The seasonality and day-to-day variation of near-surface temperature patterns can greatly control nearly all physical and biological processes though temperature predictions at such scales remain challenging. This paper implements a simple analytical approach in order to generate daily average temperatures which implic itly accounts for surface heating and drivers through a comprehensive representation of station-based temperature records on a universal standard calendar propagated by the earth’s dynamics features. The modeled and observed pattern of daily temperatures exhibits a close agreement with the level of strength agreement exceeding 0.56. The extreme high and low values of the observed temperature patterns are equally well captured although model underestimates the probability of temperatures around the two modal peaks (~25.6℃ and 27.5℃). Additionally, a theoretical thermal-based division led to the identification of six seasons, including two hot and cold periods along with two pairs of mixed hot-cold. The theoretical division proposed here appears to be a good approximation for the understanding of rainfall seasonality in this area.
Near-surface temperature is considered as one of the most prominent environmental parameters, playing a critical role for both weather conditions and natural systems; human lives and activities. Any variation in surface-air temperature can affect processes related to the formation of rainfall, cloud and local circulation, which, along with temperature pattern control nearly all physical and biological processes. Plants productivity and growth, natural habitat and biodiversity, food cropping and human comfort are all very responsive to temperature variations from the daily to annual-term scales. Although atmospheric temperature information is of the utmost relevance for a wide range of practical applications, significant limitations in station-based observations in many areas of sub-Saharan Africa countries are still denoted, raising the prospect of enhanced demand in air temperature estimates for planning and informed decision making at the local scale.
Estimates of air temperature can be derived both from remote sensing and ground station based products. Among satellite remote sensing derived environmental parameters, Land Surface Temperatures (LSTs) have widely been associated with regional estimates of surface air temperature. If the spatial coverage of LST-induced air temperatures appears to be of great importance for regional impact assessments, existing LSTs over many regions are still poorly correlated with air temperatures (e.g. [
At the near surface, air temperature is controlled by surface radiation budget through a partitioning into its turbulent heat fluxes. Heating at the surface mainly relies upon a number of features including surface physiography, weather parameters and the dynamics of the earth around the sun. Current predictions of temperatures are mostly conducted by modern numerical weather models (e.g. [
A number of analytical approaches have been developed to generate weather variables, including parametric and deterministic methods. Most of these methods consist in an extensive analysis of long series of historical data from which model descriptors are extracted. While parametric methods consist in the determination of a few ensemble statistics allowing a good description of the overall behaviour, the existing explicit methods are more mathematical formulations based on a large number of Fourier series components of which coefficients do not always contain any significant meaning ( [
To theoretically describe the variation of monthly mean temperatures at a yearly basis, in a previous work by [
T ( t ) = a sin [ ω 0 ( t ) ] + b cos [ ω 0 ( t ) ] + c (1)
where ω 0 is the Earth’s frequency of revolution, c the annual mean temperature of the studied location. A nonlinear least square technique was used to fit the data obtained from meteorological measurements to the sinusoidal function.
In fact, the contributions of the Earth’s revolution and the Earth’s rotation are not observable on the curve of the monthly mean temperatures. We supposed that this could be observed on the curve of the daily mean temperatures. Therefore, a theory of perturbation was used to separate these contributions. The annual temperature being a constant for a given year, the perturbation only concerns the amplitude of the annual cycle. An expression of the temperature where the amplitude depend on the angular frequency of the Earth’s rotation was used. This expression is then given as follow:
T ( t ) = [ a 1 ( cos 2 ω t ) + a 2 ] sin ω 0 t + [ a 3 ( cos 2 ω t ) + a 4 ] cos ω 0 t + a 5 (2)
where ω is the Earth’s frequency of rotation. Again, the nonlinear least square technique is used to fit the data obtained from meteorological measurements to the sinusoidal function.
The expression of the temperature in Equation (2) can be rewritten as:
T ( t ) = [ a 1 ( 1 − sin 2 ω t ) + a 2 ] sin ω 0 t + [ a 3 ( 1 − sin 2 ω t ) + a 4 ] cos ω 0 t + a 5 = ( a 1 + a 2 ) sin ω 0 t − a 1 sin 2 ω t sin ω 0 t + ( a 3 + a 4 ) cos ω 0 t − a 3 sin 2 ω t cos ω 0 t + a 5 = ( a 1 + a 2 ) sin ω 0 t + ( a 3 + a 4 ) cos ω 0 t − ( a 1 sin ω 0 t + a 3 cos ω 0 t ) sin 2 ω t + a 5 = T r e v ( t ) + T r o t ( t ) (3)
where
T r e v ( t ) = ( a 1 + a 2 ) sin ω 0 t + ( a 3 + a 4 ) cos ω 0 t + a 5 (4)
and
T r o t ( t ) = − ( a 1 sin ω 0 t + a 3 cos ω 0 t ) ( sin ω t ) 2 (5)
Note that the revolution temperature does not evolve with the angular frequency of the Earth’s rotation while the rotation temperature fluctuate with both revolution and rotation frequencies.
The mean of the rotation temperature can be calculated for any given value of interval. Let 〈 T r o t ( t ) 〉 be the mean of T r o t ( t ) :
〈 T r o t ( t ) 〉 = 1 T ′ ∫ T i T f T r o t ( t ) d t (6)
where T f and T i are respectively the final and initial time of the chosen interval of time. T ′ = T f − T i is the length or the duration of the period of time chosen. The linearization of T r o t ( t ) and its substitution in Equation (6) leads to:
〈 T r o t ( t ) 〉 = a 1 4 1 T ′ ∫ T i T f [ sin ( ω 0 + 2 ω ) t + sin ( ω 0 − 2 ω ) t ] d t − a 1 2 1 T ′ ∫ T i T f sin ω 0 t d t + a 3 4 1 T ′ ∫ T i T f [ cos ( ω 0 + 2 ω ) t + cos ( ω 0 − 2 ω ) t ] d t − a 3 2 1 T ′ ∫ T i T f sin ω 0 t d t (7)
The integration of Equation (7) leads to:
〈 T r o t ( t ) 〉 = − a 1 4 T ′ [ 1 ω 0 + 2 ω cos ( ω 0 + 2 ω ) t + 1 ω 0 − 2 ω cos ( ω 0 − 2 ω ) t ] T i T f + a 3 4 T ′ [ 1 ω 0 + 2 ω sin ( ω 0 + 2 ω ) t + 1 ω 0 − 2 ω sin ( ω 0 − 2 ω ) t ] T i T f + a 1 2 ω 0 T ′ [ cos ω 0 t ] T i T f + a 3 2 ω 0 T ′ [ cos ω 0 t ] T i T f (8)
For a daily mean, we have:
T f = T i + T = ( n + 1 ) T (9)
where T = 2 π ω is the time period of earth’s rotation and n the ranking of the day in a given year. We then have:
〈 T r o t ( t ) 〉 = − a 1 4 π ω ω 0 ω 0 2 − 4 ω 2 ( cos 2 π ( n + 1 ) 365 − cos 2 π n 365 ) + a 3 4 π ω ω 0 ω 0 2 − 4 ω 2 ( sin 2 π ( n + 1 ) 365 − sin 2 π n 365 ) + 365 4 π ( a 1 + a 3 ) ( cos 2 π ( n + 1 ) 365 − cos 2 π n 365 ) (10)
using ω ω 0 = 365 , we obtain:
〈 T r o t ( n + 1 ) 〉 = 365 5836 a 1 π ( cos 2 π ( n + 1 ) 365 − cos 2 π n 365 ) − 365 5836 a 3 π ( sin 2 π ( n + 1 ) 365 − sin 2 π n 365 ) + 365 4 π ( a 1 + a 3 ) ( cos 2 π ( n + 1 ) 365 − cos 2 π n 365 ) (11)
with n ≥ 0 . Finally, the mean value of the rotation temperature as a function of a day’s ranking n is derived from Equation (12) as:
〈 T r o t ( n ) 〉 = 365 5836 a 1 π ( cos 2 π n 365 − cos 2 π ( n − 1 ) 365 ) − 365 5836 a 3 π ( sin 2 π n 365 − sin 2 π ( n − 1 ) 365 ) + 365 4 π ( a 1 + a 3 ) ( cos 2 π n 365 − cos 2 π ( n − 1 ) 365 ) (12)
with n ≥ 1 .
Evaluated for one year that is for T i = 0 and T f = T 0 = 2 π ω 0 which is then the Earth’s period of revolution, we obtain:
〈 T r o t ( t ) 〉 = a 1 ω 0 2 4 π ( 4 ω 2 − ω 0 2 ) cos ( 4 π ω ω 0 ) + a 3 ω ω 0 2 π ( 4 ω 2 − ω 0 2 ) sin ( 4 π ω ω 0 ) (13)
That is:
〈 T r o t ( t ) 〉 = a 1 ω 0 2 4 π ( 4 ω 2 − ω 0 2 ) = a 1 2131596 π (14)
note that a 1 is generally a small value therefore the rotation temperature is weak. This parameter, which accounts for the contribution of the Earth’s rotation to the daily mean temperature of a given location, is of small magnitude compared to that of the revolution temperature. Although its small magnitude, this contribution is critical to capture the short-term and small magnitude variations.
The in situ observations used in this study are daily average temperature data obtained from a synoptic meteorological station of the Douala international Airport, a coastal location of Cameroon in the central Africa region (
To evaluate the skill of the proposed approach of predicted air temperatures, we used a feature-based similarity method which account for both the deterministic features covering the raw-value shape of the time series and the stochastic features representing the distributions. Thus, the model was assessed by the means of the following: root mean square difference (RMSD, absolute and relative values), scatterplots of the estimated values as a function of the observed ones, the probability density distributions, and the concordance correlation coefficient ( ρ c ). The following expressions for RMSD (˚C and %: absolute and relative value) were used:
RMSD ( ˚ C ) = [ 1 N ∑ i = 1 N ( T i , m o d − T i , o b s ) 2 ] 1 / 2 (15)
RMSD ( % ) = 100 T ¯ o b s [ 1 N ∑ i = 1 N ( T i , m o d − T i , o b s ) 2 ] 1 / 2 (16)
where T i , m o d is the modeled temperature value, N is the number of data points, T i , o b s is the observed temperature value and T ¯ o b s is a long-term mean air temperaute value. In addition, ρ c given a measure of the level of strength agreement is computed as follows:
ρ c = 2 σ m o d , o b s σ m o d 2 + σ o b s 2 + ( μ m o d − μ o b s ) 2 = ρ C b (17)
where μ m o d and μ o b s are the means of the observed and estimated temperatures while σ m o d 2 and σ o b s 2 represent the variances of the two temperatures series respectively. σ m o d , o b s indicates the cross correlation between the estimates and observations. ρ c is also expressed as the product of the Pearson correlation coefficient ( ρ ) and the bias correction factor ( C b ) which is computed as follows:
C b = [ ( σ m o d σ o b s + σ o b s σ m o d + ( μ m o d − μ o b s ) 2 σ m o d σ o b s ) / 2 ] − 1 (18)
The value of ρ c is in the range between −1.0 and 1.0, it can be shown that ρ c = ± 1 if and only if ρ = ± 1 , σ m o d = σ o b s and μ m o d = μ o b s . Thus, ρ c = 1 (−1) if and only if T i , m o d = T i , o b s ( − T i , o b s ) that is when there is perfect agreement (disagreement). The bias correction factor C b ( 0 ≤ C b ≤ 1 ) in (18) assesses the level of bias, with smaller C b indicating larger bias. Therefore, poor agreement can result from low correlation (small ρ ) or large bias (small C b ).
Temperature estimates as generated by the newly developed approach is compared to in situ observations from 2000 to 2009 obtained from a local meteorological station.
Regarding the scatter plot of the observed daily average temperatures as a function of estimates, the diagonal line represents the ideal match between the estimated and measured values.
period the two sets quite well follow the line of 45˚, indicating that the overall relationships between the modeled and station-measured temperatures is reasonable.
At the yearly granularity, a set of indicators including root mean square error (RMSE) absolute and relative values, concordance correlation coefficient and bias were used to test and examine the performance of the developed approach.
| 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | |
|---|---|---|---|---|---|---|---|---|---|---|
| RMSD (˚C) | 1.07 | 0.80 | 0.88 | 1.17 | 2.50 | 0.92 | 0.93 | 0.92 | 0.89 | 0.73 |
| RMSD (%) | 3.95 | 3.13 | 3.26 | 4.45 | 9.68 | 3.57 | 3.39 | 3.56 | 3.29 | 2.82 |
| C b | 0.95 | 0.96 | 0.95 | 0.97 | 0.52 | 0.97 | 0.89 | 0.98 | 0.98 | 0.99 |
| CCC( ρ c ) | 0.61 | 0.74 | 0.61 | 0.56 | 0.31 | 0.69 | 0.57 | 0.57 | 0.65 | 0.79 |
To further evaluate the similarities of the two temperatures series, the probability distributions of the observed and modeled daily temperatures for 2001-2010 were illustrated in
Derived and validated in sections 2 & 3, the analytical expression of the temparature is a continuous variable of time which allows an explicit determination of the different seasons over a year. We then need first of all to find the Transition dates between the different periods. There are only three considerations to be used: 1) when the second derivative of the temperature with respect to time vanishes, the time values obtained represent the transition dates between the hot and the cold periods, and reversely. At these dates, the seasonal progression of the temperature coincides with the annual mean temperature. 2) when the progression of the seasonal temperature coincides with the half maximum during the hot period, the obtained dates correspond either to the transition between the hot and the moderate hot 2 periods or the transtion between the hot and the moderate hot 1 periods associated with a negative or positive second differential of the temperature respectively. 3) when the annual temperature cycle coincides with the half maximum during the cold period, the derived dates correspond either to the transition between the cold and the moderate cold 1 period or the transition between the cold and the moderate cold 2 period associated with a negative or positive second derivative of the temperature respectively. A detailed description of the theoretical determination of transition dates was provided in [
| Period | December 8th to March 17th | March 17th to May 20th | May 20th to June 22nd | June 22nd to October 6th | October 6th to November 6th | November 6th to December 9th |
|---|---|---|---|---|---|---|
| Duration (month) | 4.34 | 1.1 | 1.1 | 4.4 | 1.0 | 1.1 |
| Season | hot period | Moderate hot 2 | Moderate cold 1 | Cold period | Moderate cold 2 | Moderate hot 1 |
theoretical thermal-based division proposed here can be a good approximation for the understanding of rainfall seasonality.
An analytical approach was developed to generate daily temperature estimates at a yearly basis. The model contains an explicit description of pattern and variations of the daily average temperatures for a given location by the mean of a combined two-component of long and short term periodic functions. A ten years historical daily data were analyzed on a thirteen months calendar (universal standard calendar) and then the model descriptors were obtained by fitting the theoretical function to the historical temperatures. Daily average temperatures were then synthetisized for this location covering the same ten years period as the historical data. A set of performance measures were used to assess how well the model reproduces the observed station data.
Overall, the relationship between model and observations is reasonable. The generated temperatures capture well the multi-year observed pattern although the day-to-day variation still exhibits some discrepancies. Relative RMSE values vary from 2.82% to 4.45% which is quite well lower compared to that obtained in a comparative assessment comprising multi analytical temperature models. The probability distribution of the generated temperatures is also similar to that of the climate being modeled. In addition, a theoretical thermal-based division led to the identification of six seasons, including two hot and cold periods along with two pairs of mixed hot-cold periods. The theoretical division proposed in this work appears to be a good approximation for the understanding of rainfall seasonality in this area.
This approach is powerful in that, by using the earth’s dynamics features, it implicitly accounts for the others influencing factors of temperature. The approach is simple and requires only small computing effort. This points to the suitability of these results for real time applications over areas of low density observations, including for instance decision on how many or what mixed of seasonal products to manufacture and store; what crop if any to plant or what actions should be taken to prevent from vector-borne diseases such as malaria.
The authors wish to thank the Agency for the Safety of Air Navigation in Africa and Madagascar (ASECNA) for making the station-based temperature data freely available. The third author acknowledges support for this work from the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste Italy) under the OEA-12 projects.
The authors declare no conflicts of interest regarding the publication of this paper.
Bell, J.-P., Modi-Mbog, E.C., Djiedeu, N. and Nana, L. (2023) Modeling Daily Average Temperatures in a Coastal Site of Central Africa: An Analysis of Seasonal Divisions. Atmospheric and Climate Sciences, 13, 341-352. https://doi.org/10.4236/acs.2023.133019