In 1978, while working as a young researcher in the Soviet Union, I developed an optimisation-based method for estimating the parameters of low-amplitude faulting from seismic reflection data. Although constrained by the computational and modelling limitations of the time, the approach embodied several ideas that later became central to full-waveform inversion (FWI), including waveform misfit minimisation and sensitivity to small structural variations. In this historical note, I revisit the original study, place it within its scientific context, and compare its conceptual foundations with those of modern adjoint-state FWI. The discussion highlights both the limitations of the early algorithm and the forward-looking insights it contained, illustrating how foundational ideas from the late 1970s contributed to the evolution of quantitative seismic imaging.
Introduction
The development of quantitative seismic interpretation has been shaped by a series of conceptual advances that gradually transformed the field from qualitative pattern recognition to physics-based inversion. Among the early contributions to this transition is a 1978 study I conducted as a young researcher in the Soviet Union, in which I proposed an optimisation method for estimating the geometry of low-amplitude faults using seismic reflection data (Landa, 1978). At the time, seismic interpretation relied heavily on visual analysis, and the detection of subtle structural features, particularly faults with amplitudes smaller than the dominant seismic wavelength, posed a significant challenge.
The scientific environment of the late 1970s played an important role in shaping this work. Advances in digital processing and the growing availability of computational resources at research institutes in Novosibirsk created conditions in which new analytical approaches could be tested on real geological problems. In hydrocarbon exploration areas such as Western Siberia, the ability to detect and characterise low-amplitude faults became increasingly important, as these subtle displacements often controlled reservoir compartmentalisation and fluid migration. Traditional interpretation methods were well suited for large structural features but struggled with small offsets comparable to, or smaller than, the dominant seismic wavelength.
Within this context, I began exploring the idea of interpreting seismic data through quantitative optimisation of physically meaningful parameters. In this formulation, seismic inversion is defined as the process of adjusting model parameters to minimise the misfit between observed and computed wavefields, a data-fitting principle that later became the foundation of full-waveform inversion. The theoretical basis for computing synthetic seismograms in my 1978 study drew on the modelling framework developed by K. D. Klem-Musatov (1971), whose formulations provided a practical and widely used foundation for seismic diffraction modelling. My thinking about inverse problems was strongly influenced by A. S. Alekseev (1967), who emphasised the role of objective functions and systematic parameter search in geophysical inference. For the optimisation step, I relied on standard nonlinear programming techniques, for which Himmelblau (1972) served as a general methodological reference.
Nearly a decade later, in 1987, when I became a postdoctoral researcher with Albert Tarantola, I finally realised how closely my early ideas paralleled the concepts he and Patrick Lailly were developing for full-waveform inversion. Looking back, I now see that the approach I developed in 1978 anticipated several ideas that would later become central to modern FWI. Modern FWI, introduced conceptually by Lailly (1983) and formalised mathematically by Tarantola (1984), uses numerical solutions of the wave equation and adjoint-state gradients to update high-dimensional subsurface models. Although the computational tools available to me in the late 1970s were far too limited to support such methods, the essential principles of waveform-based inversion, namely, misfit minimisation, sensitivity to small waveform differences, and the use of forward modelling as the core of interpretation were already present in that early work.
Figure 1 reproduces the cover of my original 1978 paper as it appeared in Geologiya i Geofizika, a reminder of the scientific environment in which these early ideas were first published.
The purpose of this historical note is threefold. First, I place the 1978 study in its historical and scientific context, highlighting the challenges and motivations that shaped its development. Second, I provide a modern summary of the original method and its key results. Third, I compare the 1978 approach with contemporary adjoint-state FWI at a conceptual level, identifying both the limitations of the early algorithm and the conceptual breakthroughs that link it to modern inversion theory. By doing so, I aim to show that the 1978 study represents not only a technical contribution to fault parameter estimation but also an early step in the evolution of waveform-based seismic inversion.
Summary of the 1978 Paper
The 1978 paper proposes an optimisation-based method for estimating the key parameters of a low-amplitude fault using seismic reflection data. The author formulates the problem explicitly as an inverse task in which the optimal model is obtained by minimising the misfit between observed and computed seismic wavefields. For each candidate set of geometric parameters, a synthetic seismogram is generated and compared trace-by-trace with the observed data, and the model that produces the smallest discrepancy is selected. This formulation, iteratively adjusting model parameters to reduce waveform misfit, is conceptually identical to the core principle that later became central to full-waveform inversion (FWI), even though the 1978 implementation operates in a low-dimensional parameter space and uses simplified forward modelling.
A simplified geometric model of the fault is introduced (shown here as Figure 2), defined by depths to the fault edges, the amplitude of the displacement, distances along the profile to the projection of the fault edges, and the dip angles of the fault wings.
Synthetic seismograms are computed for various combinations of these parameters (Figure 3), and the difference between theoretical and observed traces is quantified using a target (misfit) function. The optimisation procedure seeks the parameter set that minimises this misfit.
The structure of the misfit function is illustrated in Figure 4, where contour plots reveal the presence of multiple minima, elongated valleys, and parameter tradeoffs. These features reflect the inherent nonuniqueness of the inversion and the sensitivity of the waveform to subtle geometric variations.
The study demonstrates that even small variations in fault geometry produce measurable changes in the seismic response. Through numerical experiments, the author shows that the method can reliably recover both the position and amplitude of low-amplitude faults, even when the displacement is significantly smaller than the dominant wavelength.
Relation of the 1978 optimisation approach to modern Full-Waveform Inversion
Although developed decades before the emergence of full-waveform inversion, the 1978 optimisation method anticipates several core principles that later became foundational in FWI. The comparison presented here is conceptual rather than numerical; the goal is not to reproduce FWI results but to highlight the intellectual continuity between early model-based interpretation and modern inversion theory.
The core principle of the 1978 method, namely minimising the misfit between observed and synthetic wavefields, anticipates the formulation later established in FWI. In particular, Lailly (1983) demonstrated that the gradient of the waveform misfit can be expressed as the reverse‑time migration of the data residuals, while Tarantola (1984) provided the general adjoint‑state framework that underpins modern FWI algorithms. Although the 1978 study operates in a low‑dimensional parameter space and uses simplified forward modelling, it embodies the same optimisation philosophy that these later works formalised.
The 1978 study embodies several ideas that resonate strongly with contemporary FWI:
- Use of a misfit (objective) function: The method defines a target function measuring the discrepancy between observed and theoretical seismograms, analogous to the least-squares misfit used in FWI.
- Sensitivity of waveforms to small structural variations: The study recognises that even low-amplitude faults produce measurable waveform differences, a principle central to FWI’s ability to resolve finescale features.
- Synthetic testing as a validation tool: The use of synthetic seismograms to explore parameter sensitivity and inversion robustness remains a standard practice in FWI.
- Exploration of the inversion landscape: The 1978 analysis examines the structure of the misfit function, anticipating modern concerns about local minima, parameter coupling, and nonlinearity.
- Forward-modelling based interpretation: The study explicitly links interpretation to forward modelling, a conceptual foundation shared with FWI. In this sense, the 1978 work can be viewed as an early precursor to the philosophy and methodology that underpin contemporary waveform inversion.
Methods: comparing the 1978 algorithm with adjoint-state FWI
The 1978 algorithm and modern adjoint-state FWI share a common conceptual foundation: both treat seismic interpretation as an optimisation problem driven by the physics of wave propagation. However, the two approaches differ significantly in model complexity, numerical formulation, and computational strategy.
Unlike modern adjoint‑state FWI, which computes gradients efficiently using the formulations of Lailly (1983) and Tarantola (1984), the 1978 method evaluates the misfit explicitly across a small parameter grid. This difference reflects the computational limitations of the time rather than a conceptual gap.
Parameterisation and model space
The 1978 method operates in a low-dimensionality parameter space, describing the fault using a handful of geometric variables such as depth, displacement amplitude, and wing dip angles. This compact parameterisation allows the inversion to explore the model space explicitly. In contrast, adjoint-state FWI works in a high-dimensional space where each grid cell of the subsurface model represents an independent parameter.
Forward Modelling
Forward modeling in the 1978 study is based on simplified theoretical seismograms derived from the formulations of Klem-Musatov (1971). These models capture the essential waveform perturbations caused by small faults but do not solve the full wave equation. Modern FWI, by contrast, relies on numerical solutions of the acoustic or elastic wave equation using finite-difference, finite-element, or spectral methods.
Gradient Computation
The 1978 algorithm does not compute gradients analytically; instead, it evaluates the misfit for different parameter combinations and identifies the minimum. This approach is feasible only because the parameter space is small. Adjoint-state FWI uses the adjoint method to compute gradients efficiently, enabling optimisation in high-dimensional spaces.
Optimisation Strategy
In implementing the optimisation, I relied on derivative-free methods, since computing analytical gradients of the misfit function was impractical with the tools available to me in 1978. The approach I used was closely related to the Rosenbrock method (as described in Himmelblau, 1972), a nonlinear search technique that adjusts parameters through coordinated steps without requiring precomputed derivatives. This class of methods was well suited to the low-dimensional parameter space of my problem and to the limited computational resources of the time, allowing me to explore the misfit surface efficiently even when its structure contained multiple local minima.
Modern FWI, of course, uses gradient-based optimisation schemes such as steepest descent, conjugate gradients, or quasi-Newton methods. These algorithms enable rapid convergence, provided the initial model is sufficiently accurate to avoid cycle skipping.
Despite these differences, the intellectual lineage is clear: my 1978 study embodies the essential idea of waveform-based inversion, albeit in a simplified and computationally accessible form.
Discussion
When I look back at the 1978 optimisation approach through the lens of modern inversion theory, its limitations are clear. The simplified forward modelling restricted my ability to capture complex wave phenomena, and the low-dimensional parameterisation limited the range of geological scenarios I could address. The absence of gradient information also constrained the efficiency of the optimisation, making the method impractical for larger or more detailed models.
Yet these limitations should not overshadow the conceptual breakthroughs that emerged from the work. At a time when seismic interpretation was dominated by qualitative analysis and visual pattern recognition, I was proposing a fundamentally different paradigm: interpreting seismic data through quantitative optimisation of physically meaningful parameters.
Many of the issues I explored qualitatively in 1978, such as the structure of the misfit surface, parameter tradeoffs, and the sensitivity of waveforms to small perturbations were later formalised in the adjoint-state framework of Tarantola (1984) and the gradient interpretation of Lailly (1983).
The sensitivity of the synthetic seismograms to subtle structural variations and the structure of the misfit function foreshadowed modern concerns about nonlinearity, cycle skipping, and inversion stability. These insights remain relevant today, particularly in the context of FWI, which leverages small waveform differences to resolve finescale features.
In this sense, my 1978 work represents an early step toward the development of full-waveform inversion, even though the computational tools and theoretical frameworks of the time were not yet capable of supporting largescale implementations.
Conclusion
The 1978 optimisation method I developed for estimating low-amplitude fault parameters occupies a unique place in the evolution of seismic inversion. Grounded in the modelling of Klem-Musatov (1971) and the inversion philosophy of Alekseev (1967), and implemented using standard nonlinear programming techniques described by Himmelblau (1972), the study introduced several principles that later became central to quantitative seismic imaging.
Modern FWI extends these principles dramatically, leveraging numerical wave propagation, adjoint-state gradients, and high-performance computing to recover detailed subsurface models. Yet the conceptual bridge between the two approaches is unmistakable. My 1978 study can be viewed as an early precursor to waveform inversion, demonstrating that even simple optimisation techniques can extract meaningful geological parameters from seismic data.
By situating the original work within the broader historical trajectory of inversion research, I hope to show its role not only as a technical contribution but also as a conceptual milestone that helped to pave the way for the development of modern FWI.
References
- Alekseev, A.S. [1967]. Inverse dynamic problems of seismology. In: Some Methods and Algorithms for Interpretation of Geophysical Data [in Russian], Nauka, Moscow, 9-84.
- Himmelblau, D.M. [1972]. Applied Nonlinear Programming. McGrawHill.
- Klem-Musatov, K.D. [1971]. Theoretical Foundations of Seismic Methods. Nedra. (in Russian). Engish transation: Theory of Seismic Diffractions. By Kamill Klem-Musatov. SEG, 1994.
- Lailly, P. [1983]. The seismic inverse problem as a sequence of beforestack migrations. Conference on Inverse Scattering: Theory and Application, SIAM, 206-220.
- Landa, E.I. [1978]. On the estimation of smallthrow fault parameters by optimization. Geology and Geophysics, 7, 79-86. (in Russian)
- Landa, E.I. [1978]. Estimation of the parameters of a fault with a small throw by the optimization method. Geology and Geophysics, 7, 80-89, (English translation by Allerton Press Inc.).
- Tarantola, A. [1984]. Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49(8), 1259-1266.
