MAGNITUDES OF NEW ZEALAND EARTHQUAKES, 1901-1993

Determinations of surface-wave magnitude (Ms) are made on a consistent basis for 202 selected New Zealand earthquakes over the period 1901-1993, including most post-1942 events with local magnitude not less than 6.0 and centroid depth less than 45 km. These determinations have led to a reassessment of magnitudes and locations of some earlier events in the New Zealand Seismological Observatory Catalogue of local magnitudes (ML), in some cases with substantial revisions. The surface-wave magnitudes are compared with local magnitudes and moment magnitudes (Mw), where available, and the relations between these three variables and centroid depth are examined through regression models. The absence of surface-wave observations for some earthquakes allows an upper limit to be placed on their likely moment magnitudes. The analysis shows that estimates of Mw derived from Ms will have a standard error of about 0.15 and Mw derived from ML a standard error of about 0.3.


INTRODUCTION
This paper describes the estimation of the magnitudes of larger New Zealand earthquakes for the period 1901 to 1993.
The surface-wave, local and moment magnitudes, Ms, ML and Mw, their inter-relationships, and the influence of depth are considered.
The above subjects were first addressed about six years ago [1,2] in an attempt to provide magnitudes estimated on a consistent basis for a set of 70 New Zealand earthquakes, based largely on surface-wave magnitudes.The need to extend the study to include more earthquakes, to include more Mw data, and to improve the inter-relationship modelling, has rapidly become pressing.This is partly due to the inadequacy of the New Zealand implementation of the M1, scale, but more fundamentally there is a need to model seismic hazard in the moment magnitude scale, because it has a clearer physical basis (as distinct from other magnitude scales) and is the scale most commonly used internationally in hazard modelling.The moment magnitude database for the present study is much greater than that assembled for the previous study [l], being enlarged by the subsequent five years of earthquakes, and by two special local studies of larger events which have since been carried out [3,4] in addition to the ongoing Harvard moment determinations [5].
In all, 260 earthquakes have been considered in this study.A major part of the present study was the detennination of the surface-wave magnitudes of 133 earthquakes, additional to the 69 Ms values in the previous study.A further 67 events, for which no surface wave data were found (in addition to one such event in the previous study), were also examined, and the implications of this are discussed.
Earthquakes with magnitudes down to about Mr 5 and depths down to 300 km have been considered.While no attempt has been made at considering all earthquakes with some nominal minimum magnitude, a majority of most post-1940 events 1 Institute of Geological & Nuclear Sciences, Lower Hutt 2 Fellow 3 Member with depth he< 45 km and magnitude M1_ ;:: 5.5 located in or near the New Zealand land mass have been included in this study.

THE ASSESSMENT OF SURFACE-WAVE MAGNITUDES
Except for a few observations made on Milne instruments ( discussed later), the surface-wave magnitude Ms was determined for each observation using the so-called 'Prague formula' of Vanek et al. [ 6] as follows Ms = log10 (A;T)max + 1.66 logl0 (D) in which (Aff) max is the maximum ratio of the ground displacement amplitude, A, of the surface waves in micrometres to the associated period, T, in seconds.D is the epicentral distance in degrees, and C is the station correction (discussed later).For horizontal motions the amplitude and period were taken as where N and E refer to the orthogonal components (generally north-south and east-west).If only one horizontal component was available, an increment of 0.1 was added to that magnitude to allow for the likely contribution of the missing component. were

lvf., determined from undamped Milne seismograph data
In a correlation of Milne data against that from other seismographs, Ambraseys and Melville [8] found that Ms should be determined from Milne data using the expression Ms= log10(2A 1) + l.25log 10(D) in which (2A1) is the double trace amplitude (peak-to-peak) in millimetres on Milne seismograms, D is the epicentral distance in degrees, C is the station correction and q is a constant with a value of 4.06 for his data set (where 15° ~ D ~ 80°, N. N. Ambraseys, personal communication).
Of the new data considered in this study, only six Milne station observations were found (from Adelaide, Sydney and Suva), foreventsintheyears 1912, 1929, 1931 and 1943.As in the previous study [1,2], the Ms values from these observations lie within the scatter of the rest of the data.

EARTHQUAKE SOURCE LOCATIONS
The source locations of the earthquakes studied here are given in Table 1 and Figures I and 2, and are of varying reliability depending on the date and location of any given event.
Firstly, consider the geographical co-ordinates.Traditionally these have been thought of as the epicentre, and an epicentre is preferred for the co-ordinates given.However, instrumentally determined epicentres were not very accurate until at least the early 1960's.The plan locations for most pre-1940 events are macroscismic determinations made from the best intensity maps available.The pre-1940 instrumental determinations [9] are inherently not very accurate, expecially those for the ailershocks of the 1929 Buller earthquake.
The epicentres listed were adopted, in approximate order of preference (depending on availability), from: Where reliable instrumentally determined centroid depths were not available, a range of other information had to be considered, notably the focal depth.The focal depth was usually assumed to be an acceptable approximation to the centroid depth, especially for small deep events, except where it conflicted with models of the seismogenic slab.The centroid depths listed were adopted, in order of preference, from: Centroid depths from (i) above, Derived from focal depths from (ii) above, Derived from focal depths from (iii) above, Derived from focal depths from (iv) above, Macroseismic estimates, Depths from ( 2) to ( 5) were all checked for validity against the seismogenic crust or slab depth range below the "epicentre" (using the seism1c1ty hypocentre models of Ansell and Bannister [II] and Anderson and Webb [12]), and adjusted if necessary by Dowrick.
A scatterplot of events in terms of magnitude versus centroid depth is shown in Figure 2. The deepest event (he = 300 km) at Mw 7.3 is the third largest in the data set, while the two largest events have depths of IO and 17 km respectively.Our data is much more complete compared to the New Zealand catalogue for shallow earthquakes (he < 45 km) than for deeper earthquakes, as indicated in Table 2.This comes about mainly because, for deeper events, ML tends to overestimate 1\1w and fewer Ms estimates are possible.

Earthquakes with Substantial Relocations
Revisions of estimates of magnitudes and source locations of past earthquakes is an ongoing process, as modelling techniques are refined.Such revisions mostly involve quite small changes, but a substantial number of the earthquakes involved in this study have recently been the subject of changes to their estimated locations which are large enough to warrant specific mention.As seen from Table 3, these events all date from earlier decades of this century when determination of source parameters was much less reliable than it is now.
The locations of a substantial number of the earthquakes under consideration were reviewed for either or both of two reasons.
First, the discrepancy between Ms and ML demanded a much greater depth.Second, the locations of these earthquakes, as listed in the New Zealand Seismological Observatory catalogue, did not accord with the felt intensity data.In some cases they were not felt at all, although they would have been if located as listed in the catalogue.In other cases the intensity patterns were very different from those of intensity models.
In view of the above findings, 23 of the events in  I.
While the accuracy of the source locations is important in general, for the purposes of this paper only the depth is used (as described in Section 5 below).For regional seismicity studies, however, the epicentral locations are also important.
In this respect some of the relocations in Table 3   for eanhquakes listed in Table 1.Epicentres marked with hash (#) are macroseismic estimates, those marked G are graphical, the remainder are instrumental computational.same procedure was adopted in the present study, the method used and the resulting station corrections being discussed in a separate paper [15].With a larger data set than used in 1990, the corrections for the stations in general differ from the previous results, although such differences are mostly small.This means that the corrected magnitudes of events considered in the 1990 study may differ slightly in the two studies even when no additional station observations are used for a given earthquake.However, the larger differences tend to occur when additional station observations are used.
Examples of larger differences are provided by the and also the number of non-New Zealand stations reporting the event.This is usually the number of stations listed by the International Seismological Centre, or its predecessors, though for a few of the earlier events their lists of overseas stations were supplemented by the authors, as it was found that those events were recorded at Riverview and/or Melbourne.

RELATIONSHIPS BETWEEN DIFFERENT MAGNITUDES
We examine relationships between Ms, Mw and Mr using graphical and regression techniques.
Because we are interested in predicting one magnitude from another, ordinary least squares regression is used, rather than any technique, such as orthogonal regression, which allows for uncertainties in the predictor variables.

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Both approximations, of course, apply only within the range of the present data.In Eq. ( 5) it can be seen that the quadratic term contributes significantly to the regression because the coefficient of this term is more than twice its standard error.It is of interest to note that although their expression is different from ours, Ambraseys & Free [17] obtained a coefficient for their depth term of 0.0036, which is very similar to the coefficient of0.0034 in Eq. ( 4) above.
A local regression fit of Mw against Ms for earthquakes restricted to he :,; 50 km was calculated.This provides a robust smooth trend curve against which the linear and quadratic parametric fits can be compared.As for other local regression analyses presented in this paper, it was fitted using the Splus local regression function "loess" [18].As can be seen from Figure 3, the trend curve lies within about 0.1 magnitude units of the linear and quadratic fits, being slightly closer to the quadratic fit for most of the range of Ms.The quadratic fit in Eq. ( 5) is thus considered adequate, and is ~ used to derive an inferred value of Mw, denoted M w, for earthquakes for which Ms but no value of Mw is known.Thus the "actual or inferred" moment magnitude is available for any earthquake for which either the moment magnitude or surface-wave magnitude is known.This magnitude is plotted in Figures 1 and 2 6)).
Also shown in Figure 3 is the relation of Ekstrom and Dziewonski [19], derived from global data, between log M0 and Ms for events with h < 50km.In terms of Mw this relation is equation ( 6

Mi versus Ms
The relation between Mi, Ms and he, using all available Mi/Ms data (i.e. from 1940 March 19 onwards), is shown in Figure 4.In general the correlation between Ml and Ms is weak.Below magnitude 6, Ml tends to exceed Ms; above magnitude 6, the reverse is the case.Despite the wide scatter of the data, the effect of depth is quite marked, with a strong tendency for Ml to exceed Ms for deep earthquakes.The regression of Ml on Ms and he explains only 54% of the variance of Mi, and is given by 7.0 6.5 As seen in Figure 3, there is no great difference between this relation and the linear and quadratic fits ofEqs (4) and ( 5) for shallow events over the magnitude range of the data, but the latter also describe the effect of depth.In Figure 5 are plotted the residuals (ie. the differences between the actual and fitted values of ML in Equation ( 9)) as a function of earthquake location.There is some regularity in the distribution of the residuals with respect to location, as In comparing ML and Ms, two tsunami producing earthquakes should be mentioned, i.e. those of 25 March 1947 (ML 6.0, Ms 7.2) and 17 March 1947 (ML 5.6,Ms 7.2).Their very large discrepancies between ML and Ms were noted by Eiby [20,21 ], but in these two events the intensity patterns correspond to ML rather than Ms.The anomalously high Ms values are thought [18,19] to be the consequence of these earthquakes having "slow" ruptures, ie.rich in long-period energy, but poor in short-period energy.These two events have therefore been omitted from the above analyses and the figures in this paper. .

ML versusMw
The graph of ML against Mw is also quite scattered and the pattern of the shallow event data and the trend line are somewhat curvilinear (Figure 6).Depth again has a marked effect, with ML tending to exceed Mw for deep earthquakes.Above magnitude 6, Mw tends to exceed ML, especially for shallow earthquakes.This is consistent with the tendency of ML to saturate at values of about 7. For Mw in the range 5.0-5.5, the scatter of ML values is particularly wide.The regression of ML on Mw and he, using data from 1964 onwards, explains only 65% of the variance of ML, and is given by    In Figure 6 the above linear and quadratic relations are compared with a local regression trend curve of ML against Mw for earthquakes with he ::; 50 km.The quadratic curve is seen to lie within about 0.1 magnitude unit of the trend curve and, given the scatter of the data, is nearly as adequate a fit as can be obtained over the present data range of Mw.
A similar regression to the above was also carried out, using all available ML!Mw data, i.   ,4).The corresponding moment magnitudes are denoted Mw (Harvard), defined by Kanamori [22,23) as where M0 is expressed in Nm.
There are small differences between Mw determined from body-wave modelling and Mw from CMT inversion (ie Mw(Harvard) as illustrated in Figure 7, but no apparent systematic differences.The correlation between them is 0.98.
A regression of one variable on the other does not vary significantly from the line of equality and has a residual standard deviation of0.08 magnitude units.

Magnitude versus number of observations
The number of stations outside of New Zealand (Iv') at which an earthquake of given Mw was observed has increased steadily over the period 1901-1990.In Figure 8 a local regression fit of N against actual or inferred Mw and year of occurrence is presented, with curves showing the expected number of stations observing an earthquake as a function of magnitude Mw in a given year.Also shown are the ± 2cr (approximate 95%) tolerance limits for the actual number of stations when the expected number of stations observing an earthquake is I 0.
These curves and tolerance limits can be used to infer an (approximate) upper limit on the magnitude of any earthquake which was not recorded at any non-New Zealand station.The moment magnitude of such an earthquake is most unlikely to have been so large that the expected number of observations was 10 or more.On this basis the moment magnitude of a shallow earthquake which was not observed at any station outside of New Zealand is almost certainly less than 6.0 if it occurred after I 920, 5.8 if it occurred after 1930, 5. 7 if it occurred after 1940, 5. 5 if it occurred after 1950 and 5.2 if it occurred after 1960.Thus, for example, for the crustal earthquake of 1940 Feb 26, which is assigned a magnitude of M*6 in the New Zealand Seismological Observatory Catalogue, the moment magnitude was almost certainly less than 5.7.
In some cases the upper limits derived by this method arc highly conservative.For example, in 1922 between June 9 and September 5 a series of very shallow events, known as the Taupo swarm, occurred at about 38.60°S and l 76.10°E.Only three of these events were recorded at any non-New Zealand station, and the largest Ms was for the event of 1922 A September 5, at 5.14, giving an M w of 5.40.The events with N = 0, on June 9 and 18 and July 14, 16, 17 and 26, may therefore safely be given upper limits for Mw of 5.4, rather than 6.0 as arises out of the analysis described above.In addition, the earthquakes of 1913 April 12 and I 928 March, not reported overseas, were originally assigned M* 5¾ and 6 respectively.
From their intensity maps (Dowrick, unpublished work) these events are estimated to have been of magnitude -5.8 and -5.5 respectively.

CONCLUSION
The data assembled for this study have allowed surface-wave magnitudes to be estimated on a consistent basis for 192 New Zealand earthquakes in the period 1901-1993, including 69 earthquakes for which surface-wave magnitudes have previously been presented based on a smaller data set.The number of usable station observations of earthquakes of a given size has increased steadily over this period, with a corresponding improvement in the quality of the estimates over time.
Comparisons of Ms, Mw and ML have clarified the relations between these different types of magnitude estimate.Depth has been shown to contribute significantly to the relations, and epicentre location also appears to affect the relation between ML and Ms. However there remains much unexplained variation in the relations between ML and Ms, and between Mr and Mw.Since ML is likely to remain, in the foreseeable future, as the only magnitude routinely available for small earthquakes, it is important to try to gain a greater understanding of the reasons for this variation.
The possibility that Mw may become routinely estimated in New Zealand for events of Mw 5 is to be encouraged.At present, estimates of Mw from Ms and Mw from ML have standard errors of about 0.15 and 0.3 respectively.
As a bi-product of the study, a sizeable number of earthquakes have had their source locations substantially relocated, and upper limits have been placed on the lowest magnitude of other events.This has implications for regional seismic hazard modelling, particularly for the low seismicity Auckland/Coromandel area.

Mw
Scatter plot of centroid depth h c against actual moment (solid square) or inferred moment magnitude M w 4] (Harvard), if available, is given in round brackets under M 0 .Standard error of Ms estimate.Number of observations used in estimating Ms. Number of non-New Zealand stations recording earthquake.Note: a station may yield more than one observation. .

Figure 3 :
Figure 3: Scatter plot of moment magnitude Mwagainst surface-wave magnitude Ms/or earthquakes listed in Table 1, distinguishing events in different classes of centroid depth ho Also shown are the linear (Eq.(4)) and quadratic (Eq.(5)) regression fits for Mw evaluated at he= 25 km, a local regression trend curve of Mw on Ms/or events with he :S: 50 km, and the relation of Eckstrom and Dziewonski (Eq.(6)).

s
Figure5shows.The larger positive values of RL tend to occur in a central region incorporating the western parts of the northern South Island and southern North Island.Both shallow and deep earthquakes occur in this region.On the other hand, the largest negative values tend to occur in an eastern North Island region which has only relatively shallow events.It appears then that ML tends to be Ms in the former region and underestimated in the latter, with a systematic difference of up to a half-unit of magnitude.

s
Figure 5: ML= 1.65[ ±0.41] + 0.71[ ±0.07] Mw + 0.0065[ ±0.0013] (he -25) (9) the residual standard deviation being 0.29.It can be seen from the size of the standard error of the coefficient of Mw that this coefficient is significantly less than unity, again reflecting the tendency of ML to saturate for larger earthquakes.Adding a quadratic term in Mw improves the fit 7,-,.-:-1.1:::,.

Map of locations of earthquakes listed in Table 1 showing, for each event, the actual or inferred moment magnitude Mwand approximate centroid depth he, The inferred moment magnitudes were determined from the surface wave magnitude Ms and he (see text).
measured from seismograms by the first author.Where stations reported both vertical and horizonal component data, the Ms values obtained from both components were used as if they were from different stations, with separate station corrections being applied to each component.The above-seismograms were recorded at Riverview, in Sydney, and at New Zealand offshore stations, i.e.Scott Base and Hallett (Antarctica), Afiamalu, Rarotonga, and Suva. mentioned

Table 3
In addition they were collectively the largest, and hence statistically dominant, events in the historical catalogue for this region of !ow seismicity.The fact that they were actually very deep events in a different region (offshore Bay of Plenty) greatly reduces the apparent historical seismicity of the Auckland/Coromandel region.

TABLE 1 : Location and Magnitude Estimates of New Zealand Earthquakes yr mth dy hr
LMw4Mos Ms s.e. 6 n7 NB

TABLE 2 : Number of earthquakes in present study of those listed in New Zealand catalogue of local magnitudes, for epicentres between 37°S and 47°S latitude and up to 179°E longitude.
[2]as practised in the ISC Bulletin.This would exclude Ms data from many stations (in Europe), reducing our data set for New Zealand earthquakes from 2040 observations of Ms to 1577.However, as found previously[2], data at distances> 160° is made admissible by correcting for bias by calculating station corrections using analysis of variance.The earthquakes of 1975 June 10 and 1988 June 3.The former event was previously estimated to have an Ms of 5.3 based on two observations; now it is estimated at 5.14 based on 10 observations.The latter event was estimated at 6.7 based on 11 observations, and now at 6.50 based on 48 observations.The corrected mean Ms values and associated standard errors are given in Table1.For the purposes of comparison, macroseismic magnitudes (designated M*), local magnitudes (ML), moments (Mo) and moment magnitudes (Mw) are also given, where available.The Ms and Mw values are given to two decimal places for more precise comparisons, and also for more accurate depth correction for Ms. Table I also includes the number of station Ms values used for each event,

TABLE 3 : Earthquakes which have had substantial relocations arising from this study
Ms and Mw are close to being equal above magnitude 6.5.Ms in relation to MB for various parts of Europe, while Ambraseys & Free [17] have recently estimated a focal depth correction term for Ms in relation to log Mo for European earthquakes.As we are interested in estimates of M w rather than Ms our approach is to examine the relationship of Mw to Ms and he-First, the linear regression of Mw on Ms and he is At lower magnitudes Ms is consistently smaller than Mw and is as much as a quarter-unit smaller between magnitude 5.0 and 5.5.Depth also influences the discrepancy between Ms and Mw; for deep New Zealand earthquakes (he> 50 km) Ms is about a half-unit smaller than Mw between magnitude 5.0 and 5.5.The results from the tendency for Ms to decrease with depth for earthquakes of a given seismic moment.In depth correction term for Ml against Ms for earthquakes with he s 50 km.The closeness of these two lines shows that the linear relation in Eq. (7) is about as adequate as any fit could be.

Scatter plot of local magnitude Mi against surface-wave magnitude Ms/or earthquakes listed in Table 1, distinguishing events in different classes of centroid depth ho Also shown are the regression fit for Mi evaluated at he = 25 km and a local regression trend curve of ML on Ms/or events with he::: 50 km.
The residuals from model (7) are denoted RL , i.e., s RL =ML -[3.13 + 0.47 Ms+ 0.0059(hc-25)] s

Scatter plot of local magnitude ML against moment magnitude Mw for earthquakes listed in Table 1, distinguishing events in different classes of centroid depth he-Also shown are the linear and quadratic regression fits for ML evaluated at he = 25 km and a local regression trend curve of ML on Ms for events with hc~50 km.
For estimating Mw from ML and he, the best linear regression is

Comparison of moment magnitudes Mw(Harvard) and Mw (determined from body-wave modelling)for earthquakes listed in Table 1.
Figure 7: