Empirical regional models for the short-term forecast of M 3000 F 2 during not quiet geomagnetic conditions over Europe

Twelve empirical local models have been developed for the long-term prediction of the ionospheric characteristicM3000F2, and then used as starting point for the development of a short-term forecasting empirical regional model ofM3000F2under not quiet geomagnetic conditions. Under the assumption that the monthly median measurements ofM3000F2are linearly correlated to the solar activity, a set of regression coefficients were calculated over 12 months and 24 h for each of 12 ionospheric observatories located in the European area, and then used for the long-term prediction ofM3000F2at each station under consideration. Based on the 12 long-term prediction empirical local models ofM3000F2, an empirical regional model for the prediction of the monthly median field of M3000F2over Europe (indicated asRM_M3000F2) was developed. Thanks to theIFELM_foF2models, which are able to provide short-term forecasts of the critical frequency of the F2 layer (foF2STF) up to three hours in advance, it was possible to considerer the Brudley–Dudeney algorithm as a function of foF2STF to correctRM_M3000F2and thus obtain an empirical regional model for the short-term forecasting of M3000F2(indicated asRM_M3000F2_BD) up to three hours in advance under not quiet geomagnetic conditions. From the long-term predictions of M3000F2provided by the IRI model, an empirical regional model for the forecast of the monthly median field of M3000F2over Europe (indicated asIRI_RM_M3000F2) was derived. IRI_RM_M3000F2predictions were modified with the Bradley–Dudeney correction factor, and another empirical regional model for the short-term forecasting of M3000F2 (indicated asIRI_RM_M3000F2_BD) up to three hours ahead under not quiet geomagnetic conditions was obtained. The main results achieved comparing the performance of RM_M3000F2, RM_M3000F2_BD, IRI_RM_M3000F2 , and IRI_RM_M3000F2_BDare (1) in the case of moderate geomagnetic activity, the Bradley–Dudeney correction factor does not improve significantly the predictions; (2) under disturbed geomagnetic conditions, the Bradley–Dudeney formula improves the predictions of RM_M3000F2in the entire European area; (3) in the case of very disturbed geomagnetic conditions, the Bradley–Dudeney algorithm is very effective in improving the performance of IRI_RM_M3000F2; (4) under moderate geomagnetic conditions, the long-term prediction maps ofM3000F2generated byRM_M3000F2 can be considered as short-term forecasting maps providing very satisfactory results because quiet geomagnetic conditions are not so diverse from moderate geomagnetic conditions; (5) the forecasting maps originated by RM_M3000F2, RM_M3000F2_BD, andIRI_RM_M3000F2_BDshow some regions where the forecasts are not satisfactory, but also wide sectors where the M3000F2forecasts quite faithfully match the M3000F2 observations, and therefore RM_M3000F2, RM_M3000F2_BD, and IRI_RM_M3000F2_BDcould be exploited to produce short-term forecasting maps of M3000F2over Europe up to 3 h in advance.

IRI_RM_M3000F2_BD are (1) in the case of moderate geomagnetic activity, the Bradley-Dudeney correction factor does not improve significantly the predictions; (2) under disturbed geomagnetic conditions, the Bradley-Dudeney formula improves the predictions of RM_M3000F2 in the entire European area; (3) in the case of very disturbed geomagnetic conditions, the Bradley-Dudeney algorithm is very effective in improving the performance of IRI_RM_M3000F2; (4) under moderate geomagnetic conditions, the long-term prediction maps of M3000F2 generated by RM_M3000F2 can be considered as short-term forecasting maps providing very satisfactory results because quiet geomagnetic conditions are not so diverse from moderate geomagnetic conditions; (5) the forecasting maps originated by RM_M3000F2, RM_M3000F2_BD, and IRI_RM_M3000F2_BD show some regions where the forecasts are not satisfactory, but also wide sectors where the M3000F2 forecasts quite faithfully match the M3000F2 observations, and therefore RM_M3000F2, RM_M3000F2_BD, and IRI_RM_M3000F2_BD could be exploited to produce short-term forecasting maps of M3000F2 over Europe up to 3 h in advance.

Introduction
Ionospheric models providing a full specification of the three-dimensional (3-D) electron density profile are very important because they can be considered as an essential starting point to carry out a lot of other research (Bilitza, 2002;Cander, 2008).Recently the ISP model capable of providing a 3-D electron density profile representation of the Published by Copernicus Publications on behalf of the European Geosciences Union.
ionosphere in real time for quiet and disturbed geomagnetic conditions (Pezzopane et al., 2011(Pezzopane et al., , 2013) ) was developed and then used for calculating a 3-D ray tracing in the ionospheric medium on the base of measured oblique ionograms over a given radio link (Settimi et al., 2013).
Moreover, the development of models that are able to provide reliable predictions of the most important ionospheric characteristics is very important to ensure successful radio communications.For this reason, many global models, such as the International Reference Ionosphere (IRI) (Bilitza, 2001;Bilitza and Reinisch, 2008) and NeQuick (Radicella, 2009), and regional models (Zolesi et al., 1993(Zolesi et al., , 1996;;De Franceschi et al., 2000;Bradley, 1999;Hanbaba, 1999) have been developed over the years to predict the monthly medians of the main ionospheric parameters such as the highest frequency reflected by the F2 layer, (foF2), and the secant of the optimum angle at which to broadcast a signal that is to be received at a distance of 3000 km (M(3000)F2).
Several different techniques have been developed for forecasting the ionospheric characteristics.Artificial neural networks (Francis et al., 2001;Cander et al., 2003;Chen et al., 2008), the multiple linear regression method (Mikhailov et al., 1999), the autocorrelation method (Liu et al., 2005), and autoregression method (Koutroumbas et al., 2008) are some of the techniques available to forecast the ionospheric parameters significant for purposes of radio communication.These techniques give reliable predictions essentially for a quiet ionosphere, but they are not successful in the case of disturbed ionospheric conditions (Cander, 2003).A high degree of reliability during quiet ionospheric conditions is also provided by the two global climatological models IPS-ASAPS (IPS-Radio and Space Services, 2013) and ICEPAC (Stewart, 2013), capable of offering good guidelines for the selection of maximum usable frequencies to be used for radio communications for not disturbed ionospheric conditions (Zolesi et al., 2008), but they fail when disturbed ionospheric situations associated with geomagnetic storm events occur (Pietrella et al., 2009).
During magneto-ionospheric storms events considerable variations can occur in electron density content altering dayto-day F region ionospheric variability so that the long-term prediction models for foF2 and M3000F2 are not able to supply reliable forecasts.
The use of these models would provide HF operators with real-time or quasi-real-time support in selecting the best possible frequencies in order to guarantee an efficient Table 1.List of ionospheric observatories used for the development of the short-term forecasting procedures: the interval of years for which the monthly median measurements are taken into account to calculate the set of regression coefficients (column A) and the interval of years considered to test the reliability of the models (column B) are shown for all the stations.

MLS
Latitude Longitude A B Tortosa 40 1955-1986 1987- 1957-1985 1986-1998 maintenance of radio links, also in the case of not quiet magneto-ionospheric conditions.More recently a short-term ionospheric forecasting empirical regional model to predict the critical frequency of the F2 layer (IFERM_foF2) during moderate, disturbed and very disturbed geomagnetic conditions over the European area was developed by Pietrella (2012).
When geomagnetic storms occur, the Earth's magnetic field strength varies significantly from place to place.As the intensity of the phenomena observed in the F region of the ionosphere during ionospheric storms is strictly related to the magnetic field strength, the main element for discerning the various effects that a storm has on the behaviour of the F region is the difference in latitude of a place to another one.
This means that N local models for the prediction of M3000F2 adequately scattered in latitude, each of which having the skill to properly capture the local effects of a storm on M3000F2, could be used all together at the same time to reproduce the effects of the storm on the F region over a spatial scale larger than the local one.
Consequently, the predictions of M3000F2 obtained by N local models at a given epoch can be suitably utilized to generate forecasting maps of M3000F2 during geomagnetic storm events over the region comprising the N models.
With these considerations in mind, and inspired by the work of Pietrella (2012), two different empirical regional models for the short-term forecasting of M3000F2 over the European area were derived on the basis of 12 short-term forecasting empirical local models which have been developed following two diverse procedures.The 12 local observatories considered for the development of the forecasting procedures are Tortosa, Rome, Poitiers, and Lannion (Fig. 1a); Dourbes, Slough, Juliusruh, Kaliningrad, and Uppsala (Fig. 1b); and Lyckesele, Sodankyla, and Kiruna (Fig. 1c).
The first procedure followed for the achievement of the short-term forecasts of M3000F2 takes into account the monthly median measurements of M3000F2, i.e. those monthly median values that were obtained from the measurements of M3000F2 recorded at each ionospheric observatory over the years as shown in column A of Table 1.
Using the linear relationship between the 12-month smoothed mean value of the monthly sunspot numbers (R 12 ) and the monthly median measurements of M3000F2 (see CCIR Report 340-6, 1991), a set of statistically significant regression coefficients were established for each observatory over 12 months and 24 h, and utilized as input to calculate the monthly median predictions of M3000F2 for each local station.To the monthly median values of M3000F2 predicted with the long-term prediction models which are discussed below, we refer hereafter also as monthly median field.
Based on these 12 local models providing long-term predictions of M3000F2, an empirical regional model for the achievement of the monthly median field of M3000F2 over the European sector (indicated as RM_M3000F2) was also derived.Subsequently the monthly median field of M3000F2 predicted at each ionospheric observatory was modified by means of the Bradley-Dudeney algorithm (Bradley and Dudeney, 1973), which depends on the critical frequency of the E layer (foE), and the hourly short-term forecasts of foF2 (foF2 STF ), provided by the 12 short-term forecasting empirical local models (IFELM_foF2) developed by Pietrella (2012).As foF2 STF can be forecasted up to three hours in advance under different geomagnetic conditions, the Bradley-Dudeney correction factor confers characteristics of shortterm forecast for M3000F2, and therefore its application to the monthly median field of M3000F2 was used to generate 12 empirical local models for the short-term forecasting of M3000F2 up to three hours ahead.Based on these 12 short-term forecasting empirical local models, an empirical regional model for the short-term forecasting of M3000F2 (indicated as RM_M3000F2_BD) up to three hours in advance under different geomagnetic conditions was also obtained.
The second procedure followed to obtain M3000F2 predictions over Europe takes into account the monthly median predictions of M3000F2 provided by the IRI model at each local station.From these predictions an empirical regional model for the achievement of the IRI monthly median field of M3000F2 over Europe (indicated as IRI_RM_M3000F2) was also implemented.The IRI monthly median field of M3000F2 was corrected with the Bradley-Dudeney algorithm at each ionospheric observatory to generate 12 empirical local models for the short-term forecasting of M3000F2 up to three hours ahead.Based on these 12 short-term forecasting empirical local models, an empirical regional model for the short-term forecasting of M3000F2 (indicated as IRI_RM_M3000F2_BD) up to three hours in advance under different geomagnetic conditions was also obtained.
The geomagnetic index used in this study to distinguish the different ranges of geomagnetic activity is the ap(τ ) index introduced by Wrenn (1987).Based on previous research (Wrenn et al., 1987;Perrone et al., 2001, and references therein;Pietrella, 2012), many epochs occurring over the years shown in column B of Table 1 were chosen and thereafter classified on the basis of three different ranges of geomagnetic activity: moderate (7 < ap(τ ) ≤ 20); disturbed (20 < ap(τ ) ≤ 32); and very disturbed (ap(τ ) > 32).
Afterwards, these epochs have been grouped together and the predictions of M3000F2 provided by RM_M3000F2, RM_M3000F2_BD, IRI_RM_M3000F2, and IRI_RM_M3000F2_BD were calculated and binned by single month.The hourly measurements of M3000F2 relative to the epochs selected were taken into account to test the performance of each model for all the months in terms of global root mean square deviation (g.r.m.s.) error under moderate, disturbed, and very disturbed geomagnetic activity.
Some comparisons between the maps based on M3000F2 measurements and the maps obtained by the forecasts generated by the RM_M3000F2, RM_M3000F2_BD, and IRI_RM_M3000F2_BD models are also presented for a few days characterized by moderate, disturbed, and very disturbed geomagnetic activity.
Section 2 describes the data used and how the models were developed.Section 3 outlines the testing procedure and illustrates the related comparisons and results.The discussion concerning the RM_M3000F2/RM_M3000F2_BD/IRI_RM_M3000F2/ IRI_RM_M3000F2_BD approach, as well as some final remarks on possible future developments, is given in Sect. 4.
2 Data used and description of the procedures followed for the achievement of the regional forecasting models of M3000F2 As can be seen in Table 1, the 12 ionospheric observatories considered in this study have been separated as follows: Tortosa, Rome, Poitiers, and Lannion located at middle latitudes (40 , hereafter also referred to simply as MLS; Dourbes, Slough, Juliusruh, Kaliningrad, and Uppsala located at middle-high latitudes (50 hereafter also referred to simply as MHLS; and Lyckesele, Sodankyla, and Kiruna located at high latitudes (60 • N < λ ≤ 70 • N), hereafter also referred to simply as HLS.
The parameters utilized for the achievement of the shortterm forecasting models were (a) the monthly median measurements of M3000F2 acquired in 12 ionospheric observatories over a large period of years (Table 1, column A); (b) the monthly median predictions of M3000F2, and the predictions of the critical frequency of the E region (foE) provided by the IRI model at each ionospheric observatory; and (c) the hourly short-term forecasts of foF2 up to 3 h in advance, provided by the 12 short-term forecasting empirical local models (IFELM_foF2) developed by Pietrella (2012).In addition, the hourly time-weighted accumulation series derived from the geomagnetic planetary index ap (ap(τ )) were used to select the different ranges of geomagnetic activity.
As regards the first procedure followed for the achievement of a short-term forecasting empirical regional model, as the first step, the linear relationship between the monthly median measurements of M3000F2 (M3000F2 MED,MEAS,s,m,hh ), recorded over the years reported in column A of Table 1, and the 12-month smoothed mean value of the monthly sunspot numbers (R 12 ) was considered for each local station: where s, m, and hh indicate the station, month, and hour respectively.
From Eq. (1) a set of 288 (12 months × 24 h) pairs of statistically significant coefficients (A * s,m,hh , B * s,m,hh ) were established by means of a linear regression analysis for each observatory, and utilized as input to calculate the monthly median predictions of M3000F2 at each local station.
Each set of 288 pairs of coefficient represents a model for the prediction of the monthly median values of M3000F2 (M3000F2 MED_PRED,s,m,hh ) for a given local station s, at a well-specified future epoch (year, month, and hour): where R 12,PREV is the 12-month smoothed mean value of the monthly sunspot numbers predicted at the epoch under consideration.As in Eq. ( 2) the s index varies from 1 to 12; Eq. ( 2) represents 12 empirical local models providing longterm predictions of M3000F2, and as such they can be considered as a single empirical regional model for the long-term prediction of M3000F2 to which henceforward we refer as RM_M3000F2.M3000F2 MED_PRED,s,m,hh values predicted by each local model were then modified through the correction factor M_BD STF,s,m,d,hh provided by the empirical formula of Bradley and Dudeney (1973): where STF stands for "short-term forecast" and s, m, d, and hh indicate the station, month, day, and hour respectively.At this point, it is very important to point out that in Eq. ( 3) correcting the monthly medians predictions of M3000F2, the values of the critical frequency of the E layer (foE s,m,d,hh ) were obtained by the IRI model, whereas the values of the critical frequency of the F2 layer (foF2 STF, s,m,d,hh ) are those predicted by the 12 local models developed by Pietrella (2012) providing the short-term forecast of foF2 up to three hours in advance under moderate (7 < ap(τ ) ≤ 20), disturbed (20 < ap(τ ) ≤ 32), and very disturbed (ap(τ ) > 32) geomagnetic conditions.This means that the Bradley-Dudeney correction factor confers characteristics of shortterm forecast for M3000F2, and hence its application corrects the predicted monthly median field of M3000F2 at each local station, thus generating 12 empirical local models providing short-term forecasting of M3000F2 up to three hours ahead.
Therefore, depending on whether the short-term forecast of foF2 (foF2 STF,s,m,d,hh ) was acquired in the case of moderate, disturbed, and very disturbed geomagnetic activity, M_BD STF,s,m,d,hh can be seen as the correction factor of the monthly median field of M3000F2 under moderate, disturbed, and very disturbed geomagnetic conditions respectively, because foE s,m,d,hh in Eq. ( 3) is not affected by the magneto-ionospheric storms.
The correction of the predicted monthly median values of M3000F2 by means of M_BD STF,s,m,d,hh , constitutes a model that is able to provide short-term forecasts of M3000F2 for each observatory in diverse geomagnetic conditions because the predictions provided by Eq. ( 4) are connected with the short-term forecasts of foF2.Given that the s index varies from 1 to 12, Eq. ( 4) represents 12 1657 Fig. 2 Fig. 2. Example of a flowchart describing the procedure followed to select the forecasting models providing the best performance in Tortosa under moderate geomagnetic conditions.short-term forecasting empirical local models that all together can be considered as a single short-term forecasting empirical regional model of M3000F2 to which henceforward we refer as RM_M3000F2_BD.
Concerning the second procedure carried out for the achievement of another short-term forecasting empirical regional model, it is based on the predictions of the monthly median values of M3000F2 provided by the IRI model at each ionospheric observatory.These predictions, indicated as IRI_M3000F2 MED_PRED,s,m,hh where the s index is ranged between 1 and 12, constitute all together a single empirical regional model for the long-term prediction of M3000F2 to which henceforward we refer as IRI_RM_M3000F2.(5) Given that in Eq. ( 5) the s index ranges between 1 and 12, Eq. ( 5) represents 12 short-term forecasting empirical local models able to provide short-term forecasts of M3000F2 in diverse geomagnetic conditions because the predictions provided by Eq. ( 5) are connected with the short-term forecasts of foF2.These models all together can be regarded as a single short-term forecasting empirical regional model to which henceforward we refer as IRI_RM_M3000F2_BD.

Description of the testing procedure comparisons and results
With a procedure analogous to that followed in Pietrella (2012), many epochs were selected for each ionospheric observatory over the years reported in column B of Table 1, and then grouped together.For these epochs the predictions of M3000F2 calculated at each ionospheric observatory using the models represented by Eq. ( 2), Eq. ( 4), the IRI model, and Eq. ( 5) were binned in terms of three different ranges of geomagnetic activity: moderate (7 < ap(τ = 0.8/0.9)≤ 20), disturbed (20 < ap(τ = 0.8/0.9)≤ 32), and very disturbed (ap(τ = 0.8/0.9)> 32), where τ = 0.8 is selected for the HLS, and τ = 0.9 is preferred for the MLS and MHLS (the reason for which two different values of τ are used can be found in the paper of Pietrella, 2012).Subsequently, these data sets were binned by single month, and the performances of the forecasting procedures, Eq. (2), Eq. ( 4), the IRI model, and Eq. ( 5), were calculated and compared for each local station and for all the months in terms of global root mean square deviation (g.r.m.s.) error under moderate, disturbed and very disturbed geomagnetic conditions.An example of a flowchart describing the procedure followed to select the forecasting models providing the best performance in Tortosa for moderate geomagnetic conditions is shown in Fig. 2.
Figure 3 shows evidence of some ionospheric observatories and the related models providing the best performance for some months under moderate geomagnetic conditions.
Starting from tables such as that shown in Fig. 3, the sites for which each of the four forecasting models provides the best performance have been tabulated for each month under moderate (m), disturbed (d), and very disturbed (vd) geomagnetic activity to better realize how these models could be used for an operative approach (see Tables 2-5).
Since, as mentioned above, the set of forecasting local models can be regarded as a single forecasting empirical regional model, based on Tables 2-5, the percentages of the best performances given by the regional models RM_M3000F2, RM_M3000F2_BD, IRI_RM_M3000F2, and IRI_RM_M3000F2_BD have been also calculated for the three latitude intervals (40 The sites where the best performance is provided by the prediction model ( 2) are indicated with the full circle and shown for each month under moderate (m), disturbed (d), and very disturbed (vd) geomagnetic conditions.In these sites the different local models can be considered simultaneously operative for forecasting M3000F2 over Europe.The columns labelled with MLS, MHLS, and HLS show the number of sites operating simultaneously located at middle (40 The empty cells indicate cases that were discarded because the performance was worse than that of the others models.The values in the last column (T) indicate the total number of sites operating at the same time.The symbol $ indicates the cases in which the number of sites (N < 4) operating simultaneously is not considered satisfactory to cover the area under investigation.The term nda indicates the cases for which it was not possible to evaluate the performance of the prediction model ( 2) with respect to the other models because no data were available to calculate the g.r.m.s.error.

(a)
Tor The results thus obtained are presented in the form of histograms in Fig. 4. By adopting the criterion according to which a number N ≥ 4 of operating stations is considered sufficient to adequately cover the area under investigation providing simultaneous predictions of M3000F2, it emerges that there are several cases where it is never possible to use operatively the models, the number of sites being N < 4 (see cases labelled with $).Therefore, from a careful inspection of Tables 2-5, we find that during moderate geomagnetic  3b) although in some cases also IRI_RM_M3000F2_BD could be used (see Table 5b).
As regards very disturbed geomagnetic conditions, also in this case RM_M3000F2 and IRI_RM_M3000F2 provide an absolutely inadequate coverage for the operative use (see Tables 2c and 4c), while IRI_RM_M3000F2_BD seems to be more appropriate for the operative use, assuring almost always a good coverage of the area under study (see Table 5c), even if in some cases also RM_M3000F2_BD could be used (see Table 3c).
Ultimately, on the basis of the results presented in Tables 2-5, it emerges that the more appropriate choice of the models to be considered for obtaining M3000F2 predictions over the European area is (1) RM_M3000F2 for moderate geomagnetic conditions (Table 2a); (2) RM_M3000F2_BD for disturbed geomagnetic conditions (Table 3b); and (3) IRI_RM_M3000F2_BD for very disturbed geomagnetic conditions (Table 5c).
Nevertheless, the observation of Tables 2a, 3b, and 5c shows also that such models are subject to the following two limitations: (a) the lack of MLS and HLS means that the model cannot provide a coverage of the area in the latitude ranges 40 • N ≤ λ ≤ 50 • N and 60 • N < λ ≤ 70 • N respectively, and when this occurs the models provide M3000F2 predictions in a more restricted area; (b) a totally inadequate coverage of the area emerges for some months, and in these cases it is not possible to provide M3000F2 forecasts over the region under study.
Based on these considerations, Table 6 was built to clarify which models should be used and to better show evidence of their limits.2 but for the sites where the best performance is provided by the prediction model ( 5), indicated with the square.The term nda indicates the cases for which it was not possible to evaluate the performance of the prediction model ( 5) with respect to the other models because no data were available to calculate the g.r.m.s.error.

(a)
Tor  The limitations shown in bold in Table 6 and described at points (a) and (b) inherent to a given model could be removed or reduced using appropriately the other models.Under moderate geomagnetic conditions, RM_M3000F2_BD could be applied to provide M3000F2 forecasts for the month of March, even if in a more restricted area because the HLS are missing.
Analogously, under disturbed geomagnetic conditions, RM_M3000F2 and IRI_RM_M3000F2_BD could be used to calculate the values predicted of M3000F2 for the months of April and July respectively; also in this case the coverage of the region under consideration will be limited by the lack of HLS (in April) and MLS (in July).
Similarly, under very disturbed geomagnetic conditions, RM_M3000F2_BD could replace IRI_RM_M3000F2_BD for providing M3000F2 predictions for the months of April (in a more restricted region because the MLS are missing) and May.Therefore, in the light of these latest considerations, a possible table for an operative use was built (see Table 7).
Figure 5 shows the comparison between the map based on M3000F2 measurements (Fig. 5a) and the map based on the Table 6.Use of RM_M3000F2, RM_M3000F2_BD, and IRI_RM_M3000F2_BD under (a) moderate, (b) disturbed, and (c) very disturbed geomagnetic conditions respectively.The months for which M3000F2 predictions are provided over a more restricted area because of the lack of MLS or HLS, and the months where the model is not able to assure a coverage of the area under consideration because of the lack of an adequate number of sites operating at the same time, are marked in bold and underlined bold respectively.

(a)
Tor  long-term predictions of M3000F2 (Fig. 5b) obtained with the 9 forecasting local models simultaneously operative for the month of May (see Table 7a).The epoch under consideration, characterized by moderate geomagnetic activity, is 18 May 1991 at 01:00 UT.Figures 6 and 7 show comparisons between the maps based on M3000F2 measurements (Figs.6a-7a) and the maps based on the short-term forecasting of M3000F2 under disturbed/very disturbed geomagnetic activity (Figs.6b-7b) obtained with the RM_M3000F2_BD /IRI_RM_M3000F2_BD regional models respectively.
To get the short-term forecasting map of M3000F2 during disturbed geomagnetic conditions (Fig. 6b), RM_M3000F2_BD has taken into account the 8 forecasting local models simultaneously operative for the month of November shown in Table 7b.The epoch under consideration is 16 November 1991 at 01:00 UT.
The short-term forecasting map of M3000F2 under very disturbed geomagnetic conditions (Fig. 7b) was derived from

Discussion of the results and future developments
The very small number of sites providing the best predictions observed in Tables 2b-c/3a,5a/4a-c indicates that in these cases it is not possible to consider the forecasting local models as a single forecasting empirical regional model able to offer an adequate coverage of the area when disturbed and very disturbed/moderate/moderate, disturbed and very disturbed geomagnetic conditions occur.Anyway from the observation of Tables 2-5 emerges that it is never possible to work with all the 12 stations simultaneously.
Nevertheless, the strong point of this method is that, even not including certain stations, it is almost always possible to find a number N < 12 of local models offering an adequate coverage of the European area.
Therefore by adopting the criterion according to which a number N ≥ 4 of operating stations is considered sufficient to adequately cover the area under investigation providing simultaneous predictions of M3000F2, Tables 3a/5a show that, Fig.       7) with (see Table 7c)  7b).7c).
except for the month of March where 5/4 stations are simultaneously operative, in all the other cases (marked with $) it is never possible to use operatively the prediction models represented by Eqs. ( 4) and ( 5).This occurs because the Bradley-Dudeney algorithm is not effective in improving the monthly median field generated by Eq. ( 2) and the IRI model during moderately disturbed conditions.This implies that moderate geomagnetic conditions cannot be considered so different from quiet geomagnetic conditions, and hence the short-term forecasting of M3000F2 under moderate geomagnetic activity can be practically represented by the long-term predictions of M3000F2.Therefore under moderate geomagnetic conditions it is sufficient to use the forecasting models operating at the same time in the sites shown in Table 2a in order to get for all the months (except for March) the best short-term forecasts of M3000F2.
However, under disturbed geomagnetic activity, the Bradley-Dudeney correction factor is able to improve significantly the performance of the local models whose predictions are provided by Eq. (2).In fact for disturbed geomagnetic conditions, excluding the two cases relative to April and July, in the remaining ten cases there are always a sufficient number of sites that can operate simultaneously providing an adequate coverage of the European area (see Table 3b).With regard to very disturbed geomagnetic activity, again the Bradley-Dudeney formula improves noticeably the IRI performance given that, excluding the three cases relative to April, May, and July, in all the remaining cases the forecasting models operating at the same time in the sites shown in Table 5c can work all together providing simultaneous predictions of M3000F2 in the considered area.
Even though IRI local models provide the worst performance, as can be deduced from the observations of Table 4 and Fig. 4, nevertheless it is noteworthy that, under very disturbed geomagnetic conditions, the Bradley-Dudeney formula adopted to correct IRI model predictions improves considerably the performance of IRI local models especially for the MHLS and HLS (see Table 5 and Fig. 4).
As already explained in Sect.3, the restrictions highlighted in Table 6 have been mitigated and a new table which could be adopted for a possible operative use was obtained (see Table 7).This means that the forecasting local models represented by Eqs. ( 2), (4), and (5) could be used as a single forecasting empirical regional model RM_M3000F2/RM_M3000F2_BD/IRI_RM_M3000F2_BD for generating forecasting maps of M3000F2 up to 3 h in advance, under moderate/disturbed/very disturbed geomagnetic conditions over the European area on the basis of M3000F2 predictions produced by those local stations that can be considered as simultaneously operative (see Table 7).
Just to clarify better the question, the RM_M3000F2 regional model utilized to generate at the same time M3000F2 forecasts in May for moderate geomagnetic activity is constituted by N = 9 stations leaving out the workstations of Rome, Uppsala, and Kiruna (see Table 7a); the RM_M3000F2_BD regional model employed to produce simultaneous predictions of M3000F2 in November for disturbed geomagnetic activity is formed by N = 8 stations excluding the stations of Tortosa, Kaliningrad, Uppsala, and Kiruna (see Table 7b); and the IRI_RM_M3000F2_BD regional model used to get M3000F2 forecasts in February for very disturbed geomagnetic activity could work with N = 8 sites operating at the same time leaving inoperative the stations of Rome, Poitiers, Lannion, and Juliusruh (see Table 7c).
Table 7 shows 1 case for disturbed geomagnetic conditions (July) and 3 cases for very disturbed geomagnetic conditions (April, September, and December) for which it is not possible to rely on the MLS for the prediction of M3000F2.This implies that the remaining sites operating at the same time will provide forecasts of M3000F2 over a more limited European area extending in latitude from 50 • 1 N to 67 • 8 N and in longitude from −0 • 6 W to 26 • 6 E. Analogously, in Table 7 are shown 5 cases for moderate geomagnetic activity (January, February, March, November, and December) and 2 cases for disturbed geomagnetic conditions (February and April) in which it is not possible to rely on the HLS for the prediction of M3000F2.Therefore in these cases the remaining sites operating simultaneously will provide forecasts of M3000F2 over a more restricted European sector extending in latitude from 40 • 8 N to 59 • 8 N and in longitude from A careful analysis of the performance of the various models shown in Fig. 4, leads to the following conclusions.
As regards the MLS of the European area under consideration (Tortosa, Rome, Poitiers, and Lannion), extending in latitude from 40 • 8 N to 48 • 1 N and in longitude from 0 • 3 W to 12 • 5 E, overall the following considerations can be done: in the case of moderate geomagnetic activity, the performance of RM_M3000F2 is markedly better than RM_M3000F2_BD, IRI_RM_M3000F2, and IRI_RM_M3000F2_BD being the predictions better in 63, 21, 12, and 4 % of the cases analysed respectively.This means that the local models are much more reliable than IRI models and the correction by the Bradley-Dudeney formula does not contribute to improve the performance.
In the case of disturbed geomagnetic activity, RM_M3000F2, IRI_RM_M3000F2, and IRI_RM_M3000F2_BD do not provide good performance because their predictions were better only in 27, 18, and 6 % of the cases analysed respectively, but the Bradley-Dudeney formula applied to correct the predictions of the local models seems to be effective in improving the predictions given that the RM_M3000F2_BD performance is better in 49 % of cases.
In the case of very disturbed geomagnetic activity, RM_M3000F2, IRI_RM_M3000F2, and IRI_RM_M3000F2_BD provide a very modest performance, with their predictions being better in only 17, 14, and 19 % of the cases analysed respectively; conversely, the RM_M3000F2 performance is improved with the Bradley-Dudeney correction given that RM_M3000F2_BD provides better predictions in 50 % of cases.
With regard to the MHLS of the European area under study (Dourbes, Slough, Juliusruh, Kaliningrad, and Uppsala), extending in latitude from 50 • 1 N to 59 • 8 N and in longitude from −0 • 6 E to 20 • 6 E, in general the following considerations can be drawn: in the case of moderate geomagnetic activity, RM_M3000F2 performance is noticeably better than RM_M3000F2_BD, IRI_RM_M3000F2, and IRI_RM_M3000F2_BD because the predictions provided by RM_M3000F2 were better in 60 % of cases, while the predictions obtained with RM_M3000F2_BD, IRI_RM_M3000F2, and IRI_RM_M3000F2_BD were better in only 10 %, 17 %, and 13 % of the cases analysed respectively.This means that the local models are much more reliable than IRI models and the correction by the Bradley-Dudeney formula does not contribute significantly to improve the performance.
In the case of disturbed geomagnetic activity, RM_M3000F2 and IRI_RM_M3000F2 do not provide good performance, their predictions were better only in 9 and 23 % of the cases analysed; the Bradley-Dudeney formula improves slightly the predictions because the RM_M3000F2_BD and IRI_RM_M3000F2_BD performance is better in 39 % and 29 % of the cases analysed respectively.
In the case of very disturbed geomagnetic activity, RM_M3000F2 and IRI_RM_M3000F2 provide a very poor performance, with their predictions being better only in 5 and 10 % of the cases analysed respectively; contrarily, RM_M3000F2_BD and IRI_RM_M3000F2_BD performance improves considerably by means of the Bradley-Dudeney correction factor especially for IRI_RM_M3000F2_BD.The predictions provided by RM_M3000F2_BD/IRI_RM_M3000F2_BD were better in 31 %/54 % of the cases analysed.
Regarding the HLS of the European area under consideration (Lyckesele, Sodankyla, and Kiruna), extending in latitude from 64 • 6 N to 67 • 8 N and in longitude from 18 • 8 E to 26 • 6 E, the following general considerations can be done: in the case of moderate geomagnetic activity, even though the RM_M3000F2 performance is superior with respect to RM_M3000F2_BD/IRI_RM_M3000F2/IRI_RM_M3000F2 _BD, it is not significantly better than in the previous cases, providing better predictions in 31 % of cases in comparison with 22 %/28 %/19 %.In the case of disturbed geomagnetic activity, RM_M3000F2/IRI_RM_M3000F2 provide a very poor performance because their predictions were better in only 0 %/11 % of the cases analysed, but the predictions greatly improve by using the Bradley-Dudeney formula because RM_M3000F2_BD/IRI_RM_M3000F2_BD forecasts were better in 34 %/55 % of the cases analysed.
A very poor performance is also given by RM_M3000F2 and IRI_RM_M3000F2 in the case of very disturbed geomagnetic activity, because their predictions were better in only 3 % and 0 % of the cases analysed respectively; conversely, the Bradley-Dudeney correction factor used to correct RM_M3000F2/IRI_RM_M3000F2 has proved very efficient to get better predictions given that the RM_M3000F2_BD/IRI_RM_M3000F2_BD performance increases noticeably, providing better predictions in 29 %/68 % of the cases analysed.
Figure 5 shows the comparison between the map based on M3000F2 measurements (Fig. 5a) and the map based on the long-term predictions of M3000F2 (Fig. 5b), obtained with the 9 forecasting local models simultaneously operative for the month of May (see Table 7a).It must be said that at the considered epoch (18 May 1991 at 01:00 UT), M3000F2 measurement at Sodankyla is missing and it cannot be compared with the corresponding prediction.Since a comparison between measurement and prediction for each station is needed, to compare directly the map of the measurements with the map of the predictions, the prediction of M3000F2 at Sodankyla was not considered; for this reason the maps in Fig. 5 are presented in a more limited area.
To get the short-term forecasting map of M3000F2 during disturbed geomagnetic conditions (Fig. 6b), RM_M3000F2_BD has taken into account the 8 forecasting local models simultaneously operative for the month of November shown in Table 7b.It must be noted that at the considered epoch (16 November 1991 at 01:00 UT), M3000F2 measurements at Rome and Sodankyla are not available and therefore they cannot be compared with the corresponding predictions.Because of this the predictions of M3000F2 at Rome and Sodankyla were not considered, and for this reason the maps in Fig. 6 are presented in a more restricted area.
Under very disturbed geomagnetic conditions, the shortterm forecasting map of M3000F2 was obtained from the M3000F2 predictions derived from the 8 forecasting local models operating at the same time for the month of February and shown in Table 7c.M3000F2 predictions at Sodankyla and Kiruna were not considered to build the map because the respective M3000F2 measurements at the considered epoch (4 February 1992 at 17:00 UT) are missing; consequently also in this case the maps in Fig. 7 are represented in a narrower region.
The cells of these maps (see Figs. 5-7), depicted with a step in latitude and longitude of 2 • , were scrupulously examined to evaluate the reliability of the models on the spatial regional scale.
Under moderate geomagnetic conditions (Fig. 5a-b), it emerged that there is a very large area extending in latitude from 40 • 8 N to 64 • 6 N and in longitude from −0 • 6 W to 15 • 4 E, where the RM_M3000F2 performance can be considered very satisfactory because 83 % of this area shows that the differences between M3000F2 measurements and M3000F2 predictions differ by no more than 0.06.The performance of RM_M3000F2 goes down in the sector extending in latitude from 40 • 8 N to 64 • 6 N and in longitude from 15 • 4 E to 20 • 6 E where the differences between M3000F2 measurements and M3000F2 predictions are no larger than 0.12.In terms of the three latitudinal ranges, RM_M3000F2 performance can be considered good because the 53 %, 73 %, and 64 % of sectors located at middle, middle-high and high latitudes respectively show differences between M3000F2 measurements and M3000F2 predictions no greater than 0.06.
From the comparison between the map obtained with the M3000F2 measurements and the short-term forecasting map generated by RM_M3000F2_BD under disturbed geomagnetic conditions (Fig. 6a-6b), two sectors, one extending in latitude from 46 • 6 N to 52 • 6 N and in longitude from −0 • 6 W to 18 • 8 E, and the other one extending in latitude from 58 • 6 N to 64 • 6 N and in longitude from 7 • 4 E to 18 • 8 E, show a very satisfactory performance of RM_M3000F2_BD because in 70 % of the area covered by these sectors the differences between the M3000F2 measurements and M3000F2 forecasts are no larger than 0.06.
The performance of RM_M3000F2_BD deteriorates slightly in the zone extending in latitude from 52 • 6 N to 58 • 6 N and in longitude from 7 • 4 E to 18 • 8 E where, except in some small areas, the differences between M3000F2 measurements and M3000F2 forecasts are no greater than 0.12.RM_M3000F2_BD performance worsens further in the sector extending in latitude from 52 • 6 N to 64 • 6 N and in longitude from −0 • 6 W to 7 • 4 E where 70 % of this area shows differences between M3000F2 measurements and M3000F2 predictions no bigger than 0.18.Reasoning in terms of the three different latitudinal sectors the following considerations can be done: the 85 % of the sector situated at middle latitudes shows a very satisfactory performance because M3000F2 measurements and M3000F2 forecasts differ by no more than 0.06; the performance is not very good in the area sited at middle-high latitudes where only 31 % of this area shows differences between measurements and predictions no greater than 0.06; an improvement of the performance can be observed at high latitudes because 42 % of this sector shows differences no larger than 0.06.
Under very disturbed geomagnetic conditions (Fig. 7ab), in a relatively large area extending in latitude from 40 • 8 N to 64 • 6 N and in longitude from −0 • 6 W to 5 • 4 E, IRI_RM_M3000F2_BD performance is not very good because only 49 % of this area shows differences between M3000F2 measurements and M3000F2 predictions no larger than 0.06.
However, a very satisfactory performance is observed in the region extending in latitude from 40 • 8 N to 64 • 6 N and in longitude from 5 • 4 E to 20 • 6 E, because 70 % of this region shows variations between M3000F2 measurements and M3000F2 predictions no larger than 0.06.In terms of the three latitudinal ranges, a high performance is observed in the area situated at middle latitudes where the 75 % of this area shows differences between M3000F2 measurements and M3000F2 forecasts no greater than 0.06.The performance worsens, but it can be still considered good at middle-high and high latitudes because the 55 %, and 64 % of these sectors show differences between M3000F2 measurements and M3000F2 forecasts no greater than 0.06.
According to the criterion that has been adopted, in the special case of July under very disturbed geomagnetic conditions (see Table 7c), the number of stations operating at the same time is not considered sufficient to ensure an adequate coverage, and therefore in this unique case it is not possible to provide M3000F2 forecasts for the European area.
As a rule, when A workstations are put aside, the M3000F2 forecasts are calculated in the remaining (N − A) workstations.Starting from the predictions of M3000F2 generated at certain epochs by the (N − A) local models, and conceiving the area considered in this study as a grid of equispaced points in latitude and longitude, it is possible using a suitable interpolation algorithm to compute the values of M3000F2 also at the A workstations that were initially rejected as well as the values of M3000F2 at each grid point.The elaboration of M3000F2 data thus obtained permits the achievement of short-term forecasting maps of M3000F2 at the epochs under consideration.
With regard to the three M3000F2 forecasting maps considered in this study , in some regions located at middle, middle-high, and high latitudes, the performance of RM_M3000F2, RM_M3000F2_BD, and IRI_RM_M3000F2_BD does not give fine results (see in Figs.5b-7b the cells labelled with the white circle).This probably occurs because the prediction algorithm is composed by a monthly median model corrected with a factor that depends on the short-term forecasting of foF2, so that the prediction of M3000 is inevitably affected by an error that is the sum of the errors committed by the monthly median model and short-term forecasting model of foF2.Nevertheless, from the forecasting maps produced by RM_M3000F2, RM_M3000F2_BD, and IRI_RM_M3000F2_BD it is also possible to note wide zones situated at middle, middle-high, and high latitudes where the RM_M3000F2, RM_M3000F2_BD, and IRI_RM_M3000F2_BD predictions agree quite well with the M3000F2 measurements (see in Figs.5b-7b the cells labelled with the black circles).This is a satisfactory result because it is not easy to yield reliable predictions when not quiet geomagnetic conditions occur, particularly at high latitudes.
With regard to future developments, an operative use of Table 7 would allow the regional models RM_M3000F2, RM_M3000F2_BD, and IRI_RM_M3000F2_BD to generate short-term forecasting maps of M3000F2 up to 3 h in advance over Europe on the basis of M3000F2 predictions produced by those local stations that can be considered as simultaneously operative.
Moreover, in spite of some limitations described above, the short-term forecasting models of M3000F2 developed in this work could be used together with the IFERM_foF2 model providing short-term forecasts of foF2 (Pietrella, 2012).The predictions of M3000F2 and foF2 thus obtained, given as input parameters to the IRI model, can provide a short-term forecasting of 3-D electron density mapping of the ionosphere over the European area following a technique similar to that recently utilized to achieve quasi-real-time maps of electron density over the Mediterranean region (Pezzopane et al., 2011).
The realization of short-term forecasting maps of M3000F2 together with 3-D matrices of electron density up to three hours in advance over Europe is one of the aims to be achieved in the future.In addition, the values of M3000F2 predicted with these models and the values of foF2 predicted by IFERM_foF2 could also be used to calculate shortterm forecasts of the height of the maximum electron density of the F2 layer (hmF2) in all the operative sites.The achievement of short-term forecasting maps of hmF2 based on the predictions of hmF2 during moderate, disturbed and very disturbed geomagnetic conditions is another target to be achieved in the future. Fig.1

Fig. 3 .
Fig.3.Example of models providing the best performance (marked with a different symbol) for some ionospheric observatories and for some months, under moderate geomagnetic conditions. Fig.4

Fig. 4 .
Fig. 4. The percentages of the best performances offered by RM_M3000F2, RM_M3000F2_BD, IRI_RM_M3000F2, and IRI_RM_M3000F2_BD deduced from Tables 2-5 are shown in blue, red, green, and purple respectively in the case of (a) moderate, (b) disturbed, and (c) very disturbed geomagnetic activity for MLS, MHLS, and HLS. Fig.5

Fig. 5 .
Fig. 5. (a) Map obtained from M3000F2 measurements and (b) long-term prediction map for M3000F2 generated using the 9 forecasting local models simultaneously operative for the month of May (see Table 7a).The epoch under consideration, characterized by moderate geomagnetic activity (ap(τ = 0.8) = 17.9; ap(τ = 0.9) = 18.1), is 18 May 1991 at 01:00 UT.The white circles indicate the cells where the predictions of M3000F2 are less satisfactory.The black circles indicate the cells where the predictions of M3000F2 quite faithfully match the M3000F2 measurements.The cells labelled with both the circles indicate that the performance can be considered satisfactory only in part of the cell.
Fig. 5. (a) Map obtained from M3000F2 measurements and (b) long-term prediction map for M3000F2 generated using the 9 forecasting local models simultaneously operative for the month of May (see Table 7a).The epoch under consideration, characterized by moderate geomagnetic activity (ap(τ = 0.8) = 17.9; ap(τ = 0.9) = 18.1), is 18 May 1991 at 01:00 UT.The white circles indicate the cells where the predictions of M3000F2 are less satisfactory.The black circles indicate the cells where the predictions of M3000F2 quite faithfully match the M3000F2 measurements.The cells labelled with both the circles indicate that the performance can be considered satisfactory only in part of the cell.

Table 3 .
Same as Table2but for the sites where the best performance is provided by the prediction model (4), indicated with the triangle.The term nda indicates the cases for which it was not possible to evaluate the performance of the prediction model (4) with respect to the other models because no data were available to calculate the g.r.m.s.error.
and 5a).With regard to disturbed geomagnetic conditions, it is evident from Tables2b and 4bthat RM_M3000F2 and IRI_RM_M3000F2 are not appropriate for the operative use, because a more adequate coverage is provided by RM_M3000F2_BD (see Table

Table 4 .
Same as Table2but for the sites where the best performance is provided by the IRI model, indicated with the empty circle.The term nda indicates the cases for which it was not possible to evaluate the performance of the IRI model with respect to the other models because no data were available to calculate the g.r.m.s.error.

Table 5 .
Same as Table

Table 7 .
Same as Table6, but with some improvements for a possible operative use.