J. Phys. Earth, 21, 19-26, 1973 WHAT IS A STANDARD GRAVITY? H. TAKEUCHIand K. YAMASHINA Geophysical Institute, University of Tokyo, Tokyo (Received February 8, 1973) The Scandinavian region is uplifting and the Indian region is a stable area so that a negative gravity anomaly can be found in the former region and no systematic anomaly can be found in the latter. Against these expec- tations there is no systematic negative anomaly in the Scandinavian region and a negative anomaly in India if the anomalies are calculated on the basis of a normal gravity formula including only (n=2 and 4, m=0) terms in the spherical harmonic expansion. It may be, however, that processes responsi- ble for the Scandinavian uplift and Indian geology occur in shallower depths (probably in the B layer, asthenosphere) and that mass anomaly responsible for the anomaly of rather long wavelengths (or smaller wave numbers n and m) exists deep (probably in the C and D layers, mesosphere) within the earth. If such is the case, in order to get gravity anomaly closely related to geology, we must take as the standard gravity not only (n=2 and 4, m=0) terms but also higher order terms in the spherical harmonic expansion. In short we must choose the standard gravity formula so as to get gravity anomaly which is agreeable with other geophysical and geological observations. Studies are made on the basis of the above point of view. Gaposchkin and Lambeck's data on the spherical harmonic expansion of geoidal undula- tions are used. It is our tentative conclusion that by including terms up to (n=8, m=8) we get anomalies which are more agreeable with geology than the normal gravity formula. 1. Introduction In studying crustal and upper mantle structures, we sometimes use gravity anomalies. Subtraction of a standard gravity from the observed gravity yields the gravity anomaly. But the question is what is an optimum standard gravity. We usually use the international gravity formula as a standard. In the gravity anomaly thus calculated, are included effects due to the crust and upper man- tle as well as the lower mantle and core. Geological phenomena are related mainly to what occurs in the crust and upper mantle. Therefore in order to obtain gravity anomalies of geological interest, it is desirable to eliminate from them the parts due to the lower mantle and core. Is there a sound way to make this kind of separation? 19 20 H. TAKEUCHI and K. YAMASHINA 2. Reference Ellipsoid Gravity anomalies of lower and higher wave numbers are usually due to anomalous masses at deeper and shallower depths, respectively, and this prop- erty allows us to make the above separation. As the gravity formula for this separation, we use that due to GAPOSCHKINand LAMBECK(1971). Their ex- pression for the gravity potential V is (1) where a is the equatoria1 radius, M the mass of the earth, r distance of a point from the center of the earth, λ longitude, φ geocentric latitude, and G the gravitational constant. Corresponding gravity on a geoid is (2) where re is the distance of a point on an ellipsoidal geoid, from the earth's center. The constants of the reference e11ipsoid adopted by Gaposchkin and Lambeck and used here also are The (n=2 and 4, m=0) terms in (1) will give the following standard gravity formula (3) as against the normal gravity formula usually used (4) Table 1. Corrections to the normal gravity formula (4). What is a Standard Gravity? 21 In the attached table the correction for this normal gravity formula (4) to get(3)b y Gaposchkin and Lambeck is shown. The -14.0 mgal correction to be applied to the normal gravity formula (4) in the Potsdam system is also taken into account in the table. We see from this table that the corrections are +6 mgal and -7 mgal at the equator and the poles, respectively. 3. Analysis The Scandinavian region is rising and the southern Indian region is a stable shield, so we may expect negative gravity anomalies in the form er region and no systematic anomalies in the latter . Contrary to these expecta- tions, as we see in Figs. 1 and 5 , we have no systematic negative anomaly in Fig. 1. Free air gravity anomalies (in mgal) with respect to the reference ellipsoid . Fig. 2. Free air gravity anomalies (in mgal) with respect to the n=3 geoid . 22 H. TAKEUCHI and K. YAMASHINA Scandinavia and negative anomaly in India, if the anomalies are calculated based upon the gravity formula (3) which includes only (n=2 and 4, m=0) terms in the spherical harmonic expansion (2). By using (3) instead of (4), we automatically apply the -14.0 mgal correction to the original data. Probably, however, processes responsible for the Scandinavian uplift and Indian geology occur at shallower depths (perhaps in the B layer, or astheno- sphere). The gravity anomalies shown in Figs. 1 and 5 include significant effects due to mass anomalies at deeper depths (probably in the C and D layers, or mesosphere). If such is the case, in order to get gravity anomalies closely related to geology, we must take as the standard gravity not only (n=2 and 4, m=0) terms as in (3) but also higher order terms in the spherical expansion (2) Fig. 3. Free air gravity anomalies (in mgal) with respect to the n=8 geoid. Fig. 4. Free air gravity anomalies (in mgal) with respect to the n=22 geoid. What is a Standard Gravity? 23 In short we must choose the standard gravity formula so as to obtain gravity anomalies which agree with other geophysical and geological observations . In order to check the above idea, let us first divide Δg in (2) into two parts, one with wave numbers up to (n, m) and the other with wave numbers larger than this. Let us interpret the former as standard gravity and the latter as the new gravity anomaly, respectively. In Figs. 2-4 and 6-8 gravity anomalies in Scandinavia and India calcu- lated based upon the standard gravity formula including terms up to (3, 3), (8, 8) and (22, 22) in (2), respectively, are shown. It must be noted that the (n=4, m=0) term, which is included in (3), is rather exceptional one. Even in the (3, 3) cases shown in Figs. 2 and 6, the (4, 0) term is included in the standard gravity. Fig. 5. Free air gravity anomalies (in mgal) with respect to the reference ellipsoid . Fig. 6. Free air gravity anomalies (in mgal) with respect to the n=3 geoid. 24 H. TAKEUCHI and K. YAMASHINA As we see from Figs. 2-4, the standard gravity formula including terms up to (3, 3) gives anomalies about 10 mgal more negative than those in Fig. 1. The (8, 8) standard gravity formula gives almost no change to this result. The gravity anomalies in Figs. 2 and 3 have close correlations with the extent of uplift and other geophysical and geological observations in Scandinavia. The gravity anomaly based upon the (22, 22) standard gravity formula shows no such correlation. As we see from Figs. 5-8, the (3, 3) standard gravity formula gives very small gravity anomalies in southern India. The (8, 8) standard gravity formula gives almost no change to the result. These agree with the fact that India is a shield area of small recent geological activity. The (22, 22) standard gravity formula again gives rather large gravity anom- alies in India. Fig. 7. Free air gravity anomalies (in mgal) with respect to the n=8 geoid. Fig. 8. Free air gravity anomalies (in mgal) with respect to the n=22 geoid. What is a Standard Gravity? 25 Thus we see that (3, 3) to (8, 8) standard gravity formulae give gravity anomalies which are more closely consistent with other geophysical and geo- logical observations than the (n=2 and 4, m=0) one. One explanation is that the (3, 3) to (8, 8) parts in the gravity formula (2) are due to anomalous masses existing deep within the earth. In this connection, a result obtained by M. SAITO(1972) is worth noticing. If we assume an anomalous mass with a white spectrum at a depth d from the earth's surface r=a, we have at the earth's surface (5) Therefore if we calculate the left-hand side of (5) as a function of wave number n and plot the result on semi-log graph paper, we have a straight line from the gradient of which we can determine the depth d of the assumed anomalous mass of white spectrum. The result obtained by Saito is two straight lines as shown in Fig. 9. The corresponding depths for the two straight Fig. 9. Amplitude spectrum for the gravitational potential (in 10-7), SAITO (1972). lines, one for the wave number 2 to 8 and the other for 8 to 22, are 2000 and 300km, respectively. These two depths are within the mesosphere and asthenosphere, respectively. This depth correspondence may be one reason why we get better results for the (3, 3) to (8, 8) standard gravity formulae than for the (n=2 and 4, m=0) one. 4. Conclusions Our tentative conclusion is that the (3, 3) to (8, 8) standard gravity for- mulae provide gravity anomalies which correlate more closely with other geophysical and geological observations. The reason is that the anomalous 26 H. TAKEUCHI and K. YAMASHINA masses corresponding to (3, 3) to (8, 8) terms are within the mesosphere which has no strong connection with these observations at the surface of the earth.