Editorial

Natural Resources Research -     

Pitfalls in Nonlinear Inversion

— We discuss and illustrate graphically with simple 2-D problems, four common pitfalls in geophysical nonlinear inversion. The first one establishes that the Lagrange multiplier, used to incorporate a priori information in the geophysical inverse problem, should be the largest value still compatible with a reasonable data fitting. This procedure should be used only when the interpreter is sure about the importance of the a priori information used to stabilize the inverse problem relative to the geophysical observations. Because this is rarely the case, the user should use the smallest Lagrange multiplier still producing stable solutions. The second pitfall is an attempt to automatically estimate the Lagrange multiplier by decreasing it along the iterative process used to solve the nonlinear optimization problem. Consequently, at the last iteration, the Lagrange multiplier may be so small that the problem may become ill-posed and any computed solution in this case is meaningless. The third pitfall is related to the incorporation of a priori information by a technique known as “Jumping.” This formulation, from the viewpoint of the class of Acceptable Gradient Methods, is incomplete and may lead to a premature halt in the iteration, and, consequently, to solutions far from the true one. Finally, the fourth pitfall is an inadequate convergence criterion which stops the iteration when the data misfit drops just below the noise level, irrespective of the fact that the functional to be minimized may not have attained its minimum. This means that the a priori information has not been completely incorporated, so that this stopping criterion partially neutralizes the effect of the stabilizing functional, and opens the possibility of obtaining unstable, meaningless estimates.     

The CP 2 Analytical Plotter

This paper examines the rationale behind the design and development of the CP 2 analytical plotter and shows how the final product, in terms of both hardware and software, has been influenced by the needs of a commercial air survey company.     

Tropical soils could dominate the short-term carbon cycle feedbacks to increased global temperatures

Results of a simple model of the effects of temperature on net ecosystem production call into question the argument that the large stocks of soil carbon and greater projected warming in the boreal and tu ndra regions of the world willlead to rapid efflux of carbon from these biomes to the atmosphere. We show that low rates of carbon turnover in these regions and a relatively greater response of net primary production to changes in temperature may lead to carbon storage over some limited range of warming. In contrast, the high rates of soil respiration found in tropical ecosystems are highly sensitive to small changes in temperature, so that despite the less pronounced warming expected in equatorial regions, tropical soils are likely to release relatively large amounts of carbon to the atmosphere. Results for high-latitude biomes are highly sensitive to parameter values used, while the net efflux of carbon from the tropics appears robust.     

Climate spectra and detecting climate change

Part of the debate over possible climate changes centers on the possibility that the changes observed over the previous century are natural in origin. This raises the question of how large a change could be expected as a result of natural variability. If the climate measurement of interest is modelled as a stationary (or related) Gaussian time series, this question can be answered in terms of (a) the way in which change is estimated, and (b) the spectrum of the time series. These computations are illustrated for 128 years of global temperature data using some simple measures of change and for a variety of possible temperature spectra. The results highlight the time scales on which it is important to know the magnitude of natural variability. The uncertainties in estimates of trend are most sensitive to fluctuations in the temperature series with periods from approximately 50 to 500 years. For some of the temperature spectra, it was found that the standard error of the least squares trend estimate was 3 times the standard error derived under the naïve assumption that the temperature series was uncorrelated. The observed trend differs from zero by more than 3 times the largest of the calculated standard errors, however, and is therefore highly significant.     

Efficient time truncation in two-dimensional bem analysis of transient wave propagation problems

An efficient time truncation algorithm for transient two-dimensional wave propagation analysis by the boundary element method is proposed. First the complete (without any truncation) formulation is reviewed and subsequently time truncation concepts are discussed. Three applications are presented, where the efficiency of complete and truncated algorithms is compared. The most important conclusions inferred from the numerical analyses are: (i) time truncation errors can be controlled and made as small as required and (ii) time truncation algorithms lead to considerable savings in both CPU time and storage area.