Forward and Inverse Modelling

Forward modelling

Forward modelling is the method by which solutions for simulations of a physical phenomenon are used to predict the data expected for a given subsurface model. These models can be expressed as geological models assuming sufficient knowledge of the relevant physical property/properties of the units to be simulated, but also often can be specified simply in physical property terms. In the latter case, interpretation must be made of the models with reference to likely geological units with those properties.

The first consideration for modelling is the level of spatial fidelity required. Models can be conducted which vary only in depth (1D models), vary across the model (2D models), have some finite strike extent but still vary mostly along the model (2.5D models) or fully represent 3D space. Added computational complexity and challenges in the methods used to simulate the physics to predict data from survey results are the main factors that dictate model choice. Geological environment can also dictate this choice, e.g. for the near surface in a well characterised terrane, 1D model to produce a layered-earth solution might be sufficient, but for highly complex geological environments around ore bodies it might require a full 3D model to correctly characterise the subsurface.

Models can also vary in their complexity around simulating the physics. Simple geometric shapes in a potential field forward model have analytic solutions, allowing for fast computation. More complex physics, such as electromagnetic wave propagation, require approximate solutions to the relevant governing physics e.g. the use of finite difference solutions to solve differential equations. Understanding the limitations of the physics being simulated is key to correctly interpreting forward modelling results.

Forward modelling cannot ever prove a subsurface model is a true representation of the subsurface. If a model predicts data which match geophysical observations then the subsurface model is a plausible candidate for the subsurface property distribution, however, there remain a multitude of subsurface models that would predict the same response. Forward modelling is therefore best used as a form of hypothesis testing to rule out potential subsurface properties e.g. a regolith layer cannot be thicker than a particular thickness, or, a fault must dip to a particular direction given the known architecture of the region and stratigraphy.

Inverse modelling

Inverse modelling is the opposite of forward modelling: it aims to produce subsurface model(s) that explain the data recorded from geophysical observations. They are termed inverse models as they derive mathematically from the operation of calculating inverse matrices to solve sets of simultaneous equations although practically inverse modelling rarely directly solves simultaneous equations.

Inverse models couple a forward model simulating physics to a mathematical solver which seeks to find solutions to an ‘objective function’. An objective function will contain elements to compare predicted versus observed data, and also generally some parameters to control properties of the subsurface model e.g. smooth solutions or some aspect of how ‘close’ a predicted model is to some starting model. The choice of solver to explore a solution space is itself an active field of mathematical and geophysical research around optimisation theory. Objective function design is another area of research with different styles of constraints placed on objective functions a component of integrated design of an inverse modelling method.

Forward physics is often simplified to assist in the task of solving the objective function. One example is looking for changes to a starting model which means that the residuals between observed and predicted data change approximately linearly with linear changes in model properties. This linearisation simplifies the method by which a solver can converge on a solution to produce a model which best fits observed data.

In calculating the relationship between changes in model parameters and how this results in changes in a) the residual difference between observed and predicted data and b) other terms in the objective function, there might be multiple invocations of the forward model. Calculation speed of the forward model is thus a key constraint on many inverse model. Many inversions, such as for complex physics like electromagnetism, will therefore reduce the dimensionality and complexity of the forward model to ensure that it is able to be computed sufficiently quickly to produce an inverse model in a timely fashion.

Inverse models may also not present just a single model for a geophysical survey, but instead may produce an entire statistical ensemble of all models that sufficiently meet constraints applied by the objective function and best fit the observed data. In these inversions, termed Monte Carlo inversions or stochastic inversions, there can be even more of a need for a sufficiently fast forward model to ensure appropriately exploration of the solution space. Some stochastic models also seek to solve for properties of the model itself, e.g. the number of layers required in output models to match observed data; these are termed ‘hyper-dimensional’ models as the mathematical dimensionality of the solution space is part of the solution itself.

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