Chelungpu Fault Research Articles

Time‐Dependent Deformation in the Damage Zone of the Chelungpu Fault System and Potential Stress Relaxation

AbstractHigh fracture density in fault damage zones not only reduces the elastic stiffness of rocks but may also promote time‐dependent bulk deformation through the sliding of fracture and thus alter the stress in fault zones. On comparing the damage zones of the three faults in the Chelungpu fault system encountered in the Taiwan Chelungpu fault Drilling Project (TCDP), the youngest damage zone showed pronounced sonic velocity reduction even though fracture density is the same for all three fault zones, consistent with the shorter time for velocity recovery in the youngest fault. Caliper log data showed a time‐dependent enlargement of the borehole wall at the damage zone. These damage zones record lower differential stress than the surrounding host rock, which cannot be explained by the reduced elastic stiffness in the damage zone. Stress relaxation caused by time‐dependent bulk deformation in the damage zone may be responsible for the observed low differential stress.

Exploring the Rheology of a Seismogenic Zone by Applying Seismic Variation

Although the study of spatiotemporal variation of a subsurface velocity structure is a challenging task, it can provide a description of the fault geometry as well as important information on the rheological changes caused by fault rupture. Our main objective is to investigate whether rheological changes of faults can be associated with the seismogenic process before a strong earthquake. For this purpose, a 3D tomographic technique is applied to obtain P- and S-wave velocity structures in central Taiwan using travel time data. The results show that temporal variations in the Vs structure in the source area demonstrate significant spatiotemporal variation before and after the Chi-Chi earthquake. We infer that, before the mainshock, Vs began to decrease (and Vp/Vs increased) at the hanging wall of the Chelungpu fault, which may be induced by the increasing density of microcracks and fluid. However, in the vicinity of the Chi-Chi earthquake’s source area, Vs increased (and Vp/Vs decreased), which may be attributed to the closing of cracks or migration of fluid. The different physical characteristics at the junctional zone may easily generate strong earthquakes. Therefore, seismic velocity changes are found to be associated with a subsurface evolution around the source area in Taiwan. Our findings suggest that monitoring the Vp and Vs (or Vp/Vs) structures in high seismic potential zones is an important ongoing task, which may minimize the damage caused by future large earthquakes.

Challenges in the Preservation of Disaster Remains – Example of the Chelungpu Fault Preservation Park

Taiwan is located along the boundary of the Eurasian and the Philippine Sea plates and experiences tens of thousands of earthquakes each year. Based on historical records, Taiwan has had several earthquakes of magnitude greater than 7.0. Notable and deadly quakes occurred in 1906 (Meishan Earthquake), 1935 (Hsinchu-Taichung Earthquake), and 1999 (Chi-Chi Earthquake). Statistically, Taiwan has had a major earthquake every 30–60 years. Therefore, earthquake museums are needed for long-term earthquake education and geoheritage exhibitions. Earthquake museums highlight disaster risks and preparedness information. The purpose of preserving earthquake remains is to educate visitors about Taiwan’s natural disasters and provide a memorable experience that inspires earthquake preparedness. The Chushan trench across the Chelungpu fault is a good example of Chi-Chi Earthquake rupture. This trench has recorded the five most important earthquake events on the Chelungpu fault. Although the Chelungpu Fault Preservation Park (CFPP) has worked to preserve these earthquake remains, they have been threatened due to seepage over the years. The aim of this paper is to analyze trench seepage and explore the development of an anti-seepage model, to provide a reference for the preservation of earthquake remains and museum development worldwide.

3D modeling of the ground deformation along the fault rupture and its impact on engineering structures: Insights from the 1999 Chi-Chi earthquake, Shigang District, Taiwan

The reactivation of the Chelungpu Fault triggered the 1999 Chi-Chi Mw 7.6 earthquake, resulting in substantial ground surface ruptures and severe damage. After the earthquake, the surveys organized by the Central Geology Survey (CGS) of Taiwan and the National Center for Research on Earthquake Engineering (NCREE) immediately investigated a variety of cases related to engineering structures damaged by substantial ground deformation. The investigations gathered valuable in-situ data before human activities disturbed the local setting and altered co-seismic features of interest, providing opportunities for studying the interaction between surface rupture and engineering structures. This study outlines the reconnaissance of damaged engineering structures in Shigang District that cover three typical types of foundations and analyzes the fault geometry and fault slip at the local scale to construct a geomechanical model for numerical simulation. In addition, 3-dimensional (3D) modeling was performed for each case study to investigate the damage mechanisms of the structures subjected to the substantial ground deformations by using PFC3D (Particle Flow Code 3D), which is based on the discrete element method (DEM). The results of the DEM analyses, which have been calibrated with the in-situ data, showed the ability to evaluate the overall ground deformation, the detection of structural movement, and the critical elements of progressive collapse during faulting. This study demonstrates that the 3D DEM simulation is a reliable analysis tool for studying earthquake fault rupture-overburden layer-engineering structure interactions and can produce useful information for hazard assessment with an acceptable computation time. The 3D DEM analysis is therefore suggested to be applied to assess the potential damage scenarios for the engineering structures located in active fault zones.

The Magnetic Fabric of Gouge Mimics the Coseismic Focal Mechanism of the Chi‐Chi Earthquake (1999, Mw 7.6)

AbstractWe present a magnetic fabric investigation of small oriented rock fragments from gouges and wall rocks from the Chelungpu fault (Taiwan), where the Chi‐Chi earthquake occurred (Mw 7.6, 1999). Samples are obtained from surface outcrops and at a depth of ~1 km, including the 3‐mm‐thick Chi‐Chi principal slip zone. Wall rocks at depth show magnetic foliation roughly parallel to bedding, whereas they are oblique at the surface. Within the gouges, the mean shape of magnetic ellipsoid is oblate, and the degree of anisotropy is up to 12%. The magnetic foliation diverges of ~40° from the fault plane, compatible with a simple shear westward. The magnetic lineation is oblique, resulting probably from coseismic and interseismic records. The reconstructed focal mechanism from the magnetic fabric closely matches the focal mechanism solutions for the Chi‐Chi earthquake. Our work demonstrates that magnetic fabric of gouge fragments can bring valuable kinematic information on modern earthquakes.

Generation of sintered fault rock and its implications for earthquake energetics and fault healing

After an earthquake, faults can recover strength through fault healing, but the mechanisms responsible are not well understood. Seismic slip may induce sintering, a bonding process between solid particles in contact under high temperatures without melting, which could produce a fault rock with elevated strength and chemical stability. Here we present results from electron microscope analyses that show a typical sintered structure in a black disk-shaped rock from the Chelungpu fault, Taiwan. This structure is experimentally reproducible in simulated fault material, prepared from the local host-rock, by heating at 800–900 °C. Through thermal and kinetic analyses of experimental materials, we show that sintering is an exothermic process which can generate energy to enhance post-slip thermochemical reactions in the fault. We propose that sintering substantially contributes to earthquake energetics and fault healing and that its occurrence can be a useful indicator of past seismic slip.

Researching significant earthquakes in Taiwan using two back-propagation neural network models

This study pertains to the Chi-Chi earthquake of 1999 (a Richter magnitude (ML) of 7.3), the Meishan earthquake of 1906 (a Richter magnitude (ML) of 7.1) and the Hualien earthquakes of 1951 (a Richter magnitude (ML) of 7.3), which were triggered by the Chelungpu, Meishan and Milun faults. Two back-propagation neural networks (BPNNs)—(1) an embedded earthquake Richter magnitude (ML) prediction BPNN model and (2) an active probability BPNN model—are used to predict recurrence times over 500 years. Recurrence times for a 500-year period have been studied previously. This study examines the three earthquakes again and compares the results with those for previous studies. This process does not use any probability model with exceedance probability. The Chelungpu fault and the Tamaopu-Shuangtung fault are shown to more strongly couple. This viewpoint agrees with previous studies, which suggests that the Chi-Chi earthquake was caused by the Chelungpu faults in 1999. Its recurrence time with a Richter magnitude (ML) of more than 7 is 210 years after the Chi-Chi earthquake, and the highest probability is more than 60%. The Meishan earthquake is confirmed to have been caused by the Meishan fault in 1906. There is a high probability of more than 60% of another Meishan earthquake with a Richter magnitude (ML) of more than 7 in 170 years. There is a high probability of more than 60% for the occurrence of an earthquake with a Richter magnitude (ML) of more than 7 in Hualien due to the Milun faults. The results for both BNNN models are more realistic than those of previous studies because only the earthquake catalog is used, so that the cost of study is reduced.

Surface traces and related deformation structures of the southern Sanyi Fault, Taiwan, as deduced from field mapping, electrical-resistivity tomography, and shallow drilling

The southern Sanyi Fault extends across a highly urbanized area in central-west Taiwan. It is within 1–2 km from and generally parallels the northern Chelungpu Fault, which last ruptured in 1999 (Mw = 7.6 Chi-Chi Earthquake) and caused about 2400 fatalities. Several questions thus arise: is the Sanyi Fault likewise active? what is its geometry and extent? and should it be zoned or otherwise designated to mitigate risk through restrictions on development or engineering design? We address these questions by mapping river-exposed outcrops, by interpreting nine, near-surface geophysical surveys (electrical resistivity tomography [ERT]), and by compiling and assessing existing shallow borehole data. Outcrops in the Dajia River show that the Sanyi Fault is mainly a low-angle thrust that juxtaposes poorly lithified sandstone over alluvial gravels. Fault branches are common both in the hanging wall and footwall. To the south, the fault is not marked by a break in topography and its trace has not been clearly mapped within the alluvial plain, in part motivating the current effort. Here the ERT surveys identify the hanging-wall sandstone by a relatively low resistivity of <100 Ωm, whereas the footwall conglomerates and alluvial gravels generally exceed ~300 Ωm and ~ 100 Ωm, respectively. Borehole data likewise constrain the extent and character of the Sanyi Fault, which we identify as extending across the eastern part of Fengyuan, a city with a population now exceeding 160,000. The fault system is clearly imaged in two of five surveyed sites, and we integrate these data with drilling data to map the Sanyi fault surface trace as a ~ 400 m wide zone in the city of Fengyuan. Compiling previously documented paleoseismic data at one site, the Sanyi Fault system likely slipped in the thousand-year range throughout the Holocene. The Sanyi Fault is therefore ‘active’ and warrants zoning or other forms of mitigation to minimize potential, seismically induced loss of life and property.

Pixel Offset Fusion of SAR and Optical Images for 3-D Coseismic Surface Deformation

This letter presents a 3-D fusion method combining with the pixel offset results of synthetic aperture radar (SAR) and optical images for measuring 3-D coseismic deformation, which suffered the limitations of huge displacement gradient, low coherence, and large temporal interval. This method takes the advantages of the pixel offset results of the two different types of satellites. We use the hanging wall deformation of the 1999 Chi-Chi earthquake as an example and generate two sets of 3-D coseismic displacements from the fusion method and radar images only. We compare the two sets of results to GPS and field survey data, and the 3-D fusion result shows more consistent 3-D displacements than one derived from radar images with decreased root mean square (rms) values of 20%-50%. The results of this letter provide a detailed information of displacement distribution in the hanging wall of the Chelungpu fault (CLPF) during the Chi-Chi earthquake. Along the fault, the vertical displacement increases from south to north gradually and the vertical pattern follows the topography. The area where uplift starts to be higher than the average of vertical displacements is consistent with the turning point of CLPF. The 3-D fusion method appears not only the potentials to describe the uplift behavior of active fault, but also the abilities to constrain the subsurface fault geometry.

Strong‐Motion Directionality and Evidence of Rupture Directivity Effects during the Chi‐Chi Mw 7.6 Earthquake

AbstractThis article investigates the spatial distribution, predominant direction, and variations in the intensity measures (IMs) with orientation for classified pulse‐like and nonpulse motions during Chi‐Chi Mw 7.6 earthquake. The results show evidence of high polarization for long‐period spectral accelerations at relatively large source‐to‐site distances (50–100 km) north of the Chelungpu fault. The polarization of long‐period motions shows a clear correlation with the directivity parameters’ isochrone directivity predictor and ξ, indicating a connection between directionality and rupture directivity. The variation in strong‐motion directionality with the period is also studied. The discrepancy in directionality caused by strong directivity increases with the period from 1 to 10 s, which confirms a clear correlation of period‐dependent directionality with directivity effects. This study finds stronger directionality of pulse‐like motions than nonpulse motions for long periods over 3 s with higher maximum‐to‐median and maximum‐to‐minimum IM ratios. For periods over 3 s, the maximum‐to‐median ratios of pulse‐like motions are higher than the mean prediction of the Shahi and Baker (2014a) model, whereas those of nonpulse motions are lower than the prediction. However, this study does not find simple and clear results for the directions of the maximum component at different periods for pulse‐like and nonpulse motions. Despite clear differences between the unidirectional fling‐step and bidirectional forward directivity pulses, the effects of fling‐step and forward directivity are actually coupled in the waveforms.

Using the high-temperature phase transition of iron sulfide minerals as an indicator of fault slip temperature

The transformation of pyrite into pyrrhotite above 500 °C was observed in the Chelungpu fault zone, which formed as a result of the 1999 Chi-Chi earthquake in Taiwan. Similarly, pyrite transformation to pyrrhotite at approximately 640 °C was observed during the Tohoku earthquake in Japan. In this study, we investigated the high-temperature phase-transition of iron sulfide minerals (greigite) under anaerobic conditions. We simulated mineral phase transformations during fault movement with the aim of determining the temperature of fault slip. The techniques used in this study included thermogravimetry and differential thermal analysis (TG/DTA) and in situ X-ray diffraction (XRD). We found diversification between 520 °C and 630 °C in the TG/DTA curves that signifies the transformation of pyrite into pyrrhotite. Furthermore, the in situ XRD results confirmed the sequence in which greigite underwent phase transitions to gradually transform into pyrite and pyrrhotite at approximately 320 °C. Greigite completely changed into pyrite and pyrrhotite at 450 °C. Finally, pyrite was completely transformed into pyrrhotite at 580 °C. Our results reveal the temperature and sequence in which the phase transitions of greigite occur, and indicate that this may be used to constrain the temperature of fault-slip. This conclusion is supported by field observations made following the Tohoku and Chi-Chi earthquakes.

Fault Dynamics of the 1999 Chi-Chi earthquake: clues from nanometric geochemical analysis of fault gouges

The 1999 Chi-Chi, Taiwan earthquake (Mw 7.6) produced surface ruptures for about 90 km along the north-south trending Chelungpu fault, with surface displacements of up to 12 m. Based on the combination of nanoscopic investigation and geochemistry analysis of core samples from a 450 m long inclined borehole drilled through the slip zone, we suggest the dynamical processes that likely occurred in the northern portion of the Chelungpu fault during the faulting. Our analysis revealed that the frictional heating could have reached 1200 °C, which would cause most of the siderite in the fault gouge to evaporate, resulting in a large amount of nano-size siderite grains with a mean diameter 20 nm. These nano grains could have acted as a mechanical lubricant to reduce the dynamic frictional resistance during sliding, giving rise to the large but smooth type of slipping seen in the north.

Detection of a fracture zone using microtremor array measurement

We have conducted microtremor array measurements to estimate shallow S-wave velocity ([Formula: see text]) structures at two sites (the 921 Earthquake Museum of Taiwan and the Taiwan Provincial Consultative Council) located near surface ruptures of the Chelungpu Fault. Ten stations, consisting of three different-aperture triangles and a central station, are adopted for each array deployment. Using the array data, we calculate dispersion curves of Rayleigh waves using the frequency-wavenumber spectrum method and then estimate [Formula: see text] structures by the surface-wave inversion technique. The obtained 2D [Formula: see text] profiles could clearly show compressive and flexural deformation structures with the surface ruptures located at relatively weak (low [Formula: see text]) zones. This indicates compressive buckling as the most likely mechanism for surface rupturing along these low [Formula: see text] zones. Importantly, this study successfully depicts strata disturbances in a fault fracture zone using microtremor array measurements and forward numerical modeling of trishear fault-propagation folds.

The Effects of Zoning Regulations along Fault Zone Areas on Land Development and Property Values after the 921 Chi-Chi Earthquake in Taiwan

Earthquakes are widely recognized as unpredictable and infrequent disasters that result in serious impacts on human settlements. Land use planning is one non-structural measure used to eliminate disaster risk by steering future development away from the existing built environment and enforcing particular structural engineering measures according to the disaster risk. However, arguments have arisen about applying land use planning to earthquake risk areas, as this serves as a type of disaster risk information disclosure that might impact the willingness to develop land or property value. Therefore, this study uses the spatial autocorrelation coefficient to examine the impact of land use planning on both land use and property transactions in the Chelungpu fault zone area (15 m from each side of the fault line) in Taiwan. The overall impacts with and without zoning regulation in the fault zone area are explored. The results demonstrate that parcels that changed to building use in the earlier time period (1995–2008) are located distant from those maintaining the same building use, whereas, later, building use (2008–2014) is located on or nearby the fault zone area. In addition, the most recently constructed buildings are located in or close to the fault zone area and have a relatively higher property price. The legal zoning regulation along the fault zone for building use requires lower height and less intensive building, which might help mitigate the potential impact of future earthquakes.

Anomalous Streamflow and Groundwater-Level Changes Before the 1999 M7.6 Chi–Chi Earthquake in Taiwan: Possible Mechanisms

Streamflow recorded by a stream gauge located 4 km from the epicenter of the 1999 M7.6 Chi–Chi earthquake in central Taiwan showed a large and rapid anomalous increase of 124 m3/s starting 4 days before the earthquake. This increase was followed by a comparable co-seismic drop to below the background level for 8 months. In addition, groundwater-levels recorded at a well 1.5 km east of the seismogenic fault showed an anomalous rise 2 days before the earthquake, and then a unique 4-cm drop beginning 3 h before the earthquake. The anomalous streamflow increase is attributed to gravity-driven groundwater discharge into the creek through the openings of existing fractures in the steep creek banks crossed by the upstream Shueilikun fault zone, as a result of pre-earthquake crustal buckling. The continued tectonic movement and buckling, together with the downward flow of water in the crust, may have triggered the occurrence of some shallow slow-slip events in the Shueilikun and other nearby fault zones. When these events propagate down-dip to decollement, where the faults merges with the seismogenic Chelungpu fault, they may have triggered other slow-slip events propagating toward the asperity at the hypocenter and the Chelungpu fault. These events may then have caused the observed groundwater-level anomaly and helped to trigger the earthquake.

Experimental and numerical investigation of the temperature response to stress changes of rocks

AbstractThe temperature response to stress changes of rocks is key to understanding temperature anomalies in geoscience phenomena such as earthquakes. We developed a new hydrostatic compression system in which the rock specimen center can achieve adiabatic conditions during the first ~10 s following rapid loading or unloading and systematically measured several representative sedimentary, igneous, and metamorphic rocks sampled from two seismogenic zones (the Longmenshan Fault Zone in Sichuan and the Chelungpu Fault Zone (TCDP Hole‐A) in Taiwan) and several quarries worldwide. We built a finite element model of heat conduction to confirm the measured results of temperature response to stress changes of rocks. The results show that (1) the adiabatic pressure derivative of the temperature (β) for most crustal rocks is ~1.5 mK/MPa to 6.2 mK/MPa, (2) the temperature response to stress of sedimentary rocks (~3.5–6.2 mK/MPa) is larger than that of igneous and metamorphic rocks (~2.5–3.2 mK/MPa), and (3) there is good linear correlation between β (in mK/MPa) and the bulk modulus K (in GPa): β = (−0.068K + 5.69) ± 0.4, R2 = 0.85. This empirical equation will be very useful for estimating the distribution of β in the crust, because K can be calculated when profiles of crustal density (ρ) and elastic wave velocities (Vp, Vs) are obtained from gravity surveys and seismic exploration.

Earthquake Probability Assessment for the Active Faults in Central Taiwan: A Case Study

Frequent high seismic activities occur in Taiwan due to fast plate motions. According to the historical records the most destructive earthquakes in Taiwan were caused mainly by inland active faults. The Central Geological Survey (CGS) of Taiwan has published active fault maps in Taiwan since 1998. There are 33 active faults noted in the 2012 active fault map. After the Chi-Chi earthquake, CGS launched a series of projects to investigate the details to better understand each active fault in Taiwan. This article collected this data to develop active fault parameters and referred to certain experiences from Japan and the United States to establish a methodology for earthquake probability assessment via active faults. We consider the active faults in Central Taiwan as a good example to present the earthquake probability assessment process and results. The appropriate ＂probability model＂ was used to estimate the conditional probability where M ≥ 6.5 and M ≥ 7.0 earthquakes. Our result shows that the highest earthquake probability for M ≥ 6.5 earthquake occurring in 30, 50, and 100 years in Central Taiwan is the Tachia-Changhua fault system. Conversely, the lowest earthquake probability is the Chelungpu fault. The goal of our research is to calculate the earthquake probability of the 33 active faults in Taiwan. The active fault parameters are important information that can be applied in the following seismic hazard analysis and seismic simulation.

Insights from heterogeneous structures of the 1999 Mw 7.6 Chi‐Chi earthquake thrust termination in and near Chushan excavation site, Central Taiwan

AbstractWe describe and analyze the surface and subsurface deformation of the 1999 Chi‐Chi earthquake thrust termination of the Chelungpu fault in and near the Chushan excavation site, central Taiwan. In order to minimize damage to engineering structures within a deformation zone and formulate regulations for earthquake fault zones and fault setbacks, one needs to know the characteristics of the deformation zone and the connection between surface and subsurface deformation. The surface deformation zone of high strain induced by the earthquake faulting ranges from 15 to 70 m in width, and characterized by a 0.5 to 2 m high escarpment is much wider in the hanging wall. Exposures in the trench, 40 m long and 10 m deep, excavated across the earthquake ground rupture, show on the one hand the heterogeneous structure of a steep, monocline‐like fold. On the other hand, strikingly different surface deformation profiles on either side of the 14 m wide trench, i.e., semiparabola‐like and monoclinal, do not reflect in any obvious way the subsurface structure. The analysis of our detailed mapping results along with well logs suggests that the significant width variation of the surface deformation zone is attributed to secondary faults branching from the dominant fault at several tens of meters depth. The surface profile of the escarpment is controlled by the depth of the relevant fault tip, and the subsurface heterogeneous structure formed during at least four earthquakes.

Ultrafine spherical quartz formation during seismic fault slip: Natural and experimental evidence and its implications

In recent works on the determination of pseudotachylyte within the principal slip zone (PSZ) of the Chelungpu fault (Taiwan), we demonstrated that frictional melting occurred at shallow depths during the 1999 Mw 7.6 Chi-Chi earthquake. Thus, the characteristics of melts are of paramount importance to investigate processes controlling dynamic fault mechanics during seismic slips. We conducted rock friction experiments on siltstone recovered from the Taiwan Chelungpu fault Drilling Project (TCDP) at a slip rate of 1.3m/s and a normal stress of 1MPa. Here we not only target to characterize experimental pseudotachylyte and evaluate the associated frictional behavior, but also compare it with natural frictional melts of TCDP. Our results show that (1) initial shear stress drop was related to the generation of low viscosity melt patches, (2) the evolution of shear stress in the postmelting regime was congruent with frictional melt rheology, and (3) the slip strengthening presumably resulted from the extremely low content of water of the frictional melt. In particular, the state-of-art of in situ synchrotron analyses (X-ray diffraction and Transmission X-ray Microscope) determine the presence of ultrafine spherical quartz (USQ) grains (~10nm to 50nm) in the glassy matrices presumably produced at high temperature. Our observations confirm that the USQ grains formed in rock friction experiments do occur in natural faults. We surmise the USQ is the result of frictional melting on siltstone and represents the latest slip zones of the Chelungpu fault, and further infer that the viscous melts may terminate seismic slips at shallow crustal conditions.

Inversion of fault data with low diversity for stress: Principles and applications

Although a minimum of four independent, single-phase fault data are required to solve for a unique reduced stress tensor, we prove in this paper that a smaller number of fault data are sufficient in some instances to solve for part of the reduced stress tensor. One of the principal stress directions is determinable from either two faults with a common null shear direction on the fault planes or three faults with a common intersection in a principal stress plane of the fault planes. This direction is combined with the fault data to determine the possible ranges of other principal stress directions. Determining whether the direction is for the maximum, intermediate or minimum principal stress depends upon constraints provided by slip tendency or more fault data. This approach can also be applied to a set of four or more fault data with low orientation diversity. This new method is finally applied to two different sets of fault data from along the active Chelungpu fault, western Taiwan. The stress orientations determined from the method lie in acceptable ranges for the maximum/minimum principal stresses using other existing and comparable methods, such as the right dihedra/trihedra methods. They differ moderately in the maximum/minimum principal stress directions when compared to the moment tensor method for fault kinematic analysis. The new method has advantages over the right dihedra/trihedra methods in the accuracy of stress estimate and the independence of stress estimate upon the small number of faults that are not parallel to the dominant fault set(s).