Inversion of detrital zircon data to constrain spatially varying erosion rates

2020 ◽  
Author(s):  
Fien De Doncker ◽  
Frédéric Herman ◽  
Matthew Fox
Keyword(s):  
Least Squares ◽  
Detrital Zircon ◽  
Erosion Rate ◽  
Least Squares Method ◽  
A Priori ◽  
Age Distribution ◽  
Inversion Method ◽  
Provenance Analysis ◽  
Erosion Rates ◽  
Geological Unit

<p>Landscapes evolve through surface processes that are often transient in space and time. To understand the underlying geomorphic processes, one must assess how erosion rates vary spatially. This can be done using provenance analysis. Here, we introduce a formal inversion method to derive erosion patterns using detrital zircon age data as fingerprints. Zircons are omnipresent in Earth’s crust and contain information about the time since (re)crystallization in their U/Th-Pb ratio. For each geological unit having undergone a specific tectonic or magmatic history, one can find a unique age-frequency signature. Hence, erosion and sedimentation of grains originating from diverse source areas lead to a mix of the varying age-frequency signatures in sediments found at the outlet of a catchment. Considering that the age signal is not altered during erosion-transportation-deposition events, and given that recent technological advances enable precise dating of large amounts of grains, U/Th-Pb zircon ages provide an appropriate fingerprinting tool. Our inversion approach relies on the least-squares method with a priori information and model covariance to deal with non-uniqueness. We show with synthetic and natural examples that we are able to retrieve erosion rate patterns of a catchment when the age distribution for each geological unit is well known. Furthermore, relying on the nested form of catchments and their subcatchments, we demonstrate that adding samples taken at the outlet of subcatchments improves the estimation of erosion rate patterns. We conclude that the least squares inverse model applied on detrital zircon data has great potential for investigating erosion rates.</p>

A NONLINEAR LEAST‐SQUARES METHOD FOR SEISMIC REFRACTION MAPPING—PART I: ALGORITHM AND PROCEDURE

Geophysics ◽  
1972 ◽  
Vol 37 (2) ◽  
pp. 260-272 ◽  
Author(s):  
Leonidas C. Ocola
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Seismic Refraction ◽  
Vertical Plane ◽  
Nonlinear Least Squares ◽  
Inversion Method ◽  
Almost Everywhere ◽  
Isotropic Media ◽  
Time Term ◽  
Nonlinear Least Squares Method

An iterative inversion method (Reframap) based on the kinematic properties of critically refracted waves is developed. The method is based on ray tracing and assumes homogeneous and isotropic media and ray paths confined to a vertical plane through each source‐detector pair. Unlike the earlier Profile or Time‐Term Methods, no restrictions are imposed on interface topography except that it be continuous almost everywhere (in the mathematical sense). As in the preexisting methods, more observations than unknowns are assumed. The algorithm and procedure, on which the Reframap Method is based, generate apparent dips for each source detector pair at the noncritical interfaces from the slope of a least‐squares line approximation to the interface functional in the neighborhood of each refraction point. In turn, the dip and path along the critical refractor is, at every iteration, pairwise approximated by a line through the critical refracting points. The incidence angles are computed recursively by Snell’s law. The solution of the overdetermined, nonlinear multiple refractor time‐distance system of simultaneous equations is sought by Marquardt’s algorithm for least‐squares estimation of critical refractor velocity and vertical thickness under each element.

scholarly journals Generalization of level-set inversion to an arbitrary number of geological units in a regularized least-squares framework

Geophysics ◽  
2021 ◽  
pp. 1-76
Author(s):  
Jérémie Giraud ◽  
Mark Lindsay ◽  
Mark Jessell
Keyword(s):  
Least Squares ◽  
Level Set ◽  
Arbitrary Number ◽  
Tectonic Setting ◽  
Physical Property ◽  
Inversion Method ◽  
Regularized Least Squares ◽  
Signed Distance ◽  
Geological Unit ◽  
Geological Units

We present an inversion method for the recovery of the geometry of an arbitrary number of geological units using a regularized least-squares framework. The method addresses cases where each geological unit can be modeled using a constant physical property. Each geological unit or group assigned with the same physical property value is modeled using the signed-distance to its interface with other units. We invert for this quantity and recover the location of interfaces between units using the level-set method. We formulate and solve the inverse problem in a least-squares sense by inverting for such signed-distances. The sensitivity matrix to perturbations of the interfaces is obtained using the chain rule and model mapping from the signed-distance is used to recover physical properties. Exploiting the flexibility of the framework we develop allows any number of rocks units to be considered. In addition, it allows the design and use of regularization incorporating prior information to encourage specific features in the inverted model. We apply this general inversion approach to gravity data favoring minimum adjustments of the interfaces between rock units to fit the data. The method is first tested using noisy synthetic data generated for a model comprised of six distinct units and several scenarios are investigated. It is then applied to field data from the Yerrida Basin (Australia) where we investigate the geometry of a prospective greenstone belt. The synthetic example demonstrates the proof-of-concept of the proposed methodology, while the field application provides insights into, and potential re-interpretation of, the tectonic setting of the area.

scholarly journals An Optimum Deconvolution Method

1979 ◽  
Vol 49 ◽  
pp. 287-290
Author(s):  
C.R. Subrahmanya
Keyword(s):  
Least Squares ◽  
Prior Knowledge ◽  
Least Squares Method ◽  
A Priori ◽  
Statistical Treatment ◽  
Stable Solution ◽  
Brightness Distribution ◽  
Deconvolution Method ◽  
Explicit Recognition ◽  
Optimum Solution

An optimum solution to a deconvolution problem has to fulfil three general criteria: (a) an explicit recognition of the smoothing nature of convolution; (b) a statistical treatment of noise, e.g., using the least-squares criterion; and (c) requiring the solution to conform to all our prior knowledge about it. In the usual least-squares method, one minimises a variance of ‘residuals’, or the departures of the observed data from the values expected according to the recovered solution. However, this condition does not lead to a stable solution in the case of deconvolution, since the only stable solutions are those conforming to a criterion of ‘regularisation’ or smoothness (see, e.g., Tikhonov and Arsenin 1977). In our method, the stability is achieved by minimising the variance of the second-differences of the solution simultaneously with the fulfilment of the least-squares criterion. Such a procedure was first used by Phillips(1962). However, the solution thus obtained is still unsatisfactory since it usually does not conform to oura prioriinformation. When we seek the brightness distribution of an object, the most frequent violation of our prior knowledge is that of positiveness. This motivated us to develop an Optimum Deconvolution Method (ODM) which constrains the solution to satisfy prior knowledge while retaining the features of least-squares and smoothness criteria.

scholarly journals Use of Detrital-zircon U–Pb age-distribution Comparison to Infer Source Location

2021 ◽  
Author(s):  
Jialin Wang ◽  
Chaodong Wu ◽  
Yue Jiao ◽  
Bo Yuan
Keyword(s):  
Detrital Zircon ◽  
Age Distribution ◽  
Junggar Basin ◽  
Source Area ◽  
Source Location ◽  
Upper Triassic ◽  
Late Triassic ◽  
Provenance Analysis ◽  
Source Characteristics ◽  
Field Evidence

Abstract Provenance analysis for volcanism without field evidence remains a major challenge. Detrital zircon grains from 13 samples of the Middle–Upper Triassic Xiaoquangou Group in the Southern Junggar Basin (SJB) were analyzed using U–Pb geochronology to constrain the location and characteristics of Triassic volcanism in the area as well as to understand its tectonic implications. A comparison of the distribution of detrital zircon U–Pb ages reveals Triassic zircon ages predominate in northern Bogda Mountains, with subordinate contributions also in southern Bogda Mountains, and no or minimal input in North Tianshan piedmont. The geochronology data combined with the euhedral and angular zircon grains suggest that the Triassic zircons probably originate from Bogda Mountains. A comparative provenance analysis reveals varied sources for Xiaoquangou Group in the SJB, with sediments of the Bogda Mountains area derived mainly from North Tianshan, Central Tianshan, and Bogda Mountains. The supply of sediments from Bogda Mountains started in the Late Triassic, and is indicative of the initial uplift of Bogda Mountains. This study proves the effectiveness of the comparison of detrital zircon U–Pb age distributions for inferring source characteristics and is applicable in similar situations, particularly when the source area is poorly preserved.

A least-squares method for estimating the correlated error of GRACE models

2020 ◽  
Vol 221 (3) ◽  
pp. 1736-1749
Author(s):  
John W Crowley ◽  
Jianliang Huang
Keyword(s):  
Least Squares ◽  
Spherical Harmonic ◽  
Climate Model ◽  
Least Squares Method ◽  
A Priori ◽  
Temporal Trends ◽  
Standard Form ◽  
New Method ◽  
Correlated Errors ◽  
Global Trends

SUMMARY A new least-squares method is developed for estimating and removing the correlated errors (stripes) from the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) mission data. This method is based on a joint parametric model of the correlated errors and temporal trends in the spherical harmonic coefficients of GRACE models. Three sets of simulation data are created from the Global Land Data Assimilation System (GLDAS), the Regional Atmospheric Climate Model 2.3 (RACMO2.3) and GRACE models and used to test it. The results show that the new method improves the decorrelation method by Swenson & Wahr significantly. Its application to the release 5 (RL05) and new release 6 (RL06) spherical harmonic solutions from the Center for Space Research (CSR) at The University of Texas at Austin demonstrates its effectiveness and provides a relative assessment of the two releases. A comparison to the Swenson & Wahr and Kusche et al. methods highlights the deficiencies in past destriping methods and shows how the inclusion and decoupling of temporal trends helps to overcome them. A comparison to the CSR mascon and JPL mascon solutions demonstrates that the new method yields global trends that have greater amplitude than those produced by the CSR RL05 mascon solution and are of comparable quality to the JPL RL06 mascon solution. Furthermore, these results are obtained without the need for a priori information, scale factors or complex regularization methods and the solutions remain in the standard form of spherical harmonics rather than discrete mascons. The latter could introduce additional discretization error when converting to the spherical harmonic model, upon which many post-processing methods and applications are built.

A Study on the Galerkin Least-Squares Method for the Oldroyd-B Model

2018 ◽  
Vol 18 (2) ◽  
pp. 181-198
Author(s):  
Tsu-Fen Chen ◽  
Hyesuk Lee ◽  
Chia-Chen Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimate ◽  
Least Squares ◽  
Viscoelastic Fluid ◽  
Least Squares Method ◽  
A Priori ◽  
Fluid Flows ◽  
Planar Channel ◽  
A Priori Error Estimate

AbstractWe consider a reduced Galerkin least-squares finite element method for the Oldroyd-B model of viscoelastic fluid flows. Model problems considered are the flow past a planar channel and a 4-to-1 contraction problems. An a priori error estimate for the reduced Galerkin least-squares method is derived and numerical results supporting the estimate are presented.

scholarly journals Extracting information on the spatial variability in erosion rate stored in detrital cooling age distributions in river sands

2018 ◽  
Vol 6 (1) ◽  
pp. 257-270 ◽  
Author(s):  
Jean Braun ◽  
Lorenzo Gemignani ◽  
Peter van der Beek
Keyword(s):  
Steady State ◽  
Spatial Variability ◽  
Erosion Rate ◽  
A Priori ◽  
River System ◽  
A Priori Knowledge ◽  
Brahmaputra River ◽  
Erosion Rates ◽  
Priori Knowledge

Abstract. One of the main purposes of detrital thermochronology is to provide constraints on the regional-scale exhumation rate and its spatial variability in actively eroding mountain ranges. Procedures that use cooling age distributions coupled with hypsometry and thermal models have been developed in order to extract quantitative estimates of erosion rate and its spatial distribution, assuming steady state between tectonic uplift and erosion. This hypothesis precludes the use of these procedures to assess the likely transient response of mountain belts to changes in tectonic or climatic forcing. Other methods are based on an a priori knowledge of the in situ distribution of ages to interpret the detrital age distributions. In this paper, we describe a simple method that, using the observed detrital mineral age distributions collected along a river, allows us to extract information about the relative distribution of erosion rates in an eroding catchment without relying on a steady-state assumption, the value of thermal parameters or an a priori knowledge of in situ age distributions. The model is based on a relatively low number of parameters describing lithological variability among the various sub-catchments and their sizes and only uses the raw ages. The method we propose is tested against synthetic age distributions to demonstrate its accuracy and the optimum conditions for it use. In order to illustrate the method, we invert age distributions collected along the main trunk of the Tsangpo–Siang–Brahmaputra river system in the eastern Himalaya. From the inversion of the cooling age distributions we predict present-day erosion rates of the catchments along the Tsangpo–Siang–Brahmaputra river system, as well as some of its tributaries. We show that detrital age distributions contain dual information about present-day erosion rate, i.e., from the predicted distribution of surface ages within each catchment and from the relative contribution of any given catchment to the river distribution. The method additionally allows comparing modern erosion rates to long-term exhumation rates. We provide a simple implementation of the method in Python code within a Jupyter Notebook that includes the data used in this paper for illustration purposes.

scholarly journals Implications for sedimentary transport processes in southwestern Africa: a combined zircon morphology and age study including extensive geochronology databases

2021 ◽  
Author(s):  
Andreas Gärtner ◽  
Mandy Hofmann ◽  
Johannes Zieger ◽  
Anja Sagawe ◽  
Rita Krause ◽  
...  
Keyword(s):  
Detrital Zircon ◽  
Transport Processes ◽  
Provenance Analysis ◽  
River Sand ◽  
Zircon Age ◽  
Modified Model ◽  
Additional Tool ◽  
Erosion Rates ◽  
Zircon Morphology ◽  
Age Studies

AbstractExtensive morphological and age studies on more than 4600 detrital zircon grains recovered from modern sands of Namibia reveal complex mechanisms of sediment transport. These data are further supplemented by a zircon age database containing more than 100,000 single grain analyses from the entire southern Africa and allow for hypothesising of a large Southern Namibian Sediment Vortex located between the Damara Orogen and the Orange River in southern Namibia. The results of this study also allow assuming a modified model of the Orange River sand highway, whose origin is likely located further south than previously expected. Moreover, studied samples from other parts of Namibia give first insights into sediment movements towards the interior of the continent and highlight the potential impact of very little spatial variations of erosion rates. Finally, this study points out the huge potential of detrital zircon morphology and large geo-databases as an easy-to-use additional tool for provenance analysis.

Recent Improvements of the Algorithm of Mode Isolation

2003 ◽  
Author(s):  
Jerry H. Ginsberg ◽  
Matthew S. Allen
Keyword(s):  
Frequency Response ◽  
Least Squares ◽  
Natural Frequencies ◽  
Least Squares Method ◽  
A Priori ◽  
Single Mode ◽  
Computational Effort ◽  
Complex Frequency ◽  
Linear Least Squares ◽  
Mode Isolation

The Algorithm of Mode Isolation (AMI) identifies the natural frequencies, modal damping ratios, and mode vectors of a system by proceesing complex frequency response data. It uses an iterative procedure based on the fact that a general frequency response function is a superposition of modal contributions. The iterations focus successively on a singel mode. The mode that is in focus is isolated by subtracting the other modal contributions using prior estimates of their modal properties. This process leads to a self-contained identification of the number of modes that participate in any frequency band, whereas other techniques require a priori guesses. This paper describes modifications intended to improve AMI’s accuracy and reduce its computational effort. These involve the use of a new linear least squares method for identifying the natural frequency and dmaping ratio of a single mode, a linear least squares global fit of the data in order to identify mode vectors. Results are presented for a model of a cantilever beam with suspended spring-mass-dashpot system. This system was used by Drexel, Ginsberg, and Zaki [Journal of Vibration and Acoustics, 2003 (forthcoming)] to assess the prior version’s ability to identify weakly excited modes and modes having close natural frequencies in the presence of high noise levels. Application of the modeified version of AMI to the same system is shown to lead to significantly more accurate damping ratios are mode vectors, with equally good natural frequencies.

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