Topography can affect linearization in tomographic inversions

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1797-1803 ◽  
Author(s):  
Sean M. Wiggins ◽  
LeRoy M. Dorman ◽  
Bruce D. Cornuelle
Keyword(s):  
Surface Topography ◽  
Velocity Gradient ◽  
Radius Of Curvature ◽  
Model Data ◽  
Velocity Models ◽  
Gradient Model ◽  
Small Model ◽  
Tomographic Inversions ◽  
Gradient Models ◽  
Constant Gradient

Linearized inverse techniques commonly are used to solve for velocity models from traveltime data. The amount that a model may change without producing large, nonlinear changes in the predicted traveltime data is dependent on the surface topography and parameterization. Simple, one‐layer, laterally homogeneous, constant‐gradient models are used to study analytically and empirically the effect of topography and parameterization on the linearity of the model‐data relationship. If, in a weak‐velocity‐gradient model, rays turn beneath a valley with topography similar to the radius of curvature of the raypaths, then large nonlinearities will result from small model perturbations. Hills, conversely, create environments in which the data are more nearly linearly related to models with the same model perturbations.

Overlapping Allografts Provide Superior and More Reliable Surface Topography Matching Than Oblong Allografts: A Computer-Simulated Model Study

2021 ◽  
pp. 036354652110030
Author(s):  
Hailey P. Huddleston ◽  
Atsushi Urita ◽  
William M. Cregar ◽  
Theodore M. Wolfson ◽  
Brian J. Cole ◽  
...  
Keyword(s):  
Articular Cartilage ◽  
Surface Topography ◽  
Osteochondral Allograft ◽  
Cartilage Defects ◽  
Radius Of Curvature ◽  
Cartilage Surface ◽  
3 Dimensional ◽  
Cloud Models ◽  
And Cartilage ◽  
Osteochondral Allografts

Background: Osteochondral allograft transplantation is 1 treatment option for focal articular cartilage defects of the knee. Large irregular defects, which can be treated using an oblong allograft or multiple overlapping allografts, increase the procedure’s technical complexity and may provide suboptimal cartilage and subchondral surface matching between donor grafts and recipient sites. Purpose: To quantify and compare cartilage and subchondral surface topography mismatch and cartilage step-off for oblong and overlapping allografts using a 3-dimensional simulation model. Study Design: Controlled laboratory study. Methods: Human cadaveric medial femoral hemicondyles (n = 12) underwent computed tomography and were segmented into cartilage and bone components using 3-dimensional reconstruction and modeling software. Segments were then exported into point-cloud models. Modeled defect sizes of 17 × 30 mm were created on each recipient hemicondyle. There were 2 types of donor allografts from each condyle utilized: overlapping and oblong. Grafts were virtually harvested and implanted to optimally align with the defect to provide minimal cartilage surface topography mismatch. Least mean squares distances were used to measure cartilage and subchondral surface topography mismatch and cartilage step-off. Results: Cartilage and subchondral topography mismatch for the overlapping allograft group was 0.27 ± 0.02 mm and 0.80 ± 0.19 mm, respectively. In comparison, the oblong allograft group had significantly increased cartilage (0.62 ± 0.43 mm; P < .001) and subchondral (1.49 ± 1.10 mm; P < .001) mismatch. Cartilage step-off was also found to be significantly increased in the oblong group compared with the overlapping group ( P < .001). In addition, overlapping allografts more reliably provided a significantly higher percentage of clinically acceptable (0.5- and 1-mm thresholds) cartilage surface topography matching (overlapping: 100% for both 0.5 and 1 mm; oblong: 90% for 1 mm and 56% for 0.5 mm; P < .001) and cartilage step-off (overlapping: 100% for both 0.5 and 1 mm; oblong: 86% for 1 mm and 12% for 0.5 mm; P < .001). Conclusion: This computer simulation study demonstrated improved topography matching and decreased cartilage step-off with overlapping osteochondral allografts compared with oblong osteochondral allografts when using grafts from donors that were not matched to the recipient condyle by size or radius of curvature. These findings suggest that overlapping allografts may be superior in treating large, irregular osteochondral defects involving the femoral condyles with regard to technique. Clinical Relevance: This study suggests that overlapping allografts may provide superior articular cartilage surface topography matching compared with oblong allografts and do so in a more reliable fashion. Surgeons may consider overlapping allografts over oblong allografts because of the increased ease of topography matching during placement.

scholarly journals Study on the limitations of travel-time inversion applied to sub-basalt imaging

Solid Earth ◽  
2013 ◽  
Vol 4 (2) ◽  
pp. 543-554 ◽  
Author(s):  
I. Flecha ◽  
R. Carbonell ◽  
R. W. Hobbs
Keyword(s):  
Oil Industry ◽  
Velocity Model ◽  
Time Inversion ◽  
Wide Angle ◽  
Seismic Reflection Data ◽  
Additional Information ◽  
Velocity Models ◽  
Reflection Data ◽  
Travel Time Inversion ◽  
Tomographic Inversions

Abstract. The difficulties of seismic imaging beneath high velocity structures are widely recognised. In this setting, theoretical analysis of synthetic wide-angle seismic reflection data indicates that velocity models are not well constrained. A two-dimensional velocity model was built to simulate a simplified structural geometry given by a basaltic wedge placed within a sedimentary sequence. This model reproduces the geological setting in areas of special interest for the oil industry as the Faroe-Shetland Basin. A wide-angle synthetic dataset was calculated on this model using an elastic finite difference scheme. This dataset provided travel times for tomographic inversions. Results show that the original model can not be completely resolved without considering additional information. The resolution of nonlinear inversions lacks a functional mathematical relationship, therefore, statistical approaches are required. Stochastic tests based on Metropolis techniques support the need of additional information to properly resolve sub-basalt structures.

scholarly journals Molecular Cloud Densities from Observations of Carbon Monosulfide

1980 ◽  
Vol 87 ◽  
pp. 117-121
Author(s):  
Richard A. Linke ◽  
Paul F. Goldsmith
Keyword(s):  
Velocity Gradient ◽  
Molecular Cloud ◽  
Equal Size ◽  
Turbulent Model ◽  
Line Intensities ◽  
Gradient Model ◽  
Fractional Abundance ◽  
Carbon Monosulfide ◽  
Good Agreement

AbstractWe have made carefully calibrated measurements of the J=1→0 and J=2→1 transitions of CS in 32 molecular sources in order to obtain density and fractional abundance information from the excitation of this molecule. The antennas used provided beams of nearly equal size. The line intensities, which fall between 1 and 5K, have typically been determined with a (1σ) uncertainty of ∼10%. The two transitions show nearly equal intensities and linewidths which are, respectively, about one tenth and one half of the corresponding quantities for CO. Our observations are seen to be in good agreement with the results of excitation calculations involving either a velocity gradient model or a purely micro-turbulent model since in both cases optical depths are small (0.3 to 3.0). Values obtained for n(H2) and X(CS)/dv/dr from the velocity gradient model fall in the range 2×104 to 2×105 cm−3 and 3×10−11 to 3×10−10 (km s−1pc−1)−1 respectively.

scholarly journals Study on the limitations of traveltime inversion in the presence of extreme velocity anomalies

2013 ◽  
Vol 5 (1) ◽  
pp. 189-226
Author(s):  
I. Flecha ◽  
R. Carbonell ◽  
R. W. Hobbs
Keyword(s):  
Oil Industry ◽  
Velocity Model ◽  
Wide Angle ◽  
Seismic Reflection Data ◽  
Traveltime Inversion ◽  
Additional Information ◽  
Velocity Models ◽  
Reflection Data ◽  
Velocity Anomalies ◽  
Tomographic Inversions

Abstract. The difficulties of seismic imaging beneath high velocity structures are widely recognised. In this setting, theoretical analysis of synthetic wide-angle seismic reflection data indicates that velocity models are not well constrained. A two-dimensional velocity model was built to simulate a simplified structural geometry given by a basaltic wedge placed within a sedimentary sequence. This model reproduces the geological setting in areas of special interest for the oil industry as the Faroe-Shetland Basin. A wide-angle synthetic dataset was calculated on this model using an elastic finite difference scheme. This dataset provided travel times for tomographic inversions. Results show that the original model can not be completely resolved without considering additional information. The resolution of nonlinear inversions lacks a functional mathematical relationship, therefore, statistical approaches are required. Stochastical tests based on Metropolis techniques support the need of additional information to properly resolve subbasalt structures.

Linear inversion of body‐wave data—Part III: Model parameterization

Geophysics ◽  
1984 ◽  
Vol 49 (12) ◽  
pp. 2088-2093 ◽  
Author(s):  
M. Bée ◽  
R. S. Jacobson
Keyword(s):  
Velocity Gradient ◽  
Model Parameters ◽  
Model Resolution ◽  
Data Set ◽  
Model Parameterization ◽  
Number Of Layers ◽  
Gradient Model ◽  
Data Points ◽  
Constrained Model ◽  
Constrained Models

A velocity gradient model parameterized with the tau‐zeta inversion for seismic refraction data is examined with respect to a synthetic traveltime data set. The velocity‐depth model consists of a stack of laterally homogeneous layers, each with a constant velocity gradient. The free model parameters are the velocities of the layer bounds and the number of layers. The best velocity gradient solutions, i.e., with the least deviation from the true model, were obtained from “constrained” models in which the velocities of the layer bounds are the velocities of the observed refracted waves. An arbitrary selection of layer bound velocities was found to be a suboptimal choice of model parameterization for the tau‐zeta inversion. A trade‐off curve between model resolution and solution variance was constructed with the constrained model parameterization from examination of numerous solutions with a diverse number of layers. A constrained model with as many layers as observed data points represents a satisfactory compromise between model resolution and solution variance. Constrained models with more layers than observed data points, however, can increase the resolution of the velocity gradient model. If model resolution is favored over solution variance, a constrained model with many more layers than observed data points is therefore the best model parameterization with the tau‐zeta inversion technique.

Darryl E. Metzger Memorial Session Paper: The Streamwise Development of Go¨rtler Vortices in a Favorable Pressure Gradient

1996 ◽  
Vol 118 (1) ◽  
pp. 162-171 ◽  
Author(s):  
M. V. Finnis ◽  
A. Brown
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Velocity Gradient ◽  
Free Stream ◽  
Free Stream Velocity ◽  
Velocity Variation ◽  
Radius Of Curvature ◽  
Stream Velocity ◽  
Mean Flow ◽  
Favorable Pressure Gradient

Measurements are presented of the streamwise velocity variation within a laminar boundary layer on a concave surface of 4 m radius of curvature for which the free-stream velocity gradient factor (ν/U02)dU0/dx was approximately 1 × 10−6. The stream velocity variation was consistent with the presence of counterrotating vortices resulting from the Go¨rtler instability. The vortices exhibited exponential growth over the streamwise extent of the measurements to a disturbance amplitude of approximately 13 percent of the local free-stream velocity. The vortex growth rates were found to be less than those for a zero velocity gradient factor, indicating that a favorable pressure gradient stabilizes the flow with respect to the Go¨rtler instability. Boundary layer profiles at local upwash and downwash positions are compared with the linear theory for which the mean flow was modeled using the Pohlhausen approximation to the solution of the boundary layer equations. The agreement between the measured and predicted profiles indicates that the linear stability theory can provide a fair approximation to the small amplitude growth of the Go¨rtler instability.

Defined Velocity Gradient Model Flocculator

1968 ◽  
Vol 94 (4) ◽  
pp. 671-682 ◽  
Author(s):  
Francois A. van Duuren
Keyword(s):  
Velocity Gradient ◽  
Gradient Model

Assembly variation analysis of the non-rigid assembly with a deformation gradient model

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Ibrahim Ajani ◽  
Cong Lu
Keyword(s):  
Deformation Gradient ◽  
Transformation Matrix ◽  
Propagation Model ◽  
Assembly Process ◽  
Content Type ◽  
Target Value ◽  
Gradient Model ◽  
Variation Propagation ◽  
Target Values ◽  
Gradient Models

Purpose This paper aims to develop a mathematical method to analyze the assembly variation of the non-rigid assembly, considering the manufacturing variations and the deformation variations of the non-rigid parts during the assembly process. Design/methodology/approach First, this paper proposes a deformation gradient model, which represents the deformation variations during the assembly process by considering the forces and the self-weight of the non-rigid parts. Second, the developed deformation gradient models from the assembly process are integrated into the homogenous transformation matrix to model the deformation variations and manufacturing variations of the deformed non-rigid part. Finally, a mathematical model to analyze the assembly variation propagation is developed to predict the dimensional and geometrical variations due to the manufacturing variations and the deformation variations during the assembly process. Findings Through the case study with a crosshead non-rigid assembly, the results indicate that during the assembly process, the individual deformation values of the non-rigid parts are small. However, the cumulative deformation variations of all the non-rigid parts and the manufacturing variations present a target value (w) of −0.2837 mm as compared to a target value of −0.3995 mm when the assembly is assumed to be rigid. The difference in the target values indicates that the influence of the non-rigid part deformation variations during the assembly process on the mechanical assembly accuracy cannot be ignored. Originality/value In this paper, a deformation gradient model is proposed to obtain the deformation variations of non-rigid parts during the assembly process. The small deformation variation, which is often modeled using a finite-element method in the existing works, is modeled using the proposed deformation gradient model and integrated into the nominal dimensions. Using the deformation gradient models, the non-rigid part deformation variations can be computed and the accumulated deformation variation can be easily obtained. The assembly variation propagation model is developed to predict the accuracy of the non-rigid assembly by integrating the deformation gradient models into the homogeneous transformation matrix.

Reflection of rays in a constant gradient medium: CRP geometry

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 707-712
Author(s):  
Franklyn K. Levin
Keyword(s):  
Velocity Gradient ◽  
Seismic Data ◽  
Constant Velocity ◽  
Finite Range ◽  
Velocity Increase ◽  
Rate Of Increase ◽  
Constant Gradient ◽  
Gradient Medium

In a medium having a velocity that increases linearly with depth (constant gradient), rays are arcs of circles (Slotnick, 1936). A constant gradient medium is not a good approximation to a real subsurface. Not only does velocity increase without limit with depth, but the rate of increase is constant. Nonetheless, over a finite range of depths, a constant gradient medium is closer to reality than a medium having constant velocity down to reflector of interest. For that reason, a number of investigators have considered the changes in processes applied to seismic data when a constant velocity gradient other than zero is assumed.

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