Algorithm description

RAINFALL ESTIMATION 'AND DATA MERGING ALGORITHMS

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

3.3.1 Algorithm description

The accumulation technique developed here accounts for the spatial and temporal evolution of the rainfall field between successive radar (or satellite) scans. The temporal sampling frequency of these instruments is not always short enough to ensure that important evolutionary features of the field are adequately captured.

The advection of the field between successive scans is computed using an optical flow algorithm (Bab-Hadiashar et al., 1996), introduced in the context of short term rainfall nowcasting by Seed (200 I). The optical flow computation (as im- plemented here) produces a dense field of advection vectors each associated with a particular grid point (the advection algorithm is introduced in more detail in section 4.1.1). Figure 3.15 hows a portion of the optical flow field computed be- tween two con ecutive rainfall intensity images from the SAWS weather radar in

tU 1~ W l~ III IM .t' ^1~

Fig. 3.15: Advection field computed using the optical flow algorithm of Bab- Hadiashar et al. (1996) . Two consecuti ve radar sca ns are shown. with the first greyed out. The advection vectors in the right hand box are computed from the informationcontained in thesetwo scans.

Durban.South Africa,during Novemb er2000. The left hand side of Figure 3. 15, shows two superimposed radar scans taken fiveminutes apart. The firstscan is in greyscale while thesecond isin colour. A portion of thefieldof advection vec tors.

computed usingan optical flowalgorithm,is shown on the rightof thefigure. The complex nature of the advection is evide nt from the advection vectors shown in the right hand panel.

The acc umulationalgorithm isdeveloped belo w (follow ing the expo sition of Hanne en, 2002) . Forany pixeli in theradar field ofview, the accumulation on thepixel betweentimes toand t_l is

(3.2)

where R(Si,t) istherainfall inten ity atany time ^I,ona pixeli located at position Si= (Xi,Yi)on the radar grid.

R(Si,t)isonlyknownat theradarobservation times,but if weassum ethatthe

69

radarsamples the rainfall field at

to

and ^t1then it i possible to e timate R(Si,

t)

for

to

~

t

~

t

₁using a linear combination of the radar fields Raand Rbobserved at

to

and

t

₁respectively.

(3.3)

In equation 3.3, Salt is the location that pixel i would have occupied in Ra at time t, Sblt is the location that pixel i would have occupied in R_bat time t, and /:).l = II - looThe locations Salt and Sblt are determined by the pixels advection vector V;. All the Salt must be located on the path between Salto and ^Si> and all the Sbltare located along the path between Si and Sbltl.The weighting factors th~t and _tt.~Q provide asmooth transitionbetween Raand Rbwhile con erving rainfall volume over the target pixel. In order to change the time integral in equation 3.2 into a path integral, the linear weighting factors in equation 3.3 can be replaced by

(1 - t _ ISalt-

sil

/:).t /:).s

( - to _

ISblt -

si!

/:).t /:).s

where ISalt-

si!

isthe distancebetween the points Suitand Si,similarly ISbIt -

sil

isthe distance betweenSbltand Si,and /:).s

=

ISalto -

il =

ISbltl -

s.],

Substituting into equation 3.3gives

Noting that

Salt- Si ( ) Sw - Si (

R( si,t) ^{= /:).S} R; alt

+ /:).

~ ^bit) (3.4)

v

^{= ds}

I dt

we can combine equations3.2and 3.4 and remove the dependence on time, arriv-

s·

_I

v.

_I

Fig. 3.16: Schematic representation of the accumulation scheme. The figure shows the location of araincell in two consec utive radar scans(R" and Rb). The rainc ell hastranslated with thedistance and direction defined bythe motion vec- tor

Vi ,

The total rainfall on the pixel located at Si is the weighted sum of the integral salong the paths Sulto- Si and Si - Sbltl' Ifthe advection were not taken into account,thepixel atSi would not appear to receiveany rain during this scan interval.

ingat

The total depth of rainfall accumulating on any pixel betw een scan times is computed from equation 3.5, over the path defined by it's advec tion vector.

Theindividual depth scan then be summed toproduce accumulations over longer timescales, asrequired .

As a simple illustration of the concept figure 3.16shows a raincell with con- stant contours of intensity, moving past the target pixel betw een time

to

and 1₁^, Therainfall field measured at time

to

is Raand the fieldmeasured at time

t

₁is Rs;

71

100mm

Omm

Fig.3.17: Comparison of accumulated daily rainfall fields, produced by the two different methods. The greyed out regions indicate regions affected, either by ground clutter or missing data. The colour scale on the right indicates the total rain fall depthaccumu late d in 24 hours.

Since the radar gridis spatially fixed, the rain cell exhibitsa translat ioncharacter- ized by the advection vector

Vi .

The two integralsalong the pathsSulto toSi and Si

to Sbltl are linearly weighted so that the contribution from the data atSulto is 100

%at time toand the contribution from Sbltl is 100%at time t1•

The integrals in equation 3.5 are computed using a discrete Trapezoidal rule approximation. The computational effort ofthe integration is directly relate d to the num ber of interior points that must be evaluated to compute the path integral between consecutive scans. To reduce computation time the number of interior points used is based on the ratio between the length of the advection vector and the spatial resolution of the data. Thus for a zero advection case the accumula- tion reduce to asimple average between the intensity values forsuccess ivescans and for large values of advection, many interior points are chosen at a step size equivalent tothe spatialresolution ofthe data (1km inthis case) .

Figure3.17shows a qualitativecomparisonbetweenseveralaccumulateddaily rainfallfieldsusing the currently operationalsimple averaging technique (top row) and the accumulations made using the advection based technique (bottom row).

It is clear from qualitative observations that the precipitation swaths produced by the advection accumulation are smoother and more in keeping with what one

of the peak and trough values) and produces a more realistic spatial distribution of the precipitation swathes.