The Simplified Ionospheric Regional Model (SIRM) for HF Prediction: Basic Theory, Its Evolution and Applications

Abstract

This paper is a final review of the Simplified Ionospheric Regional Model (SIRM) developed as a prototype in the early 1990s and improved in the following years. By means of an algorithm based on the Fourier synthesis, the SIRM model in its prototype version provides predicted monthly median values of the main ionospheric characteristics such as: the ordinary wave critical frequencies (foE, foF1, and foF2) of the E, F1, and F2 ionospheric layers; the lowest virtual height (h’F) of the ordinary trace of the F region; the obliquity factor for a distance of 3000 km (M(3000)F2). Instead, the improved version focuses only on foF2 and M(3000)F2. The SIRM model has been largely employed in the framework of different international research projects as the climatological reference to output foF2 and M(3000)F2 monthly median predictions, but in its SIRMUP version is used also as a nowcasting model and as an intermediate step of complex procedures for a near real-time three-dimensional representation of the ionospheric electron density. In this regard, some results provided by both SIRM and SIRMUP for telecommunication applications are shown. Moreover, the mathematical treatment concerning both the phase correction of the Fourier synthesis and the fundamental steps carried out to define the SIRM algorithm in its final version, never published so far, will be described in detail in dedicated Appendices. Finally, for the first time the SIRM code is now downloadable for the benefit of users.

Appendix

1.1 Appendix 1.1: Phase Correction of the Fourier synthesis

A detailed description about how the phase of the Fourier synthesis has been corrected to consider the displacement of the reference ionospheric station from the central meridian of the time zone is here reported. In what follows R₁₂ is the 12-month running mean of the monthly sunspot number R, m is the month of the year, n is the harmonic number, and h* = (1/15°) is a conversion factor to express longitude differences in hours or fraction of hours.

Let us hypothesize the case of ionospheric stations located respectively to west or east of the central meridian of whatever time zone (Fig. 22).

Fig. 22

Sketch showing the longitude Ψ_S of a station S, that can be located eastward or westward of the central meridian, and its corresponding solar time t_ST, the longitude Ψ_CM of the central meridian and its corresponding local standard time t_CM. The bell-shaped drawings represent the hourly time series of monthly median values of the generic ionospheric characteristic at longitudes Ψ_S and Ψ_CM

It is important to note that hourly time series of monthly median values of the generic ionospheric characteristic refer to the local standard time t_CM of the central meridian, because measurements at S are recorded right at the time t_CM. Nevertheless, values are affected by the zenithal solar angle and this implies that time series must be referred to solar times t_ST. This will not imply a variation of values but only their translation in time with respect to the central meridian, meaning that Fourier coefficients of the amplitude remain unchanged. In the light of these considerations, we will show how to carry out the phase correction for a station positioned on both the west and east sides of the central meridian.

The diurnal variation of hourly monthly median values of the generic ionospheric characteristic (Ω) has a period T = 24 h and, generalizing Eq. (8) for a station S, for a given month and R₁₂, at the central meridian it can be exactly represented by the following Fourier synthesis:

$$\varOmega \left( {t_{\text{CM}} ,R_{12} ,m} \right)_{\text{S}} = A\left( {\varPsi_{\text{CM}} ,R_{12} ,m} \right)_{{{\text{S}},0}} + \sum\limits_{n = 1}^{12} {A\left( {\varPsi_{\text{CM}} ,R_{12} ,m} \right)_{{{\text{S}},n}} \sin \left( {\frac{{2\pi nt_{\text{CM}} }}{T} + Y\left( {\varPsi_{\text{CM}} ,R_{12} ,m} \right)_{{{\text{S}},n}} } \right)} ;$$

(10)

anyhow, to refer Ω to the solar time t_ST of the station S, (10) has to be corrected. At t_ST the time series is represented by the following Fourier synthesis:

$$\varOmega \left( {t_{\text{ST}} ,R_{12} ,m} \right)_{\text{S}} = A\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},0}} + \sum\limits_{n = 1}^{12} {A\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},n}} \sin \left( {\frac{{2\pi nt_{\text{ST}} }}{T} + Y\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},n}} } \right)} ;$$

(11)

in (10)–(11),Ψ_S andΨ_CM ranges in (− 180°;180].

The solar time t_ST is calculated through the mathematical function named floor, that gives the greatest integer that is less than or equal to its argument, as

$$t_{\text{ST}} = a - b \cdot {\text{floor}}\left( {a/b} \right);$$

(12)

specifically, in (12) a = [t_CM + (Ψ_S − Ψ_CM)h*] and b = T = 24 (number of hours in a day).

By using (12), independently of the station location (eastward or westward of the central meridian), t_ST is always within the range [0,24). For example, if the station is located at a longitude 30° eastward of the central meridian and t_CM = 0, then a = 2; by inserting a = 2 and b = 24 in (12), because floor(2/24) = 0, it can be easily verified that t_ST = 2. Likewise, if the station is located at a longitude 30° westward of the central meridian and t_CM = 0, then a = − 2; by inserting a = − 2 and b = 24 in (12), because floor(− 2/24) = − 1, it can be easily verified that t_ST = 22.

From (10) and (11) it has to be

$$\varOmega \left( {t_{\text{CM}} ,R_{12} ,m} \right)_\mathrm{S} = \varOmega \left( {t_{\text{ST}} ,R_{12} ,m} \right)_\mathrm{S} ;$$

(13)

(13) implies that

$$A\left( {\varPsi_{\text{CM}} ,R_{12} ,m} \right)_{{{\text{S}},0}} = A\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},0}} ,$$

(14)

$$A\left( {\varPsi_{\text{CM}} ,R_{12} ,m} \right)_{{{\text{S}},n}} = A\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},n}} ,$$

(15)

and

$$\sin \left( {\frac{{2\pi nt_{\text{CM}} }}{T} + Y\left( {\varPsi_{\text{CM}} ,R_{12} ,m} \right)_{{{\text{S}},n}} } \right) = \sin \left( {\frac{{2\pi nt_{\text{ST}} }}{T} + Y\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},n}} } \right);$$

(16)

considering (12), the right side of (16) becomes

$$\sin \left\{ {\frac{{2\pi nt_{\text{CM}} }}{T} + \frac{2\pi n}{T}\left[ {\Delta \varPsi_{\text{S,CM}} h^* - T \cdot {\text{floor}}\left( {\frac{{t_{\text{CM}} + \Delta \varPsi_{\text{S,CM}} h^*}}{T}} \right)} \right] + Y\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},n}} } \right\},$$

(17)

where ΔΨ_S,CM = (Ψ_S − Ψ_CM). From (16) and (17) it follows that

$$Y\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},n}} = Y\left( {\varPsi_{\text{CM}} ,R_{12} ,m} \right)_{{{\text{S}},n}} - \frac{2\pi n}{T}\left[ {\Delta \varPsi_{\text{S,CM}} h^* - T \cdot {\text{floor}}\left( {\frac{{t_{\text{CM}} + \Delta \varPsi_{\text{S,CM}} h^*}}{T}} \right)} \right].$$

(18)

(18) shows the relationship between phases of (10) and (11), the former related to the central meridian, the latter related to the station meridian.

1.2 Appendix 1.2: The SIRM Algorithm

In this section the SIRM prediction algorithm will be described in detail indicating with Ω the ionospheric characteristic (foF2 or M(3000)F2). Referring to Fig. 22, the regression analysis carried out month by month between hourly monthly median values of Ω and R₁₂ provides, for each hour (h) and month (m), the coefficients α(h,m)_S and β(h,m)_S for a definite station S. So, from the empirical law

$$\varOmega_{\text{S}} \left( {h,m,R_{12} } \right) = \alpha \left( {h,m} \right)_{\text{S}} R_{12} + \beta \left( {h,m} \right)_{\text{S}} ,$$

(19)

diurnal trends of hourly monthly median values are calculated for R₁₂ = 0 and R₁₂ = 100; the corresponding two time series can be represented through the Fourier synthesis as follows:

$$\varOmega \left( {t_{\text{CM}} ,R_{12} = 0,m} \right)_{\text{S}} = A\left( {\varPsi_{\text{CM}} ,R_{12} = 0,m} \right)_{{{\text{S}},0}} + \sum\limits_{n = 1}^{12} {A\left( {\varPsi_{\text{CM}} ,R_{12} = 0,m} \right)_{{{\text{S}},n}} \sin \left( {\frac{{2\pi nt_{\text{CM}} }}{T} + Y\left( {\varPsi_{\text{CM}} ,R_{12} = 0,m} \right)_{{{\text{S}},n}} } \right)} ,$$

(20a)

$$\varOmega \left( {t_{\text{CM}} ,R_{12} = 100,m} \right)_{\text{S}} = A\left( {\varPsi_{\text{CM}} ,R_{12} = 100,m} \right)_{{{\text{S}},0}} + \sum\limits_{n = 1}^{12} {A\left( {\varPsi_{\text{CM}} ,R_{12} = 100,m} \right)_{{{\text{S}},n}} \sin \left( {\frac{{2\pi nt_{\text{CM}} }}{T} + Y\left( {\varPsi_{\text{CM}} ,R_{12} = 100,m} \right)_{{{\text{S}},n}} } \right)} ,$$

(20b)

being T = 24 h.

(20a–20b) reconstruct exactly the time series Ω(t_CM,R₁₂,m)_S for R₁₂ = 0 and R₁₂ = 100. It is important to point out that time series provided by (20a–20b) are referred to t_CM, because measurements at S are recorded at the time t_CM. Nevertheless, as highlighted in the Appendix 1.1, measurements are somehow affected by the zenithal solar angle, which means that (20a–20b) must be rewritten as a function of the solar times t_ST of S. Therefore, a phase correction of the Fourier synthesis is needed, while coefficients of the amplitude remain unchanged. To this regard, exploiting (14) and (15) obtained in the Appendix 1.1, (20a–20b) become

$$\varOmega \left( {t_{\text{ST}} ,R_{12} = 0,m} \right)_{\text{S}} = A\left( {\varPsi_{S} ,R_{12} = 0,m} \right)_{{{\text{S}},0}} + \sum\limits_{n = 1}^{12} {A\left( {\varPsi_{S} ,R_{12} = 0,m} \right)_{{{\text{S}},n}} \sin \left( {\frac{{2\pi nt_{\text{ST}} }}{T} + Y\left( {\varPsi_{\text{S}} ,R_{12} = 0,m} \right)_{{{\text{S}},n}} } \right)} ,$$

(21a)

$$\varOmega \left( {t_{\text{ST}} ,R_{12} = 100,m} \right)_{\text{S}} = A\left( {\varPsi_{S} ,R_{12} = 100,m} \right)_{{{\text{S}},0}} + \sum\limits_{n = 1}^{12} {A\left( {\varPsi_{S} ,R_{12} = 100,m} \right)_{{{\text{S}},n}} \sin \left( {\frac{{2\pi nt_{\text{ST}} }}{T} + Y\left( {\varPsi_{\text{S}} ,R_{12} = 100,m} \right)_{{{\text{S}},n}} } \right)} ,$$

(21b)

where the phases in (21a–21b) are calculated by (18).

(21a–21b) refer to solar times and represent the algorithms for the prediction of hourly monthly median values, respectively, over S for R₁₂ = 0 and R₁₂ = 100; to get values of Ω for any value of R₁₂, a linear trend between Fourier coefficients and R₁₂ is supposed. Specifically, straight lines passing through points

$$\left( {R_{12} = 0;A\left( {\varPsi_{\text{S}} ,R_{12} = 0,m} \right)_{{{\text{S}},0}} } \right){\text{and}}\left( {R_{12} = 100;A\left( {\varPsi_{\text{S}} ,R_{12} = 100,m} \right)_{{{\text{S}},0}} } \right),$$

(22a)

$$\left( {R_{12} = 0;A\left( {\varPsi_{\text{S}} ,R_{12} = 0,m} \right)_{{{\text{S}},n}} } \right){\text{and}}\left( {R_{12} = 100;A\left( {\varPsi_{\text{S}} ,R_{12} = 100,m} \right)_{{{\text{S}},n}} } \right),$$

(22b)

$$\left( {R_{12} = 0;Y\left( {\varPsi_{\text{S}} ,R_{12} = 0,m} \right)_{{{\text{S}},n}} } \right){\text{and}}\left( {R_{12} = 100;Y\left( {\varPsi_{\text{S}} ,R_{12} = 100,m} \right)_{{{\text{S}},n}} } \right),$$

(22c)

are considered; (22a) provides the angular coefficient a(m)_S,0 and intercept b(m)_S,0; (22b) provides 12 angular coefficients a(m)_S,n and 12 intercepts b(m)_S,n; (22c) provides 12 angular coefficients c(m)_S,n and 12 intercepts d(m)_S,n.

These angular coefficients and intercepts are then used to calculate amplitudes and corresponding phases of each harmonic, for a generic value of R₁₂, through the following empirical laws

$$A\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},0}} = a\left( m \right)_{{{\text{S}},0}} R_{12} + b\left( m \right)_{{{\text{S}},0}} ,$$

(23a)

$$A\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},n}} = a\left( m \right)_{{{\text{S}},n}} R_{12} + b\left( m \right)_{{{\text{S}},n}} ,$$

(23b)

$$Y\left( {\varPsi_{\text{S}} ,R_{12} ,m} \right)_{{{\text{S}},n}} = c\left( m \right)_{{{\text{S}},n}} R_{12} + d\left( m \right)_{{{\text{S}},n}} .$$

(23c)

So, considering (23a–23c), the Fourier synthesis

$$\varOmega \left( {t_{\text{ST}} ,R_{12} ,m} \right)_{\text{S}} = \left( {a\left( m \right)_{{{\text{S}},0}} R_{12} + b\left( m \right)_{{{\text{S}},0}} } \right) + \sum\limits_{n = 1}^{12} {\left( {a\left( m \right)_{{{\text{S}},n}} R_{12} + b\left( m \right)_{{{\text{S}},n}} } \right)\sin \left[ {\frac{{2\pi nt_{\text{ST}} }}{T} + \left( {c\left( m \right)_{{{\text{S}},n}} R_{12} + d\left( m \right)_{{{\text{S}},n}} } \right)} \right]}$$

(24)

can predict the hourly monthly median values over S for whatever value of R₁₂. Relation (24) represents a local prediction model that can be applied only over S. To develop a regional model it is necessary to consider a definite number of ionospheric stations that have regularly operated for a relatively long period inside a given area, which is the validity area of the model. For simplicity, let us consider only six ionospheric stations distributed as in Fig. 23.

Fig. 23

Six ionospheric stations S_i (with i = 1,…,6) distributed in three different time zones: three located to west and three to east of the corresponding central meridian, whose longitude is Ψ_CMj (with j = 1,…,3).Ψ_Si, ϕ_Si and t_STi (with i = 1,…,6) are corresponding longitudes, latitudes and solar times

According to (24) we can write

$$\varOmega \left( {t_{{{\text{ST}}_{\text{i}} }} ,R_{12} ,m} \right)_{{{\text{S}}_{i} }} = \left( {a\left( m \right)_{{{\text{S}}_{i} ,0}} R_{12} + b\left( m \right)_{{{\text{S}}_{i} ,0}} } \right) + \sum\limits_{n = 1}^{12} {\left( {a\left( m \right)_{{{\text{S}}_{i} ,n}} R_{12} + b\left( m \right)_{{{\text{S}}_{i} ,n}} } \right)\sin \left[ {\frac{{2\pi nt_{{{\text{ST}}_{\text{i}} }} }}{T} + \left( {c\left( m \right)_{{{\text{S}}_{i} ,n}} R_{12} + d\left( m \right)_{{{\text{S}}_{i} ,n}} } \right)} \right]} ,$$

(25)

with i = 1,…, 6.

Assuming that ionospheric stations are distributed in a relatively restricted area, we can guess that values depend more on latitude than on longitude. Moreover, once the diurnal trends refer to solar times of ionospheric stations, time series are in “phase”, and the dependence on the latitude can be assessed by making a linear regression between amplitude and phase coefficients of the Fourier synthesis against latitudes of the ionospheric stations. Therefore, in the considered example, a linear trend between coefficients of (25) and the latitude of ionospheric stations is supposed and linear regressions (LR_s) are made considering the following datasets of points:

$${\text{LR}}_{1} \to \left( {\phi_{{{\text{S}}_{1} }} ;a\left( m \right)_{{{\text{S}}_{1} ,0}} } \right),\left( {\phi_{{{\text{S}}_{ 2} }} ;a\left( m \right)_{{{\text{S}}_{2} ,0}} } \right),\left( {\phi_{{{\text{S}}_{3} }} ;a\left( m \right)_{{{\text{S}}_{3} ,0}} } \right),\left( {\phi_{{{\text{S}}_{4} }} ;a\left( m \right)_{{{\text{S}}_{4} ,0}} } \right),\left( {\phi_{{{\text{S}}_{5} }} ;a\left( m \right)_{{{\text{S}}_{5} ,0}} } \right),\left( {\phi_{{{\text{S}}_{6} }} ;a\left( m \right)_{{{\text{S}}_{6} ,0}} } \right);$$

(26a)

$${\text{LR}}_{2} \to \left( {\phi_{{{\text{S}}_{1} }} ;b\left( m \right)_{{{\text{S}}_{1} ,0}} } \right),\left( {\phi_{{{\text{S}}_{ 2} }} ;b\left( m \right)_{{{\text{S}}_{2} ,0}} } \right),\left( {\phi_{{{\text{S}}_{3} }} ;b\left( m \right)_{{{\text{S}}_{3} ,0}} } \right),\left( {\phi_{{{\text{S}}_{4} }} ;b\left( m \right)_{{{\text{S}}_{4} ,0}} } \right),\left( {\phi_{{{\text{S}}_{5} }} ;b\left( m \right)_{{{\text{S}}_{5} ,0}} } \right),\left( {\phi_{{{\text{S}}_{6} }} ;b\left( m \right)_{{{\text{S}}_{6} ,0}} } \right);$$

(26b)

$${\text{LR}}_{3} \to \left( {\phi_{{{\text{S}}_{1} }} ;a\left( m \right)_{{{\text{S}}_{1} ,n}} } \right),\left( {\phi_{{{\text{S}}_{ 2} }} ;a\left( m \right)_{{{\text{S}}_{2} ,n}} } \right),\left( {\phi_{{{\text{S}}_{3} }} ;a\left( m \right)_{{{\text{S}}_{3} ,n}} } \right),\left( {\phi_{{{\text{S}}_{4} }} ;a\left( m \right)_{{{\text{S}}_{4} ,n}} } \right),\left( {\phi_{{{\text{S}}_{5} }} ;a\left( m \right)_{{{\text{S}}_{5} ,n}} } \right),\left( {\phi_{{{\text{S}}_{6} }} ;a\left( m \right)_{{{\text{S}}_{6} ,n}} } \right);$$

(26c)

$${\text{LR}}_{4} \to \left( {\phi_{{{\text{S}}_{1} }} ;b\left( m \right)_{{{\text{S}}_{1} ,n}} } \right),\left( {\phi_{{{\text{S}}_{ 2} }} ;b\left( m \right)_{{{\text{S}}_{2} ,n}} } \right),\left( {\phi_{{{\text{S}}_{3} }} ;b\left( m \right)_{{{\text{S}}_{3} ,n}} } \right),\left( {\phi_{{{\text{S}}_{4} }} ;b\left( m \right)_{{{\text{S}}_{4} ,n}} } \right),\left( {\phi_{{{\text{S}}_{5} }} ;b\left( m \right)_{{{\text{S}}_{5} ,n}} } \right),\left( {\phi_{{{\text{S}}_{6} }} ;b\left( m \right)_{{{\text{S}}_{6} ,n}} } \right);$$

(26d)

$${\text{LR}}_{5} \to \left( {\phi_{{{\text{S}}_{1} }} ;c\left( m \right)_{{{\text{S}}_{1} ,n}} } \right),\left( {\phi_{{{\text{S}}_{ 2} }} ;c\left( m \right)_{{{\text{S}}_{2} ,n}} } \right),\left( {\phi_{{{\text{S}}_{3} }} ;c\left( m \right)_{{{\text{S}}_{3} ,n}} } \right),\left( {\phi_{{{\text{S}}_{4} }} ;c\left( m \right)_{{{\text{S}}_{4} ,n}} } \right),\left( {\phi_{{{\text{S}}_{5} }} ;c\left( m \right)_{{{\text{S}}_{5} ,n}} } \right),\left( {\phi_{{{\text{S}}_{6} }} ;c\left( m \right)_{{{\text{S}}_{6} ,n}} } \right);$$

(26e)

$${\text{LR}}_{6} \to \left( {\phi_{{{\text{S}}_{1} }} ;d\left( m \right)_{{{\text{S}}_{1} ,n}} } \right),\left( {\phi_{{{\text{S}}_{ 2} }} ;d\left( m \right)_{{{\text{S}}_{2} ,n}} } \right),\left( {\phi_{{{\text{S}}_{3} }} ;d\left( m \right)_{{{\text{S}}_{3} ,n}} } \right),\left( {\phi_{{{\text{S}}_{4} }} ;d\left( m \right)_{{{\text{S}}_{4} ,n}} } \right),\left( {\phi_{{{\text{S}}_{5} }} ;d\left( m \right)_{{{\text{S}}_{5} ,n}} } \right),\left( {\phi_{{{\text{S}}_{6} }} ;d\left( m \right)_{{{\text{S}}_{6} ,n}} } \right).$$

(26f)

LR₁ provides the angular coefficient a(m) ¹₀ and intercept a(m) ²₀ ; LR₂ provides the angular coefficient b(m) ¹₀ and intercept b(m) ²₀ ; LR₃ provides the angular coefficient a(m) ¹_n and intercept a(m) ²_n ; LR₄ provides the angular coefficient b(m) ¹_n and intercept b(m) ²_n ; LR₅ provides the angular coefficient c(m) ¹_n and intercept c(m) ²_n ; LR₆ provides the angular coefficient d(m) ¹_n and intercept d(m) ²_n . Accordingly, the following empirical laws are obtained

$$a\left( m \right)_{0} = a\left( m \right)_{0}^{1} \phi + a\left( m \right)_{0}^{2} ,$$

(27a)

$$b\left( m \right)_{0} = b\left( m \right)_{0}^{1} \phi + b\left( m \right)_{0}^{2} ,$$

(27b)

$$a\left( m \right)_{n} = a\left( m \right)_{n}^{1} \phi + a\left( m \right)_{n}^{2} ,$$

(27c)

$$b\left( m \right)_{n} = b\left( m \right)_{n}^{1} \phi + b\left( m \right)_{n}^{2} ,$$

(27d)

$$c\left( m \right)_{n} = c\left( m \right)_{n}^{1} \phi + c\left( m \right)_{n}^{2} ,$$

(27e)

$$d\left( m \right)_{n} = d\left( m \right)_{n}^{1} \phi + d\left( m \right)_{n}^{2} .$$

(27f)

So, for a given month, (27a–27f) show how Fourier coefficients change as the latitude varies; specifically, there are four coefficients (a(m) ¹₀ ; a(m) ²₀ ; b(m) ¹₀ ; b(m) ²₀ ) for the average term and eight coefficients (a(m) ¹_n ; a(m) ²_n ; b(m) ¹_n ; b(m) ²_n ; c(m) ¹_n ; c(m) ²_n ; d(m) ¹_n ; d(m) ²_n ) for each of the twelve harmonics, thus obtaining for a given month a total number of 100 coefficients.

Thanks to (27a–27f), (25) can be then generalized in the context of a regional model so that the prediction algorithm can be synthetically written as follows

$$\begin{aligned} & \varOmega \left( {t_{\text{ST}} ,\phi ,R_{12} ,m} \right) = \left( {a\left( m \right)_{0}^{1} \phi + a\left( m \right)_{0}^{2} } \right)R_{12} + b\left( m \right)_{0}^{1} \phi + b\left( m \right)_{0}^{2} \\ & \quad + \sum\limits_{n = 1}^{12} {\left[ {\left( {a\left( m \right)_{n}^{1} \phi + a\left( m \right)_{n}^{2} } \right)R_{12} + b\left( m \right)_{n}^{1} \phi + b\left( m \right)_{n}^{2} } \right]\sin } \left\{ {\frac{{2\pi nt_{\text{ST}} }}{T} + \left[ {\left( {c\left( m \right)_{n}^{1} \phi + c\left( m \right)_{n}^{2} } \right)R_{12} + d\left( m \right)_{n}^{1} \phi + d\left( m \right)_{n}^{2} } \right]} \right\}. \\ \end{aligned}$$

(28)

It is worth mentioning once again that in (28) coefficients depend on the month, which means that SIRM is represented by 12·100 = 1200 coefficients. For example, Table 4 shows the 100 numerical coefficients used by SIRM to predict foF2 in January.

Table 4 Numerical coefficients used by SIRM to predict foF2 in January

Let us focus now on the issue about the dependence on longitude. The prediction algorithm (28), referred to the Greenwich prime meridian, for which t_ST (Ψ_CM = 0°) = t_UT, is

$$\begin{aligned} & \varOmega \left( {t_{\text{UT}} ,\phi ,R_{12} ,m} \right) = \left( {a\left( m \right)_{0}^{1} \phi + a\left( m \right)_{0}^{2} } \right)R_{12} + b\left( m \right)_{0}^{1} \phi + b\left( m \right)_{0}^{2} \\ & \quad + \sum\limits_{n = 1}^{12} {\left[ {\left( {a\left( m \right)_{n}^{1} \phi + a\left( m \right)_{n}^{2} } \right)R_{12} + b\left( m \right)_{n}^{1} \phi + b\left( m \right)_{n}^{2} } \right]\sin } \left\{ {\frac{{2\pi nt_{\text{UT}} }}{T} + \left[ {\left( {c\left( m \right)_{n}^{1} \phi + c\left( m \right)_{n}^{2} } \right)R_{12} + d\left( m \right)_{n}^{1} \phi + d\left( m \right)_{n}^{2} } \right]} \right\}. \\ \end{aligned}$$

(29)

If in (29) we set ϕ = cost, values of Ω(t_UT,ϕ,R₁₂,m) are modulated at various hours essentially by the amplitude coefficients; this means that time series obtained at different longitudes, varying in (29) t_UT from 00:00 to 23:00, will be always the same not only for the shape but also for the positioning with respect to the corresponding prime meridian, which is unacceptable.

To overcome this issue, over a restricted validity area, the dependence on longitude is considered in the following way: Ω(t_UT,ϕ,R₁₂,m) values are first referred to the prime meridian (Ψ = 0°) using (29) and then translated in time, t_UT → t_UT − Ψh*, being Ψ the longitude of the generic meridian, namely

$$\begin{aligned} & \varOmega \left( {t_{\text{UT}} ,\,\phi \,,R_{12} ,m} \right) = \left( {a\left( m \right)_{0}^{1} \phi + a\left( m \right)_{0}^{2} } \right)R_{12} + b\left( m \right)_{0}^{1} \phi + b\left( m \right)_{0}^{2} \\ & \quad + \sum\limits_{{n = 1}}^{12} {\left[ {\left( {a\left( m \right)_{n}^{1} \phi + a\left( m \right)_{n}^{2} } \right)R_{12} + b\left( m \right)_{n}^{1} \phi + b\left( m \right)_{n}^{2} } \right]\sin } \left\{ {\frac{{2\pi n\left( {t_{\text{UT}} - \psi h^*} \right)}}{T} + \left[ {\left( {c\left( m \right)_{n}^{1} \phi + c\left( m \right)_{n}^{2} } \right)R_{12} + d\left( m \right)_{n}^{1} \phi + d\left( m \right)_{n}^{2} } \right]} \right\}, \\ \end{aligned}$$

(30)

so as to refer them to times depending on the meridian under consideration.

To clarify better how the algorithm (30) works, it is convenient to make a numerical example considering foF2 time series reported in Table 5 obtained by applying the SIRM model for ϕ = 40°, R₁₂ = 100, and m = 1 (which means that numerical coefficients of Table 4 have been used) at longitudes Ψ = + 15° and Ψ = + 30°.

Table 5 foF2 (MHz) time series provided in UT by the SIRM model for R₁₂ = 100, m = 1 and ϕ = + 40° at (first column) longitude Ψ = 0°, (second column) Ψ = + 15°, and (third column) Ψ = + 30°

Let us indicate with S_PM the foF2 time series at the prime meridian (Ψ = 0°) (first column of Table 5). If we want to refer S_PM at the various meridians, it has to be moved opportunely according to the longitude. If the meridian at Ψ = + 15° is considered then t_UT → t_UT + 1 h; this means that if for example (29) provides a value of S_PM at t_UT = 12:00 equal to 9.3 MHz, the same value is obtained through (30) at Ψ = + 15° at t_UT = 11:00.

Therefore, S_PM will result translated in time one hour backward (second column of Table 5). If instead the meridian at Ψ = + 30° is considered then t_UT → t_UT + 2 h; this means that if for example (29) provides a value of S_PM at t_UT = 08:00 equal to 7.3 MHz, the same value is obtained through (30) at Ψ = + 30° at t_UT = 06:00. In this case S_PM will result translated in time 2 h backward (third column of Table 5). Similar considerations hold in case of negative longitudes. It is worth highlighting that in this way, to a first approximation, and for a narrow area in longitude, the dependence on longitude is well represented. This is testified by Table 5 showing for example that values of foF2 at 05:00 UT (i.e. foF2 (Ψ = 30°) = 4.6 MHz > foF2 (Ψ = 15°) = 3.6 MHz > , foF2 (Ψ = 0°) = 3.5 MHz) are consistent with the passage of the solar terminator at sunrise, while values of foF2 at 18:00 UT (i.e. foF2 (Ψ = 30°) = 4.4 MHz < foF2 (Ψ = 15°) = 5.1 MHz < foF2 (Ψ = 0°) = 6.3 MHz) are consistent with the passage of the solar terminator at sunset.

From this procedure is then possible to get foF2 maps in UT. Let us observe for example that, in the specific case of the bold row of Table 5, values of foF2 at longitudes Ψ = 0° (9.3 MHz), Ψ = + 15° (9.3 MHz), and Ψ = + 30° (9.9 MHz), at latitude ϕ = + 40°, are all referred to 12:00 UT. This means that fixing the value of the latitude, the longitude can be varied with a one degree spatial resolution in order to get corresponding foF2 values from (30). This procedure repeated varying also the latitude with a one degree spatial resolution allows to ultimately get a 1° × 1° grid of foF2 values evenly spaced in longitude and latitude which are all referred to the same UT. The grid is then processed through a graphical tool to obtain a foF2 map in UT over the area under consideration.