Seasonal predictable signals of east Asian summer monsoon rainfall in existing monsoon indices

Abstract

East Asian summer monsoon indices (EASMIs) have been widely used to investigate the variability and predictability of the East Asian summer monsoon rainfall (EASMR). However, the ability of existing EASMIs remains unclear to represent the interannual variability of the EASMR in predictable (P-) and unpredictable (U-) components. Based on a (co-)variance decomposition method, the fractional variance explained by a single EASMI has the highest value of 22% in both P- and U-components. A set of the best three EASMIs, together with the linear trend, is linearly independent of each other and can explain a large percentage of EASMR variance in P-component (54%). This set of EASMIs captures the main predictive circulation features in the corresponding EASMR P-modes, i.e., a low-level Philippine Sea (anti-)cyclone and an upper-level zonal wind tripole pattern for P-mode1, an East China Sea (anti-)cyclone for P-mode2, and a west–east pressure dipole pattern for P-mode3. In addition, they also have the major predictable sources from the predictors of their corresponding P-modes, i.e., the decaying and developing El Niño–Southern Oscillation, the spring Arctic Oscillation, the spring sea surface temperatures over the western North Pacific, tropical and southern Atlantic, southern Indian and Arctic oceans. By considering the predictable and unpredictable components, this work not only improves our knowledge of the physical meanings and the potential limitations of the existing EASMIs, but also helps us in selecting the most appropriate EASMIs when focusing on the issue of seasonal forecasting.

Appendix

Let ${t}_{yo}\left(j\right)={\mathbf{r}}_{yo}{{\mathbf{e}}_{\mathbf{o}}}^{\text{T}}\left(j\right)$ denote the jth T-PC time series corresponding to the jth T-EOF ${[\mathbf{e}}_{\mathbf{o}}\left(j\right),j=1,\cdots ,J]$ of the seasonal-mean EASMR ${\mathbf{r}}_{yo}$. It can be linearly predicted by the EASMI ${i}_{yo}$, i.e.,

$${\mathbf{r}}_{yo}{{\mathbf{e}}_{\mathbf{o}}}^{\text{T}}\left(j\right)=b\left(j\right){i}_{yo}+{\xi }_{y}$$

where $b\left(j\right)$ is the regression coefficient and ${\xi }_{y}$ is the regression error. Since ${\xi }_{y}$ is statistically independent of ${i}_{yo}$,

$$E\left({\mathbf{r}}_{yo}{{\mathbf{e}}_{\mathbf{o}}}^{\text{T}}\left(j\right){i}_{yo}\right)=E\left(b\left(j\right){i}_{yo}{i}_{yo}\right)$$

Here, E is the mathematical expectation operator. Then, the regression coefficient $b\left(j\right)$ can be estimated as

$$b\left(j\right)=V({\mathbf{r}}_{yo}{{\mathbf{e}}_{\mathbf{o}}}^{\text{T}}\left(j\right),{i}_{yo})/V\left({i}_{yo}\right)$$

(12)

Therefore, the variance of ${\mathbf{r}}_{yo}{{\mathbf{e}}_{\mathbf{o}}}^{\text{T}}\left(j\right)$ explained by ${i}_{yo}$ is

$$V\left(b\left(j\right){i}_{yo}\right)={V}^{2}({\mathbf{r}}_{yo}{{\mathbf{e}}_{\mathbf{o}}}^{\text{T}}\left(j\right),{i}_{yo})/V\left({i}_{yo}\right)$$

(13)

For the predictable component, conceptually, the jth P-PC ${\mathbf{r}\_\varvec{\upmu }}_{y}{{\mathbf{e}}_{\varvec{\mu }}}^{\text{T}}\left(j\right)$ can be linear predicted by the predictable component of EASMI ${i\_\mu }_{y}$ as

$${\mathbf{r}\_\varvec{\upmu }}_{y}{{\mathbf{e}}_{\varvec{\mu }}}^{T}\left(j\right)={b}_{\mu }\left(j\right){i\_\mu }_{y}+{\xi }_{y}$$

Although the predictable components do not exist numerically, following Eq. (12), ${b}_{\mu }\left(j\right)$ can be estimated as

$${b}_{\mu }\left(j\right)=V({\mathbf{r}\_\varvec{\upmu }}_{y}{{\mathbf{e}}_{\varvec{\mu }}}^{T}\left(j\right),{i\_\mu }_{y})/V\left({i\_\mu }_{y}\right)$$

(14)

Therefore, the predictable EASMR variability explained by the EASMI is

$$V\left({b}_{\mu }\left(j\right)i\_{\mu }_{y}\right)={V}^{2}({\mathbf{r}\_\varvec{\upmu }}_{y}{{\mathbf{e}}_{\varvec{\mu }}}^{T}\left(j\right),{i\_\mu }_{y})/V\left({i\_\mu }_{y}\right)$$

(15)

Similarly, the unpredictable EASMR variability explained by the EASMI can be estimated as

$$V\left({b}_{\epsilon }\left(j\right)i\_{\epsilon }_{yo}\right)={V}^{2}({\mathbf{r}\_\varvec{\upepsilon }}_{yo}{{\mathbf{e}}_{\epsilon }}^{T}\left(j\right),{i\_\epsilon }_{yo})/V\left({i\_\epsilon }_{yo}\right)$$

(16)