Meteorolojik verilerin ısıl değerlendirilmesi ve bilgisayarda simülasyonu

Abstract

- I - ÖZET Meteorolojik değişkenlerin bileşen olarak yer aldık ları sistem modelleri için girdi olarak kullanılabilecek değer lerin sağlanması amacıyla, kısa veya uzun süreli gözlem veya k ölçümlere dayanan simülasyon modelleri kurulabilir. Bu şekilde, sistem modelleri için hem yeterli uzunlukta veri elde edilebi lir, hem de meteorolojik değişkenler sınırlı sayıda parametreye sahip modeller olarak, ana modelde yer alabilir. Bu çalışmada, ısıtma problemlerine ait sistem model lerinde model bileşenleri olabilen, hava sıcaklığı, bağıl nem, güneş radyasyonu, rüzgâr şiddeti ve bulutluluk gibi meteorolo jik değişkenlere ait birer yıllık, saatlik ve günlük veriler kullanılarak, simülasyon modelleri kurulmuş ve yapay veri üre timi gerçekleştirilmiştir. Yılboyu ve ayboyu, saatlik ve günlük meteorolojik de ğerlerin oluşturduğu zaman serileri, deterministik ve stokastik bileşenlerine ayrılmış, meteorolojik verilerdeki peryodiklik har- monik analiz ile belirlenerek, durağan (stasyoner) stokastik de ğerler elde edilmiştir. Durağan stokastik seriler, otoregresif ve Monte Carlo süreçleriyle modellenerek, otoregresif süreçlerde model mertebe sinin otomatik olarak belirlenmesi için BIC kriteri kullanılmış tır.- II - Stokastik değerlerin simülasyon modellerinde, hava sıcaklığı ve güneş radyasyonu normal, bağıl nem ve bulutluluk beta, rüzgâr şiddeti Weibull dağılımıyla temsil edilmiştir. Simülasyon işleminin bilgisayar ile gerçekleştirilme si amacıyla, giriş verilerini kullanıcının yönlendirdiği doğrul tuda işleyerek, simülasyon modeline ait parametreleri bulan, ya pay veri üretimini gerçekleştiren, gerçek ve yapay verilerin karşılaştırmasını yapan, TISERS adlı genel amaçlı bir simülas yon programı geliştirilmiştir. Simülasyon sonucunda elde edilen yapay verilerle simü lasyon modellerinin geliştirilmesinde kullanılan gerçek veriler, yılboyu ve ayboyu, saatlik ve günlük değerlere ait ortalamalar, varyanslar ve frekans dağılımları bulunarak karşılaştırılmıştır. Ortalama değerler % 95 lik güven sınırları ile, frekans dağılım ları Kolmogorov-Smirnov testi ile istatistik anlamlılık yönünden test edilmiştir. Yılboyu ve ayboyu, günlük stokastik değerlerde, hava sıcaklığı hariç, zayıf bir persistans görülmüştür. Elde edilen. sonuçlar, normal dağılımlı verilerin yanısıra, normal dağılım dışı verilerce de oluşturulan durağan zaman serilerinin otoreg- resif süreçler ile modellenebileceğini ve yapay veri üretiminin bu modeller yardımı ile gerçekleştirilebileceğini göstermektedir. Saatlik değerlerin simülasyonu için, klâsik yaklaşımın yanısıra, ortalama gün, rasgele gün ve yüzdeler modeli gibi yaklaşı lar da denenmiş, bulutluluk hariç, diğer değişkenler için başarılı sonuçlar alınmıştır.

- Ill - SUMMARY The design and operation of system models, in which meteorological variables have to be included as model components, require adequate length of data to be used as inputs. In case, historical data are limited, they may be extended by synhetic generation through the use of stochastic simulation models. Simulation models eliminate the need for large data files within system models, instead models with limited number of parameters representing specific phenomena are included in such models. Generation of synthetic data may also produce critical values which may have never been recorded in the past, thus inclusion of a stochastic component within a system model, may imply variability in system's behaviour and in overall system's performance, causing changes in model' s prediction. The purpose of this study is to construct stochastic models for meteorological variables, such as air temperature, relative humidity, solar radiation, wind velocity and cloudiness, that are used as model components within system models developed for the study of heating problems, and to use these models for synthetic data generation. In literature there exists a good number of work on simulation. For example, hourly values of air temperature are simulated by Hansen and Driscoll [6] and daily values are modeled~ IV by Furman [7 J, Aprilesi and others [8 J. Bruhn and others' work is an example of Monte Carlo simulation of daily values of air temperature, solar radiation, relative humidity and precipitation [9]. References [lj, [3], [4 J, [5] are the works undertaken for the modeling and simulation of annual and monthly rainfall amounts. Time series modeling is a useful tool for understanding the behaviour and predicting the future values of natural phenomena. The additive mathematical form of such a model may be written as zt = Y(t) + H(t) + xfc where Y(t) and H(t) are the trend and periodic components respective` and x stands for the stochastic component and t represents the time [l6]. Y(t) and H(t) may well be represented by mathematical functions, such as the m th order polynomial may express the trend and the superimposed trigonometric functions may be used for the periodic part. The stochastic component may or may not be stationary and it may be modeled by linear schemes. The difference operator which is defined as v = xt - Vd where d is the degree of differencing, produces stationarity of x values [l5]. Delleur and Kavvas state that models based on differenced values can not be used for data generation since the differencing operation does not preserve the value of variance [4 J.- V - Autoregressive processes, which are linear stationary lodels, are commonly employed for the modeling of meteorological >henomena [7]. In an autoregressive scheme, x. is a linear aggregation )f past values of itself and a shock a, which may be written as / = h Vı +... + *P *t-P + at /here x ' s are deviations from mean value x of x ' s (i.e. x =x -x), t t t t I denotes the order of the scheme and § ' s are autoregressive >arameters. a values are called white noise and have normal t.. 2 lıstrıbutıon with zero mean and variance a. a Automatic order selection for an autoregressive scheme nay be achieved by Bayesian Information Criteria, which involves rhoosing the minimum value of quantities 2 BIC(p) = n £n a (p) + (p+l)£n n, p = 1,...,K 3. fhere K is the maximum potential order for the model [l2J. Estimation of autoregressive parameters may be through the Yule-Walker equations which involve the use of autocorrelation coefficients and details are given in reference [15J. A complete computer simulation program which processes the input data according to the user defined options, is developed to estimate the parameters of simulation model and to generate synthetic data. The program, named USERS, also compares the actual and simulated data with respect to mean responses, variability and frequency distributions.- VI - The data for this study consist of hourly values for a duration of one year, specifically for the year 1977, of air temperature (°C), relative humidity (%), wind velocity (knots) cloudiness (0-8) recorded at Yeşilköy Meteorological Station -2 -1 and solar radiation (cal. cm hr ) recorded at Florya Meteor< Station and both stations are located in Istanbul. The four simulation models that are used for synthetic generation of hourly values of the selected meteorological variables are explained below. In order to introduce simplicity in explanation and to avoid the repetition of common features in formulation, a procedure which is named as sim (...) will be discussed first. In using this procedure, it is assumed that, whether the time series has either trend or periodic component or both, is known beforehand. All forms of time series in this study are assumed to be trend-free. Procedure sim (Z) Given a time series Z={z : t=l,...,n}, this procedure involves separating, 1. the trend (y ), if it exists, by fitting m th order polynomial to z values by the method of least squares and/or, 2. the periodicity (h ), if it also exists, by fitting trigonometric functions to z -y values by harmonic analysis, to obtain the stochastic component. Then, 3. the stochastic values x ' s are modeled by a p th order autoregressive process. When p=0 the model reduces to a Monte Carlo process. With the help of model parameters and appropria statistical distribution, the stochastic values x'fs are generated ai gives the simulated values of time series.VII - First Model (Average Day Model) Given the actual daily values g..'s of a meteorological variable for every j th day in a month i, such that j=l,...,n., where n. is the number of days in that given month. Simulated daily values gî.'s are obtained by either applying sim(...) procedure on the year long actual daily values, which may be denoted as sim({gi-: i=l,...,12; j=l,...,ni>) or by applying the same procedure on each of the twelve distinct time series of month long actual daily values, separately, i.e. sim({gi. : ^=1,...^}), i=l,...,12 For each month i, month long actual hourly values s.., 's, where j denotes the days in that month and k denot-es the hours on any j th day, such that n < 24, are averaged over each of k hours by n. ^ik =./ Sijk/ni ' k=1`'->nk Then by step 1 and step 2 of procedure sim(...), trend and periodic values, y., Ts and h-, 's respectively, are calculated for the series {vik: k-1,...,^}. Trend and/or periodicity free time series consisting of n..n, stochastic values is formed by eijk = sijk`yik ~ hik. k = 1'--->nk for each j th day and by step 3 of procedure sim (...), the set {e/. : j=l,...,n.; k=l,...,n, } is generated. For each value of j /- VIII - xîjk = yik + hik + eîjk.. k = l `k and _ / o.. = E x!.. /il 1J k=l 1J are computed. Then s!., = x!.. - o.. + g!., k = 1,...,n. ıjk ıjk ıj öıj ' ' Ts. gives the simulated hourly values for month i. Above procedures are repeated for all i to obtain a complete set of s!., values. Second Model (Random Day Model) This model is actually similar to the first model, with respect to computation of simulated daily values gl.Ts, v., 's, simulated stochastic values e!' 's. The only distinction is as ıjk follows: For every j th day in a month i, a value I, such that 1 <^ l-^vi., is chosen randomly and for that j th day, Xljk = yik + hik + ei£k ' k = 1'-`.nk and n. o.. = E x!.. /n. 1J k=l ljk k are computed and then I s:., = x'... - o.. + g!., k = 1,...,n. ijk ijk ij &ij gives day long simulated hourly values.- IX - Repetition of the above procedure for all months produces the complete set of year long simulated hourly values. Third Model (Fragments Model) g..'s denoting the actual daily values of a meteorological variable, the simulated values g!,'s are generated as it is described in the first model. Then, for each of the j th day in the i th month, the sums n. u.. = E s.. fc-1 ij k=i ' ^ and for each k hour of the j th day, fragments are computed. f... = s... /u.. ıjk ıjk ij For a randomly chosen I value, such that 1 <^ I <_ n., s!., = n.. g!..f. nt ıjk k öıj ıilk is calculated for each k and this procedure is repeated for all j values in that month i. A complete set of simulated hourly values s!., is obtained by repetition of the above procedures for every month of the year. This model is originally due to G.G.Svanidze and it is used for the generation of monthly rainfall amounts from simulated annual rainfalls [5 J.- X - Fourth Model (Classical Model) Let s.., denote the value of a meteorological variable ljk at k hours of j th day in the i th month of the year. Then, the time series of year long hourly values {si : i=l,...,12; j=ls...,ni; k=l,,..,nk), where n. is the number of days in the i th month, or each of the twelve separate time series consisting of month long hourly values *Siik: J=1»---'ni; k=l,...,nkh i=l,...,12, are generated again by applying sim(...) procedure on the given series to give the simulated hourly values s!.,. ıjk This is the traditional approach used by many authors ir the existing literature. The frequency distributions of year long and month long, hourly and daily values are formed in order to test the goodness of fit of historical data to the theoretical distributions. Test is carried out by Kolmogorov-Smirnov one sample test statistics. Year long and month long daily values and month long hourly values of air temperature, relative humidity and wind velocity are fitte< fairly well by normal, beta and Weibull distributions. Year long and month long daily values of solar radiation and cloudiness are also representable well by the truncated normal and beta distribu tions, respectively, but frequencies of month long hourly values of these variables show discrepancies, in lower order frequency -2 -1 classes (^ 10 cal.cm hr ) for solar radiation in all months and in lowest (^ 2) and highest (^ 6) order classes for cloudines in some months.- XI - Analysis of month long actual hourly solar radiation values indicates that an exponential type of distribution may better represent these values. The above theoretical distributions are found to be inappropriate to represent the distributions of year long hourly values. Although some error is anticipated, month long hourly values of solar radiation and cloudiness are assumed to have normal and beta distributions, respectively. For all the meteorological variables, time series consisting of year long and month long, hourly and daily values are formed. The periodic components, present in these series are removed by harmonic analysis. Then the remaining stochastic values are modeled by autoregressive processes. For the hourly values maximum time lag was chosen to be 12 (for solar radiation it is 10). Maximum lag given for the' year long daily values was 6 and for the month long values, it was 3. Examination of autocorrelation and partial autocorrelation functions of year long and month long hourly values strongly justifies the use of autoregressive models. While autocorrelation values have a tail off tendency, partial autocorrelation coefficients show a cut off behaviour [l5]. In case of daily values this is not so and these values exhibit low persistence, except for the air temperature. Low or negligible persistence is a justification for the use of Monte Carlo process [5]. According to the procedure described as the fourth model, year long and month long, hourly and daily values are simulated by the use of governing parameters of autoregressive models, Zeroth order models are also employed in simulation runs.- XII - Actual data from which the models were developed are compared with simulated data. Frequency distributions of simulated data are formed to be tested for statistically significant differences with those of actual data. Test results show that the two groups of data compare favorably with few exceptions which may be of no importance, for the practical purposes. The mean values and the variances of year long and month long, hourly and daily values are also tabulated. They also compare well. Few of the mean values of simulated data lie outside the 95 % confidence limits derived from actual data, on the assumption that sample means are distributed normally as the sample size increases [3l]. The first, the second and the third models are also used for hourly data generation. Outcomes are compared with actual values in terms of means, variances and frequency distributions. They all indicate a good agreement between the historical and simulated data, except for the values of cloudiness, for which the maximum deviation percentages between frequency distributions are high for most of the months. As a final word, it will be concluded that not only the normally distributed variables, but the non-normally distributee ones too, may be well represented by autoregressive models, which may then be used for synthetic data generation.