34;True" field Kriged gauge field
4.1 Overview of nowcasting techniques and models
4.1.2 Adapting SBM for nowcasting
ux
Fig.4.2: Representationof Optical Flow Constraints. Since all of the line do not intersect at precisely the same point a best estimate must be made using a least
quaresapproach.
advection for each block is given by the solution of equation 4.2. The estimates for each block are then linearly interpolated onto the data grid.
119
Insert
Discard
Fig.4.3: An illustration of theSBM noise stack.
Rainfall field simulation
A typical simulation producesaseries of wetand dry period susingan alternating renewal process.The wet periods are the "beads" and thedry periodsthe "string"
connecting thewetperiods,hencethename"Stringof Beads" model.
The simulation of each "bead" begins with the genera tion of a stackof five Gaussian rand om fields on a rectangular grid with dimensions m x n. The val- ues ateach pointin a field are independ ent and normallydistributed with a zero mean and unit variance andeach field is independ ent ofthe others. A new field is construct ed (pixel by pixel) using an AR(5) model and placed at the first po- sitionon the stack, with the previous fields moving backwards one position and the final field falling away (Figure 4.3). The random number generation requires thought. At five minute intervals on a 128 x 12 grid, one hour of simulation requires generating 196608 independ ent random numbers. Itis thus important to ensure that ageneratorwithalong cycle period isu ed for lengthy simulations on large fields. TheSBM uses a fastalgorithm byWichmannand Hill (1982), which has a cycle length of more than 6.95x1012• This equates to approximately 4000 years ofsimulation at 5 minute resolutionfor the 128x128 km grid mentioned.
A mean field advection vector is used to maintain the appropriate temporal alignment of the pixel values in each field. A warm up period is required to en- sure that the sequence of field s used in the generation is properl y conditioned- i.e. thecorrect serial correlation structure, asdefined bytheAR(5)modelparam-
Fig. 4.4: Simulatedrainfall field s from SBM,using an increasingdegreeofspatial correlation. Each field is 256x256kmand
13
the exponentofthe power law filter in the Fou rierdom ain is for panel(a) 0.5,(b) 1.5,(c) 3,(d) 3.75, (e)5 and(j) 10.eters, exists. Once the stac k has been givena sufficie nt number ofrecursions to becorrectl y conditioned, a copy ofthe mostrecentl y generated field is made. At thispoint the copiedfield hasno spatialcorrelatio n. A suitable spatialcorrelation structure is impose d, by applyinga patial filter to increa e the spatialcorrelation to the desired level. The filter isdefined by a power-law functio n in the Fouri er domain, the struc ture and parameters of the filte r were defined by Pegram and Clothier (1999)on the basis of observedradarrainfall data in South Africa. Once the field has appropria te spa tial correlatio n, the simulated image scale statistic
"Wetted Area Ratio" (WAR) and "Image Mean Flux" (IMF) are impo ed on the field by a thresholding and scaling process. In this thesis, IMF will be referr ed to as Spatial Mean Flux (SMF) to avoidconfusion with the concept of"Intrinsic Mode Functions" (introduced later in sectio n 4.2.3). The resulting field is ex- ponenti ated to produ ce a field of simulated rainfall rates. Figure 4.4 shows six
121
different simulated rainfall field sproduced using SBM. Eachofthe field shas the same WAR, SMFand underlyin g noisefield,but thedegreeofspatial correlation increa es row-wise from top left. The degree of spatial organization exhibited by obse rved radar rainfall fields (Pegramand Clothi er, 1999) lies somewhere be- tween the third and the fourth panelsand isdefinedby the exponentofthe power law filter function in the Fourier domain.
Rainfall field forecasting
In forecast mode the processhas been strea mlined. The spatialcorrelationstruc- ture from the obse rved fields is retained and the pixel scale developm ent , now using an AR(2) model, computed directly from the obse rved fields. The image scale statistics are forecas t using the bivariate AR(S) WAR-SMF process of the simulation modebut thistime conditionedon the values from theprevious five ob- servations.Additionally, a sophisticated motion-trackin g algorithm(section 4.1.1) isused to estimate the field advection for each pixcl,ratherthan assuminga mean advec tion vectoras is done insimulation mode. A dense grid of advection vec- torsis computed ateach time tep and u ed to advec t the foreca ts a wella for maintainin g the appropriate temp oral alignment between pixels. The smoothed advection grid is updated , ateach time step, as new inform ation becomes avail- able. The computation of the advcc tionvectors is efficientenough to allow it to beused for real-tim e applications.