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Inverse modeling of geochemical and mechanical compaction in sedimentary basins through Polynomial Chaos Expansion

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Maximum likelihood estimation

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

Sedimentary basin

Sedimentary basins are region-scale depressions of the Earth's crust where subsidence has occurred and a thick sequence of sediments have accumulated to form a large three-dimensional body of sedimentary rock. They form when long-term subsidence creates a regional depression that provides accommodation space for accumulation of sediments. Over millions or tens or hundreds of millions of years the deposition of sediment, primarily gravity-driven transportation of water-borne eroded material, acts to fill the depression.

Sensitivity analysis

Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in its inputs. A related practice is uncertainty analysis, which has a greater focus on uncertainty quantification and propagation of uncertainty; ideally, uncertainty and sensitivity analysis should be run in tandem.

Old Red Sandstone

The Old Red Sandstone is an assemblage of rocks in the North Atlantic region largely of Devonian age. It extends in the east across Great Britain, Ireland and Norway, and in the west along the eastern seaboard of North America. It also extends northwards into Greenland and Svalbard. These areas were a part of the ancient continent of Euramerica/Laurussia. In Britain it is a lithostratigraphic unit (a sequence of rock strata) to which stratigraphers accord supergroup status and which is of considerable importance to early paleontology.

Likelihood function

In statistical inference, the likelihood function quantifies the plausibility of parameter values characterizing a statistical model in light of observed data. Its most typical usage is to compare possible parameter values (under a fixed set of observations and a particular model), where higher values of likelihood are preferred because they correspond to more probable parameter values.

Dirichlet boundary condition

In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.

Shale

Shale is a fine-grained, clastic sedimentary rock formed from mud that is a mix of flakes of clay minerals (hydrous aluminium phyllosilicates, e.g. kaolin, Al2Si2O5(OH)4) and tiny fragments (silt-sized particles) of other minerals, especially quartz and calcite. Shale is characterized by its tendency to split into thin layers (laminae) less than one centimeter in thickness. This property is called fissility. Shale is the most common sedimentary rock.

Uncertainty quantification

Uncertainty quantification (UQ) is the science of quantitative characterization and estimation of uncertainties in both computational and real world applications. It tries to determine how likely certain outcomes are if some aspects of the system are not exactly known. An example would be to predict the acceleration of a human body in a head-on crash with another car: even if the speed was exactly known, small differences in the manufacturing of individual cars, how tightly every bolt has been tightened, etc.

Standard error

The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM). The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance.

Boundary value problem

In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems.