Robert L. Parker

Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data

The inversion of electromagnetic sounding data does not yield a unique solution, but inevitably a single model to interpret the observations is sought. We recommend that this model be as simple, or smooth, as possible, in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models.To obtain smooth models, the nonlinear forward problem is linearized about a starting model in the usual way, but it is then solved explicitly for the desired model rather than for a model correction. By parameterizing the model in terms of its first or second derivative with depth, the minimum norm solution yields the smoothest possible model.Rather than fitting the experimental data as well as possible (which maximizes the roughness of the model), the smoothest model which fits the data to within an expected tolerance is sought. A practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models, resulting in a stable and rapidly convergent algorithm. The inversion of both magnetotelluric and Schlumberger sounding field data, and a joint magnetotelluric-resistivity inversion, demonstrate the method and show it to have practical application.

Positron emission tomography with fluorodeoxyglucose to evaluate tumor response and control after radiation therapy

PURPOSE Following radiation therapy, evaluation of viable tumor can often be difficult with anatomic imaging criteria (tumor size alone). In this study, the utility of biochemical imaging with the glucose analog 2-[F-18]fluoro-2-deoxy-D-glucose and positron emission tomography was investigated in patients treated with radiation therapy. METHODS AND MATERIALS Between 1990 and 1992, 19 patients were studied, including 15 patients with head and neck cancer, (4 oropharynx, 4 sinus, 3 larynx, 2 hypopharynx, 2 oral cavity [one patient], 1 nasopharynx), and 4 patients with breast cancer. Post-radiation positron emission tomography with 2-[F-18]fluoro-2-deoxy-D-glucose studies were done in all patients, with 9 head and neck patients receiving pre-radiation positron emission tomography with 2-[F-18]fluoro-2-deoxy-D-glucose scans as well. Results were correlated with other imaging techniques and pathology. RESULTS Positron emission tomography with 2-[F-18]fluoro-2-deoxy-D-glucose detected head and neck primary tumors and lymph node metastases in all nine pre-radiation scans, while magnetic resonance imaging failed to detect two primary tumors. Serial positron emission tomography with 2-[F-18]fluoro-2-deoxy-D-glucose showed a significant decrease in tumor activity after radiation therapy, compared to pre-radiation levels, (p < 0.05), except for two patients with increased uptake at the primary site. Biopsies of these two patients showed persistent/recurrent disease after radiation therapy, which was not detected by magnetic resonance imaging. Six additional head and neck patients, with suspicious examination and inconclusive magnetic resonance imaging, were imaged with positron emission tomography after radiation therapy only. Five patients had increased positron emission tomography activity, with corresponding biopsies positive in four patients, and negative in one patient with clinically worsening symptoms. The remaining sixth patient had minimal and stable positron emission tomography uptake, and is improving clinically. Four patients had mammogram findings suspicious for recurrence after conservation treatment for breast cancer. Positron emission tomography with 2-[F-18]fluoro-2-deoxy-D-glucose showed no focal activity in the breast in two patients, and increased activity in the area suspicious for recurrence in the other two patients. Biopsies correlated with positron emission tomography results. CONCLUSION Changes and presence of positron emission tomography with 2-[F-18]fluoro-2-deoxy-D-glucose activity correlated with pathologic findings in head and neck and breast cancer patients in this series. In patients with elevated or rising positron emission tomography activity after radiation therapy, persistent or recurrent disease was found in 89% of patients, (8/9). Magnetic resonance imaging did not detect the head and neck recurrences, and mammography was suspicious in patients with both benign and malignant breast changes after radiation therapy. In addition, our data indicate that in head and neck patients with pre-radiation positron emission tomography scans, a significant decrease in activity should occur after radiation therapy, if local control is to be expected.

A Fortran 90 library for multitaper spectrum analysis

The spectral analysis of geological and geophysical data has been a fundamental tool in understanding Earths processes. We present a Fortran 90 library for multitaper spectrum estimation, a state-of-the-art method that has been shown to outperform the standard methods. The library goes beyond power spectrum estimation and extracts for the user more information including confidence intervals, diagnostics for single frequency periodicities, and coherence and transfer functions for multivariate problems. In addition, the sine multitaper method can also be implemented. The library presented here provides the tools needed in multiple fields of the Earth sciences for the analysis of data as evident from various examples.

BEST BOUNDS ON DENSITY AND DEPTH FROM GRAVITY DATA

Gravity data cannot usually be inverted to yield unique structures from incomplete data; however, there is a smallest density compatible with the data or, if the density is known, a deepest depth of burial. A general theory is derived which gives the greatest lower bound on density or the least upper bound on depth. These bounds are discovered by consideration of a class of “ideal” bodies which achieve the extreme values of depth or density. The theory is illustrated with several examples which are solved by analytic methods. New maximum depth rules derived by this theory are, unlike some earlier rules of this type, optimal for the data they treat.

Optimal one-dimensional inversion and bounding of magnetotelluric apparent resistivity and phase measurements

Abstract The properties of the log of the admittance in the complex frequency plane lead to an integral representation for one-dimensional magnetotelluric (MT) apparent resistivity and impedance phase similar to that found previously for complex admittance. The inverse problem of finding a one-dimensional model for MT data can then be solved using the same techniques as for complex admittance, with similar results. For instance, the one-dimensional conductivity model that minimizes the χ2 misfit statistic for noisy apparent resistivity and phase is a series of delta functions. One of the most important applications of the delta function solution to the inverse problem for complex admittance has been answering the question of whether or not a given set of measurements is consistent with the modeling assumption of one-dimensionality. The new solution allows this test to be performed directly on standard MT data. Recently, it has been shown that induction data must pass the same one-dimensional consistency test if they correspond to the polarization in which the electric field is perpendicular to the strike of two-dimensional structure. This greatly magnifies the utility of the consistency test. The new solution also allows one to compute the upper and lower bounds permitted on phase or apparent resistivity at any frequency given a collection of MT data. Applications include testing the mutual consistency of apparent resistivity and phase data and placing bounds on missing phase or resistivity data. Examples presented demonstrate detection and correction of equipment and processing problems and verification of compatibility with two-dimensional B-polarization for MT data after impedance tensor decomposition and for continuous electromagnetic profiling data.

Geomagnetic intensity variations over the past 780 kyr obtained from near-seafloor magnetic anomalies.

Knowledge of past variations in the intensity of the Earths magnetic field provides an important constraint on models of the geodynamo. A record of absolute palaeointensity for the past 50 kyr has been compiled from archaeomagnetic and volcanic materials, and relative palaeointensities over the past 800 kyr have been obtained from sedimentary sequences. But a long-term record of geomagnetic intensity should also be carried by the thermoremanence of the oceanic crust. Here we show that near-seafloor magnetic anomalies recorded over the southern East Pacific Rise are well correlated with independent estimates of geomagnetic intensity during the past 780 kyr. Moreover, the pattern of absolute palaeointensity of seafloor glass samples from the same area agrees with the well-documented dipole intensity pattern for the past 50 kyr. A comparison of palaeointensities derived from seafloor glass samples with global intensity variations thus allows us to estimate the ages of surficial lava flows in this region. The record of geomagnetic intensity preserved in the oceanic crust should provide a higher-time-resolution record of crustal accretion processes at mid-ocean ridges than has previously been obtainable.

The inverse problem of resistivity sounding

The electric potential due to a single point electrode at the surface of a layered conducting medium is calculated by means of a linear combination of the potentials associated with a set of two‐layer systems. This new representation is called the bilayer expansion for the Green’s function. It enables the forward problem of resistivity sounding to be solved very efficiently, even for complicated profiles. Also, the bilayer expansion facilitates the solution of the resistivity inverse problem: the coefficients in the expansion are linearly related to apparent resistivity as it is measured and they are readily mapped into parameters for a model. Specifically, I consider models comprising uniformly conducting layers of equal thickness; for a given finite data set a quadratic program can be used to find the best‐fitting model in this class for any specified thickness. As the thickness is reduced, models of this kind can approximate arbitrary profiles with unlimited accuracy. If there is a model that satisfies...

Isostasy in Australia and the Evolution of the Compensation Mechanism

A linear transfer function analysis has been applied to gravity and topographic data from Australia to calculate the isostatic response function of Dorman and Lewis. The Australian response function is considerably different from that calculated for the United States. The differences can be explained on the basis of an apparent evolution of the isostatic compensation mechanism in which viscoelastic creep occurs in the lithosphere and relaxes the initial long-wavelength elastic stresses.

Analysis of 11 Myr of geomagnetic intensity variation

We have conducted a detailed exploratory analysis of an 11 million year long almost continuous record of relative geomagnetic paleointensity from a sediment core acquired on Deep Sea Drilling Project Leg 73, at Site 522 in the South Atlantic. We assess the quality of the paleointensity record using spectral methods and conclude that the relative intensity record is minimally influenced by climate variations. Isothermal remanence is shown to be the most effective normalizer for these data, although both susceptibility and anhysteretic remanence are also adequate. Statistical analysis shows that the paleointensity variations follow a gamma distribution, and are compatible with predictions from modified paleosccular variation models and global absolute paleointensity data. When subdivided by polarity interval, the variability in paleointensity is proportional to the average, and further, the average is weakly correlated with interval length. Spectral estimates for times from 28.77 until 22.74 Ma, when the reversal rate is about 4 Myr -1 , are compatible with a Poisson model in which the spectrum of intensity variations is dominated by the reversal process in the frequency range 1--50 Myr -1 . In contrast, between 34.7 and 29.4 Ma, when the reversal rate is about 1.6 Myr -1 , the spectra indicate a different secular variation regime. The magnetic field is stronger, and more variable, and a strong peak in the spectrum occurs at about 8 Myr -1 . This peak may be a reflection of the same signal as recorded by the small variations known as tiny wiggles seen in marine magnetic anomaly profiles.

A theory of ideal bodies for seamount magnetism

Recent studies of samples from seamounts indicate that the distribution of magnetic intensity is approximately lognormal, which implies that the commonly adopted models of interior magnetization based upon a constant vector with an isotropic perturbation are inappropriate. We develop a unidirectional model in which the direction of magnetization is fixed and the intensity is of one sign, with no upper limit on magnitude, which, if the seamount is built during a period of single magnetic polarity, is likely to be a better approximation. We show that models of this class fitting the data best in the two-norm sense conform to the ideal-body pattern comprising unidirectional, point dipoles in the surface of the seamount. Practical methods are developed for discovering the best data misfit associated with paleopole position. The methods are first tested on simple artificial magnetic anomalies and are found to be capable of recovering the true pole position with high accuracy when such a solution is possible; also when a mixed polarity artificial model is analyzed, it is found that there are no unidirectional solutions, just as would be hoped. The method is next applied to three seamount surveys. In the first it is found that every direction of magnetization is in accord with the data, so that apparently nothing useful can be learned from the survey without a stronger assumption; this result is in contrast with the results of an earlier solution based upon a statistical model, which yielded a high accuracy in the position of the paleopole. The second investigation provides a reasonably compact location of the paleopole of the seamount. The third magnetic anomaly is complex and earlier studies concluded this was necessarily the product of mixed polarity magnetization. We find that in fact unidirectional magnetizations can satisfy observation.