The Probability Perturbation Method: A New Look at Bayesian Inverse Modeling

Building of models in the Earth Sciences often requires the solution of an inverse problem: some unknown model parameters need to be calibrated with actual measurements. In most cases, the set of measurements cannot completely and uniquely determine the model parameters; hence multiple models can describe the same data set. Bayesian inverse theory provides a framework for solving this problem. Bayesian methods rely on the fact that the conditional probability of the model parameters given the data (the posterior) is proportional to the likelihood of observing the data and a prior belief expressed as a prior distribution of the model parameters. In case the prior distribution is not Gaussian and the relation between data and parameters (forward model) is strongly non-linear, one has to resort to iterative samplers, often Markov chain Monte Carlo methods, for generating samples that fit the data likelihood and reflect the prior model statistics. While theoretically sound, such methods can be slow to converge, and are often impractical when the forward model is CPU demanding. In this paper, we propose a new sampling method that allows to sample from a variety of priors and condition model parameters to a variety of data types. The method does not rely on the traditional Bayesian decomposition of posterior into likelihood and prior, instead it uses so-called pre-posterior distributions, i.e. the probability of the model parameters given some subset of the data. The use of pre-posterior allows to decompose the data into so-called, “easy data” (or linear data) and “difficult data” (or nonlinear data). The method relies on fast non-iterative sequential simulation to generate model realizations. The difficult data is matched by perturbing an initial realization using a perturbation mechanism termed “probability perturbation.” The probability perturbation method moves the initial guess closer to matching the difficult data, while maintaining the prior model statistics and the conditioning to the linear data. Several examples are used to illustrate the properties of this method.     

Contribution of allochthonous carbon to American shad production in the Mattaponi River,Virginia, using stable isotopes

Our objective was to quantify the contribution of autochthonous, locally-produced phytoplankton, and allochthonous, terrestrial-derived organic matter (OM) to the production of young-of-year (YOY) American shad(Alosa sapidissima) using stable isotopes. We measured the carbon and nitrogen stable isotope composition of YOY American shad in the tidal fresh water of the Mattaponi River, a tributary in the York River estuary, during three consecutive years. The isotopic ratios of larval American shad varied among years, indicating a switch from reliance on a primarily autochthonous food web pathway during low and moderate discharge years (50–90%; 2002, 2004) to a primarily allochthonous pathway during a high discharge year (< 35% phytoplankton; 2003). Reliance on phytoplankton by larval fish declined exponentially with increasing Mattaponi River discharge. In 2003, juvenile production was also supported by allochthonous OM, though autochthonous phytoplankton accounted for an increasingly large fraction during June through August, up to 40–55%. We also found a long-term, positive relationship between the duration of above average flow during April through June in the Mattaponi River and a corresponding index of juvenile American shad abundance. The largest American shad cohort recorded since 1967 was observed in 2003, a high discharge year. The production of this cohort was largely supported by allochthonous OM. The results suggest an important link between river discharge, energy flow, and recruitment, wherein high discharge favors reliance on terrestrial carbon by YOY American shad, owing to changes in zooplankton diet, macroinvertebrate abundance, or both, and also favors high American shad abundance.     

Stratigraphic Framework and Heavy Minerals of the Continental Shelf of Onslow and Long Bays,North Carolina

One hundred fourteen vibracores from the Atlantic continental shelf offshore of southeastern North Carolina were opened, described, and processed over several contract years (years 6-9) of the Minerals Management Service Association of American State Geologists Continental Margins program. Reports for years 9 and 10 of the program compiled the results of the work and assembled the data for release as an interactive CD-ROM report, respectively. The continental shelf of Onslow and Long Bays consists predominantly of outcropping Cretaceous through late Tertiary geologic units. Nearshore these units are covered and incised by late Tertiary and Quaternary units. From oldest to youngest, formally recognized geologic units mapped as part of this study are the Late Cretaceous Peedee Formation a muddy, fine-to medium-grained quartz sand with trace amounts of glauconite and phosphate; the Paleocene Beaufort Forma tion a muddy, fine-to medium-grained glauconitic quartz sand with locally occurring turritelid-mold biosparrudite; the middle Eocene Castle Hayne Forma tion a sandy bryozoan biomicrudite and biosparrudite; the Oligocene River Bend Formation a sandy molluscan-mold biosparrudite; and the Miocene Pungo River Formation a medium-grained, poorly sorted slightly shelly phosphatic sand. Infor mal units include a very widespread, unnamed fine-to very fine grained, well-sorted, dolomitic muddy quartz sand that is biostratigraphically equivalent to the Oligocene River Bend Formation; several large valley-fill lithosomes composed of biomicrudite, biomicrite, and biosparrudite of Plio Pleistocene age; muddy, shelly sands and silty clays of Pliocene, Pleistocene, or mixed Plio Pleistocene age; and loose, slightly shelly, medium- to coarse-grained sands assigned a Holocene age. Heavy minerals (SG>2.96) comprise an average of 0.54 wt % (on a bulk-sam ple basis) of the sediments in 306 samples derived from the 114 vibracores. Heavy-mineral content ranges from <0.01 to 3.69 wt %. The economic heavy mineral content (EHM ilmenite zircon rutile aluminosilicates leucoxene [altered ilmenite] monazite) of the bulk samples averages 0.26 wt % in a range of <0.01-1.70 wt %. As a percentage of the heavy-mineral concentrate, the average EHM value is 45.78 % in a range of 0.27-68.60 %. The distribution of heavy minerals offshore of southeastern North Carolina is controlled by the lithostratigraphic framework. The unnamed Oligocene sand unit has the highest heavy-mineral content, averaging 0.86 wt % on a bulk-sample basis. The remaining geologic units and their heavy-mineral content (in decreasing order of abundance) are Beaufort (0.64 %), Holocene sand (0.60 %), Plio-Pleistocene muddy sand and silty clay (0.59 %), Peedee (0.42 %), River Bend (0.34 %), Plio-Pleistocene carbonate (0.12 %), and Castle Hayne (0.08 %). The heavy-mineral assemblage is fairly consistent throughout the different units. Significantly smaller percentages of heavy minerals correlate with increased amounts of CaCO3 in the sediments. The sediments analyzed in this study have significantly lower overall heavymineral content, as well as lower EHM content than sediments that are known to host commercially important heavy-mineral deposits in the southeastern United States. The potential for economic deposits of heavy minerals in the area of this study, therefore, appears to be limited.     

Temporal fluctuations and anisotropy of the micrometeoroid flux in the Earth-Moon system measured by HEOS 2

The HEOS detector measures the mass and speed of micrometeoroids in the Earth-Moon system. They are detected by the plasma produced by particle impacts on the sensor. During 2 yr of data collection 384 particles have been registered. As shown earlier (COSPAR 1973), they can be divided into 3 categories according to their temporal distribution: particles that are (1) randomly distributed or (2) appear in “groups” or (3) appear in “swarms” In this paper the origin of the groups and swarms is discussed. For this purpose the article orbits with respect to the Earth and the Moon were traced back. The results imply a lunar origin of the groups, whereas the swarms are correlated with the vicinity of the Earth. In addition, the dependence of the cumulative flux upon the detector's viewing direction indicates clearly an anisotropic particle flux.     

Temporal Changes of Shallow Seismic Velocity Around the Karadere-Düzce Branch of the North Anatolian Fault and Strong Ground Motion

We analyze temporal variations of seismic velocity along the Karadere-Düzce branch of the north Anatolian fault using seismograms generated by repeating earthquake clusters in the aftershock zones of the 1999 M_w7.4 İzmit and M_w7.1 Düzce earthquakes. The analysis employs 36 sets of highly repeating earthquakes, each containing 4–18 events. The events in each cluster are relocated by detailed multi-step analysis and are likely to rupture approximately the same fault patch at different times. The decay rates of the repeating events in individual clusters are compatible with the Omori's law for the decay rate of regional aftershocks. A sliding window waveform cross-correlation technique is used to measure travel time differences and evolving decorrelation in waveforms generated by each set of the repeating events. We find clear step-like delays in the direct S and early S-coda waves (sharp seismic velocity reduction) immediately after the Düzce main shock, followed by gradual logarithmic-type recoveries. A gradual increase of seismic velocities is also observed before the Düzce main shock, probably reflecting post-seismic recovery from the earlier İzmit main shock. The temporal behavior is similar at each station for clusters at various source locations, indicating that the temporal changes of material properties occur in the top most portion of the crust. The effects are most prominent at stations situated in the immediate vicinity of the recently ruptured fault zones, and generally decrease with normal distance from the fault. A strong correlation between the co-seismic delays and intensities of the strong ground motion generated by the Düzce main shock implies that the radiated seismic waves produced the velocity reductions in the shallow material.     

Estimating crustal deformations from a combination of baseline measurements and geophysical models

The estimation of crustal deformations from repeated baseline measurements is a singular problem in the absence of prior information. One often applied solution is a free adjustment in which the singular normal matrix is augmented with a set of inner constraints. These constraints impose no net translation nor rotation for the estimated deformations X which may not be physically meaningful for a particular problem. The introduction of an available geophysical model from which an expected deformation vector \(\bar X\) and its covariance matrix \(\sum _{\bar X} \) can be computed will direct X to a physically more meaningful solution. Three possible estimators are investigated for estimating deformations from a combination of baseline measurements and geophysical models.     

Platinum group metals in common plants of northern forests: developments in analytical methods, and the application of biogeochemistry to exploration strategies

All platinum-group metals (PGM) in ashed plant tissues, from an area of platinum mineralization in Saskatchewan, have been measured by neutron activation analysis of a NiS fire assay bead. Concentrations of up to 1350 ppb (ng/g) Pd, 880 ppb Pt, 49 ppb Rh, 37 ppb Ru, 24 ppb Ir, and 15 ppb Os occur in ashed twigs of black spruce (Picea mariana), indicating that the ratios of PGM uptake are about the same as those occurring in the bedrock. Plants growing on drift-covered diabase, known to have about 100 ppb PGM, contain up to 77 ppb Pt in ash, demonstrating the potential value of biogeochemical methods in helping to delineate platiniferous zones. Twigs of black spruce, jack pine, and labrador tea appear to be the optimum sample media in this environment.A rapid multi-element neutron activation analysis by direct irradiation of tissue permits measurement of 0.05 ppb Ir in dry material, or 2 ppb Ir in ashed samples. In view of the normally very low concentrations of Ir in plants, any detectable Ir probably indicates enrichment of PGM in the substrate.Comparison of several wet-chemical analytical methods indicates that the optimum (cost-effective) technique for detecting 1–2 ppb Pt, Pd, or Rh in 2-g samples of ash is by Te co-precipitation (following fusion and dissolution), and analysis by ICP-MS with sample introduction by electrothermal vaporization. However, care must be taken to ensure complete dissolution of all PGMs in the ash.     

A Continuum Damage–Breakage Faulting Model and Solid-Granular Transitions

We present a thermodynamically-based formulation for mechanical modeling of faulting processes in the seismogenic brittle crust using a continuum damage–breakage rheology. The model combines previous results of a continuum damage framework for brittle solids with continuum breakage mechanics for granular flow. The formulation accounts for the density of distributed cracking and other internal flaws in damaged rocks with a scalar damage parameter, and addresses the grain size distribution of a granular phase in a failure slip zone with a breakage parameter. The stress–strain relation and kinetics of the damage and breakage processes are governed by the total energy function of the system, which combines the energy of the damaged solid with the energy of the granular material. A dynamic brittle instability is associated with a critical level of damage in the solid, leading to loss of convexity of the solid energy function and transition to a granular phase associated with lower energy level. A non-local formulation provides an intrinsic length scale associated with the internal damage structure, which leads to a finite length scale for damage localization that eliminates the unrealistic singular localization of local models. Shear heating during deformation can lead to a secondary finite-width internal localization. The formulation provides a framework for studying multiple aspects of brittle deformation, including potential feedback between evolving elastic moduli and properties of the slip localization zone and subsequent rupture behavior. The model has a more general transition from slow deformation to dynamic rupture than that associated with frictional sliding on a single pre-existing failure zone, and gives time and length scales for the onset of the dynamic fracturing process. Several features including the existence of finite localization width and transition from slow to rapid dynamic slip are illustrated using numerical simulations. A configuration having an existing narrow slip zone with localized damage produces for appropriate loading conditions an overall cyclic stick–slip motion. The simulated frictional response includes transitions from friction coefficient of ~0.7 at low slip velocity to dynamic friction below 0.4 at slip rates above ~0.1 m/s, followed by rapidly increasing friction for slip rates above ~1 m/s, consistent with laboratory observations.