34;True" field Kriged gauge field
3.6 Comparison and application of merging meth- ods
3.6.1 Ca se study - Compar ing the Bayesian and condi tional merging techniques usin g s imulated rainfall fi elds
cR(S).Ifca(s)andcR(S)are strongly(pos itively) correlated asone would expect since both gauges and radar are measurements of Z(s),then the variance of the error will be significantly less than (3 as suggested in equation 3.22. IfO"cc (s) =
O"cn(s) = 0" and p
>
0.5, then var [Z(s) - M(s)] = 20"2[1- p] showing that thevariance of the error between the merged and real rainfall field will be less than that ofeither field (Kriged gauge or radar estimate) alone. If p = 0 (the errors are uncorrelated ),then the error isthe sum of theerrorsof the individual fieldsas expected.
3.6 Comparison and application of merging meth-
95
algorithmsperformanceinan objective way.
A sequence of 1000 independent 128x128 pixel rainfall fields wasproduced usingthe "String of Beads"rainfall simulation model (Pegramand Clothier, 200 I).
Theserainfall fieldswere treated asthe "true" rainfall field and "observed" radar estimates, produc ed by adding bias and noise. Each pixel in the field represents an areaofsize Ix I km. The "true" field was sampledat 83 "raingauge" location s chose n randomly on the pixel grid. This givesanaverage coverage of one gauge forevery 198km",representing agauge spacing of 14 km,whichisafairly dense but realistic netwo rk (although in South Africa this gauge density is veryseldom realized). The "raingauge" measurements were assumed to be withouterror. A single realization of the "true" fieldaswellasthecorresponding "observed" radar estimate, thelocation ofthe gaugesand the Kriged gaugeestimate of the field are shown in figure3.33, whilea moredetail eddescription of theexperiment follows.
80 60
40 20
o
80 60
40 20
o
Gauge positions
. ' .
100
80 60 40 20
o
Kriged rainfall field
40 20
o
Fig. 3.33: A single realization of the modelled instantaneous rainfall fields (mmfhr) and raingauge locations. Note that the Krigedestimatecapturesthegen- eral structure quite well due to thehigh sampling frequ ency,but failsto represent the finerdetails.
i)Generate 1000 independ entrain fall field susing the "String of Beads" model.
~ o
ii) Add bias and noise to simulateradar measurem ents ofthe "true" rainfallfield.
97
Bias+Noise
...
iii)Sampl ethe "true" rainfallfield (at83randomlocations)togeta setof unbiased and errorfree rain gauge observations.
./~
.....,l'
Rain-gauge locations
Kriging
....,./'.. r=---'-- - ----.,/
v) Apply the merging procedure to estima te the original field on the basisof the
"ob erved"gauge and radar estimates.
vi) Compute the mean error Ej(i = 1 2 """,m) at each pixel over the 1000 real- izations.
whereYi(k )is the "true" value of the rainfall field at pixel i for realization kand 11lj(k )isthe merged estimate of the "true" value of the rainfall field at pixel i for realization k.
vii)Compute the variance of theerrorsat eachpixel overthe 1000 realizations.
1 N
a; = -
L
{Yj(k) - mj(k)- Ej(k )}2k=l
where
a r
isthe variance oferror between the"true" valueof the rainfallfieldYi(k) and11lj(k).the merged estimateof the "true" value at pixel i. Ej(k) is the error at pixeli for realization k.99
viii) Compare the mean and variance of the estimate'sdeviationsfrom the"true"
field, computed from steps vi) and vii) to tho e computed from the radar errors in order to quantify the improvement(relative to the radar estimate)gained from each merging technique.
• I
. "t.:.__.10.~ I~ ~
__;;l.
~
..
It' -
<It • •_ ...
.. .
"I
..
.~...
"rt<10'-"
..• 10-, .,
•...
".A single realization of the true field,radar observation together with the esti- matesvia Bayesianand conditional merging is shown in figure 3.34.
Figure 3.35 shows histograms of the mean errors on each field. The radar error mean was computed by taking the mean value (over 1000 realizations) of the residuals between the "true" field and the radarobservationat each pixel. This results in the 16384 values plotted as a histogram. The mean errors for each merging technique were computed in the same way. It is clear from the figure that both techniques provide an improvement over the radar observation and that the Conditional merging technique performs somewhat better in terms of bias reduction. The spike of values at zero in each case is due to the 83 rain gauge observationsthat are (as noted earlier) without any measurement bias.
The average variance of the errors at each pixel is reported in histogram form in figure 3.36.Once again both merging techniquesgive a considerable improve- ment relative to the "radar" , with the conditional merging technique once again performing better in terms of variance reduction.
All of the Kriging and merging computations were done on the logarithms of the variables in order to transform the log-normally distributed rainfall rate imulations to a Gaussian space. Compari ons,however, were done on the back transformed variablesin rainfall rates (mmlhr).
o
20 40 Radar rainfall field
100 80 60 40 20 0
Conditional merging
100
20
o o
40 60 100 80 100 80 60 40 20
Bayesian merging Simulated rainfall field
Fig. 3.34: Comp arison of Bayesian and Cond itional merged field s for a single reali zation . The estimated rainfall field produced by Bayesian mergin g appears very similar to the Kriged estimateshow n in the bottom right field in figure 3.33, whereas theConditi onall y merged field moreclosely resembles the "truth" in the upperleft simulated rainfall field .Thereasonfor thisisthatthe Bayesianmergin g algorithm uses the Kriged gauge estimate as the a posteriori observation in it's Kalman filter.
101
Fig. 3.35: Histograms of mean pixel errors relative tothe simulated rainfall field . The error isdefined as Ei(k), thedifference between the simulated and estimated fields at pixel i for realization k. The average over all the realization s at each pixel was computedand the resultin g 16384 average valuesfor the entire field are plottedhere ashistograms.
l r. • ..:; ;; t'~~= + + l j
~ l :.lm m m +m :~F:': +·· ·· ·· m !m mmm l mm j
! 2000[·
1000 r:::· t
:·· ·· · ·~~~~r/~~I·
~:~·~~ ..·..
1r..···:i f···
~·· ···· ··· ·! ~
~·
.~
o 8IIy••lan merging
0"----'- -
o 10 20 30 40
Mean error variance (mrrt/hr2)
50 so
Fig. 3.36: Hi tograms of the variance of the mean pixel errors relative to the simulated rainfall field. Theerroris defined as for figure 3.35. Theaverage over allrealizatio nsateach pixel was computedand theresult ing 16384 average values forthe entirefieldare plotted here ashistograms.
Fig. 3.37: The Liebenbergsvlei catchmentsuperimpo ed on the MRL5radar's 100 km radiusfootprint. The rain gauge network and numberingsyste m areshown by the red dots in the expanded view of the catchment.