Advances in Meteorology

Volume 2016, Article ID 7957490, 10 pages

http://dx.doi.org/10.1155/2016/7957490

## Markov Chain Decomposition of Monthly Rainfall into Daily Rainfall: Evaluation of Climate Change Impact

School of Civil, Environmental and Architectural Engineering, College of Engineering, Korea University, Seoul 136-713, Republic of Korea

Received 27 October 2015; Revised 4 April 2016; Accepted 7 April 2016

Academic Editor: Ji Chen

Copyright © 2016 Chulsang Yoo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study evaluates the effect of climate change on daily rainfall, especially on the mean number of wet days and the mean rainfall intensity. Assuming that the mechanism of daily rainfall occurrences follows the first-order Markov chain model, the possible changes in the transition probabilities are estimated by considering the climate change scenarios. Also, the change of the stationary probabilities of wet and dry day occurrences and finally the change in the number of wet days are derived for the comparison of current (1x CO_{2}) and 2x CO_{2} conditions. As a result of this study, the increase or decrease in the mean number of wet days was found to be not enough to explain all of the change in monthly rainfall amounts, so rainfall intensity should also be modified. The application to the Seoul weather station in Korea shows that about 30% of the total change in monthly rainfall amount can be explained by the change in the number of wet days and the remaining 70% by the change in the rainfall intensity. That is, as an effect of climate change, the increase in the rainfall intensity could be more significant than the increase in the wet days and, thus, the risk of flood will be much highly increased.

#### 1. Introduction

To assess the impacts of climate change on regional or local water resources, hydrological modeling with hypothetical climate input has been used. Earlier studies may be classified into two categories: one which used the General Circulation Models (GCMs) to predict the impact of climate change [1–5] and the other that was based on hydrological simulations with assumed hypothetical input to demonstrate changes of various components of the hydrological cycle [6–10].

Many researchers in Korea also have a great interest in the effect of climate change on water resources, agriculture, fishing industry, forestry, and so forth [11–15]. All of their research has been aimed at the time when the CO_{2} concentration becomes doubled. They also considered several GCM predictions for their research, where somewhat sophisticated interpolation or multiple regression techniques were used to scale down the GCM predictions to small scale information. Especially in meteorology, Oh and Hong [16] reported that the rainfall amount would be increased by 10 to 15% annually and by up to 24% seasonally (about a 10% increase in spring, 13% in summer, and 24% in fall and a slight decrease in winter). The annual mean temperature was also predicted to be increased by up to 3.5°C. Their results were derived from a multiple regression analysis of three GCM predictions (CCC, UI, and GFDL GCM). Also, KAIST [17] analyzed five GCM predictions (GFDL-R30, CCC, GISS, UKMO, and GFDL GCM) to estimate the possible changes of annual rainfall amounts by −5%~25% and those for the monthly rainfall amounts by −30%~35% as a result of the CO_{2} doubling.

Ideally, the climate simulations using the GCM predictions could be used to derive the impact of climate change on regional or local water resources. However, this issue is totally complicated by the incompatibility of space scales between hydrological models and the GCMs. While the GCMs are invaluable tools for identifying the climate sensitivities and changes in global climate characteristics, their grid system is too coarse to assess the impact of climate change on major hydrological parameters such as soil water, evapotranspiration, and runoff on regional scales [18, 19]. In addition, the GCM conceptualizations of atmospheric energy and moisture fluxes are limited by simplifications made in the parameterizations of cloud physics, energy transfer within the oceans, and land surface processes. Thus, some climate variables are better simulated than others. For example, mean air temperature is known to have higher precision than daily rainfall [20]. For this reason, the GCM outputs are interpreted as alternative climate scenarios rather than predictions [21].

Climate scenarios required for the assessment of water resources systems for the current condition (1x CO_{2}) are generally made using historic observations. If record length is limited, historic data are expanded using stochastic generation techniques [1, 20, 22–27]. Various statistical and stochastic characteristics are considered for the generation of the rainfall data. Examples are the wet and dry probabilities, the mean wet and dry periods, the mean number of wet days, the mean rainfall intensity, and so on. Similar factors are also considered for the generation of climate scenarios for 2x CO_{2} condition.

As mentioned before, the GCM simulations are interpreted as alternative climate scenarios to be used as input data for generating input data for hydrological analysis. Generally, the GCM simulations are those averaged spatially over large areas as well as temporally over monthly to annual bases. As air temperature has relatively low variability and is also known to have higher confidence among GCM simulations, direct use of them may not cause any serious effect on hydrological simulations. However, precipitation is totally different. Not only are the GCM simulations of precipitation for the current climate condition notoriously poor, but also the veracity of predictions for future changes in precipitation is in serious question [28–30]. Even worse is that hydrological extremes (i.e., floods and droughts) are closely related to the space-time variability of precipitation rather than its space-time average. As only the relative changes of precipitation amounts between GCM predictions of current and 2x CO_{2} conditions are generally considered for the generation of precipitation data for 2x CO_{2} condition, it is practically impossible to derive, in a quantitative manner, the possible changes of precipitation characteristics.

The above considerations have motivated us to assess the possible changes of the precipitation characteristics. To accomplish this research objective, we chose the daily rainfall for further analysis. This is mainly because much longer and accurate data are available for the daily based rainfall than for the hourly or even shorter-time based rainfall. It is also considered that the Markov chain model is generally applied to be the daily data rather than the hourly data. That is, in this study the occurrence mechanism of daily rainfall is assumed to follow the first-order Markov chain model [31–34].

To evaluate the impact of climate change on daily rainfall using the Markov chain model, we began by investigating the historic measurements of daily rainfall to quantify the transition characteristics between wet and dry days depending on the monthly rainfall amount. This is basically to estimate the changes of the transition probabilities of wet to wet, wet to dry, dry to wet, and dry to dry conditions due to the climate change. Also, climate change should be quantified as the changes of monthly rainfall amount, which are nothing but various GCM predictions for the condition of 2x CO_{2}. With estimated transition probabilities for the condition of 2x CO_{2}, we can also estimate the change in the number of wet days. Finally, the change in the number of wet days will lead us to know what the rainfall intensity should be. By comparing the number of wet days and the rainfall intensities for the 1x CO_{2} and 2x CO_{2} conditions, we may get some idea about how differently the patterns of flood or drought would be changed. That is, this study will lead us to know more detailed information about the rainfall pattern due to the climate change.

#### 2. A Markov Chain Model for Daily Rainfall Occurrence

The daily rainfall model based on a Markov chain decides rainfall occurrence (i.e., the wet or dry conditions) based on transition probabilities [31–35]. The transition probabilities, estimated from the historic measurements, represent the probabilities of wet to wet, wet to dry, dry to wet, and dry to dry conditions. If the next day is wet, then the rainfall intensity is given as a random variable following a probability density function. Exponential, Gamma, and mixed Gamma distributions are commonly used [36].

The daily rainfall model, based on the Markov chain, has several advantages. Easy parameter estimation and easy data generation are probably the most obvious advantages which make the Markov chain model more popular than the Poisson process model. Poisson process models have more complex structures as well as more difficulties in parameter estimation [37–40].

The daily rainfall model based on the Markov chain can be explained as follows. First, define as the wet or dry condition at the th day. That is,Also, the daily rainfall time series can be made by multiplying the rainfall intensity by such asThus, becomes zero when is zero (no rain or dry) and becomes when is one. Assuming that the occurrence probability of rainfall at present is dependent on the condition of the previous day, then follows the first-order Markov chain, and then the transition probability of daily rainfall can be divided into the following four cases: The above equations express the conditional probabilities of wet or dry on day depending on the condition of wet or dry on day . Therefore and . Also, these four probabilities constitute a transition probability matrix:

The transition probability matrix in (4), however, may not clearly quantify the characteristics of daily rainfall, especially when comparing the 1x CO_{2} and 2x CO_{2} conditions. This is because the possible change of the transition probability would be so small that no more than a 10% difference may be expected, even applying rather extreme GCM predictions for the 2x CO_{2} condition. Thus, we need a better statistic for comparing the 1x CO_{2} and 2x CO_{2} conditions.

A good statistic in this case may be the number of wet days, which is not only practical but also easier to compare. To derive the number of wet days, we first need to define the -step transition probabilities: The -step transition probabilities converge to certain probabilities as increases:These probabilities, and , are unique characteristics of a given system [41], which, in this case, represent the mean occurrence probabilities of dry and wet days, respectively. These are called the stationary probabilities or the extreme probabilities. Also, the inverse of these stationary probabilities represents the mean return period of dry and wet days; thus the multiplication of these stationary probabilities by the total number of days in a given month indicates the average number of dry and wet days, respectively.

#### 3. Data Characteristics

##### 3.1. Data

The data used in this study are the daily precipitation records obtained from the Seoul weather station. The Seoul weather station, as shown in Figure 1, is located in Seoul, Korea. Seoul is the capital city of Korea. It is located at the 37°34′ north latitude and the 126°57′ east longitude. Seoul receives an average annual precipitation of 1,370.5 mm. As Korea is located in the monsoon region of Far Eastern Asia, it receives more than 70% of its annual rainfall during wet summer season from June to September.