This paper is a survey of the theory and methods of photogrammetric bundle adjustment, aimed at potential implementors in the computer vision community. Bundle adjustment is the problem of refining a visual reconstruction to produce jointly optimal structure and viewing parameter estimates. Topics covered include: the choice of cost function and robustness; numerical optimization including sparse Newton methods, linearly convergent approximations, updating and recursive methods; gauge (datum) invariance; and quality control. The theory is developed for general robust cost functions rather than restricting attention to traditional nonlinear least squares.

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Bundle Adjustment - A Modern Synthesis

Paleoseismological data for the Wasatch and San Andreas fault zones have led to the formulation of the characteristic earthquake model, which postulates that individual faults and fault segments tend to generate essentially same size or characteristic earthquakes having a relatively narrow range of magnitudes near the maximum. Analysis of scarp-derived colluvium in trench exposures across the Wasatch fault provides estimates of the timing and displacement associated with individual surface faulting earthquakes. At all of the sites studied, the displacement per event has been consistently large; measured values range from 1.6 to 2.6 m, and the average is about 2 m. On the basis of variability in the timing of individual events as well as changes in scarp morphology and fault geometry, six major segments are recognized along the Wasatch fault. On the basis of the most likely number of surface faulting events (18) that have occurred on segments of the Wasatch fault zone during the past 8000 years, an average recurrence interval of 400–666 years with a preferred average of 444 years is calculated for the entire zone. Geologic data on the distribution of slip associated with prehistoric earthquakes and slip rates along the south-central segment of the San Andreas fault suggest that the M 8 1857 earthquake is a characteristic earthquake for this segment. Comparisons of earthquake recurrence relationships on both the Wasatch and San Andreas faults based on historical seismicity data and geologic data show that a linear (constant b value) extrapolation of the cumulative recurrence curve from the smaller magnitudes leads to gross underestimates of the frequency of occurrence of the large or characteristic earthquakes. Only by assuming a low b value in the moderate magnitude range can the seismicity data on small earthquakes be reconciled with geologic data on large earthquakes. The characteristic earthquake appears to be a fundamental aspect of the behavior of the Wasatch and San Andreas faults and may apply to many other faults as well.

Fault behavior and characteristic earthquakes: Examples from the Wasatch and San Andreas Fault Zones

Constraints on the seismogenic portion of the subduction thrust zone along the Cascadia margin are provided by the thermal regime. The zone of stick-slip “locked” behavior where earthquakes can nucleate may be limited downdip by a temperature of about 350°C, and the transition stable sliding zone into which coseismic displacement can extend by a temperature of approximately 450°C. The seaward limit of the stick-slip zone may be associated with the dehydration of stable sliding clays at 100 to 150°C and dissipation of high pore pressures in the area of the deformation front. Temperatures on the thrust have been estimated by numerically modelling the thermal regimes along three profiles crossing the margin with constraints provided by surface heat flow and detailed structural information, particularly at southern Vancouver Island. The models that best fit the heat flow data have negligible shear strain heating. The Cascadia subduction margin is unusually hot as a consequence of the very young plate age and the thick insulating sediment section on the incoming plate; the temperature at the top of the oceanic crust at the deformation front is about 250°C. As a result, the modelled zone of stick-slip seismogenic behaviour is restricted to a narrow zone beneath the continental slope and outer shelf, with the transition zone extending to the inner shelf. The seismogenic zone is wider off the Olympic Peninsula compared to off southern Vancouver Island because of the much shallower thrust dip angle and the slightly older incoming plate. The profile off Oregon is found to have intermediate width zones. An important assumption, well justified only off southern Vancouver Island, is that the thrust detachment is located at the top of the downgoing oceanic crust. The same modelling technique shows that more typical subduction zones with older incoming oceanic lithosphere such as central Chile have thermally restricted seismogenic zones that are much wider, commonly extending well beneath the coast. Support for the position of the Cascadia locked zone from the thermal results is provided by a comparison of the horizontal and vertical interseismic deformation predicted by simple dislocation models with the observed rates from tide gauge and geodetic surveys on adjacent coastal regions. The general agreement indicates that any seismic “locked zone” must be located offshore where the subduction thrust fault is less than about 20 km deep and where the contact is between the oceanic crust and the accreted sedimentary wedge, not between the oceanic and continental crusts. The restriction to an offshore zone provides an important limit to the maximum magnitude and to the ground motion and seismic hazard from subduction megathrust earthquakes in southwestern British Columbia, Washington, and Oregon.

Thermal constraints on the zone of major thrust earthquake failure: The Cascadia Subduction Zone

By comparing synthetic particle velocities with the near-source strong motion data we have constructed, by trial and error, a faulting model for the 1979 Imperial Valley earthquake The calculation of the synthetic seismograms takes into account the vertical inhomogeneity of the elastic parameters in the Imperial Valley and the spatial variation of the slip rate parameters on the fault plane The independent slip rate parameters are (1) the strike-slip rate amplitude, (2) the dip-slip rate amplitude, (3) the duration that slip rate is nonzero (the rise time of the slip function) and (4) the rupture time, which determines when the slip rate is initiated Our faulting model has the following principal features: (1) Faulting occurred on the Imperial fault and on the Brawley fault, rupture on the Brawley fault being triggered by rupture on the Imperial fault (2) The Imperial fault is a plane 35 km long and 13 km wide with a strike of 323°, measured clockwise from north, and a dip of 80° NE The Brawley fault is a 10 km long and 8 km wide plane with a strike of 360° and a dip of 90° (3) Faulting on the Imperial fault is primarily right-lateral strike slip with a small component of normal dip slip in the sediments at its northern end The larger strike-slip rates are generally confined between depths of 5 and 13 km with maximum values of about 10 m/s The duration varies on the fault with a maximum of 19 s, which is considerably shorter than the total time for the rupture to take place (4) The rupture velocity on the Imperial fault is highly variable Locally, it exceeds the shear wave velocity and, in one instance, the compressional wave velocity The average rupture velocity, though, is less than the shear wave velocity (5) Although the slip on the Brawley fault contributes only about 4% of the total moment, it greatly affects the ground motion at nearby stations (6) The total seismic moment is 67×1018 N m where the Imperial fault contributes 64×1018 N m and the Brawley fault contributes 27×1017 N m In the process of trying almost 300 faulting models, we found that given the elastic parameters of the medium, the synthetic seismograms were most sensitive to the specification of the rupture time Although the slip rate amplitudes are linearly related to the data, the rupture time and the duration are not The parameterization of the nonlinear variables has a strong effect on the generation of synthetic seismograms from a finite fault

A faulting model for the 1979 Imperial Valley earthquake

Statistical analysis of an earthquake-induced landslide distribution — the 1989 Loma Prieta, California event

An algorithm for improving the profile of a sparse symmetric matrix is introduced. Tests on normal equation matrices encountered in adjustments of geodetic networks by least squares demonstrate that the algorithm produces significantly lower profiles than the widely used reverse Cuthill-McKee algorithm.

Reducing the profile of sparse symmetric matrices

The National Geodetic Survey, an office within the National Oceanic and Atmospheric Administration, recently released version 3.1 of the Horizontal Time-Dependent Positioning (HTDP) utility for transforming coordinates across time and between spatial reference frames. HTDP 3.1 introduces improved crustal velocity models for both the contiguous United States and Alaska. The new HTDP version also introduces a model for estimating displacements associated with the magnitude 7.2 El Mayor---Cucapah earthquake of April 4, 2010. In addition, HTDP 3.1 enables its users to transform coordinates between the newly adopted International Terrestrial Reference Frame of 2008 (ITRF2008) and IGS08 reference frames and other popular reference frames, including current realizations of NAD 83 and WGS84. A more convenient format to enter a list of coordinates to be transformed has been added. Users can now also enter dates in the decimal year format as well as the month-day-year format. The new HTDP utility, explanatory material and instructions are available at http://www.ngs.noaa.gov/TOOLS/Htdp/Htdp.shtml .

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Introducing HTDP 3.1 to transform coordinates across time and spatial reference frames

OPUS-RS is a rapid static form of the National Geodetic Survey’s On-line Positioning User Service (OPUS). Like OPUS, OPUS-RS accepts a user’s GPS tracking data and uses corresponding data from the U.S. Continuously Operating Reference Station (CORS) network to compute the 3-D positional coordinates of the user’s data-collection point called the rover. OPUS-RS uses a new processing engine, called RSGPS, which can generate coordinates with an accuracy of a few centimeters for data sets spanning as little as 15 min of time. OPUS-RS achieves such results by interpolating (or extrapolating) the atmospheric delays, measured at several CORS located within 250 km of the rover, to predict the atmospheric delays experienced at the rover. Consequently, standard errors of computed coordinates depend highly on the local geometry of the CORS network and on the distances between the rover and the local CORS. We introduce a unitless parameter called the interpolative dilution of precision (IDOP) to quantify the local geometry of the CORS network relative to the rover, and we quantify the standard errors of the coordinates, obtained via OPUS-RS, by using functions of the form $$ \sigma ({\text{IDOP}},{\text{RMSD}}) = \sqrt {(\alpha \cdot {\text{IDOP}})^{2} + (\beta \cdot {\text{RMSD}})^{2} } $$here α and β are empirically determined constants, and RMSD is the root-mean-square distance between the rover and the individual CORS involved in the OPUS-RS computations. We found that α = 6.7 ± 0.7 cm and β = 0.15 ± 0.03 ppm in the vertical dimension and α = 1.8 ± 0.2 cm and β = 0.05 ± 0.01 ppm in either the east–west or north–south dimension.

Accuracy Assessment of the National Geodetic Survey's OPUS-RS Utility

We use geodetic data spanning the 1920–1992 interval to estimate the horizontal velocity field near the big bend segment of California's San Andreas fault (SAF). More specifically, we estimate a horizontal velocity vector for each node of a two-dimensional grid that has a 15-min-by-15-min mesh and that extends between latitudes 34.0°N and 36.0°N and longitudes 117.5°W and 120.5°W. For this estimation process, we apply bilinear interpolation to transfer crustal deformation information from geodetic sites to the grid nodes. The data include over a half century of triangulation measurements, over two decades of repeated electronic distance measurements, a decade of repeated very long baseline interferometry measurements, and several years of Global Positioning System measurements. Magnitudes for our estimated velocity vectors have formal standard errors ranging from 0.7 to 6.8 mm/yr. Our derived velocity field shows that (1) relative motion associated with the SAF exceeds 30 mm/yr and is distributed on the Earth's surface across a band (>100 km wide) that is roughly centered on this fault; (2) when velocities are expressed relative to a fixed North America plate, the motion within our primary study region has a mean orientation of N44°W ± 2° and the surface trace of the SAF is congruent in shape to nearby contours of constant speed yet this trace is oriented between 5° and 10° counterclockwise relative to these contours; and (3) large strain rates (shear rates > 150 nrad/yr and/or areal dilatation rates < −150 nstr/yr) exist near the Garlock fault, near the White Wolf fault, and in the Ventura basin.

Richard A. Snay

Papers

Reducing the profile of sparse symmetric matrices

Introducing HTDP 3.1 to transform coordinates across time and spatial reference frames

Accuracy Assessment of the National Geodetic Survey's OPUS-RS Utility

Crustal velocity field near the big bend of California's San Andreas Fault

DYNAP: software for estimating crustal deformation from geodetic data