the spatial pattern of precipitation at the surface; while the point-wise rain gauge obser- vations are used as the reference of surface precipitation. However, there could be some contradiction between the two types of data, namely, big radar quantiles are not necessarily corresponding to big rain gauge observations, and vice versa. The two datasets have differ- ent marginal distributions: radar quantiles follow an uniform distribution, while rain gauge observations usually present a non-uniform distribution. Due to the different marginals of the two datasets, the Spearman’s rank correlation (ranging from -1 to 1) is employed as the measure of the agreement, as the monotonicity of the two datasets is of interest. And the closer the rank correlation to 1, the higher the agreement of radar and gauge data.
Suppose there areK rain gauges located at{s1,· · ·, sK}, and correspondingly, there areK gauge observations, denoted as {z(s1),· · · , z(sK)} = {z1,· · · , zK}. The synchronous pre- cipitation map given by radar is transformed to a quantile map U by its empirical cu- mulative distribution function. At the collocated locations, there are K radar quantiles
{U(s1),· · ·, U(sK)} = {u1,· · · , uK}. The Spearman’s rank correlation of the two datasets is given by
ρ=rs(zk, uk) (4.2)
where,k ={1,· · ·, K}andrsdenotes the function to calculate the Spearman’s rank corre- lation.
Similar as the Pearson’s correlation coefficient, the Spearman’s rank correlation (or simply as rank correlation) is on the range of[−1,1]: with 1 indicating the exactly identical ranks of
the two datasets, and -1 indicating the exactly opposite ranks. The Spearman’s rank correla- tion measures how well the dependence of two variables can be described by a monotonic relationship. Note, the difference between the Pearson’s correlation coefficient and rank correlation can be rather small when the two datasets are relatively big in size, but the dis- tinction is amplified with small datasets.
The measure of agreement between radar and gauge data by rank correlation is far from perfect. For example, rain gauge observations of 7.7 mm and 7.8 mm are not so different in the precipitation values; but for a dataset of small size, the difference in ranks is more obvious. The small difference in the observed precipitation values is not necessarily because of the difference in the precipitation itself, but could also be induced by the limitation of the measuring device, e.g. the tipping buckets, whose malfunctions are more likely in case of strong precipitation intensity. In order to overcome the problem, a threshold of precipitation value could be defined, such that different ranks are considered only when the difference of two observations is above the threshold.
4.2.1 Application — test on the agreement of gauge and radar data
Rain gauges and weather radars are recording precipitation at different spatial scales: the former is a point scale measurement device; while the latter refers to a volume integral scale [Peleg et al., 2018]. As suggested by Peleg et al. [2018], in most cases, the ratio between the observation scale for a standard C-band radar and a standard rain gauge is in the order of
107.
Then, it is an interesting question to ask whether the wind information integrated radar rainfall accumulation proposed in Section 3.4.2 has a higher agreement with rain gauge observations, compared to the precipitation accumulation without integrating the wind in- formation. Because the proposed radar rainfall accumulation scheme refines the resolution of radar rainfall by a factor of 5, except for integrating the wind information,
To answer the question, an experiment was made to test on the agreement of the down- scaled radar rainfall accumulation (with the spatial resolution of100×100m) and the gauge rainfall accumulation, which is spatially representative for a circle with the radius of around 10 cm. As a comparison, the agreement of gauge rainfall and radar rainfall, accumulated by direct summation of 5-min-intervalled imageries (with the spatial resolution of500×500m), is also calculated. The agreement of radar and gauge data is measured by rank correlation as proposed in the previous section.
Figure 4.1:The agreement of radar and gauge data measured by rank correlation. Radar precipitation accumulation by direct summation of 5-min-intervalled imageries (left); radar precipitation accumulation with wind information integrated (right). Time: 2015/06/06 23:00 to 23:15.
An example is given in Fig. 4.1, the rank correlation between the rain gauge observations and the collocated radar data is calculated for both kinds of radar rainfall accumulations and labelled in the figure titles. In this case, the integtation of wind information does no benefit in terms of increasing the agreement of radar and gauge data, as the rank correlation is observed to decrease a little bit from0.802to0.747.
The same procedure was applied repeatedly for 100 15-min-events. The results are two rank correlation series of length 100: one for the radar rainfall accumulation of the original resolution (ρnon-wind) and the other for the downscaled and wind information integrated radar rainfall accumulation (ρwind). Some descriptive statistics evaluated from the two rank correlation series are calculated and listed in Table 4.1.
Table 4.1:Descriptive statistics of the two rank correlation series to evaluate the agreement between radar (wind information integrated/non-integrated precipitation accu- mulation) and gauge data.
mean median 1stquartile 3rdquartile stdev
ρnon-wind 0.151 0.232 -0.200 0.457 0.406
As revealed by Table 4.1, an increase in the agreement of radar and gauge data is observed in the wind information integrated radar rainfall accumulation, which demonstrates the ben- efit of integrating wind information in radar rainfall accumulation. However, the increase is not that remarkable. This is mainly due to the fact that only the information of the hori- zontal wind displacement is considered. The vertical wind displacement, which is assumed to impose a much stronger influence on the falling hydrometeors, is not considered yet. It is expected that the integration of vertical wind displacement could bring more benefit in terms of increasing the agreement of radar and gauge data.