Spatiotemporal Continuity of Sequential Rain Suggested by 3-D Variogram

Abstract

A series of rainfalls observed in central Japan from noon on the 13th to midnight on the 14th, August 1999 (36 h), has been analyzed by spatiotemporal variograms in order to reveal the continuity of rain precipitation in a 3-D space defined by geographic coordinates and time. All instances of zero precipitation are considered, but have been treated as four different cases: case 0 excludes all zero data, case 1 includes a zero datum neighboring to each finite value, case 2 includes a zero neighboring to each finite value and the next neighboring zero, and a fourth case (termed case A) includes all zeros. Hourly precipitation has a statistical distribution best approximated by a Weibull model, and somewhat less well by a normal distribution, in all four cases. A rectangular variogram of measured values of total precipitation shows that the best continuity appears approximately along the N-S direction (the ranges given by directional variograms are 500 and 80 km in the N-S and W-E directions, respectively). In contrast, temporally stacked rectangular variograms of hourly precipitation shows that the best continuity direction is W-E in all cases (the ranges in case A are 50 and 100 km along the N-S and W-E directions, respectively). A spatial variogram gives a spatial range independently of time, whereas a temporal variogram gives a temporal range. When geographic coordinates are normalized by the spatial range (here 80 km given by the temporally stacked omnidirectional variogram in case A), and time is normalized by the temporal range (here 7 h given by the spatially stacked temporal variogram), geographic coordinates and time can be treated as equivalent variables. Consequently, a spatiotemporal variogram can be calculated along a given direction in 3-D space using the normalized coordinates. The continuity direction of a series of rainfalls can be best understood by display on a Wulff net, where each range value is written at a point corresponding to the direction. The direction of the best continuity is N0°W+20° in the normalized space. A rectangular variogram in the normalized space, in which the horizontal and vertical axes represent N-S direction and time, respectively, suggests that the series of heavy rainfalls examined here had a continuity pattern that was elongated from west to east (the range values are 20–30 km and 100 km along N-S and W-E, respectively), and that migrated from south to north with a speed of 30 km/h.

Appendix

The software used in this article is written in MS-Excel/VBA, and can give three types of sample variograms: omnidirectional and directional variograms, and rectangular variogram (Shoji, 2002a). In a 2-D space, directional variogram values are given as a function of distance h and azimuth θ of lag vector h (only h for omnidirectional variogram) in polar coordinates, while rectangular variogram values as a function of W-E component ξ and N-S component η of h in rectangular coordinates. Either of a fan or a circle is applied for tolerance in the calculation of a 2-D variogram. Fan-shaped tolerance can fill up a 2-D space exclusively and thoroughly, but the shape varies depending on h. On the other hand, circle tolerance has a shape independent on h, but is partly overlapped when it fills up a 2-D space thoroughly. The interval between neighboring lag classes is given independently of tolerance, and hence neighboring lag classes are overlapped when the lag class interval is shorter than the width (given by tolerance) of each lag class. Prepared variogram models are linear, spherical, exponential, and Gaussian. Any of them is fitted for a sample variogram by the following algorithm: (1) to assume a range value; (2) to obtain nugget and sill values of a variogram model approximating sample variogram values by least squares or weighted least squares (for a linearly transformed function if possible, otherwise of successive approximation), where weight is number of pairs at each lag value, and lags longer than the range are considered to be the range value for linear and spherical models; (3) to calculate the deviation, which is square sum between the sample variograms and the variogram model; (4) to return Step 1, if the deviation is larger than the given permissible value. This fitting gives three parameters (nugget, sill, and range) of a variogram model. If nugget and sill values are given, then only range is calculated (if either nugget or sill is given, range and another parameter are calculated). When data are 3-D, the software can give 247 directional sample variograms automatically. A range value (nugget or sill value, too) of each directional variogram model obtained in a 3-D space can be represented by a figure or a symbol on a point of a Wulff net corresponding to the direction as Fig. 14, where positions of squares show 247 directions along which directional sample variograms are calculated. The software can be downloaded from the homepage of Japan Society of Geoinformatics (http://www.jsgi.org/).