Based on the analysis of the SEVIRI data used in this work see also, for example,, it is expected that wildfires are not only seasonal but also that their occurrence is greater during the day than night-time presumably due to higher human activity and better weather conditions. Hence the basic assumption of the formalism is that regularly ignited wildfires, and presumably all wildfires in general, tend to follow a double periodicity in which their diurnal cycles are periodically varying within and with respect to the annual one. In the formalism, the area-total FRP of regularly ignited wildfires is expected to obey the cycles in accordance with the day of the year and time of the day and is additionally modulated by the weather condition of the area.

The model output is the time-resolved FRP emitted from an area represented by an FRP function composed of three multiplicative terms, as follows: 1WΞ‾n;n,t∣φ,ϕr,ϕd=MΞ‾n;n∣φWrn∣ϕrWdt∣ϕd, where the terms represent the meteorological impact, reference shape curve, and daily shape curve components. The subindex r refers to the reference and d to the daily time. As for the variables, n is the Julian day of the year, t is the local time of the day, and Ξn denotes a set of meteorological variables, and the overline on it symbolizes an average over a time period and a spatial area. Lastly, φ, ϕr, and ϕd denote sets of fitting parameters of the terms, which are to be determined by fitting the predictions to the total FRP observed over the area. The model has only one dimension – time – and is made for application within a grid, in which each cell is treated separately. The indices of the grid cell in Eq. () are omitted for the sake of simplicity.

Each area (grid cell) has its own unique FRP reference curve with an anticipated bell-like shape, which is approximately fitted to monthly average maximum FRP values for the preceding year(s) characterized by ϕr. The curve is given by the FRP reference component Wr(n∣ϕr), which works as a Julian-day-dependent coefficient of Eq. () that sets reasonable estimates for average daily FRP maxima.

The other coefficient of Eq. () is the Julian-day- and weather-dependent component, which is designed to work so that when the weather conditions for fires are better than average, it increases. In general, the warmer, drier, and windier it is, the more fires are expected to show up. Fire danger is customarily indicated by fire indices: the bigger the index value, the more fires expected. Fire indices are constructed to represent a reasonable estimate for fire danger in terms of the cooperative action between relevant meteorological variables. A fire index is also harnessed for use in the meteorological component, which is simply taken as 2M(Ξ‾n;n∣φ)=I(Ξ‾n;n)I‾(n∣φ), where the numerator is a fire index appropriate to the area (or something analogous to it), and the denominator represents a function that is an average or a best-fit regression line through its values over a fire season. The input variable set Ξ‾n is given by the weather forecast of the coming days, and the parameter set φ is determined by the meteorological data of the past. Combined together, the reference and the meteorological components of the Eq. () determine the peak value for the time-dependent shape component, which is designed to produce a bell-like curve modulated by the peak value. Since both the reference and shape components are expected to have bell-like shapes, they are constructed from the combination of the logistic growth and decay functions. Specific mathematical representation can vary; the choice of logistic functions in this work is largely driven by convenience and comparative simplicity of the expressions. The two components may hence be given by 3Wi(τi∣ϕi)=∑ϵ=±ωiϵΘϵτim-τi1+ηiϵexp⁡ϵκiϵτiϵ-τi, where i= r or d, τr=n, τd=t, ωdϵ=1, and Θϵτim-τi are the Heaviside step functions which confine the growth “+” and decay “-” terms to their own sub-domains. One of the two terms in the sum is an s curve and the other one a mirror reflected s curve, and together they form either an asymmetric or symmetric bell curve by facing each other at the peak value. The parameter sets ϕi={ωi±,ηi±,κi±,τi±,τim} adjust the shapes and positions of the curves. The parameters nm and tm are the moments of the yearly and daily FRP maxima, respectively. The fit parameter values ωr± are the FRP maximum values of the reference curve components. Notice that Wr(n∣ϕr) has the dimension of power, whereas M(Ξ‾n;n∣φ) and Wd(t∣ϕd) are dimensionless.

A year's (or at least a dry season's) worth of data on weather, FRP, and cloud mask are needed to determine φ and ϕr and a day's worth to determine ϕd.

The method was tested in several different areas in south-central Africa, where the fires are quite regular. The parameterization for three different areas are given as an example. Because of the similarity of the results, only the results for one of the areas are displayed in more detail than the others. The SEVIRI FRP observations and the corresponding predictions are converted to the fire radiative energies (FREs) by temporal integration. These results are illustrated in the form of a correlation map of the association between observed and predicted FREs.

The neighbourhood of south-central Africa was chosen as the test site due to high incidence of fires in the region. Savannah fires in sub-Saharan Africa in the Southern Hemisphere contribute nearly a third (29 %) of global fire emissions . During the dry seasons, lasting roughly from spring to autumn, a big part of south-central Africa is filled with fires which are quite evenly distributed throughout a region about one-third the size of Europe. Most of the fires are presumably agricultural grass fires. According to SEVIRI data, the fires in the region cyclically maximize at daytime and minimize at night-time. During night-times, fires typically disappear from the view, either disappearing, becoming too small for satellite instruments to notice, or getting masked by night-time fog.

Since the fires in the test areas are assumed to be grassland fires, the numerator of Eq. () is taken here as an adaptation of the Grassland Fire Danger Index (GFDI) , with the overall multiplication factor and the curing contribution stripped out: 4I(Ξ‾n;n)=1.03T‾n0.80ϱ‾n3.38u‾n, where surface air temperature T is in units of degrees Celsius (°C), surface relative humidity ϱ in percent (%), and wind speed u in metres per second (m s-1) given at a height of 10 m above the ground level in the open. However, obviously, only the variable values without dimensions are to be inserted into Eq. (). The set of meteorological variables is denoted by Ξn={Tn,ϱn,un}, and the overline signifies an average over the approximate daylight hours and the area. The effect of grass curing is automatically incorporated into the reference curve, and any multiplicative coefficient of Eq. () would naturally cancel out in the ratio of Eq. (). The denominator of Eq. () is taken as the line of monthly averages of the adaptation of the GFDI Eq. (): 5I‾n∣φ=an+b, where φ={a,b}. The fit line represents the trend of the weather conditions favourable for fires over the course of the area-specific dry season. The first few dry-season monthly averages define the trend.

The African savannah terrain was chosen as the test subject because the fires in it during the dry season come in great quantities on a daily basis. The chosen region is confined by the geographic coordinates [1.5, 19.5]° S and [13.5, 33.0]° E, which represent the centre points of 182 evenly spaced areas of size 1.5°×1.5° (Fig. a). The dry season of the region is automatically included in the data used for the model fitting.

The model was parameterized using the data from 2010, and the model is applied and evaluated against the data for the fire season of 2018. The selection of distant years for training and evaluation demonstrates the stability of the method, as well as highlights the persistence of the regional fire ignition patterns, which did not change over almost a decade.

We present three examples of the model application for the three areas sized 1.5°×1.5° located in northern Angola, the southern Democratic Republic of the Congo, and southern Angola, with the centre points at 9.00° S and 18.00° E, 7.50° S and 21.00° E, and 13.50° S and 16.50° E, respectively.

Figure  illustrates the parametrizations and prediction results of the areas. The left-hand panels show the fire index Eq. () values arising from the ERA5 meteorological data. The straight green line in the panels defines the weather-condition-dependent trend of the fire severity. It acts as the denominator of the meteorological component Eq. () fitted throughout monthly averages from May to August (red dots). The right-hand panels show the values for the FRP maxima observed by SEVIRI and (the colour bar) the value of the fire index. The blue line in the panels is the reference FRP curve Wr(n∣ϕr) of the area. It is also made to pass approximately through the monthly averages represented (red dots) apart from the maximum value. The maximum depicts an average of a few neighbouring values around the highest observed value in order to highlight the FRP maximum of the reference curve. Each point on the scatter plots of the panels in the figure represents a value of a cloudless or nearly cloudless day. The maximum cloud cover of each area is limited to 13.33 pixels per area of 0.5°×0.5°, i.e. of about 5 % tops. In order to simplify and speed up the calculation procedure, the cloud mask is checked only once at 13:30 LT (local time), which is more or less (within a window of plus or minus 2 h) the moment of the maximum daily FRP. The applicability of this simplification was tested more rigorously by checking the cloud mask every 15 min throughout the daylight hours. It did not have any significant effect on the selection of the cloudless or nearly cloudless days. The data used in the panels are for the year 2010. The values of the parameters corresponding to the curves in Fig.  are given in Table .

Almost without exception, FRP of each area on each day can be (and is) characterized by a bell-shaped curve starting in the morning time, peaking somewhere around early afternoon, and ending in the early evening. The feature is seen in Fig. . These common characteristics may vary a little bit from area to area, as can be seen from t±,m values in Table . Figure  shows the predicted and observed diurnal FRP patterns for every cloudless (or nearly so) day within about 5 months of the dry season in 2018 for the area at 7.50° S, 21.00° E. The solid blue curves are the predictions given by Eq. () based on the parametrization made for 2010, and the violet dots are the SEVIRI observations for 2018. Note that the cloudless days are used only for evaluation purposes. When it comes to actual forecasting, it is irrelevant whether or not the day is cloudless as long as it is rainless. As can be seen in Fig. , in most cases, the correspondence between the predictions and observations is quite good, even though 8-year-old data were used for the parameterization. In the worst cases, the peak amplitudes differ from each other but in the majority of cases still stay within a factor of 2.

In order to assess the predictive power of the method, comparisons are made between observed and predicted daily FREs (given by the time integral of temporal FRP over the period of a day) over the course of the dry season. The diurnal FRP observations are integrated numerically by using Simpson's rule, and the predicted diurnal FRP predictions are calculated by an analytical integration. The accumulated energy within the time interval [ti,tf] is given by the temporal integral of Eq. (), which can readily be written analytically by exploiting the result 6∫titfdtWdt∣ϕd=∑ϵ=±ϵκdϵΘϵtm-tln⁡ηdϵ+exp⁡ϵκdϵ(t-tϵ)ηdϵ+exp⁡ϵκdϵtm-tϵ|titf.

The definite integral in Eq. () has three different solutions depending on the values of the integration limits. In the results, the solution that arises from the integral that crosses both the ascending and descending sides of a bell-shaped curve is used every time.

The scatter plot of Fig.  illustrates the correlation between the observed and predicted FREs of Fig. . For example, the three areas denoted by 1–3 in Table  have the r values of 0.89, 0.79, and 0.75, respectively. The map of Fig.  sums up the respective FRE correlations of the areas which meet the conditions set out in the text. Note that FRE represents the diurnal energy of fires, which is a time-independent quantity and is therefore independent of how the corresponding FRP is located on the time axis. In most cases the r value exceeds 0.6, which indicates a moderate to very strong correlation between observations and predictions and that the method possesses some true predictive power. Areas where the observed total seasonal FRE in the prediction year 2018 is less than 1 PJ are omitted because of anticipated low and/or irregular fire intensity and thus low predictability. Out of the total of 182 areas, 106 are included areas concentrated in the centre of the region, and the other 76 are excluded ones located at the periphery of the fire-prone region. The ratio of the sums of observed and predicted fire seasonal FREs over all 106 areas depicted in Fig. b is 1.39; i.e. the model over-predicts the FRE by 39 %. Some part of the over-prediction may be justified. In many cases, the observed FRP values are relatively lower on Sundays than on any other day of the week, which presumably corresponds to the lower agriculture activity and, consequently, the ignition intensity during the weekends. Also, the smoke from fires may mask them, to some extent, and hence reduce the observed FRP. Finally, the smallest predicted fires may be below the SEVIRI detection limit.

A comparison with MODIS generally confirmed the conclusions. With the better spatial resolution and lower detection limit, MODIS FRPs were predominantly consistent with those of SEVIRI, as can be seen in Fig. . MODIS satellites Aqua and Terra overpass the area on a daily basis at the same time each day. Aqua typically observes the area around noon and midnight and Terra in the morning and evening. However, the periods of high fire intensity are quite narrow in the considered areas, which practically leaves only the mid-day Aqua overpass as the main source of information. For example, in Fig. , the time interval between two subsequent MODIS observations is considerably longer than the characteristic duration of the fire in the area. Such short fires would pose a general problem for low-orbit satellites also in other parts of the globe.

With irregularly ignited wildfires, such as forest fires and other wildfires started unintentionally or otherwise for no good reason, it is more difficult to make reasonable day-by-day predictions. That is because such fires exhibit low fire density, and even if the weather conditions for fires are favourable, they do not necessarily occur within a specified area. The problem of predicting the fire ignitions becomes dominant, unlike in the above-considered regions in Africa, where the fire ignition is a daily routine due to specific agricultural practices. For regions with irregular fires, wider spatial and longer temporal averaging might be needed to achieve some predictive power. On the downside, the reduction in spatio-temporal resolution reduces the quality of predictions, which may not be sufficient for reasonable-quality emission predictions. As an illustration of the arising difficulties, the current prediction model was parameterized and applied to forest fires considering almost the entire Iberian Peninsula as a single fire-prone area.

Figure  shows FRP maximum values of two subsequent years, 2017 and 2018, and their monthly averages emitted by nearly the whole Iberian Peninsula. The Forest Fire Danger Index (FFDI, Mark 5) with the given drought factor accompanied by the Keetch–Byram drought index (KBDI) , with the correction given in , for the soil dryness is used as a reference for the incidence of wildfires in the area. The FFDI uses meteorological variables of air temperature, relative humidity, wind speed, and precipitation.

A comparison between the two panels of the figure shows the essence of the issue: the fires are not confined to a specific period of time but instead follow the complicated combination of short-term (days) and long-term (months) weather developments and human activities. The FFDI does not correlate well with the FRP strength between the 2 years, which is clearly seen especially around August when both years have several comparable values, with a wide range of values for 2017 and zeros for 2018. For such a fire pattern, the model developed above is too restrictive. One reason for this is that the method relies on the regular daily fire ignition, whereas forest fires are ignited sporadically. Another reason is that the forest fires do not exhibit as clear cycles throughout the years as the savannah fires in Africa. Moreover, extreme weather phenomena like powerful storms may further complicate the pattern. An example of this can be seen in Fig. a, in which the strong FRP peak in mid-October 2017 represents Iberian forest fires aggravated by high winds of tropical storm Ophelia as it passed by off the coast of Portugal. Especially on the 15th of October, on the day Ophelia hit the coast of Portugal the hardest, Iberia faced an exceptional firestorm. On that day, SEVIRI observed an FRP value exceeding 50 GW even though the area was heavily covered by clouds. This value is not displayed in the figure since it was not the true cloud-free observation.

The stochasticity of the fire occurrence (i.e. the number of fires or, alternatively, the amount of FRP) in an area can be decreased by extending the area and/or the time frame of the occurrence from instantaneous to longer. Nevertheless, forecasts should ideally predict the fire occurrence with the time dependence as done in this paper. Also, the area size should ideally be as small as possible. However, since the fire occurrence varies from area to area, the optimal area size into which a region should be evenly divided so that the areas remain sufficiently predictable depends on the chosen region. Due to the inherent stochasticity of the fire ignition, in the areas where the typical occurrence of fires is not exceptionally high the daily number of fires varies randomly even on consecutive days with the same weather conditions. This is the case, for example, with both in the excluded fringe areas of the region in Africa depicted in Fig. 4 and the area in Iberia depicted in Fig. 5. The more the weather favours wildfires, the greater the variation range is, but still nothing can be said with certainty about the daily fire occurrence and its temporal evolution. It is all about luck how many (if any) fires break out in the area on a day favourable to wildfires. This is in contrast to the case of African savannah fires, where these uncertainties are often within reasonable bounds due to regular ignitions. For example, as Fig.  illustrates, during the dry seasons, the African savannah fire occurrence (FRP) is always non-zero, much less arbitrary, and quite predictable. Moreover, as the Iberian case above indicates, in such conventional areas with low fire activity not only do daily occurrences vary widely but also annual ones. As an example within Fennoscandia, since the mid-1990s, the number of annual wildfires has ranged in Finland from approximately 1000 to 6000 and in Sweden from approximately 2500 to 8500 , and in the Republic of Karelia between the years 1956 and 2019, the number of annual fires has ranged from 35 to 1872 with values jumping up and down.