### Overview

The *M*_{w} 6.7 Northridge earthquake occurred on 17 January 1994 beneath the western portion of the San Fernando Valley in southern California (Fig. 2). The earthquake was well-recorded by strong motion stations throughout the greater Los Angeles region with more than 50 sites located within 30 km of the rupture. Further, over 60 GPS observations, as well as leveling-line data were available to provide additional constraints on the earthquake rupture. Numerous studies have been conducted on this earthquake, and numerous models of the rupture process have been presented (e.g., Dreger 1994; Hartzell et al. 1996; Hudnut et al. 1996; Shen et al. 1996; Wald et al. 1996; Zeng and Anderson 1996). The common elements of these models indicate primarily reverse-slip occurring on a roughly 20-km long blind-thrust fault dipping towards the southwest with the rupture initiating at a depth of about 18 km. Average rupture speed values used in the above studies range from about 75–85% of the local shear wave velocity and the down-dip fault width values range from 20 to 30 km. The average slip rise time from these studies is harder to constrain due to the different types of formulations employed; however, the values are generally around 1 s.

### Kinematic rupture models

The kinematic rupture models used for the grid search validation exercise are generated using the approach of Graves and Pitarka (2016) and include the shallow slip rate modification of Pitarka et al. (2021). For all the Northridge rupture models, I fix the magnitude at *M*_{w} 6.7, fault length at 20 km, depth to top of rupture at 5 km, fault strike at 122°, fault dip at 40°, and hypocenter at − 118.554° longitude, 34.208° latitude and 17.5 km depth. These parameter values are consistent with the consensus values listed in the NGA-West2 Database (Ancheta et al. 2014). Then, I generate a suite of rupture models using discrete combinations of average rupture speed, down-dip fault width, and average slip rise time. Specifically, I sample across six values of average rupture speed, five values of fault width, and four values of average rise time. Below, I describe how the values of these parameters are set in the grid search approach.

For the GP kinematic rupture generator, the average rupture speed (*V*_{r}) is parameterized as a fraction of the local shear wave velocity (*V*_{s}):

$${V}_{\mathrm{r}}={f}_{\mathrm{RV}}{V}_{\mathrm{s}}.$$

(1)

Within the grid search framework, I allow the scaling factor *f*_{RV} to take on the six discrete values: 0.65, 0.70, 0.75, 0.80, 0.85, and 0.90. The choice of this range is based on generally accepted values observed for past earthquakes and encompasses the values from the previous Northridge earthquake studies. Note that although the range of *f*_{RV} I have chosen limits *V*_{r} to subshear rupture speeds, these are only average values and the rupture timing perturbations in the GP implementation, which are partially correlated with the local slip, allow for faster, and lower, rupture speeds to occur over limited sections of the fault surface (see Graves and Pitarka 2016 for details).

The sampling of the down-dip fault width is done by first specifying a median value (*W*_{med}), and then setting additional values at ± 12.5% and ± 25% with respect to the median value. This results in a set of five discrete fault width values: \(\frac{3}{4}{W}_{\mathrm{med}}\), \(\frac{7}{8}{W}_{\mathrm{med}}\), \({W}_{\mathrm{med}}\), \(\frac{9}{8}{W}_{\mathrm{med}}\), and \(\frac{5}{4}{W}_{\mathrm{med}}\). Setting the maximum and minimum values at ± 25% about the median is a suitable choice based on results from previous finite-fault inversions (e.g., Mai et al. 2016). The choice for additional values at ± 12.5% represents a practical trade-off between the desire to adequately sample the range and also keep the number of simulations at a reasonable level. Based on the previous studies of the Northridge earthquake, I set *W*_{med} at 24 km. Using the above formulation then yields the following set of fault width values to be sampled for the Northridge case: 18 km, 21 km, 24 km, 27 km, and 30 km.

Combining the six discrete values of *f*_{RV} and the five discrete values fault width described above results in a grid of 30 unique pairs of these parameters, as illustrated in Fig. 3. This figure also plots sample ruptures generated using selected combinations of the parameters. These plots illustrate the effects of changing the average rupture speed or down-dip width on the resulting kinematic rupture model. In particular, they show that the total rupture time decreases with increasing average rupture speed, and the average (and peak) slip increases with decreasing fault width. Also, note that the slip distribution for each rupture is different, with each case being generated using a different random seed.

The final requirement needed to complete the kinematic rupture description is the specification of the slip-rate function. In the GP approach, the slip-rate function is based on the formulation of Liu et al. (2006). This function has a sharp initial pulse followed by a relatively long tail, similar to that seen in dynamic rupture simulations (e.g., Pitarka et al. 2021). The slip rise time is defined as the total time duration of the function.

For the GP method, the average slip rise time (*t*_{A}) is given by a modified version of the relation from Somerville et al. (1999):

$${\tau }_{\mathrm{A}}={\alpha }_{\mathrm{T}}{\times c}_{\mathrm{RT}}\times {10}^{-9}{\times M}_{\mathrm{o}}^{1/3},$$

(2)

where *M*_{o} is the seismic moment in dyne-cm, *c*_{RT} is a scaling constant, and *a*_{T} is a mechanism scaling factor, which reduces the rise time up to a maximum of 10% in the limiting case of pure reverse faulting and is given by

$${\alpha }_{\mathrm{T}}={\left[1+{F}_{\mathrm{D}} {F}_{\mathrm{R}} {c}_{\alpha }\right]}^{-1},$$

(3a)

where

$$F_{{\text{D}}} = \left\{ \begin{array}{ll} 1 - \left( {\delta - 45{^\circ } } \right)/45{^\circ } &\quad 45{^\circ } < \delta \le 90{^\circ } \\ 1 & \quad{\delta \le 45{^\circ } } \end{array} \right.,$$

(3b)

$$F_{{\text{R}}} = \left\{ \begin{array}{ll} 1 - \left| {\lambda - 90{^\circ } } \right|90{^\circ } & \quad 0 \le \lambda \le 180{^\circ } \\ 0 &\quad {{\text{otherwise}}} \\ \end{array} \right.,$$

(3c)

with \({c}_{\alpha }=0.1\), and where δ and λ are the average fault dip and rake, respectively.

While the local rise time at any given point on the rupture depends on both depth and amount of slip in the GP approach (see Graves and Pitarka 2016 for details), the slip rise time given by Eq. (2) represents the value averaged across the entire fault. Somerville et al. (1999) originally determined the scaling constant *c*_{RT} to be 2.0 based on analysis of rupture models inverted from past earthquakes. However, Graves and Pitarka (2010) found *c*_{RT} = 1.6 provided a reasonable fit in simulating broadband motions of four past earthquakes, although these simulations were restricted to a constant average rupture speed. Recognizing that the average rise time is likely variable for different earthquakes, in the present study I allow the scaling factor *c*_{RT} to sample across the four discrete values: 1.6, 2.0, 2.4, and 2.8. I chose this range based on results of preliminary simulations of the current event set that indicated the average rise time for the strike-slip events was noticeably underpredicted when using *c*_{RT} = 1.6. Figure 4 plots slip-rate functions corresponding to these four values of *c*_{RT}. These plots illustrate that as the rise time increases the peak amplitude of the slip-rate decreases and the function becomes relatively deficient in shorter period energy.

Combining the sets of six *f*_{RV}, five fault widths and four *c*_{RT} samples yields 120 unique combinations of these parameters that are used as inputs to the kinematic rupture generator. Additionally, for each of these unique combinations, I generate two slip model realizations. This then results in a total of 240 different rupture models that are considered in the validation exercise. While performing additional realizations (e.g., more densely sampling the range of discrete rupture parameter values or more slip realization per unique combination) might provide a more detailed statistical distribution of the results, the current choice affords a reasonable balance between computational burden and resolution of ground motion trends with respect to the kinematic parameters.

### Observed ground motion records

Ground acceleration waveforms for each of the recording sites shown in Fig. 2 were obtained from the NGA-West2 Database (ngawest2.berkeley.edu). These recordings come from various organizations including the U.S. Geological Survey, California Strong Motion Instrumentation Program, University of Southern California, Southern California Edison, and Los Angeles Department of Water and Power. As part of the NGA-West2 project, these data have been uniformly processed and were subsequently utilized in the development of the NGA-West2 ground motion models (Ancheta et al. 2014). Most of these records have a usable bandwidth of 0.2 to 25 Hz; although some records are reliable down to 0.1 Hz (all recordings were analog so recovery of lower frequency energy requires careful processing, see Ancheta 2014).

In order to concentrate on the effects of the rupture process in the Northridge validation exercise, I have limited the maximum rupture distance of the observations to 30 km. This yields a total of 53 sites as shown by the red and blue triangles in Fig. 2. These sites are located on a variety of geologic settings with *V*_{s30} values ranging from 191 m/s (deep basin) to 2016 m/s (hard rock) (Wills et al. 2015). A list of these sites including location, station name, distance to fault rupture, *V*_{s30}, and usable frequency bandwidth is provided in the Additional file 2. For the ground motions at each site, I have rotated the horizontal component records to orientations of 122° and 212°, which correspond to fault-parallel (FP) and fault-normal (FN) components, respectively. Using this rotated set of records, I then compute 5% damped pseudo-spectral acceleration (PSA) for the individual components, as well as the RotD50 spectral acceleration parameter for the combined horizontal components.

### Broadband simulations

The broadband simulations were run using the hybrid GP method (Graves and Pitarka 2010, 2015) as implemented in version 22.4.0 of the BBP. This implementation includes minor modifications to the high-frequency simulation code to (1) eliminate the slight dependence of the computed response to subfault size present in previous implementations, and (2) increase the sensitivity of the high-frequency motions to variations in fault area via an adjustment to the median stress parameter. Details of these modifications are provided in Appendix A. The BBP implementation is limited to plane-layered (1D) velocity structures and for the Northridge simulations I used the Los Angeles region model listed in the Additional file 3. This model was developed by averaging vertical 1D profiles taken from the SCEC Community Velocity Model CVM-S4.26.M01 at locations of seismic stations throughout the Los Angeles region with the reference V_{s30} value set at 500 m/s. Green’s functions for the low-frequency portion (*f* < 1 Hz) of the simulation are computed using a frequency–wavenumber approach (Zhu and Rivera 2002) with anelastic attenuation model as *Q*_{s} = 50 × *V*_{s} (*Vs* in km/s) and *Q*_{p} = 2 × *Q*_{s}. The 1D velocity model is also used to compute impedance and geometric spreading effects in the high-frequency (f > 1 Hz) portion of the simulation with *Q* effects modeled using the parameters of Graves and Pitarka (2010) and the high-frequency spectral decay parameter *κ*_{0} (Anderson and Hough 1984) set to 0.04 s. Additionally, the median stress parameter is set to 50 bars for all high-frequency computations. Given the maximum variation in prescribed fault width of ± 25% with respect to the median value, the resulting maximum adjustment to the median stress parameter is about ± 12% following the modification described in Appendix A.

In the hybrid simulation, the separate low- and high-frequency portions are computed independently and then combined using a match-filtering approach (e.g., Hartzell and Heaton 1995) with the matching frequency set at 1 Hz. The final step in the simulation process is to adjust the simulated motions from the reference *V*_{s30} condition of 500 m/s to the site-specific value at each location, which is taken from the NGA-West2 database. This site adjustment is done using the implementation of Graves and Pitarka (2010) with the site term from Boore et al. (2014). The result of the simulation is a broadband (0–20 Hz) three-component ground motion waveform at each site for the prescribed rupture model.

Computing the full broadband response at all 53 sites for one rupture model takes about 30 min using a standard implementation of the simulation codes. Because each run is independent, the simulations for different rupture models can be run in parallel. By distributing the computations across multiple CPUs, the total run time to complete all 240 realizations is less than a few hours.

The simulation results for each rupture model are compared with the observations using a spectral acceleration goodness-of-fit (GoF) criteria (e.g., Abrahamson et al. 1990; Schneider et al. 1993). This is done by first computing the 5% damped PSA and RotD50 values from the simulated FP and FN horizontal components of motion at each site. Then, residuals are computed for each site *j* as a function of period *T*_{i} in the natural log domain:

$${r}_{Xj}\left({T}_{i}\right)={\text{ln}}\left[{O}_{Xj}\left({T}_{i}\right)/{S}_{Xj}\left({T}_{i}\right)\right],$$

(4)

where and *O*_{Xj} and *S*_{Xj} are the observed and simulated responses, respectively, and *X* = FP, FN or RotD50 to specify the component. The model bias for each component is then given by:

$${B}_{X}\left({T}_{i}\right)=\frac{1}{{N}_{X}}{\sum }_{j=1,{N}_{X}}{r}_{Xj}\left({T}_{i}\right),$$

(5)

and the standard deviation is given by

$${\sigma }_{X}\left({T}_{i}\right)={\left\{\frac{1}{{N}_{X}}{\sum }_{j=1,{N}_{X}}{\left[{r}_{Xj}\left({T}_{i}\right)-{B}_{X}\left({T}_{i}\right)\right]}^{2}\right\}}^{1/2},$$

(6)

where *N*_{X} is the number of sites for each component. For the FP and FN components, I limit the distance to within 15 km from the rupture, yielding \({N}_{\mathrm{FP}}={N}_{\mathrm{FN}}=21\) (blue triangles in Fig. 2); and for RotD50, I include all sites out to 30 km from the rupture, yielding \({N}_{\mathrm{RotD}50}=53\). The reason for limiting the distance for the FP and FN components in this manner is to focus on the component specific response only at near-fault sites where wave propagation and scattering effects that are not captured by our assumed 1D velocity structure may be less significant.

In order to distill the large amount of GoF results down to a more manageable number, I compute the average absolute misfit value (|*M*|_{avg}) for each of the 240 realizations. This is done by first summing the absolute value of the GoF bias values over the period range 0.1 to 10 s for each of the RotD50, FN, and FP components, then combining these using a weighted sum based on the number of stations in each set. The formal expression is

$${|M|}_{\mathrm{avg}}\,=\,{w}_{\mathrm{RotD}50}{\sum }_{i=1,\mathrm{NP}}\left|{B}_{\mathrm{RotD}50,i}\right|+{w}_{\mathrm{NF}}\left\{{\sum }_{i=1,\mathrm{NP}}\left|{B}_{FN,i}\right|+{\sum }_{i=1,\mathrm{NP}}\left|{B}_{\mathrm{FP},i}\right|\right\},$$

(7)

where *NP* is the number of discrete periods sampled in the range of 0.1 to 10 s, and the weights are \({w}_{\mathrm{RotD}50}={N}_{\mathrm{RotD}50}/\left({N}_{\mathrm{RotD}50}+{N}_{\mathrm{NF}}\right)\) and \({w}_{\mathrm{NF}}={N}_{\mathrm{NF}}/\left({N}_{\mathrm{RotD}50}+{N}_{\mathrm{NF}}\right)\) and where I have also defined \({{N}_{\mathrm{NF}}=N}_{\mathrm{FP}}={N}_{\mathrm{FN}}\).

I use the absolute value in determining |*M*|_{avg} so that positive and negative biases do not cancel. However, I also compute the average bias as well, to determine if the given realization over- or under-predicts the observed motions, on average. I refer to this later quantity as the misfit trend and it is normalized to the range − 1 to 1 across the 240 realizations. Figure 5 plots the |*M*|_{avg} values for the 240 realizations as a function of fault width, *f*_{RV} and *c*_{RT} with the symbols color-coded by the misfit trend.

The lowest |*M*|_{avg} case occurs for a fault width of 27 km, *f*_{RV} = 0.90 and *c*_{RT} = 1.6. The plots indicate that there is a clearly defined minimum in |*M*|_{avg} as a function of *f*_{RV} that occurs near the upper bound value of 0.90. The minimum in |*M*|_{avg} as a function of *c*_{RT} is not as sharply defined, but does occur towards the lower bound value of 1.6. There is no apparent minimum in |*M*|_{avg} as a function of fault width. These results indicate that the GoF results are most sensitive to variations in average rupture speed, and relatively less sensitive to variations in average rise time and fault width, respectively, for the parameterization used here. Additionally, most of the realizations indicate an average bias that under-predicts the observations (blue misfit trend), with the few realizations that show an average over-prediction (red misfit trend) occurring for the smallest fault widths and highest average rupture speeds.

In order to examine possible trends and inter-correlations between the rupture parameters, I plot in Fig. 6 the misfit values as a function of discrete parameter pairings: *f*_{RV} vs. fault width, *f*_{RV} vs. *c*_{RT}, and fault width vs. *c*_{RT}. Determining the values shown at individual points in the panels of this figure is done by averaging the |*M*|_{avg} values across all simulations having ruptures with a particular combination of parameters. For example, each point in the *f*_{RV} vs. fault width plot represents the average of the misfit values across four *c*_{RT} values and two slip realizations. Points in the other panels are determined in a similar fashion.

The most apparent feature seen in the results in Fig. 6 is the strong trend of decreasing misfit with increasing *f*_{RV} (also clearly observed in Fig. 5). However, the results shown in the panels of Fig. 6 also suggest that in terms of producing a similar misfit level, *f*_{RV} is somewhat positively correlated with fault width, *f*_{RV} is somewhat positively correlated with *c*_{RT}, and fault width exhibits a weak negative correlation with *c*_{RT}. While these correlations are subtle, they make sense intuitively. That is, with all other parameters held fixed, increasing average rupture speed (*f*_{RV}) will generally increase ground motion levels, while increasing fault width (i.e., decreasing average slip) or increasing average rise time (*c*_{RT}) will generally decrease ground motion levels. Thus, as values of different parameters are either increased or decreased, their combined impacts on ground motion level can cancel one another, which result in the apparent correlations.

More detailed results for the three realizations with the lowest |*M*|_{avg} are shown in Fig. 7. The (fault width, *f*_{RV}, *c*_{RT}) combinations for these are Rup-184: (27 km, 0.90, 1.6), Rup-140: (24 km, 0.90, 2.4) and Rup-035: (18 km, 0.85, 2.0), and the |*M*|_{avg} values are 0.086, 0.094, and 0.095, respectively. The top panels of Fig. 7 show the rupture models for each of the realizations, and the other panels show the detailed GoF results across the period range 0.1 to 10 s for each of the RotD50, FP (122°), and FN (212°) components. This period range encompasses the vast majority of engineered structures where seismic loads are a concern. For each realization, the bias of the RotD50 component is near zero across the full period range, indicating that all of these models provide a good fit to the observations when averaged over the 53 sites. All of the RotD50 GoFs exhibit a slight over-prediction for periods shorter than 0.2 s, which then increases somewhat at the near-fault sites (FP and FN components). This is likely caused by under-estimating the strength of nonlinear effects through the use of the Boore et al. (2014) site terms in these simulations. Additionally, while the GoFs for the FP and FN components show more variability as compared to the RotD50 component (due in part to the fewer number of stations used to compute the GoF), they all show a systematic tendency to under-predict the FP component and over-predict the FN component for periods greater than 1 s. These systematic features at the longer periods are likely due to over-estimating the strength of rupture directivity effects at the near-fault sites in the simulations by using a 1D velocity structure. It is known that the geologic structure in the region just up-dip of the Northridge rupture has strong lateral heterogeneities (e.g., Hartzell et al. 1999) and these 3D structures can scatter the propagating wave field and diminish the coherence of radiation pattern effects.

While the GoF plots shown in Fig. 7 provide an assessment of how well the simulations perform when averaged across all sites, it is also important to examine if there are any underlying trends with respect to propagation distance or site type. This is done in Fig. 8, which plots the residuals as a function of distance to rupture (*R*_{rup}) and V_{s30} for selected oscillator periods. For brevity, I only show results for the lowest misfit case (Rup-184) in Fig. 8; however, results for the other low |*M*|_{avg} cases are quite similar to these. Additionally, for this analysis I have divided the sites into two groups: one for sites with V_{s30} > 400 m/s (22 sites) and the other for sites with V_{s30} < 400 m/s (31 sites). I chose 400 m/s for this division since it is roughly near the median of the V_{s30} values of the sites considered in the analysis. I then perform linear regression for each of these V_{s30} groups at each period to find the best-fitting line along with its 95% confidence bounds.

The results in Fig. 8 show that there are no statistically significant trends as a function of distance for either V_{s30} group at periods of 1 s and shorter. At periods of 2 s and longer, the residuals for sites with V_{s30} > 400 m/s still show little trend with distance; however a slight trend of increasing residuals with increasing distance is apparent for the V_{s30} < 400 m/s site group. This trend is dominated by sites having large positive residuals (i.e., simulation under-predicts observation) at *R*_{rup} distances greater than about 20 km. These sites having relatively low V_{s30} at *R*_{rup} > 20 km are all located in the northern portion of the Los Angeles basin just south of the Santa Monica Mountains (Fig. 2). Previous studies (e.g., Graves et al. 1998; Olsen et al. 2003) have noted that sites within the Los Angeles basin experienced significant amplification of longer period (*T* > 1 s) motion during the Northridge earthquake due to the generation of surface waves along the northern margin of the basin. Hence, I surmise that the underprediction of these motions in the current analysis results from the use of 1D Green’s functions, which are not able to capture this 3D wave propagation effect.

Figure 9 compares simulated three-component ground velocity waveforms for the lowest misfit ruptures with observations at the three selected sites indicated in Fig. 2. In general, all of the simulated waveforms do reasonably well in matching the amplitude level, shaking duration, and frequency character of the observed waveforms. This includes features such as the strong pulse-like motions on the fault-normal component at station SYL due to rupture directivity (and to a lesser extent at station CNP), and the extended duration, relatively higher frequency motions at station CWC, which is located to the side of the rupture. One notable feature that is not reproduced by the simulations is the large amplitude, late arriving, low-frequency pulse observed at CWC. This is likely a basin phase that is generated by the interaction of the direct waves with the 3D structure of the San Fernando basin, and which is not present in our 1D Green’s functions.

While the general character of the simulated motions shown in Fig. 9 is quite similar, there are differences among these cases that are apparent in the waveforms. For example, the difference in relative amplitude of the initial pulses at CNP, the amplitude of the fault-parallel motions at CWC, and the timing and amplitude of the large directivity pulse at SYL. These differences do not have a strong effect on the spectral acceleration GoF results; however, they demonstrate that the use of a single metric such as PSA can be non-unique in terms of the waveform features produced by a particular rupture.