The empirical relationships between M s, m b and M L for China and vicinity

Data from 753 earthquakes are used to determine a relationship between surface-wave magnitude (M _s) and bodywave magnitude (m _b), and from 541 earthquakes to determine a relationship between surface-wave magnitude (M _s) and local magnitude (M _L) for China and vicinity: M _s=0.9883 m _b-0.0420, M _s=0.9919 M _L-0.1773. The relationship of M _s versus m _b is obtained for 292 events occurred in the Chinese mainland in the time period from 1964 to 1996, 291 events occurred in Taiwan in the time period from 1964 to 1995 and 170 events occurred in the surrounding area. Standard deviation of the fitting is 0.445. Relationship of M _s versus M _L is obtained for 36 events occurred in the Chinese mainland, 293 events occurred in Taiwan, China and 212 events occurred in the surrounding area. The total amount is 541 events. Standard deviation of the fitting is 0.4673. The uncertainties of the converted M _s in different magnitude intervals can be estimated using complementary cumulative distribution function (CCDF). In the relationship of M _s versus m _b, taking ±0.25 as a range of uncertainties, in magnitude interval m _b 4.0–4.9, the probabilities for the converted M _s taken value less than (M _s-0.25) and more than (M _s+0.25) are 17% and 27% respectively. Similarly, we have probabilities for m _b 5.0–5.9 are 34% and 20% and that for m _b 6.0–6.9 are 11% and 47%. In the relationship of M _s versus M _L, if the range of uncertainties is still taken as ±0.25, the corresponding probabilities for magnitude interval M _L 4.0–4.9 are 22% and 38%, for M _L 5.0–5.9 are 20% and 15% and for magnitude interval M _L 6.0–6.9, are 15% and 29%, respectively. The relationships developed in this paper can be used for the conversion of one magnitude scale into another magnitude scales conveniently. The estimation of uncertainties described in this paper is more accurate and more objective than the usual estimation expressed by deviation. The estimations described in this paper indicate various dispersions in different magnitude intervals of original data. The estimations of uncertainties described by probabilities can be well connected with the total estimations of uncertainties in seismic hazard assessment.