Jose Pujol

The solution of nonlinear inverse problems and the Levenberg-Marquardt method

Although the Levenberg-Marquardt damped least-squares method is an extremely powerful tool for the iterative solution of nonlinear problems, its theoretical basis has not been described adequately in the literature. This is unfortunate, because Levenberg and Marquardt approached the solution of nonlinear problems in different ways and presented results that go far beyond the simple equation that characterizes the method. The idea of damping the solution was introduced by Levenberg, who also showed that it is possible to do that while at the same time reducing the value of a function that must be minimized iteratively. This result is not obvious, although it is taken for granted. Moreover, Levenberg derived a solution more general than the one currently used. Marquardt started with the current equation and showed that it interpolates between the ordinary least-squares-method and the steepest-descent method. In this tutorial, the two papers are combined into a unified presentation, which will help the reader gain a better understanding of what happens when solving nonlinear problems. Because the damped least-squares and steepest-descent methods are intimately related, the latter is also discussed, in particular in its relation to the gradient. When the inversion parameters have the same dimensions (and units), the direction of steepest descent is equal to the direction of minus the gradient. In other cases, it is necessary to introduce a metric (i.e., a definition of distance) in the parameter space to establish a relation between the two directions. Although neither Levenberg nor Marquardt discussed these matters, their results imply the introduction of a metric. Some of the concepts presented here are illustrated with the inversion of synthetic gravity data corresponding to a buried sphere of unknown radius and depth. Finally, the work done by early researchers that rediscovered the damped least-squares method is put into a historical context.

Basement seismicity beneath the Andean precordillera thin-skinned thrust belt and implications for crustal and lithospheric behavior

Data from a digitally recording seismic network in San Juan, Argentina, provide the first images of crustal scale basement faults beneath the Precordillera. This seismicity is near the boundary between the Precordillera (a thin-skinned thrust belt) and the Sierras Pampeanas (a region of thick-skinned basement deformation), two seismically active tectonic provinces of the Andean foreland. The seismicity data support models for this region in which crustal thickening, rather than magmatic addition or thermal uplift, plays the dominant mountain building role. The Precordillera seismicity occurs in three segments distributed north to south. The southern segment is an area of diffuse activity extending across the Precordillera and eastward into the Sierras Pampeanas that shows no patterns in map or cross section. The northern and central segments have well-defined dipping planes that define crustal scale faults extending from 5 to 35 km depth. It is clear from the relative fault geometries that the overlying Precordillera is not simply related to the basement activity. The seismicity here may result from reactivation of an ancient suture between the Precordillera and Pampeanas terranes or be occurring in basement of unknown affinity west of the suture. The seismicity provides the first constraints on basement fault geometries, and we present models integrating this information with the surface geology. These basement faults may have been responsible for the 1944 Ms 7.4 earthquake that destroyed the city of San Juan. The imaging of these faults suggests that seismic risk estimates for San Juan made on the basis of surface geologic studies may be too low.

Seismic wave attenuation in volcanic rocks from VSP experiments

Seismic wave attenuation in the Columbia Plateau basalts and Snake River Plain volcanics was analyzed using vertical seismic profiling (VSP) data. The computation of attenuation coefficients is based on fitting a straight line to the logarithm of amplitude ratios computed for fixed values of frequency and variable depth. This approach does not require any assumptions on the dependence of Q on frequency. For the Columbia Plateau basalts, the attenuation coefficients obtained from the field data are smaller than those computed from the synthetic VSP generated using the sonic and density logs, indicating that the observed attenuation is related to scattering effects and is substantially larger than the intrinsic attenuation of basalt. Therefore, it is concluded that only a lower bound for Q can be established, in agreement with recent findings by other authors. The effective attenuation of seismic energy in basalts (about 1.1×10-4dbs/m for the peak frequency) is comparable to the effective attenuation observ...

Real-Time Seismic Warning with a Two-Station Subarray

One of the most basic problems in seismology is earthquake location. In particular, the ability to quickly locate a large, potentially devastating earthquake is of fundamental importance in a real-time warning system where speed is a key factor in determining the level of success of such systems. We have developed a simple method that uses only the two earliest P -wave arrival times in a seismic array. Assuming a simple velocity model, these arrivals are used to construct a hyperbolic curve on which the approximate epicenter of the earthquake is expected to lie. Epicentral location along this hyperbola is further constrained by using the fact that the P waves arriving at the other stations in the array are not first arrivals. When applied to P -wave seismic data from the Hector Mine earthquake in California and a smaller event in the central United States, model results show agreement with actual earthquake locations. Although there is an inherent uncertainty in the subarray method of locating large earthquakes, this may be an acceptable trade-off in an early warning system in view of the time (a few to tens of seconds) saved by not waiting for other P arrivals. Whereas the main goal of this report is to present the location method, we also show that the station closest to the Hector Mine earthquake had recorded about 0.3 and 1 mm of ground motion within 2 and 3 sec, respectively, of the arrival of the P waves, thus indicating that a large event had occurred. Manuscript received 18 September 2003.

Three-Dimensional Geometry of the Reelfoot Blind Thrust: Implications for Moment Release and Earthquake Magnitude in the New Madrid Seismic Zone

Mapping of the Reelfoot blind thrust using portable array for numerical data acquisition (PANDA) seismicity suggests that it is a complex fault that changes its geometry along strike. The thrust appears to be bounded to the north by an east-trending strike-slip fault. The southern end of the thrust is defined by seismicity and does not terminate at a known transverse fault. The northern portion of the thrust steepens at shallow levels, forming a listric (concave upward) shape in cross section. The southern segment of the thrust is interpreted to flatten near the top of the Precambrian basement. Although some segmentation of the blind thrust is observed from the mapping of 3-km-wide strips of seismicity oriented perpendicular to the fault, it does not appear to be significant enough to prevent rupture along its entire length. The area of the blind thrust (1301 km 2 ), coupled with several estimates of displacement in the 7 February 1812 event, is used to determine the moment released during the event. The values of M 0 range from 6.8 × 10 26 to 1.4 × 10 27 dyne cm, with preferred values between 6.8 × 10 26 and 8.7 × 10 26 dyne cm. Similar calculations for an earthquake in A.D. 1450 yield a moment release of 1.0 × 10 27 dyne cm. Computed moment magnitude for the 7 February 1812 event ranges from M w 7.2 to 7.4, with preferred values between M w 7.2 and 7.3. The moment magnitude for the A.D. 1450 event is computed as M w 7.3. These values for the magnitude of the 1812 earthquake are lower than previous estimates based on historical records of shaking. However, this does not imply a lowered seismic risk in the New Madrid seismic zone, as site effects in the Mississippi embayment may significantly amplify the ground motion caused by earthquakes of a given magnitude.

Refinement of thrust faulting models for the Central New Madrid seismic zone

Abstract We present the results of the joint relocation of events recorded during 1989–1992 by the PANDA network in the central New Madrid seismic zone. The near-surface material in the study area is a gently-dipping layer of poorly consolidated sediments with low P-wave velocity and high V p / V s (estimated values: 1.8 km s −1 and 3). The sediments are underlain by high-velocity Paleozoic rocks. Under the network the difference in sediment thickness is only 0.6 km, but because of the low velocities the location of the events using layered models is affected by errors. Application of the joint hypocentral determination (JHD) technique to a subset of 580 events shows that the single-event locations may be in error by as much as 1 km in depth, depending on where the events are located. Analysis of synthetic data generated for a realistic 3-D velocity model supports the JHD results. The analysis of synthetic data also suggests that a V p / V s ≤ 2.3 is more appropriate for the post-Paleozoic Mississippi embayment sediments. Based on the JHD locations we present a new interpretation of the seismicity, with two en-echelon SW-dipping thrust faults connected by a west-dipping thrust fault. These faults appear associated with the Reelfoot scarp and its northern extension, the Kentucky bend scarp.

Joint hypocentral location in media with lateral velocity variations and interpretation of the station corrections

Abstract The application of a new implementation of the joint determination of hypocenters, origin times and station correction terms (JHD) technique to data from the Arkansas swarm (USA), the Campi Flegrei (Italy), and the Nazca plate (Argentina) throws new light onto the problem of earthquake location. In all these cases good-quality P- and S-wave arrival times were available. The main results are as follows. (1) Large lateral velocity variations (LLVV) are generally associated with relatively large station corrections. These corrections, in turn, are usually much larger than the average of the station residuals determined when the same set of events is located using a single-event location (SEL) method. (2) The locations determined by the JHD and SEL methods may be substantially different. (3) Since the effect of LLVV on JHD locations is not well established, simple 3-D velocity models are constructed so that the JHD technique applied to synthetic data generated using these models produce station corrections similar to those observed. (4) The analysis of the synthetic data also shows that the locations determined by the JHD method are much closer to the true locations than the locations determined with the SEL method. The use of the weighted average of the SEL station residuals as ‘station corrections’ does not give better located events, although there is a large decrease in the r.m.s. residual computed for each event. Therefore, the use of r.m.s. residuals to estimate the quality of event locations is not always warranted. In view of these results it is possible to conclude that the JHD station corrections can be used to detect lateral velocity variations and to produce locations substantially less affected by these variations than the locations determined by the SEL method. Furthermore, under appropriate conditions the JHD station corrections can also be used in a forward modeling scheme to get a semiquantitative velocity model. Such a model may be the only information that can be obtained from a data set when the station and event distributions are not adequate for 3-D velocity inversion.

Superdeep vertical seismic profiling at the KTB deep drill hole (Germany): Seismic close-up view of a major thrust zone down to 8.5 km depth

The lowermost section of the continental superdeep drill hole German Continental Deep Drilling Program (KTB) ( south Germany) has been investigated for the first time by vertical seismic profiling (VSP). The new VSP samples the still accessible range of 6 - 8.5 km depth. Between 7 and 8.5 km depth, the drill hole intersects a major cataclastic fault zone which can be traced back to the Earths surface where it forms a lineament of regional importance, the Franconian line. To determine the seismic properties of the crust in situ, in particular within and around this deep fault zone, was one of the major goals of the VSP. For the measurements a newly developed high-pressure/high-temperature borehole geophone was used that was capable of withstanding temperatures and pressures up to 260 degreesC and 140 MPa, respectively. The velocity-depth profiles and reflection images resulting from the VSP are of high spatial resolution due to a small geophone spacing of 12.5 m and a broad seismic signal spectrum. Compared to the upper part of the borehole, we found more than 10% decrease of the P wave velocity in the deep, fractured metamorphic rock formations. P wave velocity is similar to 5.5 km/s at 8.5 km depth compared to 6.0 - 6.5 km/s at more shallow levels above 7 km. In addition, seismic anisotropy was observed to increase significantly within the deep fracture zone showing more than 10% shear wave splitting and azimuthal variation of S wave polarization. In order to quantify the effect of fractures on the seismic velocity in situ we compared lithologically identical rock units at shallow and large depths: Combining seismic velocity and structural logs, we could determine the elastic tensors for three gneiss sections. The analysis of these tensors showed that we need fracture porosity in the percent range in order to explain seismic velocity and anisotropy observed within the fault zone. The opening of significant pore space around 8 km depth can only be maintained by differential tectonic stress combined with intense macroscopic fracturing. VSP reflection imaging based on PP and PS converted reflected waves showed that the major fault system at the KTB site is wider and more complex than previously known. The so-called SE1 reflection previously found in two- and three-dimensional surface seismic surveys corresponds to the top of an similar to1 km wide fault system. Its lower portion was not illuminated by surface seismic acquisition geometry. VSP imaging shows that the fault zone comprises two major and a number of smaller SE dipping fault planes and several conjugate fracture planes. The previously recognized upper fault plane is not associated with a strong velocity anomaly but indicates the depth below which the dramatic velocity decrease starts. Regarding the complexly faulted crustal section of the KTB site as a whole, we found that fluctuation spectra of rock composition and seismic velocity show similar patterns. We could verify that a significant amount of P wave energy is continuously converted into shear energy by forward scattering and that multipathing plays an important role in signal formation. The media behaves effectively smoothly only at wavelength larger than 150 m. It was shown by moving source profiling that the media is orthorhombic on are regional scale. The tilt of the symmetry axes of anisotropy varies with depth following the dip of the geological structure.

Polarity Reversal of Active Plate Boundary and Elevated Oceanic Upper Mantle beneath the Collision Suture in Central Eastern Taiwan

The active collision between the Eurasia and Philippine Sea plates in eastern Taiwan has been explored from the recently determined 3D velocity images and relocated hypocenters. A north-northeast-south-southwest-trending high- velocity zone corresponding to the oceanic upper mantle is narrowly defined under- neath the collision suture from Hualien to Taitung. This elevated and hot oceanic upper mantle must have played an important role in the tectonic evolution/mountain- building process of the adjacent continental crust. A northwest-dipping seismic zone can be identified in the northern collision zone extending from the surface to 30 km depth, which can be correlated with the northern Longitudinal Valley Fault (LVF). This zone marks a transitional plate boundary separating the high VP and high VP/ VS oceanic crust to the east and the high VP and VS upper crust and low VP and low VP/VS mid-to-lower continental crust to the west. A significant amount of plate con- vergence along the suture has been accommodated by the high-angle thrusting along the northern LVF. In contrast, a southeast-dipping seismic zone can be identified extending from the surface to 25 km depth near Taitung in the southern collision zone. This zone coincides with a region of high VP and high VP/VS, suggesting that earthquakes occurred within a highly fractured or fluid-rich zone. The reverse polarity of active-plate boundary faults marks two distinguished transition boundaries, one from eastward subduction in southern Taiwan to east-west collision in the southern collision zone corresponding to the early phase of plate collision, and the other from east-west collision to northwest subduction in the northern collision zone corre- sponding to the advanced phase of plate collision. The central collision zone is creep- ing and aseismic, which can be attributed to the high heat flow and geothermal activity during an interseismic period since the 1951 Taitung earthquake.

Joint Event Location — The JHD Technique and Applications to Data from Local Seismic Networks

The joint hypocenter determination (JHD) technique is a relatively simple but efficient way to account for lateral velocity variations not included in the one-dimensional velocity models used to locate seismic events. The basic idea behind the technique is the simultaneous location of a group of events and the determination of a common set of station corrections. Under appropriate conditions the station corrections absorb the unmodeled velocity variations, thus improving the locations of the events. From the experience gained so far it appears that JHD always improves relative locations, while absolute locations may or may not be affected by some amount of error, depending on the nature of the lateral velocity variations. When they are very large, as when sedimentary basins and crystalline rocks are juxtaposed, the JHD locations may be affected by a systematic shift from the true locations. Here we discuss in detail four data sets which show different aspects of the JHD analysis. One is a synthetic data set based on actual data from the flat portion of the Nazca plate recorded in the Andean foreland in Argentina. These data are used to show the effect on location of exact and approximate implementations of the JHD technique. The other data sets come from the Loma Prieta, California, mainshock-aftershock sequence, from the New Madrid, central U.S.A., seismic zone, and from the Campi Flegrei, a volcanic area in Italy. The Loma Prieta and New Madrid data illustrate the improvements in absolute and relative locations that can be achieved when using JHD, while the Campi Flegrei data show the significant effect of complicated velocity variations on earthquake location. Finally, analyses of actual and synthetic data show that the JHD station corrections can be used for a semi-quantitative assessment of the 3-D lateral velocity variations under a local network.