Verification of the Bayesian merging implementation

RAINFALL ESTIMATION 'AND DATA MERGING ALGORITHMS

3.3 An algorithm for accumulating spatial rainfall fields measured instantaneously and intermittently

3.4.2 Verification of the Bayesian merging implementation

-

I ^I

t

¹

-

- - Latticepoint

km

1 km

Fig. 3.2 1: A Schematicofthelattice showing thepositionsof the gauge measure- ment points.

and accumulate themerged field swhen using thisalgorithm.

In the following twosections, Todini's (200 I)experiment isreplicated(section 3.4.2), then repeated (section 3.4.3) using moving windows of single pixel size, for comparison .

81

Compute the matrix B where BB?'isdefined by

r ll

isthe covariance matrixamongstthe latticepointsand iscomputed from equa- tion 3.14 usingan isotropicGaussian covariance function (e.g. Cressie, 1991).

(3.14)

where a²isthe simulated field variance,p isthe nugget,wthesill,atherangeand hthe distancebetweenpoints.

B is computed by singular value decomposition (SVD), which states (e.g.

Pressetal., 1992) thatany matrixA maybedecomposed in the followingway:

A = UWyT

if we define A = BB^T,then it followsfrom equation 3.15 that

B = UW4yT

We can easilyverify by multiplication that BB^T

=

A asrequired

(3. 15)

since U = Y and V?'= y -I if A is a square symmetric matrix. The simulated rainfall fieldYtis computed from the49-element vector of (0,1) random noise

~t bypremultiplingby B to imposethe correlationstructure.

The gauge measurements are considered unaffectedby measurement errors, but a noise field (with a different covariance structure) is generated on the lattice and added to the observations to produce a simulation of radar measurements. The

50

10 20 30 40

Latticepoint

">_11I

"

_'E

IQ

"

_S^c::

I/)

50

10 20 30 40

Latticepoint

Fig. 3.22: Statistics of the simulated rainfall field Yt generated on the lattice points. Since the lattice pointsare numbered row-wise,the apparent cyclic struc- ture is due to thespatial correlationstructure imposed for each realization.

noisefield isgenerated in thesame wayasthe"observed" field,but with different variogram parameters (the parameters for both casesareshown in table3.2).The Mea n Variance Nugget Sill Range

(J.L) ^(a2) (P) (w)

(a)

Rainfall field parameters 0 10000 0 10000 10⁷ Radar noisefield param eters 40 3000 0 3000 10⁶

Tab. 3.2:The Gaussian Variogram parameters used for the numerical experiment.

statistics of the simulated rainfall field and the noisy radar measurement of the true field aresummarized in figure s3.22 and 3.23. Figure 3.22shows the average mean and standard deviation for each pixel on the lattice computed over 2000 realizations of the simulated rainfall field Yt. The apparent cyclic behaviour in the means is due to the consistentspatialcorrelation structure imposed for each realization. Figure 3.23 shows the mean and standard deviations for the noise affected radar observations

y f

of the simulatedrainfall field.

Figure3.24, shows a single reali zation of a pure Gaussian noise field, a spa- tially correlatedsimulated rainfall fieldand a contaminated "radar" field respec- tively,for visualization purposes.

The coding and implementation of the Bayesian merging algorithm wascare- fully checked forerror. ,prior to its use.The first check was of the Kalman filter-

83

80

10 20 30 40

Lattice point

.§100

ftj"> 80 CIJ

"C 60

'E

..

"C

:lc:

Ul

50

10 20 30 40

Lattice point

8O~---, 40

c:³⁰

..

_CIJ

~ 20

Fig.3.23: Statistics ofthe noise affected radar obse rvatio ns

y r

^ofthe simulated rainfallfield. As noted for figure 3.22, the apparentcyclic structure isdue to the consistently imposed spatial structure.

Gaussianrandom field Correlated random field

2O: : ^L 11 : \\ \ ,

Contaminated field

Fig. 3.24: A single realization of a spatially uncorrelated gaussianrandom field, the simulated rainfall field (spatially correla ted) and the simulated radar rainfall field(contaminated field).

ing routine on its own. The simulatedYt was substituted in place of the Kriged gauge field

yf.

In this way the efficiency of the Kalman filter in reducing the a posteriori estimationerror

(yf -

y~/)was tested.

Figure3.25shows a planview of the simulated rainfall fieldYe(top left) and a noisy radar estimate

y f

(top right) of the true field for a single realization on the lattice. The figurealsoshows the priory~ (bottom left) and posterior

Y Z

^(bottom

right)estimatesofYI.The priorfield isobtained from equation3.8 by subtracti ng

J-lE:~' the mean of theradar noise, from the radarfield

y f.

^{The plots}in figure 3.25

.' OG-I50

-"00

<I)

.. -

...

^~

D-1~1[J)

D- '<1)

..2tD-203 ...3OJ-Hl

~

~.'50-201)

.HIO·I50 - ..00

-<I)

... -

..

^,~

Q.IS)- IOO a-2m-ISJ ..251-203 3OJ-Hl

Fig. 3.25: Verification of algorithm implementation: Example fields representing asingle time realization, with the Krigedgauge field replaced by the true field.

indicate that a large part of the bias is removed by subtraction of the radar error mean, to produce the prior estimate of the field. The application of the Kalman filter removes all traces of bias and also reduces the standard deviation between the posteriorestimateand the true field to zero, as itshould do in thisea e, where thestatistics of theettime seriesare know a priori and an"error free"observation of the rainfall field is available.

Asecond check wascarried out to verify that the Kriging portion of the code did not contain any errors. In this case the set of gauge

x f

^{measurements was}

replaced with the 49 simulated rainfall fieldvalues on the7 x 7lattice.The Krig- ing procedure was employed to Krige these values back onto the lattice points.

Evidentlythe original fieldshould be recovered exactly,as the matrix ofKriging weightsA mustbecome the identity matrix in thiscase.Thiswasachieved and for thesingle realization of the two field compared in figure 3.26,there is no visible

02"'''''

0200=

.' 50-200

.'00·'SO

o~tOO

'"

8·SO<l a·' CO-SO o·HiO_t OO 0·200-, 50 .·~200 a-300- 250

True field

025o..Dl OlOOJOO at SG-Dl .' 1)0.150 QSJ·t(D all-Sll 8.s04

.·,oo-SO

0,'50-'00 0·200-' 50 .·250-200 a-300- 250

85

Fig. 3.26: Verification of Kriging algorithm implementation: Example fields rep- resent a single time realization and

x f

was set to be the "true" simulated rainfall field.

difference between them.

The mean and standard deviation for the Kriged residuals (the difference be- tween Ytand the Kriged gauge field

y f)

were also computed for 2000 applications of the Kriging routine. The resulting mean over 2000 realizations on the lattice has a maximum standard deviation, about a zero mean, of 2x 10-⁶over the 49 lat- tice points. The negligible errors of estimation can be attributed to floating point precision errors in the computation, therefore there is a high degree of confidence that the code performs as designed by Todini (200 I).

Application of Bayesian merging on the latt ice

Having carefully checked the coding of the algorithm, the Bayesian merging tech- nique was then applied on the 7x 7 lattice described earlier, the mean and covari- ance structure of the residuals were measured a priori. The time series of residuals etbetween the radar estimate of the true field

yf

and the Kriged field estimate

y f

was examined and the mean /-Let and covarianceVetcomputed from these residu- als.The Kalman filter equations were then applied with

: llllllllllllll III mnrfUIlH ¹¹ I ¹¹¹¹ llllllll ^{: ;::1}

3000 2000 1000

oLaJ...ca

comoartson ofvariance

_ radar

I

_ prior

40 20

o

comcartscn of means

llllllllllllll lllllllllllllllllll IIll llll IIll

11 _

_{_} ^radar_posterior

I

Fig.3.27: Comparisonofstatistics of residual s.The entire lattice was used in the calculatio nofy~/. ote the significant reduction in the posteriorerrorvariance.

since the Kriged estimate

y f

is assumed unbiased . The known covariance struc- ture (see table 3.2) was used for VeG. The re ults for 2000 realizations on the

I

lattice are summarized in figures 3.27and 3.28. From the e figures it is shown that the mean errors becom e close to zero and that the variance of their errors shows a significantreduction in magnitude. Figures 3.27 and 3.28 qualitatively match (the random number sequences are different ) those presented by Todini (200 I) very well,verify ingandcorroborating the impleme ntation oftheBayesian mergingtechniqu e.