# Algorithms in Tomography and Related Inverse Problems--A Review
## 6 Global Seismic Tomography: The Inverse Problem and Beyond  
Global seismic tomography has remained an active area of research since its initial systematic exploration in the early 1980s. Seismic waves generated by sufficiently large earthquakes propagate across the globe, and as they traverse the Earth's interior, they carry valuable information about the medium through which they travel, including their arrival times and waveform characteristics [138; 139]. As explained by [PERSON] [69], the fundamental challenge in seismic tomography is to reconstruct the three-dimensional elastic velocity distribution within the Earth using extensive datasets of arrival times, waveforms from both body and surface waves, and free oscillations. A recent comprehensive assessment encompassing various research studies proposed by [PERSON] and [PERSON] [140] has demonstrated a robust convergence of information regarding the Earth's structural characteristics, employing diverse datasets and distinct mapping techniques. Regions characterized by lower seismic velocities exhibit correlations with geoid highs and the locations of numerous global hotspots. These three-dimensional velocity models hold promise in establishing a solid foundation for comprehending the driving mechanisms underlying plate tectonics. Seismic tomography plays a crucial role in mapping the current thermodynamic and compositional characteristics of heterogeneities within the convicting mantle, imposing stringent constraints on potential models of mantle convection [141; 142; 143]. Key geological features, such as the thickness of continental roots, the depth range of mid-ocean ridge signals, and variations in lithospheric velocity with age [142; 144], offer valuable insights into the genesis and evolution of continental and oceanic lithospheric structures. Additionally, the relative changes in P-wave velocities compared to S-wave velocities provide substantial mineralogical constraints on the mantle's composition [141]. The thermal conditions prevailing in the lower mantle establish critical boundary conditions for potential geodynamo models aiming to explain the Earth's magnetic field dynamics [139]. Notably, a long-standing correlation between the geoid and seismic models has been acknowledged [138], suggesting that both gravity and seismological data can collectively contribute to our understanding of three-dimensional density fluctuations within the Earth. Seismic tomography primarily revolves around the reconstruction of the Earth's three-dimensional velocity field by analyzing surface observations of elastic waves. The forward problem entails the prediction of a seismogram. The tomographic inverse problem is inherently characterized by an uneven distribution of seismic sources (earthquakes) and receivers (limited to seismic stations on continents and oceanic islands). This results in an irregular sampling of the Earth's interior using elastic waves, where some regions are over-sampled while others remain undersampled. In such circumstances, the inverse problem becomes ill-posed, typically exhibiting a rapidly declining eigenvalue spectrum. This steep decline signifies that minor data errors can lead to substantial variations in the solution, rendering the problem ill-conditioned. Both ill-posedness and ill-conditioning are associated with substantial null spaces, indicating non-unique solutions. The remedy for these challenges involves the application of implicit or explicit regularization techniques. Regularization serves the purpose of either constraining the potential model space or selecting a specific solution from the multitude of possible solutions. The propagation of seismic waves within an elastic medium is primarily governed by the elastodynamic equations. Assuming a known source excitation and instrument response, this leads to a nonlinear relationship between the observed data and the provided Earth model through the underlying physics of wave propagation encapsulated in the forward theory.