Due to the limited database and model purpose, we use one of the simplest form of GMPEs with only 3 parameters (Berge-Thierry et al., 2003):

This functional form implies many hypothesis. In particular, a radial distribution of ground motion around a point source, neglecting geological heterogeneities, tectonic origin, source extension and radiation pattern. Fukushima (1996) also points out that a linear $log(D)/M$ formulation is not verified for magnitudes $\ge 6.5$ for which a $M^2$ term should be necessary. This concerns magnitudes out of our study range, but we will keep in mind that accelerations should be underestimated at long distance for large magnitudes.

To inverse the three parameters, we use seismic data recorded at 14 strong-motion permanent stations in Guadeloupe (see Fig. 1), with mixed site conditions, rock and soil (details about the seismic stations can be found in (Bengoubou-Valérius et al., 2008)), in the period from November 21 to December 28, 2004. The dataset includes about 400 earthquakes associated to 1430 triggers of 3-component acceleration waveforms. These events correspond to Les Saintes main shock $M_w=6.3$ and mostly the associated aftershocks, but also some regional events that we voluntarily kept in the database.

Locations and magnitudes come from the seismic catalog of the Guadeloupe observatory (OVSG-IPGP). Magnitudes were computed using the classical formula of duration magnitude from Lee et al. (1975) for events $M_d\le 4.5$ (Clément et al., 2000; Feuillard, 1985), and we imposed the moment magnitude from worldwide networks for greater events. This allows us to overcome the problem of duration magnitude saturation for magnitude greater than 4.5. The consistency of magnitude scale ($M_d$ versus $M_w$) has been checked by Bengoubou-Valérius et al. (2008).

For each event, a value of PGA is calculated as the maximum amplitude of horizontal acceleration signals, using the modulus of a complex vector defined by the two horizontal and orthogonal components $x(t)$ and $y(t)$. The PGA dataset is presented in Figs. 2 and 3. Magnitudes range from 1.1 to 6.3, hypocentral distances from 2 to 450 km, and PGA from 16 $\mu$g to 0.36 g.

To calculate the 3 parameters in Eq. (1), we minimized a misfit function using the $L_2$-norm. Due to the inhomogeneous dataset (magnitudes follow a power-law and there is more short-distance values), we applied a simple weighting function by multiplying the misfit by the magnitude and a power of the hypocentral distance. This gave more weight for large magnitudes and long distances.

The inversion scheme yields the following parameters: $a=0.61755$, $b=-0.0030746$, and $c=-3.3968$. It produced an RMS residual on $log(\mathit{PGA})$ of 0.47 (a factor of 3 in PGA, see Fig. 4). This value is higher than classical published GMPE results (around 0.3, see Strasser et al. (2009)), and it confirms the observation of Douglas et al. (2006) about abnormal data variability in Lesser Antilles. However, interestingly, this factor corresponds to the average ratio between rock and soil conditions in the observed PGA (Bengoubou-Valérius et al., 2008). This might also reflect the wide range of magnitudes and distances in a too simple functional form. In order to follow some of the key considerations used to develop GMPEs (Bommer et al., 2010), we checked medians and sigmas of PGA residuals (Fig. 4): it shows a very consistent distribution in the full magnitude range (from 2 to 6), while we observe a significant PGA underestimation (median around $+0.5$ so a factor 3 in amplitude) for $D<15$ km.

Eq. (1) with the parameters found is represented as an abacus in Fig. 5 showing calculated PGA as a function of hypocentral distance (from 3 to 500 km) and magnitudes 1 to 8.

Note that we voluntarily limited the minimum hypocentral distance for each magnitude, as we do not take into account the near fault saturation term. It is reasonable to assume that this minimum hypocentral distance is greater than rupture size. Earthquake magnitude reflects the seismic moment which is proportional to the total displacement averaged over the fault surface (Aki, 1972; Kanamori, 1977). Many authors propose a simple formula to express the relationship between magnitude and fault length or rupture area (Liebermann and Pomeroy, 1970; Mark, 1977; Wells and Coppersmith, 1994; Wyss, 1979). Here we use Wyss's formula (Wyss, 1979):

Fig. 6 shows representative events with observed PGA compared to our model predictions. We do not limit examples to the events from the dataset which reflects the previous residual analysis (Fig. 4), but present events in the period 2004 to 2007 with various depths, in crustal or subduction context, and for which sufficient triggers were available. As seen in Fig. 6, most of PGA values are predicted within the model uncertainty. Medians of $log(\mathit{PGA})$ residuals are equal to $+0.15$, $+0.28$, $+0.10$, $+0.19$, $+0.24$, and $-0.01$ for Fig. 6a to f events, respectively. We denote, for these 6 particular examples, a light tendency for PGA underestimation, which seems independent from magnitude. This is consistent with Fig. 4 residual analysis. The only significant PGA misfit appears for one soil condition station in the near field ($\approx 15$ km) for Les Saintes aftershocks (Figs. 6b and c), that is systematically underestimated by a factor of about 10.

We also compare these results with two published GMPE adapted to shallow crustal events: Sadigh et al. (1997) and Ambraseys (1995). The Sadigh et al. (1997) model is very similar to our PGA model for magnitudes $\ge 5.0$ (Figs. 6a, c and f) but has poor fitting for lower magnitudes (Figs. 6b, d and e) with a systematic overestimation. The Ambraseys (1995) model has a globally poor fitting, with an overestimation of PGA, particularly for $M<5.0$.

Although we know that the spectral frequency content of ground acceleration and peak velocity have important implications on building damage, establishing a direct relation between a single PGA value and macroseismic intensity has proved its efficiency in many cases (Chiaruttini and Siro, 1981; Margottini et al., 1992; Murphy et al., 1977; Wald et al., 1999). For the Lesser Antilles, we follow the suggestion of Feuillard (1985) who studied the historical and instrumental seismicity using the simple empirical relation of Gutenberg and Richter (1942):

Note that following the MSK scale, intensity must be an integer value. In this article, we decided arbitrarily to round $I$ to the nearest and smallest integer (e.g., $I=6.0$ to $6.9$ correspond to intensity of VI).

The resulting model for intensities is presented as right $Y$-axis in Fig. 5. Following Eq. (3), the 0.47 uncertainty on our predicted $log(\mathit{PGA})$, would imply an uncertainty on $I$ of $\pm 1.4$, on which we should add the uncertainty of Eq. (3) itself, which is unknown.

We test our model on a database of 20 recent earthquakes for which we have intensity reports (a total of 254 observations) as well as instrumental magnitudes and hypocenter locations. Events are from various origins with magnitudes 1.6 to 7.4, distances from 4 to 500 km, and observed intensities from I to VIII. This wide panel of event characteristics allows us to check our model applicability.

We present in Fig. 7 the intensity residuals versus observed intensity and hypocentral distance. Global standard deviation equal 0.8, with a near zero median value. Residuals are also well distributed over the intensity and distance ranges. Since this database is not statistically sufficient, we will keep uncertainty on intensities deduced from the PGA residuals, i.e., $\sigma =1.4$ corresponding to 68% confidence interval. We also checked that maximum observed intensity for each event is strictly below this probability level (see Fig. 7 solid circles).

In Fig. 8, we detail eight examples of the most significant events with observed and predicted intensities (see epicenters in Fig. 1).

Fig. 8a shows the October 10, 1974 “Antigua” earthquake (McCann et al., 1982; Tomblin and Aspinall, 1975), $M_s=7.4$, a shallow 30 km-depth with normal-fault mechanism, $M_s$ from NEIC USGS, location and MSK intensities from McCann et al. (1982). Maximum intensities and distance of observations vary from VIII at 45 km in Antigua to II at 400 km in Virgin Islands. All the observations (9 sites) are within the $B^3$ prediction uncertainty limits. The median of intensity residuals equals $-0.6$, sigma is $0.5$. This is an unexpected positive result since the model is extrapolated for magnitudes larger than Les Saintes ($M_w=6.3$); so this magnitude 7.4 is formally out of our interval of validity. Note also that near-field intensities (at 45 km) seem correctly fitted by the model while this hypocentral distance is very close to our limit defined by Eq. (2), which gives $L=42$ km.

Fig. 8b shows the March 10, 1976 earthquake, a magnitude $M_b=5.9$, 56 km-depth on subduction interface north of Guadeloupe ($M_b$ from USGS-NEIC, location and MSK intensities from Feuillard (1985)). Maximum intensities and distances of observations vary from V in Le Moule (Guadeloupe) at 85-km, to II in Martinique at 150 km distance. Most of the 22 observed intensities are underestimated (median of residuals is $+0.4$) but still within one sigma uncertainty ($\mathit{RMS}$ equals 0.5).

Fig. 8c shows the January 30, 1982 earthquake, a magnitude $M_w=6.0$, 63 km-depth on subduction interface north of Guadeloupe ($M_w$ and location from Global CMT Project, MSK intensities from Feuillard (1985)). Maximum intensities and distances of observations vary from V in various urban districts of Guadeloupe and Antigua at 90 km distance, to II in Barbuda (130 km). Most of the 34 observed intensities are within the $B^3$ uncertainty limits, with a zero median and $\mathit{RMS}$ on intensity residuals equal to 0.7.

Fig. 8d shows the March 16, 1985 “Redonda” earthquake (Feuillet et al., 2010; Girardin et al., 1991), a magnitude $M_w=6.3$, 10 km-depth normal-fault ($M_w$ and location from Global CMT Project, MSK intensities from Feuillard (1985)). Maximum intensities and distances of observations vary from VI at 30 km in Montserrat to II at 300 km in Martinique. We added a supposed intensity of VII-VIII (light gray dashed rectangle) because important cliff collapses have been observed in the Redonda island, at 10 km-distance from epicenter. All the 23 observed intensities are within the $B^3$ uncertainty limits ($\mathit{RMS}=0.7$) with zero median. Note a very local amplification effect that occurred in the region of Pointe-à-Pitre (Guadeloupe) with an intensity of V to VI at 120 km from the hypocenter.

Fig. 8e shows the November 21, 2004 Les Saintes main shock earthquake of magnitude $M_w=6.3$, $M_w$ from Global CMT Project, location from Bazin et al. (2010), EMS98 intensities (see definition in Grunthal et al. (1998)) from an official survey by the BCSF (Cara et al., 2005). Maximum intensities and distances of observations vary from VIII at 20 km in Les Saintes to IV at 140 km in Martinique, and correspond to detailed studies carried on by BCSF in 33 different urban districts. All the 29 observed intensities are within the $B^3$ uncertainty limits ($\mathit{RMS}=0.6$, median = $-0.9$).

Fig. 8f shows the largest Les Saintes aftershock, on February 14, 2005 of magnitude $M_w=5.8$, located south of Terre-de-Haut ($M_w$ and location from Global CMT Project, MSK intensities from OVSG-IPGP). Maximum intensities and distances of observations vary from VII at 14 km in Les Saintes to IV at 74 km in Anse-Bertrand (Guadeloupe). All the 25 observed intensities are within the $B^3$ uncertainty limits ($\mathit{RMS}=0.3$, median = $-1.0$) with a global light overestimation.

Fig. 8g shows one of the numerous Les Saintes aftershocks, on December 22, 2005 of magnitude $M_d=4.2$, located north of Terre-de-Bas ($M_d$, location and MSK intensities from OVSG-IPGP, unpublished). Maximum intensities and distances of observations vary from V at 15 km in Basse-Terre to II at 58 km in Saint-François (Guadeloupe). All the 7 observed intensities are within the $B^3$ uncertainty limits ($\mathit{RMS}=0.6$, median = $-0.3$).

Fig. 8h shows the November 29, 2007 Martinique intermediate-depth (152 km) intraslab earthquake of magnitude $M_w=7.4$, $M_w$ and location from Bouin et al. (2010) and Global CMT Project, with EMS98 intensities from an official survey by the BCSF (Schlupp et al., 2008). Maximum intensities and distances of observations vary from VII at 150 km in Martinique to II at 400 km in St-Barthelemy, and correspond to detailed studies carried on by BCSF in 70 different urban districts in Guadeloupe and Martinique, plus other islands reports. Most of the 74 observed intensities are within the $B^3$ uncertainty limits ($\mathit{RMS}=0.83$, median = $-0.1$), but we note three underestimated intensities at long distances: V in Saint-Vincent (250 km) and Trinidad (500 km), and IV in Anguilla (443 km). This may be due to local site amplifications because of low frequency content of the seismic waves.

These eight examples confirm that $B^3$ model seems able to predict average intensities within a global residual of $\sigma =1.4$ degree in the MSK scale, for events of magnitudes up to 7.4 in Lesser Antilles context with various hypocentral distances. This value corresponds to 68% of confidence interval and gives a convincing maximum possible intensity even when local site effects are observed.