# Algorithms in Tomography and Related Inverse Problems--A Review
## 6 Global Seismic Tomography: The Inverse Problem and Beyond
In summary, global seismic tomography has achieved significant advancements in mapping three-dimensional elastic wave velocity fields. This progress has yielded valuable insights that have sparked interdisciplinary discussions within the field of Earth sciences. Moreover, seismic full waveform inversion (FWI) for imaging Earth's interior was introduced in the late 1970s. As highlighted by [PERSON] [145], its goal is to use all of the information in a seismogram to understand the structure and dynamics of Earth, such ashydrocarbon reservoirs, the nature of hotspots, and the forces behind plate motions and earthquakes [145] (see Figure 6). FWI in seismology starts by choosing seismic sources, ranging from earthquakes to controlled sources like air guns or explosions. It involves comparing observed and simulated seismograms to optimize model parameters, such as wave speeds and densities. However, FWI focuses on specific waveform characteristics, often using time windows where seismograms align [146; 147]. Selecting an accurate starting model is crucial but challenging, particularly in the exploration of seismology. Forward simulations calculate synthetic seismograms, vital in comparison with observed data, using various numerical methods. FWI needs a fast solver for wave propagation, with a shift toward frequency-domain approaches [148]. Comparison of observed and simulated seismograms requires a chosen misfit function, measuring differences like phase and amplitude. Inversion aims to minimize this misfit using the adjoint-state method, calculating gradients of model parameters [149; 150; 151]. Yet, challenges like cycle skipping and data volume remain [152; 153; 154]. FWI algorithms use optimization techniques like L-BFGS but face issues such as local minima and data quality mismatches. This method distinguishes itself from ray-based methods, as it involves comprehensive 3D numerical simulations and iterative updates of Earth models, allowing for a detailed study of seismic-wave propagation [155; 156; 157]. FWI finds applications in controlled-source exploration, earthquake studies, and ambient-noise seismology, including hydrocarbon exploration and seismic interferometry. Notable examples demonstrate its diverse use in various seismic contexts worldwide. In the following, a summary of six key points is provided for seismic tomography, focusing on the inverse problem and the associated challenges and techniques involved in reconstructing the Earth's three-dimensional velocity field:  
A. The Inverse Problem basic stages  
Forward Problem: In seismic tomography, the forward problem predicts a seismogram \(s(t,\;\Delta)\) at a distance \(\Delta\) from the seismic source over time t based on a prescribed velocity field \(v(r)\). Integral Formulation: The forward problem is represented as an integral equation involving the velocity field, spatial location \(r\) and the underlying principles of elastic wave propagation \(s_{i}(t,\Delta)=\int_{\Omega[v(r)]}g_{i}[t,\Delta,v(r)]dr\). Inverse Problem Challenges: The inverse problem involves reconstructing the velocity field \(v(r)\) from observed seismograms, and it is inherently ill-posed due to the uneven distribution of seismic sources and receivers. Figure 6: FWI accurately computes highly detailed, data-driven models of subsurface velocity, absorption Q, and reflectivity for use in seismic imaging and interpretation by minimizing the difference between observed and modeled seismic waveforms. Regularization: Regularization techniques, whether implicit or explicit, are applied to address ill-posedness and ill-conditioning issues in the inverse problem. These techniques constrain the model space or select a specific solution. B. Forward Theory  
Seismogram Characteristics: A typical seismogram exhibits sequences of P and S body waves, along with dispersed surface waves, providing information about the Earth's interior structure. Propagation Equations: The propagation of seismic waves within an elastic medium is governed by elastodynamic equations.