Predictable and unpredictable modes of seasonal mean precipitation over Northeast China

Abstract

This study investigates the patterns of interannual variability that arise from the potentially predictable (slow) and unpredictable (intraseasonal) components of seasonal mean precipitation over Northeast (NE) China, using observations from a network of 162 meteorological stations for the period 1961–2014. A variance decomposition method is applied to identify the sources of predictability, as well as the sources of prediction uncertainty, for January–February–March (JFM), April–May–June (AMJ), July–August–September (JAS) and October–November–December (OND). The averaged potential predictability (ratio of slow to total variance) of NE China precipitation has the highest value of 0.32 during JAS and lowest value of 0.1 in AMJ. Possible sources of seasonal prediction for the leading predictable precipitation EOF modes come from the SST anomalies in the Japan Sea, as well as the North Atlantic during JFM, the Indian Ocean SST in AMJ, and the eastern tropical Pacific SST in JAS and OND. The prolonged linear trend, which is seen in the principal component time series of the leading predictable mode in JFM and OND, may also serve as a source of predictability. The Polar–Eurasia and Northern Annular Mode atmospheric teleconnection patterns are closely connected with the leading and the second predictable mode of JAS, respectively. The Hadley cell circulation is closely related to the leading predictable mode of OND. The leading/second unpredictable precipitation modes for all these four seasons show a similar monopole/dipole structure, and can be largely attributed to the intraseasonal variabilities of the atmosphere.

Appendix

Let ${x_{ym}}$ represent sample monthly values, within a season, in month m (m = 1, 2, 3) and in year y (y = 1,…, Y, where Y is the total number of years). The annual cycle is firstly removed from the data. Following ZF2004 and Frederiksen and Zheng (2007), the monthly time series of each climate anomaly can be conceptually decomposed into two components consisting of a seasonal “population” mean and a residual departure from this mean, as

$${x_{ym}}={\mu _y}+{\varepsilon _{ym}},$$

(1)

Here, ${\mu _y}$ is the seasonal population mean in year y, and ${\varepsilon _{ym}}$ is a residual monthly departure of ${x_{ym}}$ from ${\mu _y}$ and arises from intraseasonal variability. The vector $\{ {\varepsilon _{y1,}}{\varepsilon _{y2,}}{\varepsilon _{y3}}\}$is assumed to comprise a stationary and independent annual random vector with respect to year. Equation (1) implies that month-to-month fluctuations, or intraseasonal variability, arise entirely from $\{ {\varepsilon _{y1,}}{\varepsilon _{y2,}}{\varepsilon _{y3}}\}$ (e.g., $({x_{y1}} - {x_{y2}}={\varepsilon _{y1}} - {\varepsilon _{y2}}).$) We represent an average taken over an independent variable (i.e., m or y) by replacing that variable subscript with “o”. With this notation, a seasonal mean can be expressed as

$${x_{yo}}={\mu _y}+{\varepsilon _{yo}},$$

(2)

Suppose now that we have two climate variables ${x_{ym}}$ and ${x'_{ym}}$ that satisfy Eqs. (1) and (2). The interannual covariance of the intraseasonal component could be estimated as

$$\hat V({\varepsilon _{yo}},\varepsilon _{{yo}}^{\prime }) \approx {\hat \sigma ^2}(3+4\hat \varphi )/9,$$

(3)

where,

$${\hat \sigma ^2}=\frac{a}{{2(1 - \hat \phi )}},$$

(4)

$$\hat \phi =\left( {\frac{{a+2b}}{{2(a+b)}}} \right),$$

(5)

$$a=\frac{1}{2}\left\{ {\frac{1}{Y}\sum\limits_{y=1}^Y {[{x_{y1}} - {x_{y2}}][x_{{y1}}^{\prime } - x_{{y2}}^{\prime }]+\frac{1}{Y}\sum\limits_{y=1}^Y {[{x_{y2}} - {x_{y3}}][x_{{y2}}^{\prime } - x_{{y3}}^{\prime }]} } } \right\},$$

(6)

$$b=\frac{1}{2}\left\{ {\frac{1}{Y}\sum\limits_{y=1}^Y {[{x_{y1}} - {x_{y2}}][x_{{y2}}^{\prime } - x_{{y3}}^{\prime }]+\frac{1}{Y}\sum\limits_{y=1}^Y {[{x_{y2}} - {x_{y3}}][x_{{y1}}^{\prime } - x_{{y2}}^{\prime }]} } } \right\}.$$

(7)

Then, the covariance matrix of the slow or predictable component can be derived from the total interannual covariance. In particular, the covariance between two slow or predictable components can be estimated as,

$$\hat V({\mu _y},\mu _{y}^{\prime }) \approx \hat V({x_{yo}},x_{{yo}}^{\prime }) - \hat V({\varepsilon _{yo}},\varepsilon _{{yo}}^{\prime }),$$

(8)

where the total variances $V({x_{yo}},{x'_{yo}})$ can be calculated directly from two seasonal means. It is worth emphasizing that the difference between the total and the intraseasonal variances, in general, consists of not only the covariance between ${\mu _y}$ and $\mu _{y}^{\prime }$, but also their interaction terms with ${\varepsilon _{yo}}$ and ${\varepsilon '_{yo}}$. In the case where the intraseasonal and slow components are independent, the residual covariance reduces to the covariance of the slow component. When this is not the case, $V({x_{yo}},x_{{yo}}^{\prime }) - V({\varepsilon _{yo}},\varepsilon _{{yo}}^{\prime })$ may still be better related to the covariance between the two slow components than is $V({x_{yo}},{x'_{yo}})$.

We define the potential predictability as the ratio between the variance of the predictable component and the variance of the total component; that is,

$$p=\frac{{V({\mu _y})}}{{V({x_{yo}})}}.$$

(9)

This quantity represents the fraction remaining after the removal of the intraseasonal component from the total (Madden 1976; Zheng et al. 2000). The larger the value, the more closely the seasonal mean precipitation anomalies or the precipitation corresponding PC time series are related to external forcing and very low-frequency internal dynamics, and the more likely the seasonal mean precipitation or the precipitation PCs can be predicted.

The statistical significance of the covariance between the associated PC time series of the slow precipitation modes over NE China and the slow component of SST and atmospheric circulations (including height field, moisture flux and convergence, in Sect. 3.2) is able to be estimated through a Chi square test, with one degree of freedom,

$${\chi ^2}= - 2\sum\limits_{y=1}^Y {\left( {L{H_{0,y}} - L{H_{A,y}}} \right)} ,$$

(10)

where LH _0,y and LH _A,y are the log-likelihoods of the null hypotheses $\hat V({\mu _y},\mu _{y}^{\prime })=0$ and the alternative hypotheses respectively, given the observations $\left( {{x_{yo}},x_{{yo}}^{\prime }} \right)$; and they are calculated using the multivariate normal distribution assumption (see Wilks 2006; Grainger et al. 2017), with the zero means and the covariance matrices V ₀ and V _A ,

$${V_0}=\left[ {\begin{array}{*{20}{c}} {\hat V({x_{yo}},{x_{yo}})}&{\hat V({\varepsilon _{yo}},\varepsilon _{{yo}}^{\prime })} \\ {\hat V({\varepsilon _{yo}},\varepsilon _{{yo}}^{\prime })}&{\hat V(x_{{yo}}^{\prime },x_{{yo}}^{\prime })} \end{array}} \right],$$

(11)

$${V_A}=\left[ {\begin{array}{*{20}{c}} {\hat V({x_{yo}},{x_{yo}})}&{\hat V({x_{yo}},x_{{yo}}^{\prime })} \\ {\hat V({x_{yo}},{{x'}_{yo}})}&{\hat V(x_{{yo}}^{\prime },x_{{yo}}^{\prime })} \end{array}} \right].$$

(12)

The significance of the intraseasonal covariances between two climate variables can be estimated by a Student’s t test. The t statistic, with the degree of freedom of Y-2, for each pair of intraseasonal covariance is,

$$t=\frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{Y - 2}}} }},$$

(13)

$$r=\frac{{\hat V({\varepsilon _{yo}},\varepsilon _{{yo}}^{\prime })}}{{\sqrt {\begin{array}{*{20}{c}} {\hat V({\varepsilon _{yo}},{\varepsilon _{yo}})}&{\hat V(\varepsilon _{{yo}}^{\prime },\varepsilon _{{yo}}^{\prime })} \end{array}} }}.$$

(14)

here, r is an estimation of the correlation between the associated PC time series of the intraseasonal precipitation modes over NE China and the intraseasonal component of atmospheric circulations.