Mwpd¶

Duration-amplitude moment magnitude Mwpd (Lomax & Michelini 2009), ported from Early-est 1.2.9. The vertical broadband is restituted to displacement, high-passed at the BRB-HP corner and integrated over the source duration T0; T0 is estimated from the high-frequency envelope. The amplitude processor emits the displacement integral (nm*s) and carries T0 as the amplitude period; the magnitude processor forms M0 and Mwpd with the distance, PREM depth and duration corrections.

Description¶

$M_{wpd}$ is a per-station duration–amplitude moment magnitude for large earthquakes, determined from teleseismic P waveforms. It integrates the broad-band P-wave ground displacement over the apparent source duration $T_0$ , after Lomax & Michelini (2009), and is a port of the Mwpd implementation in Early-est (A. Lomax). Because it integrates over the whole source duration rather than taking the peak (as $M_{wp}$ does), it does not saturate for events with long rupture duration, such as tsunami earthquakes and great earthquakes.

The plugin contributes both an amplitude processor and a magnitude processor of type Mwpd. Both use the vertical broad-band component only.

Method¶

Given the far-field P displacement $u(t)$ for a source of duration $T_0$ , the scalar seismic moment is (Tsuboi et al. 1995; Lomax & Michelini 2009)

$M_0 = C_M \int_{t_P}^{t_P + T_0} u(t)\, \mathrm{d}t ,$

where $t_P$ is the P-arrival time. The implementation accumulates the running displacement integral separately over its positive and negative lobes (to separate the direct P wave from later reflected/secondary phases of opposite polarity, eq. 3 of Lomax & Michelini 2009) and uses the larger of the two at $T_0$ . The moment magnitude follows the standard relation

$M_{wpd}^{\mathrm{raw}} = \tfrac{2}{3}\left(\log_{10} M_0 - 9.1\right) - C_\Delta(\Delta, h) ,$

with a distance correction $C_\Delta$ (INGV/Early-est, subtracted; zero for $h>100$ km), then a PREM step depth correction $C_h(h)$ and a moment-scaling term for great/slow events:

$M_{wpd} = M_{wpd}^{\mathrm{raw}} + C_h(h) + r\,(M_{wpd}^{\mathrm{raw}} - 7.2)\cdot 0.45 , \qquad r = \mathrm{clip}\!\left(\tfrac{T_0-90}{110-90},\,0,\,1\right).$

The duration ramp $r$ engages only for $T_0 > 90$ s (the moment scaling for large interplate-thrust / tsunamigenic events, eq. 5a in Lomax & Michelini 2009).

Amplitude¶

For each P pick on the vertical broad-band, the amplitude processor:

restitutes counts to ground velocity (sensitivity gain) and high-passes with a Butterworth filter at a 0.005 Hz corner (the Early-est BRB-HP filter); a low corner is essential to retain the long-period displacement that carries the moment of great earthquakes;
single-integrates to displacement and accumulates the running double integral into positive/negative lobes;
estimates the apparent source duration $T_0$ from the high-frequency envelope: the velocity is band-passed 1–5 Hz, squared and boxcar-smoothed, and $T_0$ is derived from the times at which this envelope last drops below 90/80/50/20 % of its peak;
integrates over $\min(T_0,\ t_S-t_P,\ \mathtt{signalEnd})$ — the $S\!-\!P$ cap (from a travel-time table) keeps the integral free of S and surface-wave energy.

The amplitude carries the displacement integral (unit nm*s); $T_0$ is carried as the amplitude period. In real-time application, the computation is progressive: the processing is updated as new data arrives and finalised once $T_0$ is resolved.

Magnitude¶

The magnitude processor reads the displacement integral (nm*s) and $T_0$ (period) and applies the relations above, with the moment constant

$C_M = 4\pi\,\rho\,V_p^{3}\,F_p \cdot \tfrac{10000}{90}\cdot 1000 \approx 4.68\times10^{21}$

(Tsuboi constant; $\rho=3400$ , $V_p=7900$ , $F_p=2$ ). Since $M_{wpd}$ is already a moment magnitude, no further $M_w$ conversion is applied (SeisComP’s per-station estimateMw step is the identity here); the network magnitude is a robust (median / trimmed-mean) average of the station $M_{wpd}$ values.

Calibration¶

The implementation was validated against GCMT moment magnitudes for 972 events ( $M_w$ 6.0–8.8, 2015–2026), processed through this algorithm with broad-band stations from global networks. $M_{wpd}$ tracks $M_w^{\mathrm{CMT}}$ along the 1:1 line with no saturation and no significant depth dependence.

Mean difference $M_{wpd} - M_w^{\mathrm{CMT}}$ by magnitude band:

$M_w$	mean	sigma	N
6.0 – 7.0	+0.07	0.14	831
7.0 – 7.5	-0.05	0.13	94
7.5 – 8.0	-0.04	0.20	40
8.0 – 9.5	-0.09	0.16	7

and by source depth:

depth (km)	mean	sigma	N
0 – 70	+0.07	0.16	702
70 – 300	-0.02	0.13	175
300 – 800	+0.01	0.10	95

These results are consistent with operational Early-est (slope ~1.01, $\pm 0.13$ ) and with Lomax & Michelini (2009), which reports $M_{wpd}$ matching $M_w^{\mathrm{CMT}}$ within $\pm 0.2$ .

A small residual bias remains at the low-magnitude end (mean $M_{wpd}-M_w^{\mathrm{CMT}}\approx +0.07$ for $M_w$ 6.0–7.0, within the $\sigma\approx 0.14$ scatter). Because $M_{wpd}$ is reported as a moment magnitude, no correction is applied by default, but any residual trend can be flattened per network without code changes using the standard scmag mechanisms: a linear station correction $M_{wpd}' = a\,M_{wpd} + b$ via magnitudes.Mwpd.multiplier / magnitudes.Mwpd.offset, or a piecewise-linear proxy mapping $M_w(M_{wpd})$ via magnitudes.Mwpd.MwMapping (as used for other $M_w$ proxies). Operators should derive these from a regression against their own reference magnitudes.

Early-est itself is deliberately conservative with $M_{wpd}$ at smaller magnitudes: it is not reported/alerted below $M_{wpd}\approx 7.0$ (alert.mwpd_min, default 6.95) and is only set as the event’s preferred magnitude once $M_{wpd}\gtrsim 8.0$ (preferred.min_value.mwpd; A. Lomax, pers. comm., 2026, suggests this could be lowered to approximately 7.5). SeisComP’s own preferred-magnitude selection (scevent) is a static per-type priority list rather than a value-gated threshold, so operators wanting equivalent behaviour should configure the $M_{wpd}$ priority accordingly: treat it with caution below $M_w\approx 7$ and avoid letting it override other magnitude types unless the event is confirmed large.

Configuration¶

Add mwpd to the plugins parameter and enable the Mwpd amplitude and magnitude in scamp / scmag (or scolv). The defaults reproduce Early-est; the most important amplitude parameter is amplitudes.Mwpd.highpassCorner (0.005 Hz — keep low). See the binding parameter descriptions for the full list. Because $M_{wpd}$ integrates over the source duration, it requires a long post-P window (up to signalEnd), so the final value has higher latency than $M_{wp}$ . For faster tsunami early warning, Lomax & Michelini (2013) describe $M_{wpd}\mathrm{(RT)}$ , a real-time variant that uses a smaller minimum station distance and avoids some of the duration estimation and re-scaling; it is not implemented in this plugin.

References¶

Lomax, A. & Michelini, A. (2009). $M_{wpd}$ : a duration–amplitude procedure for rapid determination of earthquake magnitude and tsunamigenic potential from P waveforms. Geophys. J. Int. 176, 200–214, doi:10.1111/j.1365-246X.2008.03974.x
Lomax, A. & Michelini, A. (2013). Tsunami early warning within five minutes. Pure Appl. Geophys. 170, 1385–1395, doi:10.1007/s00024-012-0512-6
Tsuboi, S. et al. (1995). Rapid determination of $M_w$ from broadband P waveforms. Bull. Seismol. Soc. Am. 85, 606–613.

Mwpd¶

Description¶

Method¶

Amplitude¶

Magnitude¶

Calibration¶

Configuration¶

References¶

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