164
7 Integration of Functionals, PCM and Stochastic IntegralEquations
that the exponentials in the bottom two equations tend to unity, and the expressions
within the parentheses in all three equations tend to the same approximate value of
−(1/2)(δ/L) 2 , as can be verified by expanding each expression in a Taylor series.
This means that in the limit of small δ/L, G mn = δ 2 for all −N ≤ m, n ≤ N .
Hence, this matrix is singular, and has a constant (normalized) eigenvector, v n =
1/
√
2N for all −N ≤ n ≤ N, and a single nonzero eigenvalue, λ 2 = 2Nδ = 2b,
where 2b is the total length of the anomalous region. Thus, this process is fully
correlated, and the expansion, (7.16), has only a single term with just one random
variable. From the perspective of (7.9) the operator equation on the left-hand side is
a simple integrator, which means that the eigenfunction must be a constant.
On the other hand, in the limit of vanishing correlation length, for which δ/L
become very large, we have the opposite situation. In this case the off-diagonal
terms in (7.20) vanish to exponential order, and we are left with a diagonal matrix
that is a multiple of the identity matrix. Actually, the expression for G mm in (7.20)
vanishes as L → 0, because the double-exponential covariance function vanishes
everywhere except at x = x , where it is assigned the value of unity. The integral of a
function that vanishes everywhere except at one point is zero, unless at that one point
the function becomes ‘infinite’ in some manner. If we normalize the covariance by
dividing by 2L, then as can be seen from (7.20) the diagonal elements all take on
the value of δ as L → 0.
Thus, there are 2N identical eigenvalues, and 2N orthogonal eigenvectors that
can be chosen arbitrarily. The expansion, (7.16), therefore, has the full number of
terms with uncorrelated random variables; there will be no decay of the eigenvalues
with n. This is an example of an uncorrelated process, that we could call ‘discrete
white noise’; each voxel in the VIC-3D® grid is uncorrelated with each other voxel.
This is essentially the way that we are currently treating random problems with
VIC-3D®.
Figure 7.2 shows the normalized eigenvalue spectrum for a 32 × 32 matrix (7.20)
for three values of the ratio, L/δ. Clearly, our surmise is confirmed: the spectrum
dies out faster for larger values of this ratio, and this will have a profound effect on
the profile of conductivity, as will be discussed next.
In the preceding discussion of the eigenvalue spectrum, and in what follows,
we assume a one-dimensional grid that is 6.4 mm long and contains 32 cells, each
of length δ = 0.2 mm. Further, we assume that the correlation function shown
in Fig. 7.1 is multiplied by a variance, σ 2 , to produce the covariance function
associated with the random surface.
The random conductivity profiles to be shown next are computed from (7.16),
with {ξ n } being independent, identically distributed, zero-mean, unit-variance random variables with a uniform density function. These are generated using the
Fortran RANDOM_NUMBER subroutine, as described in Appendix 2. The profile
shows the (uniform) conductivity of each cell, which is the basis for defining anomalous regions in VIC-3D®. Sample functions for the case, L/δ = 0.1, are shown in
Fig. 7.3, while Fig. 7.4 shows three sample functions for the conductivity profile
when L/δ = 1, and Fig. 7.5 shows three sample functions for the conductivity
profile when L/δ = 3.
7 Integration of Functionals, PCM and Stochastic IntegralEquations
that the exponentials in the bottom two equations tend to unity, and the expressions
within the parentheses in all three equations tend to the same approximate value of
−(1/2)(δ/L) 2 , as can be verified by expanding each expression in a Taylor series.
This means that in the limit of small δ/L, G mn = δ 2 for all −N ≤ m, n ≤ N .
Hence, this matrix is singular, and has a constant (normalized) eigenvector, v n =
1/
√
2N for all −N ≤ n ≤ N, and a single nonzero eigenvalue, λ 2 = 2Nδ = 2b,
where 2b is the total length of the anomalous region. Thus, this process is fully
correlated, and the expansion, (7.16), has only a single term with just one random
variable. From the perspective of (7.9) the operator equation on the left-hand side is
a simple integrator, which means that the eigenfunction must be a constant.
On the other hand, in the limit of vanishing correlation length, for which δ/L
become very large, we have the opposite situation. In this case the off-diagonal
terms in (7.20) vanish to exponential order, and we are left with a diagonal matrix
that is a multiple of the identity matrix. Actually, the expression for G mm in (7.20)
vanishes as L → 0, because the double-exponential covariance function vanishes
everywhere except at x = x , where it is assigned the value of unity. The integral of a
function that vanishes everywhere except at one point is zero, unless at that one point
the function becomes ‘infinite’ in some manner. If we normalize the covariance by
dividing by 2L, then as can be seen from (7.20) the diagonal elements all take on
the value of δ as L → 0.
Thus, there are 2N identical eigenvalues, and 2N orthogonal eigenvectors that
can be chosen arbitrarily. The expansion, (7.16), therefore, has the full number of
terms with uncorrelated random variables; there will be no decay of the eigenvalues
with n. This is an example of an uncorrelated process, that we could call ‘discrete
white noise’; each voxel in the VIC-3D® grid is uncorrelated with each other voxel.
This is essentially the way that we are currently treating random problems with
VIC-3D®.
Figure 7.2 shows the normalized eigenvalue spectrum for a 32 × 32 matrix (7.20)
for three values of the ratio, L/δ. Clearly, our surmise is confirmed: the spectrum
dies out faster for larger values of this ratio, and this will have a profound effect on
the profile of conductivity, as will be discussed next.
In the preceding discussion of the eigenvalue spectrum, and in what follows,
we assume a one-dimensional grid that is 6.4 mm long and contains 32 cells, each
of length δ = 0.2 mm. Further, we assume that the correlation function shown
in Fig. 7.1 is multiplied by a variance, σ 2 , to produce the covariance function
associated with the random surface.
The random conductivity profiles to be shown next are computed from (7.16),
with {ξ n } being independent, identically distributed, zero-mean, unit-variance random variables with a uniform density function. These are generated using the
Fortran RANDOM_NUMBER subroutine, as described in Appendix 2. The profile
shows the (uniform) conductivity of each cell, which is the basis for defining anomalous regions in VIC-3D®. Sample functions for the case, L/δ = 0.1, are shown in
Fig. 7.3, while Fig. 7.4 shows three sample functions for the conductivity profile
when L/δ = 1, and Fig. 7.5 shows three sample functions for the conductivity
profile when L/δ = 3.