1 Introduction

The determination of earthquake focal mechanisms is a common practice in many seismological studies that provides useful and important information on the tectonic activity of the seismogenic fault and the driving stress field. Focal mechanisms of relatively small earthquakes (normally M < 4) are determined mainly by the use of P-wave onsets (McKenzie 1969; Udias et al. 1982; Lund and Slunga 1999; among many others) or a combination of P and S waves (i.e., Kisslinger et al. 1981; DeNatale et al. 1991; Rau et al. 1996; Nakamura 2002) which are observed at local or regional stations. In the case of moderate to strong earthquakes, inversion of teleseismic and regional broadband observations is the preferred technique (several out of many, Kikuchi and Kanamori 1991; Dreger and Helmberger 1993; Zhu and Helmberger 1996). Microearthquakes are significantly more abundant than moderate to strong ones but lack long period energy which can be used to invert for the focal mechanism, and therefore most mechanisms use first P-wave arrivals and ratios of the S to P amplitudes and spectral amplitudes (i.e., Brillinger et al. 1980; Kisslinger et al. 1981; Nakamura 2002; Lund and Boovarsson 2002; Hardebeck and Shearer 2003).

Many studies applied fault plane solutions to study specific tectonic regions (several out of many, Udias et al. 1982; van Eck and Hofstetter 1989, 1990; Hardebeck and Hauksson 2001; Hauksson 2002; Salamon et al. 2003; Hofstetter et al. 2007; Bailey et al. 2010) and describe algorithms for computing first-motion mechanism solutions (i.e., Brillinger et al. 1980; Rabinowitz and Hofstetter 1992; Hardebeck and Shearer 2002). Thus, determining the accuracy and reliability of the fault plane solutions are an important prerequisite for any seismotectonic study.

Through the years, several codes for the determination of focal plane solutions were proposed (i.e., Hardebeck and Shearer 2002; Snoke 2003). However, the FPFIT software package (Reasenberg and Oppenheimer 1985) is probably the most common code for determining focal plane solutions based on the polarity of the first P-wave arrivals. The code performs a grid search over a range of strike, dip, and rake values to obtain the best fitted two nodal planes. The misfit function is the ratio between the number of inconsistent polarities and the total polarity observations, where the former are weighted by the quality of the observation and the relative distance from the nodal planes. The function may have several minima rendering into multiple solutions, which are presented as well. Here, we take advantage of this code to explain the effect of various quality estimators. Hardebeck and Shearer (2002) proposed a new program HASH, based on FPFIT, for determining first-motion mechanisms, while accounting for the possible errors in the assumed earthquake location, velocity model, and wrong polarity determinations. Salamon et al. (2003) proposed several more quality estimators such as how wide the observed stations are distributed on the focal sphere, how close the observed stations constrain the position of the calculated nodal planes, and the dependence of the solution on critical stations in the case of an uneven station distribution. Kilb and Hardebeck (2006) tested for velocity model sensitivity using the FPFIT and HASH algorithms. Several other factors that control the focal plane solution and that may affect its stability and reliability are, for example, epicentral depth, and the presence of both near and far stations.

The goal of this study is to examine how changes in the raw data affect each of the focal plane solution, and how sensitive the solutions upon the quality estimators are. This is done by carefully isolating and checking the effect of each factor. Overall, it allows for a better and concise understanding of the sensitivity and importance of these quality estimators. In the above-mentioned studies, most of the quality estimators are qualitatively described. Here, we conduct a detailed test of the effect of the quality estimators separately and all together.

2 Data and method

In order to illustrate the dependencies of the focal plane solution on the various quality estimators, we examine wherever it is relevant up to two prototypes of focal mechanisms—strike-slip, normal (or thrust) as shown in Fig. 1 and listed in Table 1, assuming double-couple orientation. The polarities of the last two are opposite to each other, but in fact they are geometrically identical. We therefore run only the normal/thrust type as the representative mode. The depths of the earthquakes are chosen arbitrarily to present upper crustal events. The investigation is conducted by using the FPFIT software package of Reasenberg and Oppenheimer (1985), with the velocity model of four layers over half space as listed in Table 2, except the cases where we add or subtract several layers.

Fig. 1
figure 1

Station distribution (X and Y coordinates are in kilometer) and two focal mechanisms that are tested in this study: normal (left side) and strike-slip (right side) mechanisms using 12, 16, and 20 stations

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Table 1 Parameters of the earthquakes and focal mechanisms that are used in this study
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Table 2 1D velocity models that are used in this study
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The application of any code for focal plane solution involves two angles: the azimuthal angle defined by the location of the station relative to the epicenter and the take-off angle that is dependent on the velocity model. The number of possibilities to distribute the stations on the focal sphere is unlimited, which implies an infinite number of take-off and azimuthal angles for any given velocity model, rendering into an infinite number of fault plane solutions. The common velocity model and the even station distribution ensure that the “original” focal plane solutions (Fig. 1) are suitable for our study, and relatively small changes of the “original” fault plane solutions may not lead to different conclusions regarding the importance of the quality estimators.

Normally, aftershock studies immediately following a strong earthquake or the operation of portable networks for a few years involves a large number of stations specifically installed in a relatively small region. Small-to-medium magnitude earthquakes are typically observed by 10–20 stations. Nevertheless, there are instances in which the station distribution may be poor, there may be noisy stations, or stations do not function properly, and consequently the total number of reporting stations is small and limited. Depending on the station distribution one can obtain a fairly reliable focal plane solution using less than 10 stations. However, in this study, we consider that 12 stations is probably the minimal number for a fair solution, as there are at least three stations in each quadrant. We therefore use this number as the threshold to test the dependency of the focal mechanism on the various parameters and by thus avoiding the problem of lack of data. These focal plane solutions are created synthetically, using three possible sets of stations: 12, 16, or 20. The locations of the stations are chosen in order to ensure as even as possible azimuthal distribution around the center at various take-off angles (Fig. 1).

The focal mechanism is dependent on the azimuth and the take-off angle, which in turn are dependent on the velocity model. For layered velocity models, if the hypocenter is located just below a boundary and is far enough above the next lower boundary, the take-off angle is larger than 90° (i.e., an upgoing ray). In the lower hemisphere projection, which is used by the FPFIT algorithm, this station is moved to the opposing quadrant with minor or no effect on the solution at all. In order to avoid such complications of the take-off angle, we set the location of the earthquake just above the boundary between two adjacent layers.

The proximity of the station to the nodal plane is normally indicated by relatively small amplitude of the first onset regardless of its polarity, whether “up” or “down” (i.e., Aki and Richards 1980; Kasahara 1981; Ben Menahem and Singh 1983). As a result, the FPFIT software can be configured to down weight readings that are located near the nodal planes. In our case, however, we assume that all first-motion observations are impulsive and can be easily picked, regardless of the fact whether the station is close or away from any of the nodal planes. Therefore, in all the tests below, we assign equal weight to all recordings. The case of stations that are close to the nodal plane is discussed in detail below; in Fig. 1, we place the stations slightly away from the nodal planes.

3 Quality estimators of fault plane solutions

The FPFIT method (Reasenberg and Oppenheimer 1985) defines two quality estimators that describe the quality and reliability of the determined mechanism: (1) misfit quality estimator. This quality estimator compares the observed polarities at the stations with those calculated for a suite of source models (0 = perfect fit, 1 = complete misfit) or the number of reverse polarity stations (RPS); (2) station distribution ratio (STDR). This quality estimator quantifies the spatial distribution of the stations on the focal sphere, relative to the radiation pattern. Robust solution involves values of STDR > 0.5, where the readings are located away from the nodal planes, while less reliable solutions are associated with STDR < 0.5, where a large number of readings are close to the nodal planes. Kilb and Hardebeck (2006) found that STDR is the best quantitative discriminator of quality focal mechanism. The FPFIT also reports the range of uncertainty expressed as a quality code, and it summarizes the range of deviation of the strike, dip, and rake in a given solution.

Hardebeck and Shearer (2002), Salamon et al. (2003), and Kilb and Hardebeck (2006) added a few more quality estimators to quantify the reliability of the fault plane solution: (1) dependence on the event location, (2) dependence on the velocity model, (3) azimuthal distribution (AD) of the stations on the focal sphere, (4) constraints on the nodal planes (CNP), and (5) critical stations (CS). There are several aspects to the proposed STDR quality estimator that are discussed in detail in quality estimators 3 and 4. We study also another quality estimator of near versus far stations. The above-mentioned studies discussed the relevance of those quality estimators. However, our analysis takes into account the combined quality estimators using exhaustive sets of possible focal plane solutions. Below, we briefly describe all these seven quality estimators and discuss their importance, and then the determined focal plane solutions are compared with the ideal ones.

4 Investigation of the quality estimators

To illustrate the importance of each quality estimator, we examine their application to the two ideal “end member” modes, which are the strike-slip, normal (or thrust) mechanisms. Regarding the number of stations, the lowest threshold for a sufficient solution is about 10, given that the stations are spread equally all around the focal sphere more or less. Here, we use the range of 12, 16, and 20 stations, which are 3, 4, and 5 stations per quadrant, respectively, simulating a real station distribution, resulting in 6 “master” variants to be considered (see Fig. 1).

The dependence of the focal plane solution on the take-off and azimuth angles is a nonlinear function and cannot be described using a simple relationship. For example, if the computed solution has a slightly tilted vertical plane to either side, then the strike changes by 180°. Thus, we rate the quality of the solution using the rms, as determined for both nodal planes of the ideal and calculated solutions

$$ rms=\frac{1}{\sqrt{6}}{\left\{{\displaystyle \sum_{i,j}\left\{{\left[{s}_i(id)-{s}_j(cal)\right]}^2+{\left[{d}_i(id)-{d}_j(cal)\right]}^2+{\left[{r}_i(id)-{r}_j(cal)\right]}^2\right\}}\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$
(1)

where s, d and r are azimuth, dip, and rake of the nodal planes (in degrees), respectively, id and cal are the ideal and calculated solutions, and i and j form two pairs of the two nodal planes for both solutions and can have two values composed of (1,1) and (2,2) or (1,2) and (2,1). The summation includes six terms. We select the minimal value between the possible two values. We can estimate the quality of the obtained solution. In the case of even station distribution over the focal sphere, the azimuthal separation is about 360/n where n is the number of stations. We get values of 30°, 22.5°, and 18° for 12, 16, and 20 stations, respectively. Thus, the value of 20° is a reasonable estimate of the average separation, and we assume that this value is suitable for the strike, dip, and rake to be used in the following cases. Throughout the study, we use the following definition, based on the rms determination in Eq. 1: a focal plane solution is acceptable, moderate, or unacceptable if rms ≤ 20, 20 < rms ≤ 40, 40 < rms, respectively.

Near versus far stations

Here, we explore the focal plane solutions using two groups of stations: a group of near stations and a group of far stations. When using the group of near stations, we shift the far stations, which are in the distance range of 100–200 km towards the epicenter by subtracting 100 km from the distance of each station. When using the group of far stations, we shift the near stations, which are in the distance range of up to 100 km, away from the epicenter by adding 100 km to the distance of each station. The shifts of the stations along the azimuthal angle cause significant changes of the take-off angles only, and obviously a change of the station distribution on the focal sphere.

Table 3 lists focal plane solutions based on near- and far-field stations as applied to the original strike-slip and thrust mechanisms (see Fig. 1 and Table 1), with hypocenters at depths of 6.9 and 14.9 km, respectively. The strike-slip mechanisms are relatively stable in both groups of near- or far-field stations, while the original thrust mechanisms are changed into different mechanisms in either case. In the case of the strike-slip mechanism, the stations are moved closer or away relative to the center along the same azimuthal direction, and they stay in the same quadrant. In the case of the thrust mechanism, the hypocenter is located at a depth of 14.9 km, the nodal planes are at a dip of 45°, and the distance on the surface between the nodal planes is about 30 km. Any station shift of 100 km definitely moves it into another quadrant, rendering a different station distribution on the focal sphere resulting in a major change of the focal mechanism.

Table 3 The dependence of the focal plane solutions (including multiple solutions) on the near and far stations as applied to strike-slip and thrust mechanisms using the sets of 12, 16, and 20 stations, as compared with the prototypes of focal mechanisms presented in Fig. 1
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A variant of this test is to determine the focal plane solution using the solely near or far stations by removing the far or near stations, respectively, thus involving 50 % of the original stations. As in the first case, the strike-slip mechanisms are relatively stable in both groups of only near-field or only far-field stations, while for the original thrust mechanisms we get different mechanisms in either case. We note that in cases of uneven station distribution, change or shift of stations may lead into significant variations of the focal plane solutions.

Event location

Here, we explore the dependence of the fault plane solution on horizontal shift of the epicenter, as applied to normal and strike-slip mechanisms. This dependence is tested using two types of horizontal shifts while the depth is held fixed at 14.9 and 6.9 km for normal and strike-slip mechanisms, respectively. In the first part, the epicenters are horizontally shifted by 5 km along the azimuthal directions 37°, 143°, 217°, and 323°, and in the second a shift of 10 km along the directions 45°, 135°, 225°, and 315° to accommodate the cases of severe mislocations. The azimuths and the depths of the events are selected to be different just to avoid repeating shifts along the same azimuthal directions.

Table 4 lists the 5 and 10 km horizontal shifts of the epicenter using the sets of 12, 16, and 20 stations as applied for normal and strike-slip mechanisms. The 5 and 10 km epicentral shifts cause relatively minor changes in the sets of 12 and 16 stations in all four directions, but there are several multiple mechanisms in the set of 20 stations. We note that the original focal plane solution of the set of 20 stations is mostly affected by the 5 and 10 km shifts as there are a large number of near stations that migrate to the opposing quadrant. As a conclusion of this test, it is obvious that the specific station distribution on the focal sphere plays an important role in the determination of the focal plane solution, taking into account that in reality the station distribution is not necessarily uniform, some stations may be very close to or far from the nodal planes, and the number of stations can be relatively small.

Table 4 The dependence of the focal plane solutions (including multiple solutions) on the horizontal shifts as applied to normal (depth of 14.9 km) and strike-slip (depth of 6.9 km) mechanisms for each set of 12, 16, and 20 stations, as compared with the prototypes of focal mechanisms of strike-slip and normal presented in Fig. 1
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Velocity model

The velocity model is a simplified picture of the real earth, and in most cases a detailed knowledge of the vertical gradient is poorly known. In general, we assume a simple model of the earth based on a layered model, normally comprising three to four layers over half-space, and the velocity increases with depth. Seismic refraction and tomography studies can help to refine the determination of the velocity model, resulting in more reliable focal plane solutions. The take-off angle depends on the velocity model and the depth of the epicenter through a non-linear relation (for example Lee and Stewart 1981), and thus the focal mechanism cannot be expressed or determined using a simple linear relationship. This nonlinear dependency is not equally divided as Hardebeck and Shearer (2002) demonstrated that the focal mechanisms are more sensitive to change in the velocity model than to change in the source depth, using four velocity models. They showed that the discrepancies between the take-off angles computed from the different models were sometimes up to 40°. Based on Snell’s law, we can show that this has a significant effect in the case of near stations, where the distances of the stations are comparable to the characteristic size of the model. For far stations, the changes of the take-off angles due to different velocity models are essentially small whether there are a few or many layers over a half space, rendering to minor changes of the fault plane solutions, as was above-mentioned in the case of far stations. Using Snell’s law and assuming that the velocity increases downwards, we find that unlike the down-going rays (take-off angle <90°), up-going rays (take-off angle >90°) are hardly affected by division of layers above the hypocenter, and so is the determined fault plane solution.

We conduct several tests using different velocity models and focal mechanisms of strike-slip and thrust at focal depths of 6.9 and 14.9 km, respectively. Table 2 lists the velocity models that are used: half-space, one layer over half-space, four layers over half-space, and seven layers over half-space. We retain the thickness of the crust to be 30 km in all the models, which represents the crust in large parts of the globe. Except for the case of stations located just above the hypocenter, the upgoing take-off angles are the results of our relatively coarse velocity model (i.e., velocity models of half-space or one layer over half-space). Table 5 lists the obtained fault plane solutions. Comparing with the original mechanisms (Fig. 1; computed using four layers over half space), there are some changes in the focal mechanisms in the cases of half space and one layer over half space, while minor changes in the cases of four and seven layers over half space. The stability of the fault plane solutions in the cases of several layers over half space is due to: (1) the even station distribution on the focal sphere and therefore the changes of the take-off angle are essentially similar to changes occurring to stations in opposing quadrants; (2) relative large number of far stations that have the same or similar take-off angles; and (3) as above-mentioned, the upgoing rays (take-off angle >90°) are less sensitive to division of layers above the hypocenter and so is the determined solution.

Table 5 The dependence of the focal plane solutions (including multiple solutions) on the velocity structure (see Table 2) as applied to strike-slip (depth of 6.9 km) and thrust (depth of 14.9 km) mechanisms for each set of 12, 16, and 20 stations, and compared with the prototypes of focal mechanisms of strike-slip and thrust presented in Fig. 1
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Misfit and RPS

It has already been demonstrated that the appearance of stations with reverse polarities in phase files is common over many years. Hodgeson and Adams (1958) and Oppenheimer et al. (1988) reported that a certain percentage of the P-wave readings were inconsistent with the solutions. Naturally, emergent phases are ambiguous and hard to distinguish especially in the case of noisy stations, but reverse polarity observations also occur for impulsive phases that can be clearly identified. This can be a severe problem, as illustrated by Hardebeck and Shearer (2002) and Shearer et al. (2003) who estimated the polarity error rate for the 1981–1998 Northridge SCSN catalog. They found that for clusters with similar earthquakes, as identified from the waveform cross-correlation, there are ∼10 % inconsistent impulsive polarities and ∼20 % inconsistent emergent phases even after accounting for station polarity reversals. Furthermore, Salamon et al. (2003) showed that in the case of the WWSSN the percentage of stations having reverse polarity was for a long period of time amounting roughly from 15 to 20 %, with an extreme case of 40 %. Obviously, the larger the number of reverse polarity stations, the lower the reliability of the solution is. Reverse polarity can be due to technical problems (i.e., wrong wire connections), erroneous phase readings normally resulting from poor signal to noise ratio, complexity in the initial part of the event, misprinting, and lateral refraction across faults due to large velocity contrasts (Oppenheimer et al. 1988). Waveform cross-correlation for clusters of similar earthquakes (e.g., Shearer et al. 2003) may at least partially overcome erroneous phase pickings. However, this useful technique is limited for a relatively small number of events and cannot be easily applied in a case of very large clusters. Another preventive measure is a routine verification of station polarities across the network using well-recorded teleseisms.

We test the sensitivity of the RPS quality estimator by intentionally reversing the polarities of up to 25 % of the total number of stations in each set of stations, or up to 3, or 4, or 5 in the sets of 12, 16, and 20 stations, respectively, and applying all the possible combinations for each station set. The number of possible permutations is computed using \( \left(\begin{array}{l}n\hfill \\ {}m\hfill \end{array}\right) \) where n is the number of stations in a given set that gets the values of 12, 16, or 20 and m is the number of reverse polarity stations that gets the values of up to 3, or up to 4, or up to 5, respectively. The minimal numbers of possible combinations where m = 1 are 12, 16, and 20, and the maximal number of possible combinations for n = 20 and m = 5 are 15,504. Other parameters like depth and velocity are kept fixed at all combinations.

As expected, many focal plane solutions, especially those with several reverse polarities, are significantly different from the original ones. Figure 2 illustrates in the upper part the rms values (abscissa) that are calculated using Eq. 1. The rms values may be different for each combination of n and m because the m values of reverse polarities relate to different selected stations. By counting the times that a given rms value is in a predetermined interval, we can create empirically the rms distribution function (rdf) for fixed values of n and m, for strike-slip and thrust mechanisms (Fig. 2). If the stations are evenly distributed on the focal sphere, then the typical spacing of the azimuthal angle is about ∼20°, which can serve as a typical interval for the rdfs. If m is kept fixed, the number of possible combinations increases as n increases following the permutation formula resulting in large rdf values. Normally, the rms distribution functions have one to two prominent peaks for the strike-slip and thrust mechanisms and some small peaks as well. The shape of the rms distribution function is irregular due to the nonlinear dependency on the take-off and azimuthal angles. Keeping n and m fixed and setting a slightly different station distribution over the focal sphere, which means changing the azimuthal and take-off angles, may lead to somewhat different rdfs. Since there are infinite possibilities of station distributions, there are also infinite numbers of rdfs. Thus, the rdfs are essentially specific network functions. The studied cases of strike-slip and thrust mechanisms illustrate the quality of the focal plane solutions, i.e., one is likely to get large values of rms for m ≥ 2. For example, for thrust mechanism with n = 20 and m = 3, the probability of having large deviations of strike, dip, and rake is significant. Alternatively, one can investigate a priori for a given station distribution and m reverse polarity stations the feasible rdfs.

Fig. 2
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Frequency of occurrences versus the RMS (up) and Kagan’s angle (bottom), respectively, due to the reversed polarity stations as applied for each set of stations (12, 16, and 20) to strike-slip (solid line) and thrust (dashed line) mechanisms, as compared with the prototypes of focal mechanisms presented in Fig. 1. The notation n_m on the upper left side, means n stations and m reverse polarity stations for example the case of 20_4 is n = 20 and m = 4

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The lower part in Fig. 2 presents Kagan’s angle (Kagan 1991), which is a 3-D rotation from one double-couple earthquake source (ideal) into another double-couple (perturbed). This angle varies from 0° to 120° which are complete agreement to complete disagreement, respectively. Small and large angle values, say <40° and >80° between the ideal and perturbed double couples indicate that the two focal mechanisms are similar or dissimilar, respectively.

Azimuthal distribution of the stations on the focal sphere

It is assumed that the wider and more even the stations are spread on the focal sphere, the more reliable the solution is. One can expect a large uncertainty in determining a focal plane solution based on a station set with two near stations having a large azimuthal gap. The AD quality estimator is not necessarily overlapping with STDR, which has higher values if the stations are distributed away from the nodal planes. We qualitatively explore the importance of this quality estimator by determining solutions involving large azimuthal gap, in the cases of strike-slip and thrust mechanisms, using 12, 16, and 20 stations in each set. We keep station 1 in Fig. 1 fixed and we move station 2 clockwise, where the azimuthal gap is gradually increased each time by 1° between these two adjacent stations, until a maximal azimuthal gap of 180° is reached. As the azimuthal gap increases, we add 180° to the azimuthal angle of the stations that are included in the gap, and they are shifted to the opposing quadrant in the case of strike-slip mechanism, or the other side of the same quadrant in the case of thrust mechanism. The resulting focal plane solutions are changed with respect to the initial solution, in both cases of strike-slip and thrust mechanisms (Table 6). The mechanisms in the upper left side of each pair of lines present focal plane solutions (including multiple solutions) in selected azimuthal gaps of 80°, 120°, and 160°. In fact, major changes occur in the strike-slip solutions, once the azimuthal gaps are increased to large values, and the resulting mechanisms are a mixture of, thrust, strike-slip and dip-slip faulting. In contrast, the changes in the thrust mechanism are small to moderate and all resulting mechanisms retain the original mechanism with relatively small changes of the odal planes. Shifting the stations to the third and fourth quadrants, by adding 180° to the azimuthal angle, creates a large concentration of stations for both mechanisms, which serves as a constraint on one of the nodal planes separating opposing P-wave onsets.

Table 6 The dependence of the focal plane solutions (including multiple solutions) on the increased azimuthal gap for selected azimuthal gaps of 80°, 120°, and 160°, for each set of 12, 16, and 20 stations, as compared with the prototypes of focal mechanisms for strike-slip and thrust presented in Fig. 1
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We add a complication in which we remove the stations that are included in the azimuthal gap instead of moving them to the opposing quadrants (upper right side of each pair of lines in Table 6). As the azimuthal gap between stations 1 and 2 increases, more stations are being removed and the focal solutions are based on 8, 10, and 12 stations instead of the original sets of 12, 16, and 20 stations, respectively. We observe major changes of the focal mechanisms relative to the original solutions, especially in the case of strike-slip mechanism. In contrast, the changes in the thrust mechanism are small to moderate and all the resulting mechanisms retain the original mechanism with relatively small changes of the nodal planes.

We add another complication to the above-mentioned cases by reversing the polarity of the station that is being shifted clockwise. Here, we simulate a solution involving a large azimuthal gap and a wrong phase onset (lower left side of each pair of lines in Table 6). In the case of strike-slip, there are major changes in the resulting mechanisms. In contrast, the changes in the thrust mechanism are small to moderate. Furthermore, as we increase the azimuthal gap, we also remove the stations that are included in that azimuthal gap instead of moving them to the opposing quadrant (lower right side in each pair of lines in Table 6). As the azimuthal gap increases, more stations are removed, resulting in sets of as low as 8, 10, and 12 stations instead of the original sets of 12, 16, and 20 stations, respectively. We observe major changes of the focal mechanisms relative to the original solutions, especially in relation to strike-slip mechanism.

CS

In the section of the quality estimator AD, we note that if the stations are evenly distributed on the focal sphere, then the reliability of the solution is high. Salamon et al. (2003) showed that in certain configurations, the fault plane solution is critically dependent on the polarity of one or a few specific stations. For example, if most of the stations are clustered in three or three quadrants and the other quadrant is populated by one station only, reversing the polarity of that particular station may change significantly the computed solution, while changing the polarity of any other station in the adjacent clusters may hardly affect the solution. This is an important condition, sometimes more influential than the RPS quality estimator as is above mentioned. For example, in many cases, there is a certain percentage of inconsistent phases and even correcting for the known reversals, e.g., Northridge SCSN catalog (Hardebeck and Shearer 2002) or WWSSN data (Salamon et al. 2003), may not stabilize a solution that contains a critical station.

We study the effect of this quality estimator by adding 180° to the azimuths of the stations in the first and second quadrants and essentially moving them to the opposing quadrants (first to third; second to fourth), except one station (on the right side) that populates the first or the second quadrants. In the case of thrust mechanism, it means moving the far stations to the other side of the same quadrant (just on the left side) and the near stations to the “moon-shape” quadrant on the left side. Then, we move the critical station clockwise from north (azimuth of 0°) in steps of 1° each time until it reaches 180° while calculating the focal plane solution at each step. The results for selected azimuths are shown in Table 7.

Table 7 The dependence of the focal plane solution (including multiple solutions) on the critical stations for each set of 12, 16, and 20 stations, as compared with the prototypes of focal mechanisms for strike-slip and thrust presented in Fig. 1
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It is clear that the new solutions, especially in the case of the strike-slip mechanism, change significantly relative to the initial one, implying a clear solution dependence on the critical station. We note that for both mechanisms, we get a large concentration of stations in the third and fourth quadrants, which is almost a constraint on one of the nodal planes that separates the quadrants with opposing P-wave onsets. We add a complication in which we remove the stations instead of moving them to the opposing quadrants (upper right side in each pair of lines in Table 7), and the focal solutions are based on 8, 10, and 12 stations instead of the original sets of 12, 16, and 20 stations, respectively. The resulting focal mechanisms are significantly different from the original mechanisms, especially in the case of the strike-slip mechanism. As in the former case, for both mechanisms there is a large concentration of stations in the third and fourth quadrants, which is almost a constraint on one of the nodal planes that separates the quadrants with opposing P-wave onsets.

The dependence of the solution on the critical station can be more severe if the station has a reverse polarity. Since in reality, the station distribution is not necessarily ideal as in our case and we may encounter reverse polarity stations (for example, rapid installation in the case of aftershock studies), the effect can be of obvious importance. Thus, to explore this effect, we add another complication to the above-mentioned cases by reversing the polarity of the station that is being shifted clockwise. Here, we simulate a solution involving a large azimuthal gap and wrong phase picking (lower left side in each pair of lines in Table 7). Furthermore, as we increase the azimuthal gap, we also remove the stations that are included in the azimuthal gap instead of moving them to the opposing quadrant (lower right side in each pair of lines in Table 7). The focal solutions are based on 8, 10, and 12 stations instead of the original sets of 12, 16, and 20 stations, respectively. We observe major changes of the focal mechanisms relative to the original solutions, especially in the case of strike-slip mechanism.

CNP

It is reasonable to assume that the closer the onsets to the nodal plane, the better it is constrained and more reliable the solution is. The FPFIT, however, has an option to down-weight readings that are close to the nodal plane for being emergent (STDR < 0.5) because the radiation amplitude approaches zero at the plane, and consider a reliable solution when STDR ≥ 0.5, with an implicit assumption that the onsets are more reliable away from the nodal planes. In general, a dataset may have a few stations that are relatively close to the nodal planes having clear P-wave onsets. Therefore, the importance of this quality estimator should be tested and verified. This parameter is explored using a series of focal plane solutions consisting of 12, 16, and 20 stations (Table 8). Here, the station distributions are identical to those presented in Fig. 1, and the stations that are near the nodal planes (azimuthal gap of ∼13–15° relative to the nodal planes) are even closer to the nodal planes (azimuthal gap ∼3°; upper left side of the first line of each station set). The focal plane solutions in Table 8 are identical to those presented in Fig. 1. Assuming strike-slip mechanism, we redistribute the stations on the focal sphere in such a way that some stations are close to the nodal planes with azimuthal angles of 0° and 90°. In addition, we select up to four pairs of stations, where each pair has up and down polarities, with azimuthal gap of about ±3° relative to the nodal planes. For each set of stations, we conduct the following tests: (1) reducing the number of pairs that constrain the nodal planes from four pairs to one pair and align the stations along the azimuthal directions: 45°, 135°, 225°, and 315° (upper line in each pair of lines in Table 8); (2) retaining four pairs near the nodal planes and reversing the polarities of part or all the other stations but with no alignment of the stations along some specific azimuthal directions. In the cases of 12, 16, and 20 stations we reverse up to 4, 8, and 12 stations, respectively (lower line in each pair of lines in Table 8).

Table 8 The dependence of the focal plane solution (including multiple solutions) on the constraints on the nodal planes as applied to strike-slip for 12, 16, and 20 stations, as compared with the prototype of focal mechanism of strike-slip presented in Fig. 1
Full size table

The first test deals with the reduction of the number of pairs that constrain the nodal planes, from four pairs to one pair near the nodal planes, and all the other stations are aligned along the azimuthal directions of 45°, 135°, 225°, and 315°. Here, we observe no or slight changes of the focal plane solutions having four or three pairs near the nodal planes and somewhat large change or multiple solutions once we have one or two pairs only. It means that the readings near the nodal planes can have major influence on the focal plane solution. In the second test, we arbitrarily reverse the polarities of several stations. We observe no or slight change of the focal plane solutions for a small number of reverse polarity stations, but for relatively large number (i.e., 4 out of 12 stations or 12 out of 20 stations) the changes of the focal mechanisms are significant and we get also several multiple solutions.

5 Perturbations of the ideal case

Above, we describe a series of difficulties that one may encounter in achieving a reliable fault plane solution. It is reasonable to assume that any given solution is likely to be biased by a few or all the above mentioned factors, in any possible combination, i.e., misfit, reverse polarity station, unfavorable station distribution, etc. As a result, the obtained solution should be consider together with the uncertainties of the parameters, which in the ideal case are expressed in the form of statistical distribution of the parameters (Tarantola 1987). Thus, those uncertainties should be transformed into confidence zones of the nodal planes. In general, this statistical distribution is unknown and any change of the parameters is not larger than some percentage of the parameter value. In order to test this, we apply the Monte Carlo approach by generating a large number of random samples of the perturbed parameters in the assumed distribution, determine the focal plane solution in each case, compare with the ideal case, and convert the obtained nodal variations into a data space. Furthermore, by comparing the original solution with the obtained solution, we get the rms distribution function that characterizes the quality of the solutions. The Monte Carlo approach, common in general in many domains of science, was used by Hardebeck and Shearer (2003). They determined focal plane solution by randomly selecting 8 to 12 stations with good-quality P-wave polarity and S/P amplitude ratio observations from a subset of 203 small events from the Northridge region.

For conducting the above-mentioned tests, we select the strike-slip and thrust mechanisms each with the sets of 12, 16 and 20 stations, and allowing perturbations of the following parameters: (1) a shift of up to ±5 km in each of the NS and EW horizontal directions (maximal shift can be up to 7 km); (2) reverse polarities of up to three, four, and five stations for the sets of 12, 16, and 20 stations, respectively; (3) increase the azimuthal gap up to 180° between the stations just east of the north direction (stations 1 and 2 in Fig. 1); (4) critical stations—if the azimuthal gap between the stations is less than 180°, we also perturb the second azimuthal gap between the station that was shifted and the nearest one. The total gap can be increased up to 180°, which may leave a single or almost single station in the two quadrants on the right side; (5) constraints on nodal planes—we perturb the azimuthal gaps near the nodal planes in the range of 3° to 30° between two stations for one to four pairs of stations; (6) for each perturbation, we select the velocity model, where the optional models are: half-space, one layer over half-space, four layers over half-space, and seven layers over half-space. For each mechanism, either strike-slip or thrust and each set of 12, 16, or 20 stations, we select the above-mentioned parameters randomly. In total, we determine 1,000,000 focal plane solutions for each set of stations. We rate the quality of the solution by the rms, as was explained before. Figure 3 (upper part) shows the rms distribution function versus rms for the strike-slip and thrust mechanisms, as applied for the sets of 12, 16, and 20 stations. In the case of strike-slip mechanisms, there are two pronounced peaks around rms ∼30–40 and ∼100–120. In the case of the thrust mechanism, there is a pronounced peak around rms ∼30–40 and a minor one at ∼110–140. For both types of mechanisms, the peaks of the rdfs are getting lower as the number of stations increases. As the number of stations increases, although some the stations are reversed, we get more relatively stable solutions and less multiple solutions. The lower part in Fig. 3 presents Kagan’s angle (Kagan 1991). The distributions of both mechanisms for the sets of 12, 16, and 20 stations are quite skewed, having medians near 30°. The clear peaks, especially in the case of the thrust mechanism, at about 20–40°, suggest that the ideal and perturbed solutions are rather similar.

Fig. 3
figure 3

Application of the Monte Carlo approach by generating a large number of random samples of the perturbed parameters in the assumed distribution. Upper line the root mean square distribution functions (see Eq. 1) of the strike-slip (solid line) and thrust (dashed line) mechanisms in the case of the perturbed input parameters (described in the text), applied for each set of 12, 16, and 20 stations (shown in the upper left inner corner), and compared with the prototypes of focal mechanisms presented in Fig. 1. Lower line Kagan’s angle (Kagan 1991) for the strike-slip (solid line) and thrust (dashed line) mechanisms in the case of the perturbed input parameters

Full size image

We apply this procedure in the case of a strong earthquake that occurred in California on 22 December 2003 at 19:15:56.24 GMT, M W 6.6, depth of 8.70 km, lat. 35° 42.03′ N, lon. 121° 06.03′ W, within the Northern California Seismic Network. The earthquake was recorded by 248 stations of this network and many other stations in the USA and worldwide. The distance range of the stations is: closest station is at 5.2 km, another 4 stations are up to 20 km, 66 stations are between 20 and 100 km, and the other 173 stations are between 100 and 380 km. There are many stations in the azimuthal directions from northwest to southeast (clockwise) and smaller number of stations in the directions from south to west (clockwise). In many cases, the station distribution is not even or uniform, and thus we use only the recordings of this network. The focal plane solution is presented in Fig. 4 (left side). Both nodal planes are well constrained, especially the NE dipping one.

Fig. 4
figure 4

Left: focal plane solution of an earthquake in California on 22 December 2003 at 19:15:56.24 GMT, M W 6.6, depth of 8.70 km, lat. 35° 42.03′ N, lon. 121° 06.03′ W, based on P-wave onsets. Right application of the Monte Carlo approach by generating a large number of random samples of the perturbed parameters in the assumed distribution. Root mean square distribution functions (Eq. 1) in the upper part and Kagan’s angle (Kagan 1991) in the lower part, applied for sets of 12 (dashed line), 16 (solid line), and 20 (dotted line) stations

Full size image

We perturb the following parameters: (1) for each determination of focal plane solution, we randomly select 12, 16, or 20 stations out of the 248 stations; (2) reverse polarities of up to 3, 4, and 5 stations for the sets of 12, 16, and 20 stations, respectively; (3) to accommodate changes in the velocity model and location errors the incidence angle and azimuthal angle are randomly changed by up to ±15° and up to ±5°, respectively. In total, we determine 1,000,000 focal plane solutions for each set of stations. The rdfs in the case of the perturbed input parameters, applied for the sets of 12 (dashed line), 16 (solid line) and 20 (dotted line) stations are shown in Fig. 4 (upper right side). The shapes of the rdfs are similar with a pronounced rms peak at about 20 and a smaller one at about 130 and a clear trough between 70 and 90. We note that the shape and the location of the peaks are similar to the rdfs of the prototype of thrust mechanism in Fig. 3. In a similar way to Fig. 3, as the number of stations increases, although some the stations are reversed, we get more relatively stable solutions and less multiple solutions. The lower right part in Fig. 4 presents Kagan’s angle (Kagan 1991). The distributions for the sets of 12, 16, and 20 stations are quite skewed, having medians near 35°. The clear peaks for the cases of 12, 16, and 20 stations, at about 25°, suggest that the ideal and perturbed solutions are rather similar.

6 Discussion and conclusions

We explore a series of quality estimators or parameters that describe the type and reliability of the focal plane solution. Those quality estimators do not necessarily have the same weight. The dependence of the focal plane solution on the take-off and azimuth angles is a nonlinear function and cannot be described using a simple relationship, or easily relate it with the sensitivity of the quality estimators. It is obvious that a more refined velocity model, especially using a series of sublayers with gradually increasing velocity immediately above and below the hypocenter, can increase the likelihood that rays are downgoing (take-off angle <90°), which helps to stabilize the solution, as was also discussed by Hardebeck and Shearer (2002). The existence of several reverse polarity stations can cause a major change of the focal plane solution, depending on their relative number and distribution over the focal sphere. We intentionally reverse the polarity of up to 25 % of the phase readings in all our cases, and thus the number of possible combinations is very large. The obtained solutions in many of these combinations are different from the ideal solution, thus implying that it is a very important quality estimator. It is obvious that the determination of the correct polarities is important and should be done carefully. There are three common ways that help significantly to minimize this problem. One is to use cross correlation of waveform for clusters of similar earthquakes (Shearer et al. 2003), the second is an ongoing verification of polarities of stations across the network using well-recorded teleseisms, and the other is to use co-located accelerometer and velocity sense. If the station distribution on the focal sphere is uneven, it may degrade the reliability of the solution. Likewise, the case of a single station in a given quadrant, which is separated from the neighboring stations on both sides by large azimuthal gaps, might harm the stability of the solution. This dependence may degrade the solution, especially if it has a few reverse polarities. Such station distribution may exist, and one should be aware of the drawbacks that it may imply on the focal plane solution.

The study deals with three “basic” types of focal mechanisms and the quality estimators that can affect the focal plane solutions. Since these are common focal mechanisms in many parts of the world, maybe with some small variations, studying such mechanisms is relevant and of significant importance. On Earth, there are several major long faults exhibiting pure or nearly pure strike-slip mechanisms, i.e., San Andreas fault, North Anatolian fault, and the Dead Sea fault. Thrust faulting is common among many parts of the subduction trenches surrounding the Pacific Ocean. In reality, one can expect some small variations of the focal mechanisms. In addition, the station distribution of the local or regional may be different from the ideal station distributions that are used here. However, given a station distribution of a permanent or temporary network, location of epicenters and plausible focal plan solutions, one can conduct a study estimating a priori the quality and reliability of those mechanisms, by testing each quality estimator separately (see Tables 2, 4, 5, 6, 7, and 8) or using a combined perturbation (see Fig. 3). When conducting such study, one should be aware that the rdfs are essentially specific network functions. Furthermore, if the station distribution is not highly uneven, then it appears that the reverse polarity station is the most important quality estimator. The importance of critical stations seems to be of second rank. The third rank of importance includes the other quality estimator, as normally the velocity model is reasonably known and the errors in the epicenter location are smaller than the extreme values used in the above examples.

Studies for the determination of fault parameter constraints using relocated earthquakes or the heterogeneity of focal mechanisms can be very useful, in example the studies of Kilb and Hardebeck (2006) and Bailey et al. (2009, 2010) in California. In this study, we assume a double couple orientation, however, if this is not the case then a more elaborated study is needed, i.e., Bailey et al. (2010) and Tape and Tape (2012). Kagan (1991, 1992, 2005) presented 3-D rotation of double-couple earthquake sources, which rotates one mechanism into another. He suggested that this algorithm can be used for investigating the focal mechanisms and the tectonic setting of a region of interest, and also for studying the stress field. Zahradník and Custódio (2012) presented a method to assess the uncertainty of earthquake focal mechanisms using the standard theory of linear inverse problems. They showed that the method is useful for network design and applied it in the case of the Iberian Peninsula. For a given region with observed seismic activity, a study of the potential focal mechanisms, similar to the one presented here, can help in identifying stations that may have a major influence on the focal plane solution. Such study can serve as an important tool in the determination of the character of seismicity and the tectonic setting.