Is the interannual variability of summer rainfall in China dominated by precipitation frequency or intensity? An analysis of relative importance

Abstract

The summer rainfall in China has a large interannual variability, which results from the concurrent variations of precipitation frequency and intensity. Using the observed daily precipitation in the 194 stations during recent 62 years, we examine the relative importance of the frequency and intensity in the variability of the rainfall. A simple method, based on linear regression, is used to estimate the relative importance. The products of the change rates of rainfall with respect to frequency and intensity, determined from the regression, and the corresponding standard deviations of the two variables, which reflect their variation scales, are defined to measure the importance of frequency and intensity. To determine the frequency, rainfall amount, and intensity from daily precipitation, we need a threshold to define the “rainy day”. In this study, we use a series of thresholds, ranging from 1 to 30 mm/day. So, while presenting the result of relative importance for each threshold, we also examine how the relative importance varies with the threshold. Results show that for the threshold of 1 mm/day, with which the rainfall may include even the light rains, the variabilities of summer rainfall in most stations are dominated by intensity. With the increase in threshold, the importance of frequency increases, while the importance of intensity decreases. When the threshold reaches 30 mm/day, with which the rainfall includes only moderate-to-heavy rains, the variabilities of the rainfall in all stations are dominated by frequency. Analysis suggests that such a change, in the dominance with the threshold, is reasonable. This reasonability, in turn, supports the reliability and robustness of the method.

Appendix

1.1 The method for estimating the dominance

Suppose quantity Z is influenced by variables X and Y. The relation is expressed as $Z = Z(X,Y)$. Generally, the relation can be nonlinear. However, if the nonlinearity is not strong, we may use a linear regression, $Z = aX + bY + c$, to approximate the relation. The coefficients a, b, and c can be determined with the data of X, Y, and Z. To be sound in statistics, the validation of the approximation needs to be verified with a significance test.

The meaning of the a and b can be illustrated by expressing them as $a = {{\partial Z} \mathord{\left/ {\vphantom {{\partial Z} {\partial X}}} \right. \kern-0pt} {\partial X}}$ and $b = {{\partial Z} \mathord{\left/ {\vphantom {{\partial Z} {\partial Y}}} \right. \kern-0pt} {\partial Y}}$. They represent, respectively, the change rates of Z with respect to X and Y, or the amount of the changes in Z corresponding to a unit increase in X and Y. Meanwhile, we can use σ _X and σ _Y , the standard deviations of X and Y determined from the data, to indicate, respectively, the scales of the variations of X and Y.

Then, as suggested in Lu et al. (2010, 2014), the products of the change rates and the corresponding variation scales, i.e., $\left| {{{\partial Z} \mathord{\left/ {\vphantom {{\partial Z} {\partial X}}} \right. \kern-0pt} {\partial X}}} \right| \cdot \sigma_{X}$ and $\left| {{{\partial Z} \mathord{\left/ {\vphantom {{\partial Z} {\partial Y}}} \right. \kern-0pt} {\partial Y}}} \right| \cdot \sigma_{Y}$, are used to measure, respectively, the scales of the changes in Z induced by the variations of X and Y. With the a and b obtained from the regression, the two measures can be defined as $S_{X} \equiv \left| a \right| \cdot \sigma_{X}$ and $S_{Y} \equiv \left| b \right| \cdot \sigma_{Y}$.

1.2 The implication of the two measures

To better understand the implication of the S _X and S _Y , we can express them in terms of the correlation coefficients. After some derivations, the two measures can be written as

$$ S_{X} = \frac{{\left| {\tilde{r}_{ZX} } \right|}}{{1 - r_{XY}^{2} }}\sigma_{Z} $$

(1)

and

$$ S_{Y} = \frac{{\left| {\tilde{r}_{ZY} } \right|}}{{1 - r_{XY}^{2} }}\sigma_{Z} , $$

(2)

where σ _Z is the standard deviation of Z, and r _XY is the coefficient of the simple linear correlation between X and Y. The $\tilde{r}_{ZX}$ and $\tilde{r}_{ZY}$, which are in form of $\tilde{r}_{ZX} \equiv r_{ZX} - r_{ZY} r_{XY}$ and $\tilde{r}_{ZY} \equiv r_{ZY} - r_{ZX} r_{XY}$, are proportional, respectively, to the coefficients of the partial correlations between Z and X and between Z and Y.

The partial correlation of an independent variable with the dependent quantity implies that the influences from other independent variables have been removed. Equations (1) and (2) suggest that what the measures S _X and S _Y reflect are the partial correlations of Z with X and Y, not the simple correlations. It is therefore more reasonable to use the measures defined than the simple correlations, to estimate the relative importance of the variables.