Yağış verilerinin autorun analizi
Autorun analysis of precipitation data
Yazar: KASIM KOÇAK
Danışman: PROF.DR. EREN OMAY
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Meteoroloji Mühendisliği Ana Bilim Dalı
Derece: Yüksek Lisans, 1992
Konu: Meteoroloji = Meteorology
Dizin: Autorun analizi = Autorun analysis ; Yağışlar = Rainfall

Özet:
Yağış; atmosferdeki nemin, kati veya sıvı fazda yeryüzüne dönmesidir. Yeryüzündeki tatlı suyun ana kaynağı olan yağışın dağılımı, zamana ve yere bağlı olarak büyük değişim gösterir- Aynı değişim yağışın şiddeti ve mik tarı için de geçerlidir. Oldukça karmaşık fiziksel süreçlerin ürünü olan yağış, hidrolojik çevrimin ve bir bölgenin ikliminin en önemli bileşenlerinden biridir. Yağış ya da belirli dönemlerde yaşanan kuraklık genel olarak canlı yaşamını, özellikle de insanoğlunun pek çok etkinliklerini doğrudan etkiler. Verilen bir bölgeye gelecekte ne kadar bir yağışın düşeceği, konuyla ilgili bilim adamlarının temel hedefi olmasına rağmen, halen öngörüsü en zor meteorolojik parametrelerden biridir. Yağışın diğer meteorolojik parametrelerle ilişkisi araştırılarak, bu parametrelere olan bağımlılığı ve bunun sonucu olarak dolaylı tahmini yapılabileceği gibi, yağış verilerini bir dizi olarak ele alıp çeşitli istatistiksel yöntemler uygulayarak ta bazı sonuçlara varmak mümkündür. Bu öngörüler, gerçekte atmosferin ortalama veya bazı Klimatolojik olayların gerçekleşme olasılığını gösterir. Hidrolojide, kurak ve yağışlı dönemlerin bilinmesi, su lama, su temini, taşkın kontrolü, kuraklığın önlenmesi ve su yapılarının planlanmasında Önemli rol oynar. Bir planlamacı, yağışlı ve kurak dönemlerin ortalama peryodunu, kurak bir dönemin maksimum uzunluğunu ve yağışlı bir peryot süresince meydana gelecek su miktarını bilmeyi; en azından güvenilir bir şekilde tahmin etmeyi ister. Bu tez çalışması, persistansin teorik olarak incelenmesinde kullanılan otokorelasyona benzer bir yöntem olan "autorun analizine" bazı katkıları ve Göztepe yağış verileri kullanılarak yapılan bir uygulamayı içer mektedir. Ayrıca Markov Olasılık Modeli kullanılarak hesaplanan olasılıklar, özellikle tarım, endüstri, inşaat, turizm ve spor alanlarında kullanılabilecek önemli bir bilgi niteliği taşımaktadır. 111

Abstract:
This study is mainly concerned with the investigation of sequential properties of hydrologic series on the basis of wet and dry spells which are directly related to the run properties. In hydrology literature it is a well- known fact that if the hydrologic series considered is dependent, then the high values tend to fallow high val ues and low values tend to fallow low values. This last statement can be interpreted in a different way by saying that in the case of dependent series, the periods of wet and dry spells tend to be grater than that in the case of independent series. Such a property has been termed "persistence" in hydrology. Hydrologists have made vari ous attempts to measure this "persistence" mainly by three analyses which are ; 1- Autocorrelation Analysis, 2- Spectral Analysis, 3- Rescaled-Range Analysis. Each one of these analyses depicts different sequential properties of the series considered. For instance, the autocorrelation analysis is a means of measuring linear dependence between any two observations. As stated by various researchers autocorrelation analysis is by no means a general measure of dependence. ûn the other hand, the spectral analysis of sequential patterns in se ries by the classical periodogram methot has dominated hydrology for years. Finally, the resacaled-range analy sis, was introduced into hydrology by H.E. Hurst (1956), has the advantage of being comparatively more robust than any other analysis. In other words, the resacaled-range analysis is not very sensitive to the form of mariginal probability distribution function (PDF) (Şen, 1978). In this study, a new mode of analysis which is referred to as "autorun technique" has been worked on. One of the principal problems in the theory of time series analysis has been the measurement of the relationship between any, two observations which are k time units apart from each other. The solution to such a problem has been achieved through autocorrelation analysis wich already appears to be one of the classical statistical technicques. The information on wet and dry spells of a hydrologic series plays a vital role in the design and operation of water- resource system. The planner would like to know, or at ivleast to predict reliably, the average periods of wet and dry spells, the maximum length of a dry spells, the amount of water over a wet period. Each one of these quantities is very important in hydrology. None of the previously mentioned analyses is capable of providing direct information about these quantities. From a classical point of view, a run is defined as a succession of the same kind of observations preceded and succeeded by at least single observations of different kind. For example, given a sequence of observations, X^'s, and a constant level, Xq, two kinds of observations relative to this constant level occur. One kind of observation is positive deviation X^-X0>0, which represents a water surplus in that particular period,!, and the other kind is a negative deviation, X^-XqIO, representing water deficit. More specifically, a run made up of succesive water surplusses is referred to as a wet period in hydrology and a positive run in statistics. Similarly, a run of water deficit is referred to as either a dry or drought period in water engineering and a negative run in statistics. In hydrology, the most important properties of runs are their lengths, such as positive run lenght. Up, and negative run lenght, rijj. Feller (1967) has given the expected values of positive and negative run lenghts of an independent sequence of infinite length as E(np) = i/q and Ed^) = 1/p (D,(2) where p = F(x£>Xq) and q = F(x^ xûjxi-1>x0) (4) In the following, r^ will be referred to ası the lag-1 au- torun coefficient. It might sometimes be convenient to write eq. (4) in terms of joint probability of x^ and xi-l as: rx = [P(xi>xû,xi-1>3c0)]/[P(xi>xû)] (5) The autorun coefficient can be defined at any truncation level, but for practical purposes it will be taken as equal to the median, m, of the sequence considered. Hence, the final form of the autorun coefficient, can be written from eqs. (4) and (5) as: ri = P(x^>mjx^_^>Hi) = 2P(x^>m,Xj_i>m) (6) similarly, the lag-k autorun coefficient can be defined as: r^ - P(xi>m|Xi_jc>m) = 2P(xi>m,Xi_jt>m) (7) from the definition of probability it is obvious that Ûlr^i.1. In the case of purely independent observations, whatever the underlying PDF, the conditional probability in eq. (7) becomes equal to 1/2. Therefore, r-^ = Û.5 shows the fact that the two observations, separated by lag-k, are independent of each other, on the other hand, if observations are perpectly correlated, this is equiva lent to saying that the conditional probability in eq. (7) becomes equal to 1, then rj^ = 1. Apart from these two extreme situation, r^ theoretically assumes any value between û and 1. The value of r^ at the origin, that is for k = 0, is equal to 1. Moreover, the plot of r^ vs. k is called the autorun function. In the case of an independent process the theoretical autorun function becomes : r0 = i ; rk = 1/2 (8) The population autorun coefficient has been defined by eq. (6). The necessary information for evaluating the population autorun coefficient is the joint PDF of obser vations lag-k apart. Unfortunately, in practice, the PDF vxX is not available - instead a sequence of past observa tions is at hand- Hence, the autorun coefficient have to be estimated from hitoric data. An estimate,3^, of r^ can simply be proposed by considering eq (7) together with the classical definition of probability in text books. Acording to this definition, the probability, P(A), of an event A is found by counting the total num ber, N, of possible alternatives of the event. If H^ in of these alternatives the event A occurs, then P(A) is given by: P(A) = NA/K (9) In eq. (7) event A corresponds to the joint event (Xi>«jx^_j->m). Hence, in a sequence of n observations, there exist n-k possible alternatives for two observations lag-k apart being simultaneously grater than m. As a result, if number of the events (x'£>tt|Xj>_]£>m) in a given sequence of length n is n^, then from eq. (9) : P(acj>>n|Xi.|C.>ia) = nk/(n-k) (10) The subtitution of which in eq. (7) leads to the small sample estimate of rj^ as: r^ = 2 nk/(n-k) (11) The nominator of eq. (11) is an integer-valued random variables, whereas the denominator is a fixed value. Hence, the random characteristics of the estimate r^ can directly be obtained from the characteristics of RV nj^. On the basis of the frequency interpretation of the prob ability, the estimate r^ can be calculated by successive execution of the following steps: 1- The median, m, of a given sequence x^,X2j.../Xn is calculated. 2- The series is truncated at the level of m; hence, sequence of surpluses (x^>m) and deficits (x^lm) are obtained. 3- The number, n^, of overlapping successive surplus pairs (observations lag-k apart) are calculated. 4- The estimate of r^ is then calculated from eq. (11). It is interesting to point out at this stage that the calculations in aforementioned four steps are all distri bution-free. Contrary to the autocorreletion analysis, the autorun analysis does not distort the dependence structure of sequence considered. On the other hand, the autorun coefficients are easier to calculate than the au tocorrelation coefficients. Furthermore, autorun analysis is directly related to run properties which play an effective role in various water-engineering problems. ~"r:"l vxiConfidence limits for autorun test are given as (Şen, 1979) : CL(rk) = 1/2 + ta^Cn-k)]-1/2 (12) In practice, the most commonly employed significance level is a=0.05 which correspond to ta=1.645 as normal deviate. Therefore, for % 5 significance level, the con- f idece limits become: CL(rk) = 1/2 ± 1.163(n-k)-1/2 (13) If rk lies within the limits, then the hypothesis that the sequence is generrated by a purely random process is accepted, otherwise it is rejected. To decide on a bet ter dependence of a series considered the lag-1 autorun coefficient plays an important role. If it falls outside the confidence limits then the strucrural independence of series is rejected at the % 5 level. On the other hand, the runograms are capable of reflecting the periodic structure of seasonal series such as monthly precipita tion, flow sequences, etc. The following conclusion can be drawn: 1- Estimates of the autorun coefficient are unbiased and consistent. 2- As far as the independence or dependence of a series is cocerned both autorun and autocorrelation analyses yield to the same result. 3- The autorun analy sis is capable of depicting the perodicity in the series such as montl y precipitation. 4- The autorun analysis is directly related to the probability of high value to fal low high value and therefore it seems to be more reveal ing to the meteorologist and hydro logist about the per sistence. Autorun technique, which has been explained above, was applied to the daily precipitation data (Göztepe, 1961- 1990). During the application, it was seen that the autorun technique had given the best result for the monthly and annual precipitation data, but some problems were revealed when daily precipitation data had been used. Therefore, in this study, Eq. (11) and (13) have been rearranged in order to obtain a general form. In the Third Chapter, by using the first order Markov chain model, the transitive probabilities of wet and dry day sequences were calculated. These probabilities can be used in the following activities: - Agriculture, - Construction, - Industry, - Tourism, - Some outdoor community activities such as fairs, pa rades, athletic events etc. vxxx