Application of Harmonic Analysis in the Preliminary Prediction of Air Temperature over Lagos and Abuja, Nigeria

Harmonic Analysis technique has been employed in predicting the hourly air temperature variations over Lagos and Abuja, Nigeria. The variations in hourly air temperatures over the two stations are periodic and thus have strong tendency of being repeated the next day, if all other atmospheric variables are constant. It was observed that the variation in hourly air temperature in the two stations is dominated by the first harmonic, thus it fluctuates by one cycle with a period of 24 hours. Invariably, harmonic equations could be applied to hourly temperature prediction even on a large scale data. The maximum hourly air temperature occurred two hours on the average after the maximum solar irradiance has occurred in each station. It was found that the temperature of the air at a particular hour is dependent on that of the previous hour.


INTRODUCTION
The climate system adjusts when one or more of external factors change, for example, global average temperatures would be expected to increase with an increase in solar output. Climatic predictions are made using climate system models as the Atmosphere-Ocean general circulation models (AOGCMs) (Cornwell and Danny Harvey, 2008;Gregory et al, Research Article 2001 andSokolov et al, 2001). These are mathematical expressions of the thermodynamics; fluid motions; chemical reactions; and radiative transfer of the complete climate system that are as comprehensive as allowed by computational feasibility and scientific understanding of their formulation. The ultimate aim is to model as much as possible the climate system, especially the complex feedbacks among the various components.
A number of models are in operation in various research institutes and universities Worldwide. Although the models are based on the same laws of physics, each has different ways of dealing with processes that cannot be represented explicitly by physical laws, such as formation of clouds and precipitation. Variations in these parameterizations lead to different regional projections of climate change, particularly for precipitation.
AOGCMs as well as many other climatic models cannot simulate all aspects of climate. The AOGCMs in particular have large uncertainties associated with clouds, it has difficulty portraying accurately precipitation patterns in mountainous regions and resolving important synoptic weather features (such as Mesoscale Convective Systems) that strongly influence precipitation patterns and amounts in many agricultural regions (Ebi and Mearns, 2002).
In the past, some researchers had employed harmonic analysis technique in studying temperature variations but none has been known to be carried out in Nigeria, especially with the Nigerian Environmental Climatic Observing Program (NECOP), real time data. Lidija (2007) used harmonic analysis to investigate the seasonal cycle in temperature at five locations over the Mount Biokovo region (Croatia): namely, Makarska, Opuzen, estanovac, Imotski, and Vrgorac. The monthly averages of temperature as well as monthly means of minimum and maximum temperatures from 1961 to 1980 were subjected to harmonic analysis. The results were reported to have a good implication for botanical investigations. Atsu and David (2000) used mean monthly average daily values of global irradiation for seven stations in Oman to develop harmonic models. Pertinent amplitudes and phase angles were obtained for each station. The results show the dominance of the first two harmonics in the southern stations, Salalah and Marmul and the dominance of only the first harmonic in the north, Buraimi and Seeb.
The mean annual variations of the air temperature over European and Mediterranean area have been studied using Harmonic Analysis. Basic data consist of the mean monthly values of air temperature from 1961-1990. It was found that the first 2 harmonic terms contribute, altogether, to the total variance of over 95%. (Makrogiannis and Balafoutis, 2001).
Harmonic analysis along side with other statistical methods has also been applied to study seasonal variability of the air temperature at Mlynany using mean hourly, monthly and annual values of the air temperature during the period of 1962-2002. The results showed that the air temperature trend at Mlynany has an increasing tendency and that the mean annual air temperature increased by about 1.4 °C during the investigated period, i.e., approximately by 0.34 °C per decade (Ostrozlik, 2003).
In a recent study, air-water temperature relationship in the Illinois River (Peoria) was studied using Harmonic analysis. Its application to daily mean air, water temperature records for this location, indicates that the first harmonic accounts for a major portion of the total variance in the records. It was discovered that water temperature residuals are well correlated with air temperature residuals. This result enabled the development of a mathematical model whose parametric values were used for predicting water temperatures from air temperature records and this was seen to be stable from year to year (Kothandaraman, 2007). According to FAA (1975), the amount of solar energy received by any region of the earth varies with time of day, with seasons, and with latitude. These differences in solar energy create temperature variations. These temperature variations create forces that drive the atmosphere in its endless motions.
Since the challenges of climate change persist globally, the need for climatic models and better statistical methods for analyzing climatic variables will continue to grow. This study in its own way of improving on existing methods of analyzing climatic variables, used the descriptive statistics (e.g. mean) in the analysis, as well as developed a computer program for predicting the average hourly temperature over Lagos and Abuja, Nigeria. The identification and implementation of methods that will effectively enhance the study of atmospheric variables has posed a challenge over time. Thus the aim of this work is to employ the use of harmonic analysis technique in fitting a periodic function or model to carry out a preliminary prediction of hour to hour air temperature over Lagos and Abuja, Nigeria. Hence the work will help in providing useful information that will improve human and agricultural activities in these areas especially in this era of global erratic climatic change.

SOURCE OF DATA AND METHOD OF ANALYSIS
Lagos is in the south western part of Nigeria and lies approximately between longitude 2° 42' and 3° 22' east and between latitude 6° 22' and 6° 42' north. It stretches over along the guinea coast of the Bight of Benin on the Atlantic Ocean. On the other hand, Abuja is in the northern part of Nigeria at latitude 8° 56'north and longitude 7° 06'east. So the locations of these two cities make them liable to climatic variations due to ocean and desert effects respectively The air temperature and solar irradiance data used for this work were obtained from Nigerian Environmental Climatic Observing Program (NECOP). NECOP is a new programme, about three years old, designed to establish a network of meteorological and climatological observing stations spatially located across Nigeria. NECOP's main objectives among other things is to make real time data available for meteorological and climatological research which will serve as a veritable tool for decision makers involved in emergency management, natural resources management, transportation and agriculture. The size of the NECOP real time data obtained in these stations is small; Abuja and Lagos have 8 and 12 months' data respectively. This does not allow for a long time climatic prediction of the area, hence, this research serves as a preliminary investigation to the climatic prediction in these areas.

Fundamental Theoretical Concepts
This work was carried out based on the harmonics series equation adapted from Panofsky and Brier (1960): where N is the number of observations, the time series, = t X the time series, X = arithmetic mean, P= period of observation, (2) i A and i B are coefficients X = Observed value, t = time and i = number of harmonics.
In other words the time series equals the mean plus the sum of all N/2 harmonics.The first harmonic (or fundamental) has a period equal to the total period studied. The second harmonic has a period equal to half the fundamental period, the third harmonic a period of one -third of the fundamental and so on. It is not always required to determine all the N/2 harmonics; usually the first two, or at most three, harmonics describe the variation of the periodic function sufficiently well. The equation (1) can be re-written as; is the time at which the ith harmonic has a maximum.
The variance accounted for by the ith harmonic is except for the last harmonic where it is. If the ratio of this quantity to the total variance is formed, we have as the contribution of the ith harmonic. It can also be expressed in percentage. A harmonic with an overwhelming contribution would definitely account for most of the periodic variation in the data, while the contributions of other harmonics would be considered negligible.

Data Analysis
The method adopted for the analyses consists of the following statistical methods; harmonic analysis, spectral analysis and autocorrelation analysis. It is considered appropriate because, the temperature distribution is periodic. In other words, the hourly temperature measurements considered, has a considerable amount of cyclic variations and interdependencies over time. The following steps were taken in the data analyses: 1. Determination of the period of cyclic variation using the time series plot of the average hourly temperature observations for each station. 2. Harmonic analysis on the average hourly temperature observations for each station was performed using the sample size and period as inputs. This was carried out by fitting a periodic function of sinusoidal character to enhance the determination of the contribution of each harmonic (expressed as a percentage of total variation in the temperature measurements it accounts for), the amplitude of each harmonic and the time at which each harmonic is maximum. 3. Autocorrelation plots of the data were made in order to determine whether significant dependencies exist in the data. Hence, they were used to determine whether the air temperature at a succeeding hour depends on that of the preceding hour. The package used for this is the SPSS statistical software. 4. The time series plots of the average hourly solar irradiance observations for each station were made. The turbo Pascal for windows programming language was used to implement the computations involved in the harmonic analysis. This is required because of the enormous computational tasks it involves. The program receives as input the period of the observation (P) and the sample size (N). It proceeds to read the hourly average temperature data from the input file and automatically fit a periodic function to the data, compute the contribution, the amplitude and the time at which each harmonic is maximum.

RESULTS AND DISCUSSION
The analyses and result presentations are made for each station using their hourly average air temperature and solar irradiance measurements. The time series plots and summary of the results of the harmonic analysis program using the hourly average air temperature data for each station are presented.

Analyses and Results for Lagos Station
The time series plots of hourly average air temperature and solar irradiance measurements for Lagos station are given in Figures 1, while  (4) where Ai and Bi are the coefficients of the sine and cosine respectively and the i's are integers ranging from 1 to 3 as given in Table 1. Equation 4 was employed in the harmonic analysis program so as to make a six days forecast of hourly average air temperature measurements for Lagos station. The result of this forecast is displayed with the corresponding actual and estimates of hourly average air temperature measurements in Table 2.

Analyses and Results for Abuja Station
The same process was used to carry out the analysis for Abuja and the results for the time series plots of air temperature and solar irradiance measurements for Abuja station are given in Figure 2 where Ai and Bi are the coefficients of the sine and cosine respectively and the i's are integers ranging from 1 to 3 as given in Table 3.   The result of six days forecast of hourly average air temperature measurements for Abuja is displayed with the corresponding actual and model estimates of hourly average air temperature measurements in Table 4.
The maximum hourly average air temperature and solar irradiance and the time at which they occurred for each station are displayed in Table 5. Observations show that the equations (1) -(5) exhibit a good fit to the hourly average temperature of Lagos and Abuja, as they produced very close estimates of the actual hourly average air temperatures, yielded the same mean (average daily air temperature) as that of the actual data and very close standard deviation of the actual data and that of the model estimates as shown in tables 2 and 4 respectively. It can also be observed from Tables 1 and 3 that the first harmonic dominates the periodic components in the hourly average air temperature of all the stations since it has the highest contribution of 86.72% and 91.69%, for Lagos and Abuja stations respectively. This shows that the hourly air temperature of each station fluctuates by one cycle. This implies that the contributions of the second and third harmonics for each station is negligible thus, the information about them is discarded. This result is in agreement with the work of Kothandaraman, (2007) in which the application of harmonic analysis to daily mean air, water temperature records indicated that the first harmonic accounted for a major portion of the total variance in the records.
The six days forecasts of hourly average temperatures for each station as shown in Tables  3, 2 and 4 respectively, depict a strong indication that the hourly air temperatures across the two stations have the tendency of being repeated in every twenty four hours provided all other intervening atmospheric variables are kept constant. The results in Tables 1and 3, show that the first harmonic is maximum at times of 14.54h and 15.07h for Lagos and Abuja, respectively, which give close estimate of the time at which the actual value of the maximum hourly average air temperature occurred for each station as shown in table 5. This result further validates the fitted periodic function to the data for each station. This result also supports the periodic nature of air temperatures as ascertained by Ostrozlik, (2003), Lidija, (2007) and Makrogiannis, and Balafoutis, (2001 It could be seen from Table 5 that the maximum air temperatures is not the same across the two stations and neither did it occur at the time for which the solar irradiance is maximum for each station. The maximum average hourly temperature occurs 2hrs after the maximum average hourly solar heating had occurred in Lagos, while in Abuja, it occurs after 3 hours. This could be attributed to the fact that the stations differ in the nature of geophysical features they are endowed with. These physical features such as water bodies, hills and mountains differ in their specific heat capacities. The rate of radiation loss at the different locations could be explained by the Stefan-Boltzman law, which relates the total energy radiated per unit surface area of a body in unit time to the fourth power of its thermodynamic or surface temperature. Thus, since these surfaces differ in their surface temperatures, their respective radiation loses cannot be equal to the solar irradiance at the same time, affirming the reason why maximum air temperatures did not occur at the same time of the day across all stations and neither did it occur at the time for which the solar irradiance is maximum for each station. This could also be explained by a phenomenon of solar and terrestrial balance which asserts that the maximum air temperature occurs at the time when the solar heating and the energy lost by these surfaces (terrestrial radiation) are equal. In addition, amount of solar energy received by any region varies with the time of the day, season and latitude and this difference in solar energy creates temperature variations (FAA, 1975).

CONCLUSION
From the results of the analyses made for each station, we have drawn the following conclusions: We have deduced that the variations in hourly air temperatures across Lagos and Abuja, Nigeria are periodic and thus have strong tendency of being repeated the next day, if all other atmospheric variables are constant.
Furthermore, the hourly air temperatures and solar irradiance vary across each station, and that their respective maximum air temperatures never occurred at the time when the solar irradiance is a maximum. The maximum hourly air temperature occurs 2hrs on the average after the maximum solar irradiance has occurred in each station.
Equally observed is that the temperature of the air at a later hour depends on that of the previous hours for each station. Interestingly, it is also noted that the Nature of features (hills, water bodies, mountains, etc.) could have effects in variations in the average hourly temperature across stations.
The harmonic equations exhibit a good fit to the hourly average temperature of Lagos and Abuja and hence can be applied to hourly temperature prediction irrespective of the length of the data