Warm season heavy rainfall events over the Huaihe River Valley and their linkage with wintertime thermal condition of the tropical oceans

Abstract

Warm season heavy rainfall events over the Huaihe River Valley (HRV) of China are amongst the top causes of agriculture and economic loss in this region. Thus, there is a pressing need for accurate seasonal prediction of HRV heavy rainfall events. This study improves the seasonal prediction of HRV heavy rainfall by implementing a novel rainfall framework, which overcomes the limitation of traditional probability models and advances the statistical inference on HRV heavy rainfall events. The framework is built on a three-cluster Normal mixture model, whose distribution parameters are sampled using Bayesian inference and Markov Chain Monte Carlo algorithm. The three rainfall clusters reflect probability behaviors of light, moderate, and heavy rainfall, respectively. Our analysis indicates that heavy rainfall events make the largest contribution to the total amount of seasonal precipitation. Furthermore, the interannual variation of summer precipitation is attributable to the variation of heavy rainfall frequency over the HRV. The heavy rainfall frequency, in turn, is influenced by sea surface temperature anomalies (SSTAs) over the north Indian Ocean, equatorial western Pacific, and the tropical Atlantic. The tropical SSTAs modulate the HRV heavy rainfall events by influencing atmospheric circulation favorable for the onset and maintenance of heavy rainfall events. Occurring 5 months prior to the summer season, these tropical SSTAs provide potential sources of prediction skill for heavy rainfall events over the HRV. Using these preceding SSTA signals, we show that the support vector machine algorithm can predict HRV heavy rainfall satisfactorily. The improved prediction skill has important implication for the nation’s disaster early warning system.

Appendices

Appendix 1: Influence of data autocorrelation on Bayesian inference of Normal mixture model

Climate variables at daily scale usually contain certain autocorrelation, which violates the requirements by many statistical models. However, autocorrelation does not impact Bayesian inference of Normal mixture model, because it does not require statistical samples to be strictly independent and identically distributed (i.i.d.) (Hoff 2009).

Figure 8a shows the autocorrelation of HRV daily rainfall. According to the 52-year sample sets, the autocorrelation with 1-day lag is statistically significant ($\rho = 0.4$ for 52-year average, significant at 0.01 level), indicating that the rainfall events is highly dependent on those 1 day prior (Fig. 8a).

Fig. 8

a Autocorrelation of daily precipitation samples; the dots denote the maximum likelihood estimator of the autocorrelation coefficients, and the error bars denote one standard deviation of the autocorrelation derived from the daily precipitation during summer seasons in 1961–2012. The dashed blue lines are α = 0.05 significance level. The autocorrelation located within the regions bounded by the two lines is statistically insignificant; b, c the comparison of Bayesian inference on HRV rainfall intensity and frequency using the original samples and those without autocorrelation

To eliminate the impact of data autocorrelation on Bayesian inference, we ran the framework (Eqs. 1–6) by using two subsets of de-autocorrelated data samples. In each subset, the rainfall data are sampled sequentially at 2-day interval, when autocorrelation becomes insignificant (Fig. 8a). The comparison between the inference on the de-autocorrelated sample sets and the original sample sets is shown in Fig. 8b, c. For the three rainfall clusters, neither cluster intensity nor cluster weight (frequency) is sensitive to the sample autocorrelation. The distribution parameters derived from the two sample sets are aligned tightly to the line of $y = x$, indicating the inference generates identical results using the two different sample sets (Fig. 8b, c).

In conclusion, the Bayesian inference on Normal mixture model does not require the sample to be strictly i.i.d., making it easy to apply to climate variables, including precipitation and temperature, which usually have high autocorrelation.

Appendix 2: SVM algorithm

In this study, we apply a least square SVM, meaning that the w ^T and b in Eq. (8) are formulated to minimize the cost function:

$$ \psi_{L} \left( {w,\varepsilon } \right) = C\sum\nolimits_{i = 1}^{n} {\varepsilon_{i}^{2} } + \frac{1}{2}\left\| w \right\|^{2} $$

(9)

In the cost function, $\varepsilon_{i}^{2} = \left( {y_{i} - \hat{y}_{i} } \right)^{2}$ is the quadratic loss term. C is a positive real constant reflecting the tolerance rate of prediction errors. The larger the value of C, the lower tolerance for prediction errors, and thus higher punishment added in the cost function. The primal SVM problem becomes an optimization problem to obtain $\mathop {\hbox{min} }\limits_{{w \in R^{n} ,b \in R}} C\sum\nolimits_{i = 1}^{n} {\varepsilon_{i}^{2} } + \frac{1}{2}\left\| w \right\|^{2}$. The solutions to the optimization problem can be obtained using the Lagrangian:

$$ L\left( {w,b,\vec{\varepsilon },\vec{\alpha }} \right) = \frac{1}{2}w^{T} w + C\sum\nolimits_{i = 1}^{n} {\varepsilon_{i}^{2} } - \sum\nolimits_{i = 1}^{n} {\alpha_{i} \left\{ {\hat{y}_{i} + \varepsilon_{i} - y_{i} } \right\}^{2} } , $$

where $\alpha_{i}$ are Lagrangian multipliers. The optimal solutions must satisfy the conditions [also known as Karush–Kuhn–Tucker (KKT) conditions]:

$$ \left\{ {\begin{array}{*{20}l} {\frac{\partial L}{\partial w} = w - \sum\nolimits_{i = 1}^{n} {\alpha_{i} \varPhi \left( {\vec{x}_{i} } \right)} = 0} \hfill \\ {\frac{\partial L}{\partial b} = \sum\nolimits_{i = 1}^{n} {\alpha_{i} } = 0} \hfill \\ {\frac{\partial L}{{\partial \varepsilon_{i} }} = \alpha_{i} - C\varepsilon_{i} = 0} \hfill \\ {\frac{\partial L}{{\partial \alpha_{i} }} = \hat{y}_{i} + \varepsilon_{i} - y_{i} = 0} \hfill \\ \end{array} } \right. $$

(10)

By solving the above set of linear equations, the SVM regression model can be obtained:

$$ f\left( {\vec{x}} \right) = \sum\limits_{i,j} {\alpha^{*} } K\left( {x_{i} ,x_{j} } \right) + b^{*} , $$

(11)

where $\alpha^{*}$ and $b^{*}$ are the solutions to Eq. (10); and $K(x_{i} ,x_{j} ) = \phi (x_{i} )^{T} \phi (x_{j} )$.