by Gx_g the multiplication of the vector x_g, defined in grid point space, by the function g(s):

(Eq. 18)

where s defines the coordinate of a grid point, and S is a geographical region.

Define the function w(n) in spectral space as

(Eq. 19)

where n identifies a wave number and N is a sub-space of the spectral space.

Consider a vector x. The application of the local projection operator T defined as

(Eq. 20)

to the vector x sets the vector x to zero for all grid points outside the geographical region S. Similarly, the application of the spectral projection operator W to the vector x sets to zero its spectral components with wave number outside N.

The projection operators T and W can be used either at initial or at final time, or at both times. As an example, these operators can be used to formulate the following problem: find the perturbations with (i) the fastest growth during the time interval t-t₀, (ii) unitary E₀-norm and wave components belonging to N₀ at initial time, (iii) maximum E-norm inside the geographical region S and wave components belonging to N₁ at optimization time. This problem can be solved by the computation of the singular values of the operator

(Eq. 21)

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