Droughts century and concluding whether there has been

 Droughts are caused by not only lack of precipitation and high temperatures but by overuse and overpopulation. The aim of this report is to analyse meteorological drought using PDI and SPI.  This report will shine light onto how the standardized precipitation index (SPI) is calculated using cumulative probability and will aim to answer the question of whether there is a trend between a higher SPI and the likelihood of droughts.                         ?     1. Introduction  This report is the result of the final year project at the University of Salford for the course of applied Mathematics in the school of computing, science and engineering. It will discuss what meteorological droughts consist of and will look to assess meteorological droughts in three states (Colorado, Texas and California) of the USA using the standardised precipitation index and the palmer drought severity index over the past century and concluding whether there has been an increase in droughts over the past century. Since there are many indices to monitor droughts and to examine the severity if droughts, it would have been extremely difficult to focus on every single one of them, therefore two indices have been chosen (SPI and PDSI) given the timescale to complete this paper.  The standardised precipitation index (SPI) was developed by Mckee et al in 1993. It is a powerful, flexible index that very simple to calculate, the required input is precipitation data and allows an analyst to analyse not just wet cycles but dry periods also.  According to Guttman (1994), data that spans over 50 years is optimal and preferred in the use of the SPI. In 2009 a workshop was held at the university of Nebraska-Lincoln, United States of America, where 54 participants took part representing 22 countries from all over the world in which they reviewed the drought indices currently in use in different parts of the world to explain the different kinds of droughts and discussed the need for consensus standard indices to describe different types of droughts. The experts at the meeting recommended that the Standardised precipitation index (SPI) be used by all National Meteorological and Hydrological Services (NMHSs) around the world to examine meteorological droughts.  The Palmer drought index (PDSI) developed In 1965 by W.Palmer incorporates antecedent precipitation, moisture supply and moisture demand. It is used as a tool to monitor drought severity and a particular index value is often a clue to start preparations or discontinue elements  of a drought contingency plan. The PDSI is usually considered to be very useful primarily for agricultural drought as it takes into account the soil moisture levels which in turn can be good or bad for crops depending on the level of moisture levels. It is also beneficial for other water uses that are sensitive to soil moisture. The PDSI was intended to look at wet and dry conditions from a water balance viewpoint.  Variations of the index include the modified PDSI, which is designed for real time use, the Palmer Hydrologic drought index, which is primarily used for water supply monitoring, and the Z index, which is a measure for an individual month’s wetness or dryness.  The PDSI and its variations have been used widely for monitoring droughts.     2. Literature Review Many studies indicate that the frequency of droughts have increased in the 20th century (Dai, 2012), (Frich et al, 2002). Not only are droughts occurring more often but they are costing vasts amount of money to the USA. According to AghaKouchak  (2012), the droughts experienced by Texas in 2006 and 2011 resulted in billions of dollars in economic losses. In fact, droughts in the USA accounted for 25% of all weather related losses, accumulating a total of $180 billion between 1980 and 2010 (Hayes, 2012).  Wen Yang and many other researchers have concluded that drought can be divided into many different types such as, meteorological drought, agricultural drought, hydrological drought and socioeconomic drought. This study will mainly focus on hydrological and meteorological droughts. Estimation of drought characteristics and return period of droughts of various lengths is extremely important in drought forecasting and management. Dilek Eren Akyuzet al applied first and second order Markov chain models to dry and wet periods of annual streamflow series to reproduce stochastic structure of hydrological droughts and concluded that successive wet and dry periods of a time series of streamflows were usually considered to follow a first order Markov chain model (MC1). However, the first order Markov chain model is not adequate when auto correlation of the original hydrological series is high.  A study was carried out on the use of standardized precipitation index (SPI) for drought severity assessment, in this study monthly rainfall data from June to October for 39 years was used to generate the standardize precipitation index values for Gamma distribution for a low and a high rainfall district of Andhra Pradesh state, India. This study concluded that the SPI tended to under estimate the intensity of dryness and wetness when the rainfall was low or high respectively.  Gutmman (1998) compared historical time series of the PDI with time series of the corresponding SPI through spectral analysis. Guttman concluded that the spectral characteristics of the PDI vary from site to site through the U.S., while those of the SPI do not vary from site to site. Furthermore, it was concluded that the PDI had a complex structure with an exceptionally long memory, while the SPI was easily interpreted, simple moving average process.  Alley (1984) published a paper on the “limitations and assumptions” of the Palmer drought index and concluded that the PDSI addresses two properties of droughts, their intensity and the start and end times, however it uses rather arbitrary rules in quantifying these properties. Furthermore, the methodology is based on very limited comparisons and therefore is only weakly justified on statistical grounds. In addition to this, since the PDSI relies on the previous month’s index it is therefore bimodal.              3. Theory 4  Methodology  4.1 SPI Defined The standardized precipitation index (SPI) was developed in order to define and monitor droughts. It allows an analyst to determine a very wet period or a very dry period at a given timescale for any rainfall station with data collected from that particular station. One can not however predict droughts using SPI but in some cases it may assist in the prediction of future droughts. The SPI is based on the cumulative probability of a given rainfall event occurring at a station and then the data is fitted to a gamma distribution, as the gamma distribution has been found to fit the precipitation distribution pretty well (Thom 1966). All this is done through a process of maximum likelihood estimation of the parameters, and . This then allows the rainfall distribution to be represented by a mathematical cumulative probability function. Therefore, an analyst can then tell whether the probability of the rainfall will be less than or equal to certain amount. Consequently, the probability of rainfall being less than or equal to the average rainfall for that area will be roughly  0.5, whilst the probability of rainfall being less than or equal to an amount much smaller than the average will be lower.  This means that if the probability is low on the cumulative function then this indicates a likely drought event as rainfall will be less than average and on the other hand, if the probability is high then this means that period was very wet.  McKee et al (1993) developed the SPI and since then, Colorado climate centre amongst many others use the SPI to monitor current states of drought in the USA. 4.1.1 The Gamma Distribution The gamma distribution is defined by it probability density function:       for  x>0  where:  ?                           ?? is a shape parameter        ?                                                ? is a scale parameter x?x is the precipitation amount         ?(?) is the gamma function   To fit the distribution to the data needs ? and ? to be estimated. Edwards and McKee (1997) suggested using maximum likelihood estimation to achieve an approximation for the parameters:  ?= ?= A=  Where n represents the number of observations. If the probability density function is integrated with respect to x, this leads to the cumulative probability function:  G(x)= Letting , the equation becomes:  G(x)= The gamma function is undefined for x=0 and since it is possible to have zero values in the sample set, the cumulative probability function is modified as:    where q is the probability of zero. After this, the cumulative probability function is transformed to the standard normal distribution to obtain the SPI, where the mean is zero and the variance is one.  Abramowiz and Stegun (1965) proposed the following approximation in order to simplify the calculations for the SPI:  Z=SPI= , Where t= For 0

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