Geophysical Inverse Problems: A Comprehensive Overview

Introduction

Geophysical inverse problems constitute a fundamental aspect of geoscience, facilitating the exploration of the Earth’s internal structure by leveraging observable data. This process typically involves analysing seismic wave arrival times recorded at seismic stations to deduce the properties and composition of subsurface materials. This comprehensive overview elucidates the various facets of geophysical inverse problems, including model parameterisation, formulation of the inverse problem, linearisation methods, and strategies for addressing challenges such as non-uniqueness and data noise. The aim is to provide a rich resource for students and professionals engaged in understanding and solving geophysical inverse problems.

1. Model Parameterisation

• Definition: Model parameterisation refers to the method of mathematically representing the internal structure of the Earth within a geophysical model. This choice significantly affects the types of structures that can be resolved from the inverse problem, making it a vital consideration in the analysis.

• Early Methods:

  • Regular Tessellations: Initially, geophysical studies predominantly utilised uniform grids or regular tessellations of constant slowness or velocity blocks due to their computational simplicity. While these methods permitted straightforward calculations, they often introduced artificial discontinuities in the model that could lead to instability in solutions, thereby limiting the accuracy of the inverse problem.

• Modern Approaches:

  • Mesh/Grid Points: Contemporary techniques employ a grid of points or mesh that accurately describes the Earth’s structure in three dimensions. Interpolation functions—be they linear, cubic, or higher-order—are applied to produce a continuous representation of physical properties throughout the mesh.

  • Spectral Parameterisations: Various spectral techniques, such as truncated Fourier series or spherical harmonics, are increasingly used to expand the flexibility of model representation, permitting the inclusion of complex geological features and accommodating varying data densities.

• Grid Spacing:

  • Key Principle: The grid or mesh spacing is crucial in resolving geological features. It is essential that the grid size be smaller than the smallest feature intended for imaging. For instance, to accurately image granite bodies that measure around 10 km, the grid size must be less than 10 km. This principle ensures that finer details are captured without being masked or lost in the averaging process of grid-based calculations.

• Interface Representation:

  • Accurately defining discontinuities or interfaces, such as the Mohorovičić discontinuity (commonly referred to as the Moho), is essential for appropriately utilising reflected seismic phases during data inversion. Complex interfaces can exacerbate the challenges associated with solving both the forward and inverse problems, thus proper representation is necessary for reliability.

• Irregular Parameterisation:

  • Irregular parameterisation intelligently adjusts the grid resolution based on the density of available seismic data. Finer meshes are used in areas where data coverage is plentiful, while coarser grids are employed in data-sparse regions, thereby optimising computational efficiency while preserving model accuracy.

  • Techniques may be classified into static and adaptive types, where static methods maintain a fixed irregular grid throughout the process, while adaptive methods dynamically adjust the mesh’s resolution during the inversion, contingent on incoming data and the estimates of the model.

2. Formulation of the Geophysical Inverse Problem

• Mathematical Relationship: The fundamental relationship connecting observed data (denoted as d) and model parameters (denoted as m) can be articulated by the equation:

  [ d = g(m) ]

  where g represents a potentially non-linear operator that encapsulates the complexities involved in the physical process being modelled.

• Data and Model Vectors:

  • Data Vector (d): The data vector consists of a series of observations, such as the arrival times of P-waves detected across multiple seismic stations. A comprehensive dataset might involve numerous stations and earthquake events, creating a data vector containing thousands of elements.

  • Model Vector (m): This vector characterises the model parameters that define the subsurface properties, such as slowness values assigned to quantified grid points within the three-dimensional model. In an illustrative example, a grid comprising 10 points in depth and 20 points in both latitude and longitude results in 2,000 model parameters to estimate.

• Types of Inverse Problems:

  • Linear Inverse Problems: These problems can be efficiently expressed in matrix notation, whereby the equation is transformed to:

  [ d = Gm ]where G represents a matrix denoting the system’s characteristics. An example of this would be the inversion of gravity data to discern density variations within subsurface materials.

  • Weakly Non-linear Problems: These can be approached by assuming local linearity around a sufficiently accurate initial model. A case in point, global traveltime tomography often operates under this assumption, thus simplifying the inversion process while retaining enough fidelity for meaningful interpretations.

  • Fully Non-linear Problems: Unlike weakly non-linear problems, these situations cannot safely adopt local linearity assumptions. Receiver function inversion serves as a classic example of fully non-linear issues that require sophisticated handling techniques.

• Focus on Weakly Non-linear Problems:

  • Given their ubiquitous nature and manageable complexity, weakly non-linear problems are the predominant focus of much educational discourse pertaining to geophysical inversion methodologies.

3. The Newton-Raphson Method

• Concept: The Newton-Raphson method serves as a numerical approach to identifying roots of non-linear equations. Its effectiveness lies in iteratively refining an initial guess to converge upon a solution, making it particularly beneficial in situations characterised by complex relationships between variables.

• Process:

  • The method initiates with an educated guess; a tangent line is subsequently fitted to the curve at this point. The intersection of the tangent with the x-axis yields a new approximation, which can be iterated upon until a satisfactory level of convergence is achieved.

• Limitations:

  • The Newton-Raphson method can falter if the initial guess is poor, or if the function in question exhibits high levels of non-linearity. Furthermore, it may only converge to the root nearest to the original guess in scenarios where multiple roots are present.

4. Linearisation of the Geophysical Inverse Problem

• Taylor Series Expansion: The Taylor series expansion stands as a cornerstone technique in the linearisation of functions, enabling approximations at specific points based upon information from reference points. This method is essential for rendering non-linear functions manageable within the framework of inverse problems.

• First-Order Approximation:

  • In the context of geophysical inverse problems, only first-order terms are considered to streamline the calculations, allowing for simplifications based on linearised relationships.

• Mathematical Expressions:

  • When considering the relationship d = g(m), the Taylor series can be expressed as follows:

  [ g(m_0 + \delta m) = g(m_0) +
  abla g(m_0) \delta m ]

  where (
  abla g(m_0) ) indicates the gradient at the reference point ( m_0 ) and ( \delta m ) represents the perturbation applied to model parameters.

  • The difference between observed data and predicted model outputs can be expressed as a perturbation:

  [ \delta d = d_{obs} - g(m_0) ]

  Thus, linearised expressions take the form:

  [ \delta d = G \delta m ]

  where G is the Fréchet matrix, representing the sensitivity of observable data to changes in each model parameter.

• Iterative Solution Process:

  • Similar to the Newton-Raphson methodology, solving the linearised equations necessitates a reliable starting model. The perturbation ( \delta m ) that closes the gap towards a solution is discovered through iterations, converging upon a successful model through repeated adjustments until satisfactory accuracy is achieved.

5. Data Coverage and Noise

• Non-Uniform Coverage:

  • In seismic tomography, it is seldom the case that data coverage is uniform. Taking global traveltime tomography as an example, seismic stations tend to be predominantly located on land, while earthquake sources may be concentrated along tectonic plate boundaries, leading to irregular ray path coverage that complicates modelling efforts.

• Data Noise:

  • The influence of noise on seismic data is substantial and often inadequately understood. Variability in the quality of data sourced from global seismological databases presents challenges; the noise can produce non-unique solutions, making it difficult to ascertain a singular accurate representation of the Earth’s structure. Various ad hoc methodologies—such as the elimination of outliers and the averaging of travel times from seismically equivalent stations—are typically employed to improve data quality, although noise remains an intrinsic factor in the modelling process.

Example Applications

• Seismic Traveltime Tomography:

  • This method serves as the most common technique for imaging the Earth across various scales, exploiting seismic phase arrival times to reconstruct the subsurface velocity structure. It illustrates the practical application of principles laid out in this overview, reinforcing the relevance of weakly non-linear problem-solving methodologies.

• Checkerboard Tests:

  • Checkerboard tests are employed to evaluate the resolving capability of datasets; synthetic reconstructions of alternating fast and slow anomalies ascertain the fidelity of inversion outputs. Areas where the checkerboard is adequately recovered are deemed as exhibiting high resolving power, which is vital for validating the effectiveness of the dataset.

Conclusion

An intricate understanding of the principles underpinning geophysical inverse problems is crucial for the advancement of accurate Earth imaging techniques. By mastering concepts associated with model parameterisation, careful formulation of inverse problems, and considerations surrounding noise and data quality, geophysical scientists can extract invaluable insights into the Earth’s internal structure. As computational methodologies continue to evolve, so too will opportunities for enhanced geophysical tomography, significantly contributing to our understanding of geological processes and structures.