PERIODICAL CHANGES OF PRECIPITATION AND THE NUMBER OF PRECIPITATION DAYS IN CRACOW

Transkrypt

1 PRACE GEOGRAFICZNE, zeszyt 115 Instytut Geografii i Gospodarki Przestrzennej UJ Kraków 2005 Robert Twardosz, Marta Cebulska PERIODICAL CHANGES OF PRECIPITATION AND THE NUMBER OF PRECIPITATION DAYS IN CRACOW Abstract: The article presents an attempt at assessing the usefulness of the method of Fourier s harmonic analysis to forecast precipitation. To achieve the purpose use was made of annual precipitation totals and of the number of precipitation days from a meteorological station of Astronomical Observatory in Cracow over Verification was also made of the accordance of precipitation values over obtained as an interference of statistically significant cycles (at an adopted level of 10%) on the basis of the Fourier method with (real) measured precipitation values. Key words: Fourier s harmonic analysis, periodicity, precipitation, Cracow Introduction Research in the evolution of climate has been intensified in recent years. Most often studied are thermic and precipitation trends. Forecasting of climate changes is not easy because only few stations are in possession of long measurement series of full observation documentation metadata and only some can be regarded as homogenous. Scientists take up various attempts at reconstructing climatic conditions in the past. For this purpose they use intermediate data covering a wide range of social and economic events to conclude on climate oscillations. Many expectations in climate reconstruction are associated with a rapid development of historical climatology. Great chances in the field of climate reconstruction are provided by rich sources of archival data. Such data from the territory of Europe were gathered and worked out by C. Pfister within the Programme CLIMHIST (Pfister et al., 1999). Attempts at such a reconstruction of the climate of Cracow were also taken by D. Limanówka (2001). The author, by using chronicles unique on a global scale in which recorded were data on weather phenomena, has reconstructed climatic conditions in part of

2 50 PRACE GEOGRAFICZNE, ZESZYT 115 the XVI th century. Attempts at reconstructing precipitation in Cracow were also taken by R. Twardosz (1999). For this purpose he made use of the number of days with precipitation. High values of the coefficient of correlation of the course of precipitation and of the number of days with precipitation made it possible for the author to reconstruct precipitation totals over Records in the observation books over contain gaps which causes that precipitation totals are not accounted for in the analyses. Therefore, the available continuous series of data starts with Owing to that reconstruction, the Cracow series of precipitation is one of the longest in Europe and on account of the homogeneity of the observation place and time it constitutes a perfect material for analyses and comparisons to know the past climatic conditions as well as to forecast in the future. Recent results of investigations of that series show the existence of fairy regular fluctuations with a 55-year repetition of the annual precipitation maxima (Twardosz 1999, Cebulska 2002). The earlier studies (Kożuchowski, Trepińska 1986) in which much more modest observation material was used also found a periodicity (25.7-year cycle of annual precipitation). Investigations by K. Kożuchowski (1985) in spatial variability of precipitation in Poland have shown the existence of 2-3 year, 10-year, and year cycles. The first comprehensive study of the variability of precipitation in an aspect of periodicity in Poland was carried out by Z. Kaczorowska (1962). The present paper is an attempt at assessing the usefulness of the Fourier s harmonic analysis method to forecast precipitation. Verification was also attempted of the accordance of precipitation amounts as an interference of statistically essential cycles with the measured values. Source materials and the method of studies To implement the purpose of the paper precipitation data from were employed from a research station Astronomical Observatory of the Department Climatology of the Institute of Geography and Spatial Management, Jagiellonian University. Two characteristics were adopted of precipitation for the purposes of the analysis. One of them are annual precipitation totals, the other annual number of days with precipitation ( 0,1 mm). The source material adopted for the analysis is homogenous as was proved by R. Twardosz (1997) on the basis of the used statistical Barlett s test. In order to find statistically essential periods the method of Fourier s harmonic analysis method was employed the use of which consists in approximating time series by means of trigonometric polynomials. Any given time series subject to periodical changes can be shown by employing the Fourier extension into a trigonometric series in the form; ); (1)

3 PERIODICAL CHANGES OF PRECIPITATION where: y(t) function approximating given measurement sequence; mean value of the approximated sequence; k degree of trigonometric polynomial fulfilling the relationship 2k+1<N;, for n odd for n even, n numerical force of sequence, a i, b i Euler-Fourier coefficients as calculated on the basis of the formula: 2ϖji 2ϖji ; for i = 1, 2,..., k; j = 1,2,...,n where: y j (t j ) successive values of the sequence. The analysis of the cyclicity of the relevant sequences was begun by presenting the course of their real variability in the shape of the described trigonometric polynomial, which leads to obtain a set of harmonic components. The next stage included a selection from that set of statistically essential harmonic components on the adopted level of 10%. For this purpose use was made of the Fisher-Snedecor test by calculating its statistics with n 1 = 2 and n 2 = 3 degrees of freedom from the formula (2), by comparing it with critical values determined on the basis of the positions by R. and W. Zieliński (1990). in which: n numerical force of series; R coefficient of multiple correlation as calculated from the formula; (2) ; where: S 2 variance of an investigated sequence; ε 2 residual variance.

4 52 PRACE GEOGRAFICZNE, ZESZYT 115 Analysis of the results obtained On analysing the changeability of annual precipitation over in Cracow (Fig. 1) it is possible to notice a cyclicity of some 60 years (Tab. 1). In the analysed sequence of precipitation a clear repeatability is conspicuous of the largest values of some 1000 mm, successfully in the years: 1855, 1912, In turn, the values of the lowest precipitation i.e. some 450 mm appeared in 1859, 1920, 1981, which confirms the statistically essential period of 56.3 years obtained in the harmonic analysis. The same statistically essential period was designated in the case of the annual number of days with precipitation despite the fact that the extreme values of these Fig. 1. The course of annual precipitation totals in Cracow over with a forecast of precipitation amounts over on the basis of Fourier analysis Ryc. 1. Przebieg rocznych opadów atmosferycznych w Krakowie w latach wraz z prognozą wartości opadów w latach na podstawie analizy Fouriera explanations: 1 observation data, 2 interference of statistically significant cycles at a level of 10%, 3 forecast on the basis of Fourier analysis over objaśnienia: 1 dane obserwacyjne, 2 interferencja istotnych statystycznie cykli na poziomie 10%, 3 prognoza na podstawie analizy Fouriera w latach

5 PERIODICAL CHANGES OF PRECIPITATION characteristics do not coincide over time (Fig.2). As far as the number of days with precipitation is concerned the highest values i.e. some precipitation days appeared in successive years: 1843, 1899, Besides, in the course of variabilities of the compared characteristics also designated were identical long statistically essential cycles of 42.3 and 33.8 years as well as short cycles of some 3 years. The periods of 12.1 years of annual precipitation and of 8.89-year of the annual number of days with precipitation can be recognized as cycles revealing a relationship with changeable solar activity. The course of precipitation and of the number of days with precipitation shown in Fig. 1 and 2 was additionally enriched by an interference of statistically essential Fig. 2. The course of the annual number of days with precipitation in Cracow over with a forecast of precipitation amounts over on the basis of Fourier analysis Ryc. 2. Przebieg rocznej liczby dni z opadem w Krakowie w latach wraz z prognozą wartości opadów w latach na podstawie analizy Fouriera explanations: 1 observation data, 2 interference of statistically significant cycles at a level of 10%, 3 - forecast on the basis of Fourier analysis over objaśnienia: 1 dane obserwacyjne, 2 interferencja istotnych statystycznie cykli na poziomie 10%, 3 prognoza na podstawie analizy Fouriera w latach

6 54 PRACE GEOGRAFICZNE, ZESZYT 115 Tab. 1. Statistically significant component cycles of the annual precipitation totals and the number of days with precipitation in Cracow over Tab.1 Istotne statystycznie cykle składowe rocznych sum opadów oraz rocznej liczby dni z opadem w Krakowie w latach (on the adopted level of 10%) harmonic components designated in the Fourier analysis on the basis of sequences reduced to the period and by a forecast of considered variables, as obtained on the basis of the found statistically significant cycles. On comparing the results obtained, especially the values forecast over with values drawn from instrumental observations it is possible to note that: on the adopted level of significance (10%) the calculated (on the basis of putting together the statistically significant cycles) annual precipitation totals reveal a good fitting to the measured values. It was calculated that in the relevant period the real precipitation values are higher by some 9% with regard to the forecast values, in the case of the annual number of days with precipitation the result obtained is not sufficiently satisfactory on account of considerable differences (on average 16%) between the real and forecast values in particular years of the period Conclusions On account of the length and homogeneity of the observation place the Cracow series of precipitation constitutes a perfect material for analyses and comparisons serving the knowledge of the past climatic conditions. The present article presents exemplary applications of harmonic analysis to the description of a periodical variability of time series. The results obtained may confirm the faults and limitations of the method shown by other authors which postulates among other things the completeness of the analysed sequences and assumes a priori lengths of the periods resulting from the length of the investigated series. In such a situation it is necessary to verify the obtained results through the application of other tools of cyclicity analysis e.g. spectral analysis, the J. Boryczka s method of regression sinusoids, the method of Falk s analysis and others.

7 PERIODICAL CHANGES OF PRECIPITATION References Cebulska M., 2002, Wykorzystanie analizy harmonicznej do badania okresowości rzeczywistych ciągów opadowych w wybranych stacjach dorzecza górnej Wisły, Czasop. Techniczne, PK, 4, Kraków, Kaczorowska Z., 1962, Opady w Polsce w przekroju wieloletnim, Instytut Geografii PAN, Prace Geogr., 33, Kożuchowski K., 1985, Zmienność opadów atmosferycznych w Polsce w stuleciu , Acta Geographica Lodziensia, 48, Kożuchowski K., Trepińska J., 1986, Fluktuacje opadów atmosferycznych w Krakowie w okresie , Zeszyty Naukowe UJ, Prace Geogr., 64, Limanówka D., 2001, Rekonstrukcja warunków klimatycznych Krakowa w pierwszej połowie XVI wieku, Materiały Badawcze IMGW, seria Meteorologia, 33, Pfister C., Brazdil R., Glaser R., A. Bokwa., Holawe F., Limanówka D., Kotyza O., Munzar J., Racz L., Strommer G., Schwarz-Zanetti G., 1999, Daily weather observations in sixteenthcentury Europe, Climatic Change 43, Twardosz R., 1997, Homogenizacja serii pomiarów opadów atmosferycznych na stacji meteorologicznej w Krakowie, [w:] J. Trepińska (red.) Wahania klimatu w Krakowie ( ), Instytut Geografii UJ, Kraków, Twardosz R., 1999, Warunki pluwialne w Krakowie w latach , Czasop. Geogr., 2, Zieliński R., Zieliński W., 1990, Tablice statystyczne, PWN, Warszawa. Zmiany okresowe opadów atmosferycznych i liczby dni z opadem w Krakowie Streszczenie W artykule podjęto się próby oceny przydatności metody analizy harmonicznej Fouriera do prognozowania opadów atmosferycznych. Dokonano także weryfikacji zgodności wartości opadów uzyskanymi jako interferencja istotnych statystycznie cykli (na przyjętym poziomie 10%) na podstawie tej metody z wartościami pomierzonymi. Do realizacji celu wykorzystano roczne wartości sum i liczby dni z opadem ( 0,1 mm) ze stacji naukowej ("Obserwatorium Astronomiczne") Zakładu Klimatologii Instytutu Geografii i Gospodarki Przestrzennej UJ w Krakowie z lat W przebiegu wieloletnim obu charakterystyk opadów (Ryc. 1) wykryto kilka istotnych statystycznie cykli (tab. 1). Wykazano dobre dopasowanie obliczonych (prognozowanych) na podstawie założenia istotnych statystycznie cykli na poziomie 10% sum rocznych opadów w latach do wartości zmierzonych opadów (rzeczywistych) w tym okresie. W przypadku rocznej

8 56 PRACE GEOGRAFICZNE, ZESZYT 115 liczby dni z opadem, otrzymany wynik nie jest dostatecznie zadowalający, ze względu na znaczne różnice rzeczywistych i obliczonych wartości w poszczególnych latach okresu Robert Twardosz Department of Climatology Institute of Geography and Spatial Management Jagiellonian University Cracow Poland Marta Cebulska Institute of Water Engineering and Water Management Cracow University of Technology Poland