Using Bayesian Statistics to Capture the Effects of Modelling Errors in Inverse Problems

When the parameters of a numerical model are adjusted, so that the predictions of the model match measurements from the real system, we need to take account of two sources of errors. These being measurement errors and modelling errors. Measurement errors are commonly considered, and a number of different approaches are in general usage, the most common being the weighted sum of squares method. In this paper the standard Bayesian equation, used for inverse problems, is reformulated so as to make it more intuitive to use. This allows the inclusion of both a modelling error and correlations between measurements to be carried out easily. The results are tested on a simple one-parameter numerical model and a cross-sectional model of a petroleum reservoir. In the first case the proposed error model appears to work well. In the second case it appears that the objective function is multimodal, leading to multiple acceptable solutions. The results of this paper are important to those whose numerical models are thought to contain significant modelling error. This encompasses many areas of modelling related to earth science and engineering.     

Timing of Precambrian melt depletion and Phanerozoic refertilization events in the lithospheric mantle of the Wyoming Craton and adjacent Central Plains Orogen

Garnet peridotite xenoliths from the Sloan kimberlite (Colorado) are variably depleted in their major magmaphile (Ca, Al) element compositions with whole rock Re-depletion model ages generally consistent with this depletion occurring in the mid-Proterozoic. Unlike many lithospheric peridotites, the Sloan samples are also depleted in incompatible trace elements, as shown by the composition of separated garnet and clinopyroxene. Most of the Sloan peridotites have intermineral Sm–Nd and Lu–Hf isotope systematics consistent with this depletion occurring in the mid-Proterozoic, though the precise age of this event is poorly defined. Thus, when sampled by the Devonian Sloan kimberlite, the compositional characteristics of the lithospheric mantle in this area primarily reflected the initial melt extraction event that presumably is associated with crust formation in the Proterozoic—a relatively simple history that may also explain the cold geotherm measured for the Sloan xenoliths.

The Williams and Homestead kimberlites erupted through the Wyoming Craton in the Eocene, near the end of the Laramide Orogeny, the major tectonomagmatic event responsible for the formation of the Rocky Mountains in the late Cretaceous–early Tertiary. Rhenium-depletion model ages for the Homestead peridotites are mostly Archean, consistent with their origin in the Archean lithospheric mantle of the Wyoming Craton. Both the Williams and Homestead peridotites, however, clearly show the consequences of metasomatism by incompatible-element-rich melts. Intermineral isotope systematics in both the Homestead and Williams peridotites are highly disturbed with the Sr and Nd isotopic compositions of the minerals being dominated by the metasomatic component. Some Homestead samples preserve an incompatible element depleted signature in their radiogenic Hf isotopic compositions. Sm–Nd tie lines for garnet and clinopyroxene separates from most Homestead samples provide Mesozoic or younger “ages” suggesting that the metasomatism occurred during the Laramide. Highly variable Rb–Sr and Lu–Hf mineral “ages” for these same samples suggest that the Homestead peridotites did not achieve intermineral equilibrium during this metasomatism. This indicates that the metasomatic overprint likely was introduced shortly before kimberlite eruption through interaction of the peridotites with the host kimberlite, or petrogenetically similar magmas, in the Wyoming Craton lithosphere.     

Light attenuation and submersed macrophyte distribution in the tidal Potomac River and estuary

Changing light availability may be responsible for the discontinuous distribution of submersed aquatic macrophytes in the freshwater tidal Potomac River. During the 1985–1986 growing seasons, light attenuation and chlorophylla and suspended particulate material concentrations were measured in an unvegetated reach (B) and in two adjacent vegetated reaches (A and C). Light attenuation in reach B (the lower, fresh to oligohaline tidal river) was greater than that in reach A (the recently revegetated, upper, freshwater tidal river) in both years. Reach B light attenuation was greater than that in reach C (the vegetated, oligohaline to mesohaline transition zone of the Potomac Estuary) in 1985 and similar to that in reach C in 1986. In reach B, 5% of total below-surface light penetrated only an average of 1.3 m in 1985 and 1.0m in 1986, compared with 1.9 m and 1.4 m in reach A in 1985 and 1986, respectively. Water column chlorophylla concentration controlled light availability in reaches A and B in 1985, whereas both chlorophylla and suspended particulate material concentrations were highly correlated with attenuation in both reaches in 1986. Reach C light attenuation was correlated with suspended particulate material in 1986. The relationship between attenuation coefficient and Secchi depth was K_PAR=1.38/Secchi depth. The spectral distribution of light at 1 m was shifted toward the red portion of the visible spectrum compared to surface light. Blue light was virtually absent at 1.0 m in reach B during July and August 1986. Tidal range is probably an important factor in determining light availability for submersed macrophyte propagule survival at the sediment-water interface in this shallow turbid system.     

Role of weather and water quality in population dynamics of submersed macrophytes in the tidal Potomac River

Weather and water-quality data from 1980 to 1989 were correlated with fluctuations in submersed macrophyte populations in the tidal Potomac River near Washington, D.C., to elucidate causal relationships and explain population dynamics. Both reaches were unvegetated in 1980 when mean growing-season Secchi depths were <0.60 m. Macrophyte resurgence in the upper tidal river in 1983 was associated with a growing-season Secchi depth of 0.86 m, total suspended solids (TSS) of 17.7 mg l^?1, chlorophyll a concentrations of 15.2 μg l^?1, significantly higher than average percent available sunshine, and significantly lower than average wind speed. From 1983 to 1989, mean seasonal Secchi depths <0.65 m were associated with decrease in plant coverage and mean seasonal Secchi depths >0.65 were associated with increases in plant coverage. Changes in mean seasonal Secchi depth were related to changes in mean seasonal TSS and chlorophyll a concentration; mean Secchi depths >0.65 generally occur when seasonal mean TSS is <19 mg l^?1 and seasonal mean chlorophyll a concentration is ≤15 μg l^?1. Secchi depth is highly correlated with plant growth in the upper tidal river and chlorophyll a and TSS with plant growth in the lower tidal river. Wind speed is an important influence on plant growth in both reaches.     

Mapping the state of fracture around cavities

The state of stress around cavities is heterogeneous. Consequently, the state of fracture may also vary from point to point. Under compressive loading, cavity may be in one of four possible states: pre-fracture (pre-microfracture initiation), microfracture propagation (initiation to the onset of dilata dilatancy to failure) and post-failure. These four states are separated by the crack initiation stress, the crack damage stress and the failure stress. fracture events with confining pressure is examined with reference to three intact rocks: a brittle granite, a semi-brittle limestone and a ductile sal

The maximum principal stress (σ₁) at crack initiation, crack damage (onset of dilatancy), yielding and failure are established as a function stress (σ₃). For a single intact rock, all four fracture events can be represented using one function (the Rocker function) with a single fac various fracture states.

The proposed fracture criteria, based on the experimental data, are combined with the existing state of stress to prepare a fracture map around an elli intact Lac du Bonnet granite. The state of stability is expressed through a newly defined stability factor, the unconfined strength ratio (USR), wh to the traditional safety factor. In contrast to the conventional safety factor in rock mechanics (SFstrength/σ₁), which is σ₃ 3–σ₁, space of the stability diagram.     

Fitting strength criteria to intact rock

Summary Rock strength data covering the full range of possible stress conditions are presented for three rocks: a granite, a limestone and a salt rock. The Hoek and Brown square root parabola and the Johnston criteria are fitted to the strength data coming from around 500 laboratory tests. The fitting procedure is facilitated by a specially built PC code, ROCKER, which is available to anyone on request.The Hoek and Brown criterion is modified through the inclusion of a third parameter to account for the low tensile strength of the salt rock. A new criterion, the Rocker function is formulated to follow strength data closely in the tension-low confining pressure region.     

Overwash occurrence consequent on morphodynamic changes following lagoon outlet closure on a coarse clastic barrier

The reduction in sediment volume of an ebb-tide delta, as a consequence of lagoon outlet closure on a coarse clastic barrier in southeast Ireland, initiated a sequence of beachface and barrier changes downdrift. Elimination of sediment supply to the ebb-delta caused a cessation of downdrift longshore swash bar welding, and led to beach volume reductions which in turn allowed a temporal sequence of beach and nearshore morphodynamic domains to develop. These domains then controlled the sedimentation regime of the barrier adjacent to the old outlet. A temporal sequence of (a) dissipative barrier; (b) reflective barrier; and (c) inner reflective (barrier face)/outer dissipative (subtidal) wave regimes match respectively periods of (a) barrier crest build up by crestal dune development; (b) barrier crest instability (barrier width increasing, barrier height decreasing) due to rhythmic overwash; and (c) a return to barrier crest stability with limited aeolian accretion. Two barriers at different stages in this sequence are discussed.