Gordon J. F. MacDonald

The rotation of the earth : a geophysical discussion

This book gives an account of certain observed irregularities on the rotation of the Earth, both in its rate of rotation (giving a variable length of day) and in the position of its axis. These irregularities are caused by events on and within the Earth and provide a means of studying a number of geophysical problems. Seasonal shifts in air masses and variable winds are causes of short-period fluctuations in the rotation. Climatic changes and their attendant sea levels are in part responsible for long-term fluctuations. Modern observations of the Moon and descriptions of ancient elipses both establish a secular increase in the length of day. The interpretation involves atmospheric, oceanic and bodily tides. The book provides a unified treatment of the rotation of the Earth, making this method of studying geophysical phenomena more readily accessible to geophysicists and others.

Relative Contributions of Uranium, Thorium, and Potassium to Heat Production in the Earth

Data from a wide variety of igneous rock types show that the ratio of potassium to uranium is approximately 1 X 104. This suggests that the value of K/U ≈1 X 104 is characteristic of terrestrial materials and is distinct from the value of 8 X 104 found in chondrites. In a model earth with K/U ≈ 104, uranium and thorium are the dominant sources of radioactive heat at the present time. This will permit the average terrestrial concentrations of uranium and thorium to be 2 to 4.7 times higher than that observed in chondrites. The resulting models of the terrestrial heat production will be considerably different from those for chondritic heat production because of the longer half-life of U238 and Th238 compared with K40

Stochastic Optimization of Insurance Portfolios for Managing Exposure to Catastrophic Risks

A catastrophe may affect different locations and produce losses that are rare and highly correlated in space and time. It may ruin many insurers if their risk exposures are not properly diversified among locations. The multidimentional distribution of claims from different locations depends on decision variables such as the insurers coverage at different locations, on spatial and temporal characteristics of possible catastrophes and the vulnerability of insured values. As this distribution is analytically intractable, the most promising approach for managing the exposure of insurance portfolios to catastrophic risks requires geographically explicit simulations of catastrophes. The straightforward use of so-called catastrophe modeling runs quickly into an extremely large number of “what-if” evaluations. The aim of this paper is to develop an approach that integrates catastrophe modeling with stochastic optimization techniques to support decision making on coverages of losses, profits, stability, and survival of insurers. We establish connections between ruin probability and the maximization of concave risk functions and we outline numerical experiments.

A system approach to management of catastrophic risks

There are two main strategies in dealing with rare and dependent catastrophic risks: the use of risk reduction measures (preparedness programs, land use regulations, etc.) and the use of risk spreading mechanisms, such as insurance and financial markets. These strategies are not separable. The risk reduction measures increase the insurability of risks. On the other hand, the insurance policies on premiums may enforce risk reduction measures. The role of system approaches, models and accompanying decision support systems becomes of critical importance for managing catastrophic risks. The paper discusses some methodological challenges concerning the design of such models and decision support systems.

The internal constitutions of the inner planets and the moon

The internal structures of the moon, Mars, Venus, and Mercury are examined in the light of what is known about the constitution of the earth. The gravitational figure of the earth as obtained from orbits of artificial satellites is used to estimate the possible deviations from hydrostatic equilibrium on other planets. Observations of the orbital and rotational motion of the moon are consistent with the hypothesis that the interior of the moon supports density inhomogeneities of the same order as those supported by the earth. The available data on the moon are insufficient to determine whether or not the moon is differentiated. The orbits of Phobos and Deimos yield an adequate value for the moment of inertia of Mars. The moment of inertia and the mass are consistent with a metallic core containing about 10 per cent of the mass of Mars. The observations of the possible magnetic field of Mars would be of importance both to the understanding of planetary magnetic fields and elucidating the internal structure of that planet. Seismic investigations on the earth yield an equation of state for silicates to pressures of about 1 × 106 bars. This equation of state is used in determining density variation within Mars.The surface heat flow for the earth is consistent with the hypothesis that the concentration of radioactive elements is the same as that in chondritic meteorites. The observed ratio of potassium to uranium in surface and near-surface rocks is not consonant with the chondritic hypothesis. The moon can be of chondritic composition only if it is differentiated with the radioactivity concentrated in the upper few hundred kilometers. A chondritic composition for Mars would require a differentiation in excess of that consistent with its mass and moment of inertia. It is concluded that a chondritic composition is not a satisfactory chemical model for the inner planets.

Stress history of the Moon

Abstract The character of the lunar surface indicates that surface faulting has not been an important mechanism for the build-up of the lunar surface. If the radioactive content of the Moon is of the same order as that of chondritic meteorites, then the absence of major surface faults can be explained in a number of ways. A near-surface concentration of radioactivity will provide an equality of heat production and surface heat flow necessary for the maintenance of a constant lunar radius. Alternatively, the radioactivity could be deeply buried, with the radius still remaining constant over the past 2,000,000,000 years. Heat transported by mechanisms other than radiation and thermal conduction will also tend to keep the radius of the Moon at a constant value. Even though the radius of the Moon remains constant, there is a major build-up of strain energy throughout the Moon. The rate is such that, on the average, something on the order of 10 24 –10 25 ergs of distortional energy should be released per year throughout the Moon, provided the radioactivity is uniformly distributed. A near-surface concentration of the radioactivity might decrease this rate of energy release but certainly by no more than an order of magnitude. Under all circumstances it would appear that a Moon of chondritic composition would have strong Scismic activity.

Simultaneous presence of orbital inclination and eccentricity in proxy climate records from Ocean Drilling Program Site 806

Ocean Drilling Program Site 806 in the western Pacific shows evidence of a remarkably constant average sedimentation rate. This feature allows us to analyze ancient climate proxies without the need for “orbital tuning,” a standard procedure in prior work, but one that can lead to biased results. Spectral analysis of stable oxygen isotope ratios at this site, a proxy for global ice volume, shows a single narrow peak with a period ≈ 100 k.y., a result that supports our model which links glacial cycles to variations in the inclination of the Earths orbit. In contrast, spectral analysis of the coarse component fraction of the sediment (primarily foraminifera) shows a structure characteristic of standard Milankovitch theory, with a triplet of peaks with periods near those expected from the Earths eccentricity: 95, 125, and 400 k.y. Bispectral analysis confirms these linkages but suggests that orbital inclination also plays some role in the coarse fraction. From the clear presence of both signals in different proxies at the same site, we conclude that although eccentricity affected the local climate, it is orbital inclination that drove the variations in the global ice volume for the past million years.

A Model for Estimating Future Emissions of Sulfur Hexafluoride and Perfluorocarbons

Sulfur hexafluoride (SF6), perfluoromethane (CF4) and perfluoroethane (C2F6) are strong greenhouse gases with long (>1000 year) atmospheric residence times. We derive emission factors for the major anthropogenic sources and project future emissions for 5 regions and the world. Although firms in many industrialized countries are already limiting emissions, without further policy intervention global emissions will rise 150% (CF4 and C2F6) and 210% (SF6) between 1990 and 2050; radiative forcing will increase 0.026 W m-2. Full application of available low-cost and costless policies in industrialized nations would cut that radiative forcing by one-quarter. Increased forcing due to these gases is small (<2%) relative to other gases but permanent on the timescale of human civilization. We also quantify plausible manipulations to governmental data that will be used to determine compliance with the 1997 Kyoto Protocol, which includes commitments for industrialized countries to regulate these and other greenhouse gases. More complete and transparent data are urgently needed. West European nations, for example, can ‘cut’ their emissions of these gases by half by 2010 simply by manipulating emission factors within the current bounds of uncertainty.

Tritium and Helium-3 in Solar Flares and Loss of Helium from the Earth's Atmosphere

Analysis of the data gathered by the Discoverer XVII satellite on the constituents of solar flares leads to results that have broad implications in geophysics and solar physics.

Insurability of catastrophic risks: The stochastic optimization model

Catastrophes produce losses highly correlated in space and time, which break the law of large numbers. We derive the insurability of dependent catastrophic risks by calculating conditions that would aid insurers in deliberate selection of their portfolios. This paper outlines the general structure of a basic stochastic optimization model. Connections between the probability of ruin and nonsmooth risk functions, as well as adaptive Monte Carlo optimization procedures and path dependent laws of large numbers, are discussed