b'Coping with ambiguityFeatureFigure 5.(a) Rays from a distant sun impinging on a spherical earth. (b) Rays from a nearby sun and a flat earth.calculations might be construed as having been due to a lucky guess. But he was also an astronomer, having measured the angle between the ecliptic and the rotational plane, and was probably aware of the writings of Pytheas, who, in the fourth century BCE voyaged to northern Britain and probably Iceland, (Cunliffe, 2001), returning with stories of the midnight sun and a congealed (frozen) sea. Eratosthenes would also have been aware of the lines of latitude proposed by Hipparchus (c.180125BCE). He would have had suspicions from these and other bits of independent information (like a ship just over the horizon that shows first its sails and only later its hull when it gets closer), that allowed him to select the correct solution from a myriad of possibilities. His deeper thoughts on the matter, if indeed they were recorded, probably went up in smoke along with many other ancient writings when the Library of Alexandria, having stood for 700 years, suddenly burned (391CE) just when Theodosius I (347395CE) was expressingFigure 6.Sphere diameter required to generate the observed anomaly amplitude for both magnetics and gravity.his preference for Christian dogma over ancient knowledge and was shutting down pagan institutions.merge the horizontal density and susceptibility scales for the range of calculated sphere diameters.Ambiguity resolved?One more assumption, in this case easily accepted, that the Returning to the Dubawnt Lake problem, I constructed twobody causing the gravity anomaly was the same as the body equations which I took to be self-evident. Equation 1 gives thecausing the magnetic anomaly, allowed me to infer from density of an arbitrary mixture of mafic rock and magnetite,Figure6 that the sphere that simulated both the gravity and and Equation 2 gives the magnetic susceptibility of the samemagnetic data would be characterised by the intersection mixture. Thus point of the two curves. Evidently, the source body would need to have a diameter of about 900 m, and with a coincidentally d 1 p 1 + d 2 p 2 = d m Eq. 1 similar value for the depth of the centre, the top would be at about 450 m depth. Furthermore, the abscissa of the intersection point allowed me to estimate that it would s 1 p 1 + s 2 p 2 = s m Eq. 2 comprise gabbro plus about 10 percent magnetite. The reason that the curves in Figure 5 have different shapes is because where, d and s refer to density and susceptibility, and subscriptswhen we recede from the vicinity of the sphere, the rate of 1, 2 and m refer to magnetite, mafic rock and the mixture,fall-off of the field intensity is different for gravity, 1/r2, than for respectively, while p 1and p 2refer to the proportion (0100%)magnetics which is complicated by the dipolar nature of the of each component. Eq. 2 was not used at the time but it isfield and falls off as 1/r3, see Blakely, 1995, p75.reasonable, and I present it now for later reference. In fact, I used a power law relation given by Grant and West (1965, p367)When the drilling results confirmed the interpretation, it who reference Mooney and Bleifuss (1953) to estimate magneticseemed like I had solved the ambiguity problem, and, while susceptibility as a function of magnetite content. I was happy to receive the accolades of my colleagues, deep down I knew I had made some lucky guesses.I was therefore able to tabulate the density and susceptibility for a mafic + magnetite mixture in increments of 10% magnetite,Further developmentsstarting with a mafic rock (gabbro), and then for each mixture I calculated the diameter that a sphere would need to haveFollowing this, the next decade or so was dominated by my in order to generate the anomaly amplitude observed inattempts to understand EM theory, but it did leave time to both the magnetic and gravity data. In plotting the resultsimplement an algorithm based on Talwanis (1960, 1961) shown in Figure 6, and, because I knew both the density andtheories for the magnetic and gravity responses of almost susceptibility for each mafic + magnetite mixture, I was able toarbitrary shapes - in the hope that such a modelling code 47 PREVIEW DECEMBER 2022'