Analysis of rainfall seasonality from observations and climate models

Abstract

Two new indicators of rainfall seasonality based on information entropy, the relative entropy (RE) and the dimensionless seasonality index (DSI), together with the mean annual rainfall, are evaluated on a global scale for recently updated precipitation gridded datasets and for historical simulations from coupled atmosphere–ocean general circulation models. The RE provides a measure of the number of wet months and, for precipitation regimes featuring a distinct wet and dry season, it is directly related to the duration of the wet season. The DSI combines the rainfall intensity with its degree of seasonality and it is an indicator of the extent of the global monsoon region. We show that the RE and the DSI are fairly independent of the time resolution of the precipitation data, thereby allowing objective metrics for model intercomparison and ranking. Regions with different precipitation regimes are classified and characterized in terms of RE and DSI. Comparison of different land observational datasets reveals substantial difference in their local representation of seasonality. It is shown that two-dimensional maps of RE provide an easy way to compare rainfall seasonality from various datasets and to determine areas of interest. Models participating to the Coupled Model Intercomparison Project platform, Phase 5, consistently overestimate the RE over tropical Latin America and underestimate it in West Africa, western Mexico and East Asia. It is demonstrated that positive RE biases in a general circulation model are associated with excessively peaked monthly precipitation fractions, too large during the wet months and too small in the months preceding and following the wet season; negative biases are instead due, in most cases, to an excess of rainfall during the premonsoonal months.

Appendix: Properties of the relative entropy

1.1 Relative entropy and information entropy

Given a discrete probability distribution $p=\{p_m\}_{m=1}^N$ describing a random variable, the information entropy associated with ${p_m}$ is a measure of the uncertainty of a random variable described by $p$ and it is defined as

$$\begin{aligned} {\mathcal {H}}\left( p\right) \equiv -\sum _m^N p_m\log _2 p_m \end{aligned}$$

(4)

and $0\le {\mathcal {H}} \le \max ({\mathcal {H}})$, where $\max ({\mathcal {H}})=\log _2 N$ for the uniform distribution $p_m=1/N$ (maximum uncertainty) and 0 if one out of the $N$ values of $p$ is equal to one and all the remaining are zero ($x\log x\rightarrow 0$ as $x\rightarrow 0$) (no uncertainty). In the case considered in this study, $N=12$ and $\max ({\mathcal {H}})=\log _2 12$. The relative entropy of $p$ with respect to $q,\,{\mathcal {D}}(p|q)$, is introduced instead to measure how different two probability distribution $\{p_m\}_{m=1}^N$ and $\{q_m\}_{m=1}^N$ are:

$$\begin{aligned} {\mathcal {D}}(p|q)\equiv \sum _{m=1}^N p_m\log _2\left( \frac{p_m}{q_m}\right) \end{aligned}$$

(5)

and it measures the inefficiency of assuming $q$ when instead the true distribution is $p$. It can be demonstrated (Cover and Thomas 1991) that ${\mathcal {D}}(p|q)\ge 0$ for any $p, q$ and ${\mathcal {D}}(p|q)=0$ if and only if the two probability distributions are the same. The RE is not symmetric, ${\mathcal {D}}(p|q) \ne {\mathcal {D}}(q|p)$ and therefore is not a distance in a mathematical sense. However it is still useful to think of it as a distance between two probability distributions. For introducing the seasonality index we define $D\left( p\right)$ such as

$$\begin{aligned} D(p)\equiv \sum _{m=1}^{N} p_m\log _2\left( N\,p_m \right) \end{aligned}$$

(6)

that is such as the RE of the probability distribution $p_m$ with respect to the uniform distribution $q_m=1/N$, which is taken as a reference. In the following and in the rest of this manuscript we will still refer to $D(p)$ as RE. From this definition it follows that

$$\begin{aligned} D\left( p\right) =-{\mathcal {H}}\left( p\right) +\log _2 N. \end{aligned}$$

(7)

As a consequence, for two probability distributions $p$ and $w,\,D\left( p\right) -D(w)={\mathcal {H}}(w)-{\mathcal {H}}\left( p \right)$.

1.2 Relative entropy and the spread of $p_m$

Let us assume now that $\{p_m\}_1^{N}$ are the monthly precipitation fractions ($N=12$). From what said so far, it is expected that the larger it is $D$, the less uniformly the precipitation is distributed throughout the year. So $D$ is related to the “spread” of precipitation signal. This concept can be framed in a rigorous way in information theory by defining the effective number of values of $p$

$$\begin{aligned} n^\prime \left( p\right) =2^{{\mathcal {H}}\left( p\right) }=12\, \cdot 2^{-D}. \end{aligned}$$

(8)

Mathematically $n^\prime$ defines the number of months over which $p_m$ is considerably different from zero, i.e. the support of $p_m$ (Cover and Thomas 1991). Therefore $n^\prime$ can be interpreted as the effective number of wet months in a year. Areas characterized by $D=0$ have $n^\prime = 12$, that is no significant dry period (non-seasonal rainfall regime), whereas regions featuring $D=D_{\mathrm{max}}=\log _2 12,\,p_k=1$ have their annual precipitation all concentrated in one month (extreme seasonal rainfall regime). For regions having a unimodal seasonal rainfall distribution, $n^\prime$ provides a measure of the duration of the wet season (e.g. Indian region). It has to be noted however that different measures of the wet season duration which are not based on integral properties of the rainfall distribution but on local properties – e.g. retreat minus onset dates (Sperber et al. 2013; Kitoh et al. 2013; Hasson et al. 2014), where onset and retreat are defined by the 5 mm/day threshold—may give different results.

Within this framework, let us also introduce another useful statistical indicator of rainfall seasonality, the centroid. By using circular statistics (Fisher et al. 1993), the first moment of $p_m$ (centroid) is defined as

$$\begin{aligned} C=\mathrm arg (z), \quad z=\sum _{m=1}^{12} p_m e^{i \frac{2 \pi m}{12} } \end{aligned}$$

(9)

and it is shown in Fig. 15. The centroid provides a measure of the the timing of the wet season. While it can be mathematically defined for any precipitation sequence $r_m$, it is really meaningful only for those rainfall regimes that are somewhat “localized” during the year—i.e. having a clear dry and wet period. A more extensive analysis of $C$ in present condition and future emission scenarios will be reported elsewhere.

Fig. 15

Centroid (a) and effective number of wet months (b) of the monthly precipitation sequence $p_m$ for the GPCC dataset. The centroid is not shown for regions with $D\ge 0.2$—corresponding approximately to $n^\prime$ greater than $11$

1.3 Coarse graining properties of $D$

A remarkable property of $D$ is the possibility to control its magnitude as the time resolution of the time series is coarse grained. The choice of a certain time series resolution is somewhat arbitrary and dependent on the data available. It is therefore desirable to have indicators that are stable against changes in the accumulation time bin or, at least, that vary in a controllable way. RE allows us to set lower bounds for the error associated with the loss of information due to time coarse-graining. If $D_N$ is the RE estimated from rainfall data at high time resolution $\tilde{p}_j$ (e.g. daily, $N = 365$ or pentads, $N=73$), we can aggregate sequentially $\nu$ of the $\tilde{p}_j$ (e.g. $\nu =5$ for pentads) and obtain

$$\begin{aligned} p_i=\sum _{j=\nu i-\nu +1}^{\nu i} \tilde{p}_j \end{aligned}$$

(10)

with $i=1,\ldots M$ and $M=N/\nu$. By using the log sum inequality (Cover and Thomas 1991)

$$\begin{aligned} \sum _{i=1}^{n} a_i\log {\frac{a_i}{b_i}}\ge \left( \sum _{i=1}^{n} a_i \right) \log {\frac{\sum _{i=1}^n a_i}{\sum _{i=1}^n b_i}}, \quad a_i, b_i\ge 0 \end{aligned}$$

(11)

where the equality holds only if the $a_i$ and the $b_i$ do not depend on $i$, and from the definition (6) it follows that

$$\begin{aligned} \sum _{j=1}^{N} \tilde{p}_j \log _2{\frac{\tilde{p}_j}{\tilde{q}_j}} \ge \sum _{i=1}^{N/\nu } \left( \sum _{j=\nu i-\nu +1 }^{\nu i } \tilde{p}_j \right) \left( \log _2 \frac{\sum _{j=\nu i-\nu +1}^{\nu i} \tilde{p}_j}{\sum _{j=\nu i-\nu +1}^{\nu i} \tilde{q}_j} \right) =\sum _{i=1}^{M} p_i\log _2 (p_i/q_i) \end{aligned}$$

(12)

and therefore

$$\begin{aligned} D_{N} \ge D_{M} \quad \mathrm for \quad N\ge M. \end{aligned}$$

(13)

From the definition of the DSI $S$ in Sect. 2 (Eq. 3), it is obvious that also $S_N \ge S_M$ and so information about rainfall seasonality is lost in the upscaling procedure unless the values in each temporal bin are equal. In Fig. 3 the differences $D_{73}-D_{12}$ and $S_{73}-S_{12}$ are shown as an example.