• # Earthquake Parameters

## 1 Azimuth, distance and depth

### 1.1 Estimation from the record

The foregoing section has described the appearance of the records at various distances from the focus of an earthquake. The seismologist who first looks at a record, however, may have no knowledge of the source of the disturbance and must therefore use the record itself to find out what he can.

#### 1.1.1 Azimuth

If the station has a full set of long-period and short-period instruments, look for the first observable motion on all the records. The normal situation is that P or PKP is strongest on the short-period vertical instrument. If the same phase is readable on the horizontal records, the approximate azimuth can be determined from the fact that the P motion is along the ray, either towards or away from the source. Hence note the following:

1. If vertical component is UP, vector sum of horizontals points AWAY from the epicentre.
2. If vertical component is DOWN, vector sum of horizontals points TOWARDS the epicentre.

The apparent azimuth of the horizontal motion is estimated by measuring the amplitudes of corresponding longitudinal waves on the N-S and E-W components, and determining the azimuth from the equation

• tan alpha = AE/AN.

The resulting value of tan alpha corresponds to two possible azimuths differing by 180°. The decision between these depends on the direction of vertical motion, as summarised in the following table:

The polarisation of surface waves on long-period instruments may also be used to estimate the azimuth of an earthquake. If the first clearly-recorded surface waves are on the horizontal instruments only, they will be Love waves, and their polarisation will be perpendicular to the direction of approach. Thus, if corresponding peaks are to the north and west, the Love waves will be approaching from the northeast, or southwest. This ambiguity is resolved by the Rayleigh waves, which are also recorded on the vertical component. The horizontal components will be in phase along the line of approach of the waves, but will be 90° out of phase relative to the vertical components. The direction from which the waves approach is the direction of the peak on the horizontal record which follows one quarter of a cycle after the upward peak on the vertical. Thus, if the order of successive peaks is U, N, D, S and U, E, D, W, the earthquake is in the north-eastern quadrant from the station.

The angle of incidence can also, in principle, be determined by measuring the vertical component of amplitude, and using the following formula:

• tan i = sqrt[(AN2 + AE2)/AZ].

In practice, these estimates may be severely disturbed by local crustal irregularities, and should therefore be interpreted with caution.

If the first P phase is weak or absent on the short-period instruments, the source is probably in the shadow zone, and the first motion seen on the long-period instruments is a diffracted P.

#### 1.1.2 Epicentral distance

When the P arrival has been identified and read, look for S, which is usually the first phase which is more conspicuous on the long-period horizontal instruments than on the short-period vertical. Usually the general character of the seismic signal will suggest whether we have a normal-focus or a deep-focus event. Large shallow earthquakes have extended complex body-wave signals and excite large-amplitude surface waves. This is due to near-source multiple reflections of the seismic signal and perhaps to multiple ruptures within the main shock. Large deep earthquakes have compact body-wave signals and are poor exciters of surface waves.

Now read the times of all other phases, make a list of them and subtract the time of the first arrival. Then take a piece of paper, and put it down against the time axis of the travel-time curve. Make a mark at the origin to indicate the time of the first onset, and other marks to represent the times of the other phases on the scale of the graph. Move the piece of paper over the graph, keeping the first onset mark on the P or PKP curve, until as many of the other phases as possible fit on to the lines. If no fit can be found on the normal-focus curves, try the deep-focus ones, and interpolate mentally to find the depth as accurately as possible. Then enter the phase identifications, and check the diagnostic characteristics which have been given in the section on Record Content. Epicentral distance and time of origin can be read immediately from the curve. The process is illustrated by Fig. 1.1.2.

A very practical device for preliminary identification of phases is a set of traveltime curves plotted on transparent paper or on plexiglass plates; the time scale must be selected according to the recording speed of the seismograph producing the most complete records, usually classes B, C, D, E. Again by moving the graph over the seismogram we attempt to find the best coincidence between recorded phases and travel-time curves.

#### 1.1.3 Depth of focus

Although the general appearance of the record gives some indication of focal depth, a more precise estimate is often desirable. Shallow earthquakes are far more numerous than deep earthquakes. Shallow events are more likely to produce damage or to generate tsunamis. Also, good estimates of depth of focus are of considerable geophysical importance.

Certain phase arrivals may correspond to the near-source surface reflections, pP and sP, which are the most important 'depth phases'. Figs. 1.1.3a and 1.1.3b show the time differences pP-P and sP-P as a function of observer distance and source depth. This figure is based on the tables of Jeffreys and Shimshoni (1964) who used the same data as in the preparation of the J-B Tables (Jeffreys and Bullen, 1940). Note that the time difference pP-P and sP-P are not simply related as a function of observer distance and focal depth.

The pP phase, having been reflected from the free surface, will have opposite polarity from that of the P arrival, provided that both emerge from the same focal mechanism quadrant. Depth can be unambiguously determined when both the pP and sP arrivals are present, provided that multiple rupture arrivals and suchlike are not prominent. However in the case of shallow earthquakes, radiation pattern factors often lead to the strong excitation of only one of these depth phases. Experience at a given seismic station will often allow a good guess at this phase identification, depending on the source region. For example it is nearly always true that Chilean earthquakes will show a strong sP phase and a very weak pP phase at eastern United States stations. In the distance range 70° to 80°, PcP-P has values similar to pP-P and sP-P for shallow earthquakes and care should be taken to avoid confusion in phase identification. Generally, even if positive identification of a potential depth phase cannot be made, it is important to report such a reading to central seismological agencies, where additional data may permit positive phase identification.

If the pP reflection point is oceanic, the strongest reflector will be the ocean-air interface. The subsequent pP-P times will thus reflect significant water travel path and lead to estimates of depth that can be too great by 10 to 20 km. The corresponding sP phase will be reflected at the ocean bottom. Often weak reflections of the pP type will occur at acoustical boundaries in the source-region crust. These phases will look like pP but have much lower amplitude.

Less important depth phases include PcP, pPcP, sPcP, pPKP, sPKP, (SKP-PKP), and (PKS-SKP). These can be used with standard seismological tables (Jeffreys and Bullen, 1940; Herrin and others, 1968) for estimates of focal depth.

### 1.2 Determination of azimuth and distance from a known epicentre

#### 1.2.1 Geocentric coordinates

If the coordinates of the earthquake focus have been determined by an epicentral agency, the station operator will wish to determine azimuth and distance as a basis for final interpretation. The formulae used for this purpose are based on the equations of spherical trigonometry. In a sphere, the radius vector at any point on the surface coincides with the normal to the surface at the same point, so that the directions of either of these vectors may be used in the calculations. The real Earth, however, has a noticeable degree of ellipticity. 'Geographic' latitudes are determined by measuring the angle between the vertical direction of any point and the equatorial plane, and this can differ significantly from the 'Geocentric' latitude which is the angle between the radius vector and the equatorial plane.

The use of geocentric latitude substantially simplifies the task of calculating distances over the surface of an ellipsoidal earth, or of comparing travel times along ray paths which penetrate into it. Conversions from geographic to geocentric latitude may be calculated from the formula

• tan phi = (1 - e)2 tan phi = 0.993 277 tan phi.

Or may be read from Fig. 1.2.1. The geocentric latitudes so obtained will be appropriate for all subsequent charts and formulae.

#### 1.2.2 Numerical calculation

The angular distance between a station (phi0, lambda0) and a known epicentre (phi, lambda) can be determined numerically by means of the following basic relation of spherical trigonometry:

• cos Delta = sin (phi0 sin phi + cos phi0 cos phi cos (lambda - lambda0).

In determining the azimuth from a station to an epicentre measured from north to east we use

• sin alpha = cos phi sin (lambda - lambda0)/sin Delta.

To obtain azimuth from epicentre to station, interchange phi0, lambda0 with phi, lambda.

The formulae can be simplified for a numerical procedure by introducing the direction cosines. If we take rectangular axes at the centre of the Earth, the axes x and y being directed to the points on the equator with longitudes 0° and 90° respectively, and the axis z being directed positively to the north pole, the direction cosines of the radius to a place with geographic co-ordinates phi0, lambda0 are given by

• a = cos phi0' cos lambda0
• b = cos phi0' sin lambda0
• c = sin phi0'

where phi' is the geocentric latitude. If A, B, and C are analogous quantities for the epicentre (phi, lambda) we can write

• cos Delta =aA +bB +cC.

Alternative formulae are

• cos Delta = 1 - 1/2 [(a - A)2 + (b - B)2 + (c - C)2],
• -cos Delta = 1 - 1/2[(a + A)2 + (b + B)2 + (c + C)2].

The first formula is recommended for Delta = 60°-120°, the second for Delta = 0°-60°, and the third one for Delta = 120°-180°. The values of a, b, c were tabulated by Comrie and Jeffreys (1938) and periodically by the International Seismological Centre (from 1967).

An approximate formula

• Delta2 = (phi - phi0)2 + (lambda - lambda0)2 cos2 - 1/2(phi + phi0)

can be used when Delta<6.5°. Angular distances can be converted to kilometres by converting degrees to radians, and multiplying by the local radius r of the earth, where

• r = 6371 + h km

and h is the value from Table PAR 2 at the latitude corresponding to the centre of the ray path. As an alternative, N-S and E-W distances may be taken directly from geodetic tables, and the classical approximation (1° = 111.1 km) is good to within 0.1% over most of the surface of the Earth.

#### 1.2.3 The general use of the stereographic net

The chart which is most generally applicable for approximate determinations of azimuth and distance is the 'stereographic net' which is widely used for the solution of spherical triangles in crystallography.

For use in seismology, the meridians should be numbered in both directions across the equator. In referring to points in the eastern hemisphere, the zero meridian is taken to be the left-hand half of the bounding circle of the projection, and the longitudes are the positive numbers reading from left to right. For the western hemisphere, the zero meridian is the right-hand boundary, and the longitudes are the negative numbers reading from right to left. The latitude corresponding to each parallel is entered on the bounding circle, and the angular distances of the parallels from the south pole are marked along the vertical axis. When in use, the chart is fixed to a smooth board, and covered by a sheet of tracing paper held in place by a single pin through the centre.

To find the distance and azimuth of a point P (whose latitude and longitude are phi0, lambda0) from Q (phi, lambda) first subtract the angle lambda from the longitudes of both points. Plot P on the tracing paper at phi0, (lambda0 - lambda) and Q at phi, 0, using the sign convention for longitude outlined in the preceding paragraph. (Fig. 1.2.3). Now rotate the tracing paper until Q is carried to the south pole of the projection, and let P1 and Q1 be the displaced positions of P and Q. Then the distance PQ being equal to P1Q1 can be read immediately on the vertical scale of the projection. The azimuth of P is equal to the longitude of P1, and is read on the equatorial scale of the projection.

#### 1.2.4 Specialized stereographic net for a single station

If it is sufficient to refer all azimuths and distances to a single station, the stereographic net may be used more conveniently by drawing the azimuth and distance circles on the base chart (Willmore, 1957). Assuming that the complete circle of Fig 1.2.4 represents the bounding circle of a stereographic net, the construction of the distance circles proceeds as follows:

1. Plot the latitude and longitude of the recording station R.
2. Draw the diameter AOB through R.
3. Draw CO perpendicular to AOB.
4. Draw CR and produce to D, on the circumference of the projection.
5. To draw the small circle of radius Delta about R, mark F and C on the circumference of the projection, such that angle FOD = angle DOG = Delta.
6. Draw CC and CF, and produce if necessary to cut AOB in H and I.
7. Draw the circle on HI as diameter.
8. Repeat stages 5-7 with different values of Delta. For rough work, values of Delta increasing by steps of ten degrees have been found to give sufficiently close datum lines, without congesting the diagram excessively.

In carrying out the construction, difficulty may be encountered when CG or CH becomes almost parallel with AB, for under these conditions the intersection point moves off towards infinity. To avoid this difficulty we use the relations

1. OH = OC tan (theta - Delta/2)
2. OI = OC tan (theta + Delta/2)

where theta = angle OCR. These relations can easily be proved by considering the magnitudes of the angles OCH and OCI. By yielding a value of the diameter OH-OI, the equations enable the larger circles to be constructed by using a bent spline, and therefore eliminate the requirement for a long beam compass.

The azimuth circles, being great circles through R, are orthogonal to all the small circles about R. Hence the small circle which projects as a straight line is the common diameter of the azimuth circles. Let this line cut AB at H'. As the line can be regarded as a circle of infinite diameter, we find theta from equation [2] by setting (theta + Delta/2) = 90°, and hence find OH from [ [1]. The common diameter of the azimuth circles is the normal to AB through H' .

The N-S azimuth circle is the meridian through R, and its centre O'' is the point at which the diameter through H' cuts the equator. As the stereographic projection conserves angles, the great circle which corresponds to an azimuth Z will make an angle Z with the meridian NRS. Its tangent at R is therefore constructed by laying off an angle Z from the tangent of NRS, and its diameter is perpendicular to this tangent. The centre of the E-W azimuth circle is therefore the point O'' at which the diameter through R intersects the diameter through H'. For interpreting seismograms it is often sufficient to draw only the E-W azimuth circle in addition to the meridian. This circle is shown as WRE in Fig. 1.2.4. Base nets, 20 inches in diameter, may be obtained from the Dominion Observatory, Ottawa, Canada. A typical finished chart is shown in Fig. 1.2.4a.

To find the azimuth and distance of an earthquake epicentre in the same hemisphere as the recording station, the coordinates of the epicentre are located on the base net and the required quantities are then read off from the circles. For an epicentre in the opposite hemisphere, one works from the antipode of the epicentre instead of from the epicentre itself and obtains a value of 180°-Delta instead of Delta The necessary reversal may be obtained by reading the longitudes from the opposite end of the equatorial scale when entering the coordinates of the epicentre, by reading latitude South in place of North, and by numbering the distance circles so that the smallest one corresponds to the largest value of Delta. The azimuth of the antipode of the epicentre will be in the opposite quadrant from the azimuth of the epicentre itself.

#### 1.2.5 A nomogram for azimuth

A general method for determining azimuth is given in Fig. 1.2.5. Here, one enters the difference between the epicentral and station longitude on the vertical scale, and draws a line down to the epicentral distance on the horizontal scale of the inner part of axes. A parallel line from the epicentral latitude on the horizontal scale of the outer pair of axes cuts the vertical scale at a point corresponding to the azimuth alpha. Uncertainties relating to the sign or quadrant of the azimuth may be resolved by inspection of a sphere or stereogram.

Additional nomograms and charts are reviewed in the papers of Karnik (1955) and Tsuboi (1951).

#### 1.2.6 Use of a large terrestrial globe

To measure the approximate azimuth and distance between the epicentre and the station, a large terrestrial globe can be used. If a transparent plastic cover fitting over the globe with latitude and longitude grid lines is prepared, the measurement will be much easier, putting the pole of the cover at the station or the epicentre. The globe should be held in an open-topped frame, to avoid the obstruction of the conventional fixed axis through the North and South poles. The method of making commercially available globes is, however, liable to introduce errors of one or two degrees, and the method is, therefore, less accurate than the best of the other graphical methods.

## 2 Determination of travel-time from geocentric distance

The major advantage of using geocentric coordinates instead of geographic ones is that the geometry of ray paths through an ellipsoidal earth is greatly simplified. The order of magnitude of the effects with which we are concerned can be seen from the fact that the time of propagation of P waves along an equatorial diameter of the Earth is about 4 seconds longer than that required for propagation along the polar axis. The 'Jeffreys-Bullen' travel-time tables are worked out for a fictitious spherical earth, defined so that the volume within any constant-velocity sphere is equal to that of the corresponding ellipsoid in the real earth.

The exact tabulation of the ellipticity corrections requires a triple-entry table, in which the corrections are found in terms of epicentral latitude and the azimuth and distance of the observing station (Bullen, 1937a). A much more convenient approximation (Bullen, 1937b) is obtained by putting the correction in the form

• f(Delta) (h0 + h1)

where h0 and h1 are the heights of the epicentre and recording station above the standard sphere. The values of h as a function of latitude, and of f(Delta) are given in Tables 2 and 2a below (in Bullen's original paper, the times given in the table for h should be reversed in sign). Tables for other phases are also available (Bullen, 1938a and b)).

If the latitudes used for calculating and for entering the table are geocentric, the corrections are accurate to within ±0.2 s for distances out to 90°, increasing to ±0.3 s for certain paths over greater distances. In theory, they may be reduced to ±0.08 s by the use of a 'seismological latitude' phis and slightly modified values of f(Delta). phis is defined by the equations

• phis = phic - 0.1(phig - phic)
• phis = phig + 1.1(phic - phig)
• phis = 1.1phic - 0.1phig

where phic and phig are the geocentric and geographic latitudes respectively. In fact, the theoretical improvement achieved by the use of this formula is barely significant in view of the actual departure of the earth's interior from the assumed ellipsoidal form.

For deep-focus earthquakes (Bullen, 1938c) the ellipticity corrections are almost the same as those for a surface focus at the same epicentre. The altitude of the station theoretically requires special treatment in so far as the high points of the earth's crust generally reflect the presence of great thicknesses of low velocity material. For this purpose, Bullen suggests an additional correction of 0.14 s per kilometre of station altitude for P waves, and 0.25 s per kilometre for S waves. In fact, recent evaluation of station corrections (see, for example, Herrin and Taggart, 1968) have shown that altitude is not the only factor, or even the dominant factor, in determining the station correction, and the progressive refinement of corrections based directly on observations seems to be the most appropriate course of action.

## 3 Magnitude and seismic moment

The concept of magnitude was originally introduced to provide an instrumental measure of the size of earthquakes, using measurements of earth motion adjusted to take account of epicentral distance and focal depth. Later development led to the utilisation of many different earthquake phases, and to efforts either to 'unify' the results obtained by different methods into a common measure of earthquake energy, or to utilise differences between individual estimates for a given earthquake to build up a picture of the character of the source. Magnitude statistics have thereby come to provide a basis for studying and comparing the seismic activity of different regions of the earth, or for studying the variation of activity with time in a given region.

### 3.1 The local magnitude

#### 3.1.1 The Richter local magnitude

The first approach to this problem was the 'local magnitude' ML designed by Richter (1935) for the classification of local shocks in Southern California, and defined as

• ML = log a - log a0

where a is the maximum trace amplitude recorded by the 'standard' Wood-Anderson torsion seismometer (T0 = 0.8 s, h = 0.8, V0 = 2800) at a given distance, and a0 is that for the shock of zero magnitude at the same distance.

The amplitude function a0(Delta) was determined empirically for distances Delta = 25-600 km (Richter, 1935) and later supplemented for 0 < Delta < 25 km (Gutenberg and Richter, 1942). Richter (1935) also gave log a = 3.37-log Delta (a in millimetres and Delta in kilometres) for 200 < Delta < 600 km with the hope of extending the formula to larger distances, although this extension revealed itself in later years to be unapplicable over regional distances. Outside California, the Richter scaling function of distance, and the relationship with other magnitude scales, is of provisional utility only.

It should be noted that the focal depth of California earthquakes lies in a very limited range, say 10-20 km, while in many other seismic regions the depth range is much wider than in California. Thus a considerable change in the amplitude function is expected for the deeper shocks in other regions.

The content of the amplitude tables is included in a nomogram (Fig. 3.1.1) in which the original format has been modified by Eiby and Muir (1961). To use the figure set a straight-edge between the distance or P-S interval on the left-hand scale and the maximum trace amplitude on the right-hand scale, and read ML where it crosses the centre scale. If a two-component Wood-Anderson system is available, the arithmetic mean of the two values of amplitude should be used, irrespective of the arrival times.

If a Wood-Anderson instrument is not available, records from other short-period seismographs can be used if the difference of magnification is properly allowed for. For this purpose, a magnification curve for the standard Wood-Anderson instrument is provided (Fig. 3.1.1a). If the magnification of an available seismograph is plotted on this chart, the separation of the curves for earth motion of any given period will be a measure of the amplitude ratio expected on records taken by the two instruments.

It is convenient to use the logarithmic scale of the figure like a slide rule. After measuring the amplitude and period of an appropriate phase on the seismogram, make two marks on a strip of paper to indicate the sensitivity ratio, and move the paper until the mark which represents the sensitivity of the available seismograph comes against a grid line which corresponds to the amplitude of the observed signal. The second mark on the paper will then indicate the trace amplitude which would have been recorded by a Wood-Anderson seismograph on the same pier.

In performing this operation, note that any difference between the shapes of the two response curves will affect the relative amplitudes of earthquake phases of different period. It may therefore happen that the largest amplitude recorded by the available seismograph may not correspond to the same phase as that which would give the largest record on the Wood-Anderson. For this reason, several phases should be read, and the one which gives the largest converted amplitude should be used.

#### 3.1.2 Other formulae for local magnitude

More direct methods, based on the determination of true ground motion, are now available for assigning magnitudes to near earthquakes. Calibration functions for a number of areas have been published (see, for example, Karnik, 1963; Aranovich and others, 1966, etc.). Some examples which are tied to teleseismic magnitudes are set out in Sections 3.2 and 3.3 below.

Methods of determining magnitude from intensities experienced in the vicinity of the epicentre are summarised in Section MAC 3.2.3.

### 3.2 Teleseismic estimates of magnitude

Further development led to the extension of the magnitude scale to all epicentral distances, to shocks deeper than normal and to the use of an estimate of particle velocity of the ground instead of trace amplitude.

Body and surface-wave magnitudes are defined as

• mb = log (A/T)max + sigma (Delta, h) + Sigma delta mb
• MS = log (A/T)max + sigma (Delta, h) + Sigma delta MS

where A is a ground amplitude in microns* (in the SI now called micrometers, µm), T is a period in seconds, corresponding to the particle velocity maximum (A/T)max, in a wave of a particular type; the empirical amplitude-distance function sigma (Delta, h), a so-called calibrating function variant for different types of waves, expresses a change in the (A/T)max value with epicentral distance Delta, and focal depth h, and corresponds to zero magnitude. Sigma delta mb and Sigma delta MS are station corrections or regional ones allowing for effects of a recording site, wave path, variation of depth, focal mechanism, etc. The procedure for measuring amplitude and period is given in OP 3.4.

In routine practice, the last term is often assumed to be zero, but it can be useful for individual stations to study their corrections for mb and MS in order to apply them to improve preliminary estimates of magnitudes, before obtaining information from regional and/or international centres. It is especially useful for a local tsunami warning service. Regional characteristics of the corrections for different epicentral regions may also be worthy of note.

*NEIS and ISC use A for mb in millimicrons (now called nanometres (nm) in the SI).

#### 3.2.1 Body-wave magnitudes

Magnitudes which can be determined by body-waves are mPV and mPH by P waves, mPPV and mPPH by PP waves, and mSH by S waves. The classical study is that of Gutenberg and Richter (1956) in which the calibrating functions, called Q(Delta) are given.

At the IASPEI General Assembly in Zurich (1967) the Committee on Magnitudes recommended stations to report the magnitude for all waves for which calibration functions are available, as well as publishing amplitude and period values separately.

Apart from the multiplicity of possible phases, the following problems arise in practice:

1. The early work was all undertaken on the basis of observations from medium-period instruments (Class C of INST 3.4) whereas a large proportion of modern data is coming from instruments of much shorter period (Class A of INST 3.4).
2. Significant differences arise from the method of reading, depending on whether the maximum phase amplitude was read during the first few cycles of motion, or allowed to develop for as much as 25 seconds. The procedure recommended for current work is set out in Section OP 3.
3. Many different studies of amplitude-distance functions have now been published. Important examples include Vanek and Stelzner (1960), defining a function beta(Delta), Vanek and others (1962), combining beta(Delta) and Q(Delta) under the name of sigma(Delta), Carpenter and others (1967) and Vieth and Clawson (1972) both treating short-period teleseismic P-waves from shallow events.

None of the recent studies has received world-wide endorsement for routine use, and the major international agencies are therefore continuing to apply the tables and nomograms of Gutenberg and Richter (1956) as recommended in 1967 by the Committee on Magnitudes. In the case of NEIS and ISC, the data stream is dominated by the input from short-period instruments, and the instructions for reading required operators, until recently, to concentrate attention on the first few cycles of the record. The statistics of the resulting magnitude estimates therefore differ markedly from those based on older data, and from those which depend on the instruments and practice of the USSR and affiliated networks.

It is hoped that the revised practices now recommended in sections OP and OUT of this Manual will bring about a progressive improvement of the situation. In the meantime, it is imperative to ensure that all studies based on the comparison or merging of sets of magnitude data should include adequate specification for class of instrument, reading procedure and method of data reduction.

##### 3.2.1.1 Earthquakes less than 33 km in depth
Table 3.2.1 for Q(Delta ), or Fig. 3.2.1.2a, Fig. 3.2.1.2b, or Fig. 3.2.1.2c for Q(Delta, h) are fully applicable when Delta > 20° and the readings are taken from instruments of Class C. Carpenter and others (1967) or Vieth and Clawson (1972) may yield more consistent results for short-period instrumentation.

For shallow earthquakes at distances less than 20°, the amplitude distance functions vary substantially from place to place, and regional studies must be invoked. Some examples are Vanek and Stelzner (1960) for Europe, Jordan and others (1965) for North America, Antonova and others (1967) for Central Asia and Soloviev and Solovieva (1970) for the Far East.

Miyamura (1974) proposed a different approach by using PKP, and constructed provisional calibrating functions for the New Zealand and Macquarie Islands regions.

##### 3.2.1.2 Deeper earthquakes

Calibrating functions for depths greater than 30 km are given by Gutenberg and Richter (1956). Their figures 3-5 are repeated as Fig. 3.2.1.2a, Fig. 3.2.1.2b, and Fig. 3.2.1.2c of this Manual.

#### 3.2.2 Surface-wave magnitudes

Historically the surface-wave magnitude was first calculated by Gutenberg and Richter (1936) as an extension of local magnitude at teleseismic distances using the trace maximum of the Wood-Anderson seismographs. Gutenberg (1945a) gave a formula based on maximum horizontal ground amplitudes, and from 1949 to 1959 various authors calculated similar empirical magnitude formulae for different stations (cf. Bath, 1966).

Soloviev (1955) proposed the use of the maximum ground particle velocity (A/T)max instead of the maximum ground displacement Amax and the corresponding calibrating functions were obtained by Soloviev and Shebalin (1957), Vanek and Stelzner (1959), Christoskov (1965) and others.

##### 3.2.2.1 Standardisation for horizontal components

Collaboration between research teams in Prague and Moscow (Karnik and others, 1962, and Vanek and others, 1962) yielded the standardised calibrating function

• sigma(Delta) = 1.66 log Delta + 3.3

for the epicentral distances 2° < Delta < 160°, as a conventional synthesis of the then published calibrating functions for Amax and (A/T)max. The IASPEI Committee on Magnitudes, Zurich 1967, recommended the use of this standardised formula for sigma(Delta) and the determination of the station and epicentre correction Sigma deltaM The mean periods corresponding to the maximum amplitudes of surface waves to be used for the MS determination were also given, and are reproduced in the following table:

When the period differs considerably from the values given in Table 3.2.2.1, it may be advisable not to use the data for the MS determination.

In order to find (A/T)max on the seismogram, calculate A/T for several trace maxima and select the largest value of A/T among them. Combine the two horizontal components vectorially in the form

• AH = sqrt(AN2 + AE2 )

irrespective of the types of wave (Rayleigh or Love) and the arrival times. If one of the two components is absent, use sqrt(AN) or sqrt(2AE).

Recent studies, e.g. Evernden (1971), Marshall and Basham (1973) and others, have shown that the condition of amplitude decrease with distance expressed by the term 1.66 log Delta of the standard calibrating function is not met for distances shorter than 25°.

##### 3.2.2.2 Utilisation of the vertical component

The magnitude determination from the horizontal components arose historically when seismological stations were principally equipped only with horizontal instruments. During the last 20 years the stable operation of medium and long-period vertical seismometers has become easier and amplitude and period data of the vertical components have increased considerably. In addition, it is theoretically more desirable to use the vertical components than to use the horizontal ones, because the former contain exclusively Rayleigh type waves, while the latter contain superposed Love and Rayleigh waves which introduce instability in MS values to hamper the determination procedure.

Many seismological stations and centres have published MS or MLH and MLV routinely using different calibrating functions, but current practice is to use the standard formula for all components. The NEIS and ISC collect the amplitude and period data of surface waves and calculate MLH and MLV using the standard calibrating function for (A/T)max with periods T = 20±2 s at 20 < Delta < 160°. NEIS has published estimates of MS by averaging values of MLH at individual stations, but adopted the vertical component as its standard in May 1975. The ISC takes the average of available A/T values for all components at individual stations, but does not adopt a mean value for the complete earthquake.

Most of the routine determinations of surface-wave magnitudes have been made by using (A/T)max for the period range 18-22 s, without specifying the type of waves, although the standard calibrating function is assumed valid for smaller and larger period waves. Recently there has been a tendency to determine the surface-wave magnitude specifying the type of the waves used, e.g. MLRH or MLRV from Rayleigh waves and MLQH from Love waves, over a wider range of period.

Brune and King (1967) showed that the amplitudes of the surface waves at a 100-second period may be a better measure of the energy rating for larger earthquakes than MS at 20 s. Recent studies indicate that comparing MS at 30-40 s with mb is more effective than using MS at 20 s for the discrimination between underground nuclear explosions and natural shallow earthquakes (Marshall and Basham, 1972). Considering these results, we may see more use of surface-wave magnitudes at different periods, e.g. MS20, MS40, MS100 and so on, in conjunction with multiperiod estimates of mb to describe the earthquakes more completely.

#### 3.2.3 Unified magnitude

Although historically the local magnitude ML, the surface-wave magnitude MS, and the body-wave magnitude mb were intended to coincide with each other, separate determinations revealed inconsistencies, especially between MS and mb. Gutenberg and Richter (1956a, b) finally obtained the relation

• MS - mb = a(MS - b)

where a = 1/4, b = 7 and a = 0.37, b = 6.76 were possible pairs of values.

Although Richter had thought of magnitude as only a simple measure of size, Gutenberg was insistent on the need to define a magnitude representing the earthquake energy and Gutenberg and Richter (1956b) eventually defined the unified magnitude m as 'the weighted mean of mb and mS, where mb is the body-wave magnitude as found directly from body waves and mS, the corresponding value derived from MS by applying the relation

• MS - mb = a(MS - b),

MS being the surface-wave magnitude as found directly from surface wave with period around 20 s. They also defined Mb as the value of surface-wave magnitude derived from mb and M without subscript as weighted mean between MS and mb. Using the unified magnitude m, Gutenberg and Richter (1956b) defined the earthquake energy E by the relation

• log E = 5.8 + 2.4m

which yields the often-used relation

• log E = 11.8 + 1.5M,

after applying the relation

• MS - mb = 0.37(MS - 6.76),

and putting mb = m and MS = M (Richter, 1958).

The basic thesis which underlies the concept of unified magnitude is that spectral content (which determines the partition of energy between body waves and surface waves) is a function of magnitude only. Modern seismologists, as we have seen in Section 3.2.2 above, are increasingly conscious of a multiplicity of constituents in earthquake character. Unification of magnitude data, therefore involves the loss of some of the information which was contained in the original magnitudes, and current work on the transference of magnitude estimates from one scale to another is tending to be aimed at the provision of an infill of approximate data for sets of events for which direct determinations do not provide complete cover.

### 3.3 Other amplitude-based measures of magnitude

Many agencies are faced with the problem of making magnitude estimates from the data of their own stations, independently of those listed by regional or international agencies. The general procedure is to adopt a measure of particle motion derived from available seismograms, and to fit a calibration function to magnitude data from an external agency for a set of events recorded at both levels. When this has been done, the derived formula becomes available for subsequent internal use. Some examples are as follows:

#### 3.3.1 Magnitude of Japanese earthquakes by JMA

The Japan Meteorological Agency (JMA) has been publishing the magnitudes of Japanese shallow earthquakes using the formula

• M = log (AN2 + AE2 )1/2 + 1.73 log Delta - 0.83

where Delta is the epicentral distance in kilometres, and AN and AE are the maximum ground amplitudes in microns measured on the N-S and E-W components of horizontal Wiechert seismographs in JMA stations (Tsuboi, 1954). M is the magnitude equivalent to that given Gutenberg and Richter (1949, 1954), which appears to correspond with MS as defined in Section 3.2.2. above.

For deeper earthquakes (h „ 60 km), JMA uses the formula

• M = (AN2 + AE2)1/2 + K

where K is a depth-distance factor given by Katsumata (1964).

#### 3.3.2 A short-range body-wave magnitude

A quantity called mb* has been defined for distances as short as 200 km by the formula

• mb* = log V + 2.3 log Delta - 2

where V is the maximum ground velocity in microns/second in the P-wave train, and Delta is measured in kilometres. The equation was originally produced for use of surface explosions in the United States (Navarro and Brockman, 1970), and has been checked in Europe against underwater explosions (see, for example, Jacob and Willmore, 1972) and a few earthquakes. The available data show good agreement with teleseismic estimates of mb.

### 3.4 Magnitude based on duration

Bistricsany (1958) proposed the use of the duration of the surface-wave train for the determination of magnitude, and obtained the following formulae for Wiechert seismographs at the Budapest station:

i for shallow shocks
M = 2.12 log (F - eL) + 0.0065 Delta + 2.66
ii for deeper shocks
M = 1.58 log (F - eL) + 0.0020 Delta + 0.0007 h + 4.02,

where F and eL are the end and commencement times (in minutes) of the recorded surface waves, Delta is the epicentral distance in degrees and M is the magnitude equivalent to the surface-wave magnitude determined at Praha station.

Tsumura (1967) obtained

• M = 2.85 log (F - P) + 0.0014 Delta - 2.53

for local earthquakes in Kii Peninsula, Central Japan, where F - P is the total duration time in seconds, Delta is the epicentral distance in kilometres and M is the magnitude equivalent to MJMA.

With small epicentral distances the term containing Delta may be omitted in all of the formulae.

Owing to the narrow dynamic range recorded by a single seismometer, it is often impossible to measure the maximum amplitude of strong seismic movements, and the magnitude determination by duration is a useful and simple way of overcoming this problem. Thus, the method has been adopted recently at many seismological stations with their own definition of the duration time and empirical magnitude formula. The method is especially convenient for local shocks in determining the magnitude without requiring an exact knowledge of epicentral distance.

### 3.5 Seismic moment

We have seen above that a single determination of magnitude is not enough to specify completely the spectral content of an earthquake source. ML for instance depends on the spectral level at a period of 0.8 s. Similarly, seismic moment, defined as M0Au can be thought of as representing the zero-frequency spectral level. (Here A is the area of the ruptured fault, u the displacement of the fault movement, and µ the shear modulus of the source rock.) For a simple source of the type proposed by Brune, combined measurements of M0 and ML determine the amplitude spectrum completely, and enable the associated parameters source area, fault displacement, stress drop and seismic energy to be found. Seismic moment may be determined directly from the zero-frequency level of the radiation spectrum obtained from spectral analysis of earthquake records, or from the sum of the envelopes of the earthquake records on the three components of a standard long-period seismograph (Brune and others, 1963).

### 3.6 Recommendations from the Joint General Assembly of the IASPEI/IAVCEI, Durham, 1977

In the course of the joint General Assembly of the IASPEI the Subcommission on Magnitudes of the Commission on Practice held a number of discussions on the current status of magnitude determinations, and on the nomenclature to be used in future publications. The principal conclusions are summarised below.

#### 3.6.1 Nomenclature

The increasing use of telegraphic communication and computer processing has made it necessary to express magnitude information within the constraints of the upper-case alphabetic character set, without using subscripts. The following system is therefore recommended

• ML—Magnitude determined by surface waves in general.
• ML—Magnitude determined by surface waves in general.
• MB—Magnitude determined by body waves in general.
• M—Unified magnitude, as defined by Gutenberg and Richter (1956).
• MWA—Local magnitude from Wood-Anderson seismographs, as defined by Richter (1935).
• MLH—Magnitude from horizontal surface waves.
• MLV—Magnitude from vertical surface waves.
• MLRV—Magnitude from vertical component of Rayleigh waves.
• MLRH—Magnitude from horizontal component of Rayleigh waves.
• MLQ—Magnitude from Love waves.
• MPV—Magnitude from P phase (vertical).
• MPH—Magnitude from P phase (horizontal).
• MPPV—Magnitude from PP phase (vertical).
• MPPH—Magnitude from PP phase (horizontal).
• MSV—Magnitude from S phase (vertical).
• MSH—Magnitude from S phase (horizontal).
• MB'—Body-wave magnitude converted from ML.
• ML'—Surface-wave magnitude converted from MB.
• MBN—Body-wave magnitude for near earthquakes, as in sect. 3.3.2.
• MLG—Magnitude determined from Lg (Sg) waves at distance < 10°.
• MD—Magnitude determined from the duration of earthquake signal.

When a magnitude estimate is derived from a specific instrument, it is desirable to indicate the class of instrument by adding the class letter as defined in INST 1.1, giving complete descriptions such as MPVA, MPVB, MSHC, etc. Note that this procedure is not applicable to the unified magnitude M.

#### 3.6.2 Magnitude conversions

Studies by Gorbunova and Kondorskaya (1977) have established the following relationship between MPVC and MPVA:

• MPVC = 1.34 MPVA - 1.16

The importance of this distinction stems from the fact that the early work of Gutenberg and the current determinations by Eastern countries are based largely on Class C instruments, whereas the Western agencies have relied strongly on data from Class A.

## References

### R.1 Azimuth, Distance And Depth

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### R.2 Determination Of Travel-time from Geocentric Distance

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### R.3 Determination Of Magnitude And Seismic Moment

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• ```Date created: 8/10/96