Spectral decay parameter (κ) and seismic quality factor (Qβ) in northern Baja California, Mexico
Parámetro de decaimiento espectral (κ) y factor de calidad sísmica (Qβ) en el norte de Baja California, México
Luis Munguía1,*
1 División de Ciencias de la Tierra, Centro de Investigación Científica y de Educación Superior de Ensenada, B. C., Carretera Ensenada-Tijuana No. 3918, Zona Playitas, C. P. 22860, Ensenada, Baja California, México.
* Corresponding author: (L. Munguía) This email address is being protected from spambots. You need JavaScript enabled to view it.
How to cite this article:
Munguía, L. (2026). Spectral decay parameter (κ) and seismic quality factor (Q β) in northern Baja California, Mexico: Boletín de la Sociedad Geológica Mexicana, 78(2), A111025. https://doi.org/10.18268/BSGM2026v78n2A111025
Manuscript received: June 27, 2025. Corrected manuscript received: September 23, 2025. Manuscript accepted: October 6, 2025.
ABSTRACT
Understanding the spectral attenuation parameter κ and the seismic quality factor Q β is essential for seismology and earthquake engineering. This study estimates both parameters using seismic data from northern Baja California, including earthquakes of magnitudes 4.0 to 5.8 recorded within 140 km of the epicenters at sites with diverse geological conditions. Kappa was independently estimated from horizontal acceleration spectra recorded at eight sedimentary and seven rock sites. It was derived from the slope of the high-frequency amplitude decay in smoothed spectra, using frequency bands between 25 and 42 Hz.
A linearly increasing trend of κ with distance was observed at both rock and alluvium sites, with average slopes of 0.00025 s/km and 0.00016 s/km, respectively. The zero-distance intercepts of κ (κ₀) were 0.0200 s for rock and 0.0367 s for sediment. A frequency-dependent Q β model for the Mexicali Valley was also developed using S-wave spectra from eight earthquakes. Shear-wave attenuation was modeled with the factor exp (−πf[R/(Q β(f) β) + κ₀]), which incorporates the site-specific κ₀ = 0.0367 s, estimated in this study to account for high-frequency decay near the surface. Q β was initially estimated through individual event inversions and subsequently through a joint inversion of the spectra from all eight earthquakes. In both stages, Q β was estimated separately from the spectra of each horizontal component, as well as from their averaged spectra.
All inversions utilized 20 discrete frequencies spanning the 2–40 Hz range. The first stage yielded an average Qβ = 16.4 f 1.0, while the second produced Qβ= 24.1 f 0.98. Although both results are consistent, greater confidence is placed in the model obtained from the joint inversion of all events. The κ and Q β( f) estimates obtained in this study are essential for improving strong ground motion predictions and refining regional seismic hazard assessments.
Keywords: attenuation, spectral decay parameter κ, quality factor Q β, Baja California.
RESUMEN
La comprensión del parámetro de atenuación espectral κ y del factor de calidad Q β es esencial para la sismología y la ingeniería sísmica. En este estudio se estiman ambos parámetros usando datos sísmicos del norte de Baja California, correspondientes a eventos de magnitudes 4.0–5.8 registrados a menos de 140 km de los epicentros. Kappa se estimó de espectros de aceleración horizontal registrados en ocho sitios sedimentarios y siete sitios en roca. Se derivó de las pendientes de la atenuación de amplitudes de alta frecuencia en espectros suavizados, utilizando bandas de frecuencia de entre 25 y 42 Hz.
Los resultados mostraron una tendencia lineal creciente de κ con la distancia, tanto en sitios sobre roca como en sedimentos, con pendientes promedio de 0.00025 s/km y 0.00016 s/km, respectivamente. Los valores de κ a distancia cero (κ₀) fueron de 0.0200 s para roca y 0.0367 s para sedimento. También se desarrolló un modelo de Q β dependiente de la frecuencia para el Valle de Mexicali, utilizando espectros de ondas S de ocho sismos. La atenuación se modeló mediante el factor exp(−πf [R/(Q β(f) β) + κ₀]), usando el valor κ₀ específico del sitio, igual a 0.0367 s, estimado en este estudio. Q β se estimó primero mediante inversiones por evento y luego mediante una inversión conjunta de los ocho sismos. En ambas etapas, Q β se derivó de los espectros horizontales y de su promedio a frecuencias de 2 a 40 Hz. De la primera etapa se obtuvo, en promedio, Q β = 16.4 f1.0, la segunda produjo Q = 24.1 f0.98. Aunque ambos resultados son consistentes, se prefiere el modelo obtenido de la inversión conjunta de todos los eventos. Las estimaciones de κ y Q β(f) obtenidas en este estudio son útiles para optimizar las predicciones de movimientos fuertes del suelo y las evaluaciones del peligro sísmico regional.
Palabras clave: atenuación, parámetro de decaimiento espectral κ, factor de calidad Q β, Baja California.
1. Introduction
It is well established that elastic waves radiated by seismic sources can be attenuated or amplified as they propagate from the source to a site of interest (e.g., Boore and Joyner, 1997; Cotton et al., 2006). A common and effective method for analyzing these attenuation and amplification effects on ground motion amplitudes is to use logarithmic plots of the Fourier acceleration spectra. As first described by Hanks (1982), such spectra are typically flat between the source corner frequency and a higher corner frequency (fmax), beyond which they exhibit a rapid decay. These spectra provide valuable information about both the earthquake source and the propagation of seismic waves through the Earth’s crust, making them essential tools in seismology and earthquake engineering applications.
Anderson and Hough (1984) characterized the high-frequency portion of the acceleration spectrum using the spectral decay parameter κ. They demonstrated that, for frequencies f >fE, this portion of the acceleration spectrum—when plotted on log-linear axes—can be approximated by an exponential decay of the form exp (−π κ f), where fE marks the onset of the linear decay in spectral amplitude. In this formulation, the acceleration spectrum at high frequencies is represented by the equation
where A0 is a parameter dependent on both the source characteristics and the propagation distance. A detailed understanding of κ is fundamental for applications such as strong ground motion simulations and seismic hazard assessments (e.g., Douglas et al., 2010; Campbell et al., 2009; Ktenidou et al., 2016).
Previous studies have observed a linear relationship between individual estimates of κ and the epicentral distance R at which they were obtained (e.g., Anderson and Hough, 1984; Hough et al., 1988; Anderson, 1991; Ktenidou et al., 2013; Ktenidou et al., 2016). The best-fitting lines, determined using the least-squares method, are commonly expressed in the following form:
where κ0 represents the extrapolated value of κ at zero distance (interpreted as a site-specific parameter), and κR is the slope that accounts for additional attenuation due to wave propagation through the crust (e.g., Anderson and Hough, 1984; Hough et al., 1988; Fernández et al., 2010; Douglas et al., 2010; Ktenidou et al., 2013; Ktenidou et al., 2015; Ktenidou et al., 2016; Lai et al., 2016). The parameter κ0 is independent of distance and thus free from the regional attenuation effects associated with wave propagation. Reported values of κ0 typically range around 0.06 s for sedimentary sites and approximately 0.02 s for rock sites across various seismic regions worldwide (Atkinson and Silva, 1997; Fernández et al., 2010; Castro and Ávila, 2015; Alibazi and Saffari, 2019; Biro et al., 2020). In northwestern Mexico, several studies have investigated the spectral decay parameter κ. Rebollar (1990) analyzed microearthquake data recorded at distances less than 30 km and reported average κ0 values of 0.0328 s and 0.0252 s for the two rock sites marked with white asterisks in Figure 1. He also observed that κ appeared to be independent of distance at such short distances. Fernández et al. (2010) estimated κ using earthquakes with magnitudes less than 3.5 from the southern Basin and Range province in Sonora, Mexico. Their results showed that κ increased with distance up to 70-80 km but decreased between 80 and 100 km. This pattern was attributed to stronger attenuation (lower Q) in the shallow crust and reduced attenuation (higher Q) at greater depths, which dominate longer propagation paths. Additionally, κ was found to be independent of earthquake magnitude. The average κ0 estimated for the region was approximately 0.04 s. Castro and Ávila (2015) estimated κ and κ0 using data from thirteen earthquakes with magnitudes between 5.1 and 6.6, with epicenters located in the Gulf of California. They observed κ values around 0.04 s at distances up to 60 km; however, as the distance increased to approximately 500 km, average κ values increased up to 0.18 s. For most sites, κ0 values were close to 0.03 s, except for two sites on the eastern side of the gulf, where less consolidated soils yielded higher κ0 values of 0.05 s and 0.065 s. They also found that, for most sites, κ0 did not correlate with either earthquake magnitude or back azimuth.
Anderson and Hough (1984) analyzed data from earthquakes that occurred in the Imperial Valley between 1934 and 1968, with magnitudes ranging from 5.4 to 6.8. These events were recorded on analog instruments at a single station in El Centro, California, located just north of the Mexicali Valley. At the time, the El Centro station was equipped with low natural frequency sensors (15–16 Hz), and the recordings were affected by noise introduced during the manual digitization process. Despite these limitations, the researchers observed consistent slopes in the horizontal acceleration spectra of S-waves. The estimated κ values increased linearly with epicentral distance R, following the relation: κ = 0.054 s + (0.00041 s/km) R.
In addition to the spectral decay parameter κ, the anelastic attenuation factor, commonly referred to as the seismic quality factor Q , is of considerable importance in seismology and earthquake engineering. Q measures seismic wave energy loss from anelastic attenuation, especially at higher frequencies. Accurate estimation of Q is essential for earthquake-resistant design, seismic hazard assessment, and evaluation of site-specific ground response.
Previous studies in the Mexicali Valley have addressed seismic attenuation through estimates of the quality factor Q. Singh et al. (1982) analyzed aftershocks of the 1979 Imperial Valley earthquake to determine Q along the Imperial Fault zone, while Domínguez et al. (1997) obtained Q values from coda-wave analysis of local earthquakes recorded within 8 km in the Cerro Prieto geothermal field, Baja California. These works provide an important reference framework for the present study, which extends the analysis to a broader set of seismic data and conditions.
The two main objectives of this study are: (1) to estimate the spectral decay parameter κ using high-quality digital acceleration records, and (2) to develop a frequency-dependent shear-wave quality factor model, Q β(f), for the sediments of the Mexicali Valley. To achieve these objectives, an acceleration dataset comprising 204 earthquakes recorded at sedimentary sites and 95 earthquakes recorded at rock sites was analyzed. Most events had magnitudes between 4.0 and 5.8. Although some were recorded at distances up to 140 km, the majority occurred within 80 km of the recording stations. It is worth noting that this is the first study to evaluate the spectral decay parameter κ and to develop a frequency-dependent Q β(f) model based on the analysis of such a large strong- motion dataset from the region.
2. Area of study and data
The study area is in the northern region of Baja California, Mexico (Figure 1). Geologically, this region comprises two main provinces: the Baja California Peninsular Ranges (BCPR) to the west and the sedimentary environment of the Mexicali Valley (MV) to the east. North of the Mexico–United States border, the extension of the Mexicali Valley is known as the Imperial Valley. Together, the Mexicali and Imperial Valleys form part of a tectonically active basin known as the Salton Trough, which extends from the San Gorgonio Pass in southern California southeastward to the Gulf of California. This region is characterized by high tectonic activity, with frequent small- to moderate-magnitude earthquakes occurring along major fault systems, including the Imperial–Cerro Prieto and El Mayor–Cucapah fault systems (Frez and González, 1991; Hauksson et al., 2010). These and other mapped Quaternary faults are shown as solid lines in Figure 1.
The strong-motion data used in this study were obtained from permanent stations of the Northwest Mexico Accelerograph Network (RANM, now RANOM), operated by the Center for Scientific Research and Higher Education of Ensenada, Baja California (CICESE). The locations of the stations that provided data for this study are shown as small squares in Figure 1, and their geographic coordinates, together with the corresponding instruments, are listed in Table 1. Eight stations are located within the sedimentary environment of the Mexicali Valley, while seven are situated on the granitic rocks of the Baja California Peninsular Range. All stations are equipped with 2g accelerometers and record free-field ground motions at a sampling rate of 200 samples per second. Prior to 2009, the network stations operated with AltuS-Etna and K2 accelerographs, both manufactured by Kinemetrics. Since 2009, some stations have been equipped with GEOSIG GMS-18 instruments, as listed in Table 1. Most instruments provide flat amplitude responses up to approximately 200 Hz and resolutions ranging from 16 to 24 bits. As shown in the following section, the high-resolution digital recordings obtained with these instruments ensure reliable data with minimal uncertainty.
Figure 1 shows the epicenters of the earthquakes analyzed in this study, represented by open circles, black dots, and stars. Open circles correspond to earthquakes recorded at sedimentary sites, black dots to those recorded at rock sites, and stars denote the epicenters of three significant historical events: the 2010 El Mayor–Cucapah earthquake (M 7.2), the 1979 Imperial Valley earthquake (M 6.5), and the 1980 Victoria earthquake (M 6.3). All other events occurred between 1996 and 2018 and had magnitudes ranging from 4.0 to 5.8. The selected magnitude range, combined with the high quality of the recordings, ensures adequate spectral bandwidth above the corner frequency for reliable κ estimation (e.g., Ktenidou et al., 2016; Palmer and Atkinson, 2023).
The dataset comprises 204 earthquakes recorded at sedimentary sites and 95 at rock sites. Approximately 75% of the events recorded at sedimentary sites had focal depths between 4 and 12 km, with an average depth of 8 km. Similarly, about 90% of the events recorded at rock sites had depths ranging from 4 to 15 km, also averaging 8 km. It is worth noting that nearly half of the recordings at rock sites were generated by earthquakes located within the Mexicali Valley. Although some events were recorded at distances up to 140 km, the majority were recorded within 80 km.
3. Strong-motion data selection and processing
Earthquakes recorded at sedimentary and rock sites were analyzed separately to evaluate attenuation specific to each geological setting. The horizontal acceleration components were baseline-corrected prior to computing the Fourier amplitude spectra. Only recordings with clear S-wave arrivals and high signal-to-noise ratios, especially at high frequencies, were used to compute spectra, based on 20-second windows covering the S-wave portion.
As an example, Figure 2 displays the horizontal accelerations recorded at four of the seven stations that captured a magnitude 4.1 earthquake. These time series exemplify the high signal quality that characterizes most recordings used in this study. The bars beneath the GEO North–South accelerogram indicate the 10-second noise window and the 20-second S-wave signal window used for spectral analysis. Figure 3 presents the S-wave acceleration spectra computed from the North–South and East–West components of the GEO station shown in Figure 2. The spectra are plotted using linear and logarithmic scales to highlight the spectral decay characteristics.
The use of 20-second signal windows ensures that the entire S-wave portion is included in the computation of the Fourier amplitude spectra. However, comparable results were obtained using 10-second windows, provided they encompassed the portion of the signal containing the strongest S-wave energy (e.g., Anderson and Hough, 1984; Tsai and Cheng, 2000). Prior to spectral computation, a 5% cosine taper was applied to both ends of each window to minimize spectral artifacts caused by edge effects. Additionally, the resulting spectra were smoothed using one-third octave band averaging to reduce spectral fluctuations. The smoothed spectrum is overlaid on the plots in Figure 3. As clearly shown, the spectral shapes exhibit dominant linear trends in the frequency range of approximately 10 to 40 Hz or higher.
Figures 4 and 5 present log-linear plots of Fourier amplitude spectra obtained from seismic recordings at short and long epicentral distances, respectively. Figure 4 shows the spectra obtained from S-wave signals of a magnitude 4.1 earthquake, recorded at close distances by the first and fourth stations shown in Figure 2. In contrast, Figure 5 illustrates spectra from a magnitude 5.2 event recorded at more distant stations. In both figures, dotted lines represent spectra calculated from 10-second pre-event noise windows, taken before the P-wave arrival. Notably, the signal-to-noise ratio (SNR) remains high up to approximately 25–30 Hz at longer distances (Figure 5), whereas at shorter distances (Figure 4), the SNR extends across a wider frequency range.
Figure 5 also illustrates that, although the usable frequency band for estimating κ from recordings at long distances is narrower, the spectral slopes can still be reliably determined. In our dataset, this situation applies to a limited number of events recorded at distances greater than 80 km. Furthermore, Figures 3 to 5 demonstrate that the spectral smoothing applied improves the stability and consistency of the slope estimation process.
In the top-left panel of Figure 4, f0 marks the corner frequency of the displayed acceleration spectrum. A dashed straight line superimposed on the spectrum represents the regression line that best fits the high-frequency spectral amplitudes. The regression is performed over a frequency window that begins at fE, where the flat portion of the spectrum ends and the high-frequency decay begins, and ends at fx, where the signal amplitude approaches the noise level. Both fE and fx were manually selected from the computer screen based on visual inspection. The slopes of the regression lines within these windows are denoted by the symbol γ in Figure 4. In most cases, the two horizontal components produced very similar slope values, indicating consistency in the high-frequency decay. Figure 6 displays the relationships between earthquake magnitude and the characteristic frequencies f0, fE, and fx obtained from the spectral analysis.
Note that the frequencies derived from spectra at rock sites (right column of Figure 6) are distinguished by appending an “r” to their subscripts (e.g., f0r, fEr, fxr), differentiating them from those obtained at sedimentary sites. For both site types, most acceleration spectra exhibit corner frequencies (f0 and f0r) in the range of 0.3 to 4 Hz. In most cases, the difference between f0 and fE exceeds 2–3 Hz. The observed fE values span from 3 Hz to 15 Hz, with 86% of the cases showing fE to be at least three times greater than the corresponding corner frequency f0. This significant separation between source-related ( f0) and site-related (fE) frequency parameters reduces the likelihood of trade-offs during interpretation (see Ktenidou et al., 2016; Tsai and Chen, 2000).
Depending on the type and quality of the accelerograms, fx values—marking the upper limit of reliable spectral amplitude before noise becomes significant—were measured in the range of 20 to 45 Hz. As a result, the effective frequency bandwidths used for κ estimation (fx − fE) range from 25 to 42 Hz for both sedimentary and rock site data. Following Anderson and Hough (1984), κ was calculated by dividing the slope γ of the high-frequency spectral decay by −π log(e).
4. Results for kappa (κ) and quality factor Qβ
4.1 ANALYSIS OF κ FROM SEDIMENTARY SITE DATA
Kappa values estimated from acceleration spectra recorded at sedimentary sites exhibit distinct characteristics reflecting site-specific attenuation effects. The analysis focused on identifying the high-frequency spectral decay, following the methodology outlined in Section 3, to determine κ for each event–station pair located on sedimentary deposits. Estimates of κ were derived from the two horizontal acceleration components recorded at the eight sediment sites (see inset in Figure 1), averaged, and plotted as a function of epicentral distance R in Figure 7. The dataset includes recordings from earthquakes with magnitudes ranging between 4.0 and 5.8, and epicentral distances up to 140 km. In Figure 7, data from individual sites are combined without differentiation. Notably, unlike some previous studies (e.g., Castro et al., 2000; Douglas et al., 2010; Van Houtte et al., 2011), no significant differences were observed between the κ estimates derived from the two horizontal components of individual earthquake records. Instead, a high degree of similarity between these components was commonly found, as illustrated in Figure 4. The κ values included in Figure 7 meet the criterion that the ratio between the smallest and largest estimates from the two components is at least 80%. Horizontal recordings failing to satisfy this requirement were excluded from the analysis.
A robust least squares method was applied to determine the best-fit line representing the relationship between κ and epicentral distance R. In Figure 7, this regression line is shown in black, with accompanying lines indicating ±1 standard deviation. The equation of the fitted line is also provided in the figure. The results indicate that κ increases linearly with distance, reflecting S-wave energy loss due to geometrical spreading and intrinsic anelastic damping, consistent with findings from other regions (e.g., Anderson and Hough, 1984; Anderson, 1991; Fernández et al., 2010; Douglas et al., 2010; Ktenidou et al., 2013). The observed systematic increase in κ with epicentral distance is consistent with the expected attenuation characteristics of the upper crust. Of particular interest is the extrapolated value of κ at zero distance, which reflects attenuation of vertically propagating shear waves through near-surface soft layers. This distance-independent component of κ, denoted as κ0 by Anderson and Hough (1984), plays a crucial role in controlling the high-frequency decay of the acceleration spectrum.
In this case, where data from all sites are plotted without distinction (as shown in Figure 7), κ is observed to increase with distance at a rate of 0.00015 s/km (κR), with an intercept at zero distance of 0.03756 s (κ0). To reduce the influence of data scatter—particularly at shorter distances—κ values were averaged within 20-km distance intervals, and a linear regression was applied to these averaged values. In Figure 7, the larger gray circles represent the mean κ values for each distance bin, while the dashed and dotted gray lines indicate the regression line fitted to the binned data and its ±1 standard deviation bounds, respectively.
The results show that both the slope (κR) and the intercept (κ0) remain largely unchanged, confirming the robustness and stability of the estimated κ–distance relationship.
Estimates of κ for each of the eight sedimentary sites are shown individually in Figure 8. As in the previous figure, each panel includes a regression line along with two thin lines representing ±1 standard deviation. The Figure shows that both κ and its zero-distance extrapolation, κ0, vary slightly from site to site—a result consistent with the complex crustal structure of the Mexicali Valley (McMechan and Mooney, 1980; Lira-Herrera, 2005). Besides crustal complexity, variations in κ0 may also result from differences in shallow geological conditions (e.g., sediment thickness, degree of consolidation, and material heterogeneity) or from residual source and path effects not fully eliminated during the estimation process. Acting together, these factors can introduce modest site-to-site variability in κ0 even within the same tectonic setting.
A summary of the κR (slope) and κ0 (intercept) values for the sites shown in Figure 8 is presented in Table 2. The third column of the table indicates that κ₀ ranges from 0.024 s at site MDO to 0.046 s at site CHI, while κR varies from 0.00007 s/km (SAL) to 0.00024 s/km (CHI ). In particular, the average values of κR and κ0 —0.00016 s/km and 0.0367 s, respectively—are nearly identical to those obtained when combining data from all sites (Figure 7). The relatively small inter-site variations indicate that the average κ₀ value, calculated from the eight individual site estimates, provides a reasonable representation of the sedimentary region, at least within the range of distances analyzed.
4.1.1 KAPPA ESTIMATES FOR THREE LARGER REGIONAL EARTHQUAKES
The data set analyzed here includes strong-motion records from three significant regional earthquakes: the 1979 Imperial Valley (M 6.5), the 1980 Victoria (M 6.3), and the 2010 El Mayor-Cucapah (M 7.2) events. Estimates of κ and κ0 for these earthquakes are plotted together in Figure 9 as functions of distance. Although the 1979 and 1980 earthquakes had similar magnitudes, reliable data for the 1980 event were only available from two sites. As observed for smaller magnitude earthquakes, κ also increases linearly with distance for these larger events. Comparing Figures 7 and 9 reveals that the slope (κR) and intercept (κ0) differ only slightly, by 0.00008 s/km and 0.0063 s, respectively. However, these differences may stem from the limited data available for the largest events, as well as the potentially lower quality of data from the CAF and AER sites, which were obtained by digitizing analog accelerograms. Another possible explanation is that the seismic motions recorded during strong earthquakes may exhibit nonlinear sediment response. Pérez-Arce et al. (2023) investigated soil nonlinearity by comparing H/V spectral ratios from strong and weak seismic motions in the Mexicali Valley, finding evidence of significant nonlinear behavior at sediment sites during these three large earthquakes.
4.2. ANALYSIS OF Κ FROM ROCK SITE DATA
In this section, data recorded at seven permanent strong-motion stations located on rock sites within the Peninsular Mountain Ranges of Baja California are analyzed (Figure 1). These rock sites have approximate Vs30 shear-wave velocities ranging from 330 to 620 m/s, based on McPhillips et al. (2020) (see Table 3). Accordingly, all sites except HDI—which is classified as Class CD (rigid soil)—are categorized as very dense soil (Class C) according to the National Seismic Hazard Reduction Program (2020) site classification.
In the BCPR region, significantly fewer earthquakes were recorded at rock sites than at sedimentary sites in the Mexicali Valley. The rock dataset includes acceleration records from 95 earthquakes with magnitudes ranging from 4.0 to 5.8. For the CUC and VTR sites, which recorded relatively few events with magnitudes above 4.0, data from some earthquakes with magnitudes between 3.5 and 4.0 were also included. It is worth noting that nearly half of the recordings at rock sites were generated by earthquakes located within the Mexicali Valley.
Values of κ calculated from accelerograms recorded at rock sites are plotted as a function of epicentral distance in Figure 10a. The κ estimates are well distributed over a distance range from near 0 to approximately 140 km. Although some scatter is evident between 80 and 120 km, most values fall within ±1 standard deviation of the best-fit regression line. Figure 10a also shows κ values averaged in 20-km distance bins (gray circles). The regression line fitted to the binned data, along with its ±1 standard deviation bounds, is displayed in gray. Consistent with the observations for sediment sites, the slopes of the regression lines for both the grouped (binned) and ungrouped data are nearly identical. Figure 10b includes only data from earthquakes that occurred and were recorded in the rocky environment, excluding events with epicenters in the Mexicali Valley. In those excluded events, seismic energy traveled from a sedimentary province into a rocky site, potentially influencing the κ estimates. Although these excluded events represent roughly 50% of the data recorded at rock sites, their removal results in only a minor change in the estimated κ0 value (as seen by comparing Figures 10a and 10b). Other than the noticeable data scatter between 80 and 120 km in Figure 10a—which is absent in Figure 10b—no significant differences are observed in the κ estimates between the two figures.
Estimates of κ for earthquakes with epicenters in the Mexicali Valley are shown in the graph of Figure 10c. Figures 10a–c show that the slopes of the fitted lines and the standard deviations are nearly identical across all three graphs, with only slight variations in the corresponding zero-distance intercept values (κ0). Therefore, it appears justifiable to combine data from both geological provinces for the estimation of κ, on the condition that all recordings are obtained from rock sites.
Figures 7 and 10 show that κ increases linearly with distance in both sedimentary and rocky environments. However, as indicated by the parameters κR and κ0, the linear growth trends differ slightly between the two geological provinces. The most notable difference lies in the zero-distance kappa intercept (κ0): for sediment sites, κ0 is 0.0367 s, whereas for rock sites it is 0.0200 s. This substantial difference is consistent with the contrasting near-surface crustal properties of the two regions. Similar observations—where κ₀ values are higher for sedimentary sites than for rock sites—have been reported in various other seismic regions worldwide (e.g., Anderson and Hough, 1984; Douglas et al., 2010; Ktenidou et al., 2012; Alibazi and Saffari, 2019).
Plots of κ versus distance for individual rock sites are shown in Figure 11. Although the number of earthquakes used to generate these plots is smaller than that for sedimentary sites, the mean κ estimates are well distributed over distances up to 140 km. This distribution allowed for reliable estimation of the zero-distance kappa value, κ0. Table 3 presents a summary of κ estimates derived from acceleration spectra recorded at rock sites. As shown, κ0 values for rock sites range from 0.011 s at RSL to 0.036 s at CIC, with an average value of 0.0200 s. Notably, the κ0 value at CIC is significantly higher than at the other sites, likely due to the presence of unconsolidated sediments beneath that location.
4.3. FREQUENCY-DEPENDENT SHEAR-WAVE QUALITY FACTOR (Qβ) FOR THE MEXICALI VALLEY
A commonly used approach to model Fourier acceleration spectra involves introducing a decay time parameter, t⁎, through the exponential attenuation factor e-πt*f (e.g., Cormier 1982; Singh et al., 1982; Anderson and Hough, 1984; among others). This attenuation time t⁎ represents the path-integrated effect of the inverse of the quality factor Q, effectively summarizing the anelastic attenuation of seismic waves. Singh et al. (1982) employed aftershock data from the 1979 Imperial Valley earthquake to estimate the quality factor Q along the Imperial Fault. In their analysis, they modeled the spectral attenuation of SH waves along the fault using the expression
where S(f) is the source spectrum, f is frequency, R is distance and t* is the attenuation time that accounts for energy loss due to anelastic attenuation along the propagation path.
Singh et al. (1982) explicitly separated attenuation into two depth-dependent contributions: the shallow crust (upper ~4 km), modeled by the exponential factor e-πft*, where t* represents the near-surface attenuation parameter, and the deeper crustal path (>4 km to the recording site), modeled by the factor e-πft/Q ( f). In this latter attenuation term Q(f) is the frequency-dependent shear-wave quality factor describing anelastic attenuation during propagation through the deeper crust and upper mantle. Owing to the low shear-wave velocities in the upper ~4 km of the crust, S-waves propagate nearly vertically toward the surface, making attenuation effectively independent of horizontal distance.
Moreover, if Q is assumed to be frequency-independent within this shallow segment of the ray path, then t* is also independent of frequency. Under these assumptions, the S-wave spectra are well represented by equation (3). It should be noted that the expression in square brackets in this spectral representation is analogous to κ in equation (1). By comparing with equation (2), it follows that t* corresponds to κ, and κ equals 1/(Q ). Thus, for t* = 0= 0.0367 s, as found in this study, a constant quality factor Q of approximately 73 is inferred for the upper 4 km of the crust. The κ relation in Equation (2) depends explicitly on distance rather than frequency. In this formulation, κ increases linearly with slope κR and is interpreted as the incremental attenuation associated with S-wave propagation through the Earth’s crust (Anderson and Hough, 1984). Consequently, when κ is interpreted as a site parameter describing spectral decay, it is necessary to assume that Q is frequency-independent and that β represents a uniform average shear-wave velocity along the path (see also Campbell, 2009). At regional distances, however, these assumptions may not hold, as they are strongly influenced by the crustal geological conditions. For the sedimentary region studied here, the mean slope of the regression line fitted to κ estimates is κR = 0.00016 s/km, over the frequency range of 3 to 45 Hz. Assuming an average shear-wave velocity of 3.0 km/s for sediments in the Mexicali Valley (McMechan and Mooney, 1980) and applying Q = 1/(κRβ), a regional constant Q value of approximately 2083 is obtained for depths greater than ~4 km.
The estimated constant Q of 2083 should be regarded only as a crude approximation, since the study area is characterized by a layered crust with significant variations in physical properties (Lira-Herrera, 2005), and Q is expected to vary with frequency. For instance, Singh et al. (1982) showed that, below the upper 4 km along the Imperial fault zone, Q follows the relation Q = 20 f, increasing from about 60 at 3 Hz to 500 at 25 Hz. This evidence motivated the application of a joint inversion of multiple sites and events, which yielded the more robust, frequency-dependent Q β model presented below.
In this section, taking advantage of the larger dataset available from sediment sites relative to rock sites, a frequency-dependent Q β model for the Mexicali Valley is derived using a data inversion technique. For this, we consider that for earthquakes with seismic moment M0 the acceleration spectra of shear waves at a distance R can be represented with the expression (see Boore, 1983).
where
In these expressions, C is a constant that incorporates several physical and geometrical factors. Specifically, it includes the radiation pattern Rθϕ, the free-surface amplification FS, the density ρ, and the shear-wave velocity β. The term S(f) represents the source spectrum, which is modeled here assuming a classic ω2 source model. The corner frequency of the spectrum is denoted as f0. The term D(f) accounts for the attenuation effects due to both geometrical spreading and frequency-dependent anelastic attenuation along the wave propagation path. In this study, D(f) is modeled using an additional exponential term, in contrast to the high-cut filter originally proposed by Boore (1983), which was designed to suppress high-frequency spectral amplitudes beyond a limiting frequency, fmax (see Hanks, 1982).
This exponential attenuation factor explicitly incorporates the site-specific parameter κ0, which represents the decay of high-frequency energy near the surface at zero distance, particularly for waves propagating vertically. In the following calculations, an average value of κ0 = 0.0367 s is used. This estimate, derived from the analysis of sediment-site recordings, represents one of the key results of this study.
The geometrical spreading function, G(R), used in this study follows the two-part model proposed by Singh and Ordaz (1994), which distinguishes between body-wave and surface-wave propagation regimes. Specifically, we adopt:
This formulation reflects the physical transition from geometrical spreading of body waves at shorter distances to a surface-wave dominated regime at larger distances, as observed in the region (see Castro et al., 2009). Although most accelerometric data were recorded at distances shorter than 80 km, a transition distance of R0 = 80 km was adopted for the few events at greater distances, consistent with the regional studies of Castro et al. (2008) and Castro et al. (2009).
After substituting the expressions for the constant C, the source spectrum S(f), and the attenuation function D(f) into Equation (4), taking the natural logarithm of both sides, and rearranging terms, one obtains (Joshi et al., 2010):
where
The residual term e in equation (5) accounts for all unmodeled effects, random error, or discrepancies between observed and predicted spectral amplitudes.
Using equation (5), the acceleration spectra of one or more earthquakes can be inverted to estimate the two primary unknowns: the frequency-dependent quality factor Q β(f) and the source corner frequency f0, for a set of specified frequencies. However, if the corner frequencies f0 are independently determined beforehand, as done in this study, the inversion problem is simplified, leaving Q β as the sole remaining unknown to estimate.
For the inversion process, data from eight earthquakes with magnitudes ranging between 4.2 and 4.7, recorded at four to seven sites per event, were considered (Table 4a). This dataset provided a well-distributed set of spectral observations over varying source-receiver paths, which enhanced the reliability of the inversion results for estimating Q β(f).
Initially, Q β was estimated by inverting the data from each individual earthquake and averaging the resulting values. Subsequently, a joint inversion was performed using all horizontal acceleration spectra from the eight earthquakes considered. In both stages of the analysis, Q β was estimated separately from the N–S and E–W components, as well as from their averaged spectra. All inversions were carried out using 20 frequencies in the range of 2–40 Hz. An example of the inversion results is shown in Figure 12. Panels (a) and (b) present plots of Q β versus frequency, obtained from the N–S and E–W spectra, respectively, while panel (c) displays the results derived from the averaged horizontal spectra.
All plots in Figure 12 include a robust regression line along with lines representing ±1 standard deviation from the mean. The robust regression method employed is less sensitive to outliers and thus provides a more reliable fit to the majority of the Q β estimates.
The short error bars and low standard deviation values reflect the low variability and high consistency of the resulting model. Each graph presents the fitted attenuation model in the standard form Qβ(f) = Q0fn,where Q0is the quality factor at 1 Hz, and n is the frequency-dependent exponent derived from the slope of the best-fit line in the logarithmic plots. Details of the eight processed earthquakes, along with the corresponding Q0 and n values, are summarized in Tables 4a and 4b.
Table 4b presents the variation in the model parameters Q0 and n across the individual earthquakes considered. Although some variability is observed among events, the average values of the parameters—calculated separately from the N–S, E–W, and averaged horizontal spectra—are generally consistent, as shown in the central portion of the table. It is also noteworthy that the standard errors associated with both Q0 and n are small in all cases, indicating stable and reliable estimates. Based on the results obtained in this first stage of the inversion process, the average attenuation model is given by: Q (f) = 16.4×f 1.0. This model reflects a linear dependence of Q β on frequency and serves as a representative approximation of the attenuation characteristics derived from the individual event inversions. The frequency-dependent quality factor Q β was next estimated by jointly inverting all horizontal acceleration spectra recorded for the eight earthquakes listed in Table 4a. As in the previous stage, the two horizontal acceleration components (N–S and E–W) and their averaged spectra were processed separately. The resulting Q β estimates from these three inversion approaches are shown in Figure 13. Notably, the results from all three processes are nearly identical, indicating a high level of consistency and reliability. Although the inversions were conducted over the 2–40 Hz frequency range, the derived model Q (f) = 24.1× f 0.98 is considered valid primarily for frequencies above 3 Hz. This threshold is supported by the noticeable reduction in the size of the error bars associated with the least squares fits at higher frequencies, suggesting improved model stability and reduced uncertainty in that range. This model also suggests that Q β increases nearly linearly with frequency, as the exponent n = 0.98 is very close to 1. A higher Q β implies lower attenuation, so attenuation decreases with increasing frequency according to this relationship. This behavior is consistent with typical seismic wave attenuation observed in many crustal regions, where higher frequencies are less attenuated compared to lower frequencies. The frequency-dependent Qβ model derived from multiple spectra inversion, along with the average model obtained from single-event data inversions, is shown in Figure 14. These are compared with previous models by Singh et al. (1982) and Domínguez et al. (1997). The model by Domínguez et al. (1997) was based on the analysis of coda waves from earthquakes recorded at distances less than 8 km within the Cerro Prieto geothermal field, Baja California, Mexico. In their case, the model parameters Q0 and n were estimated for the 6–24 Hz frequency range, but with significantly higher standard errors than those obtained in this study. Among the models obtained in this study, the relationship Q = 24.1 f 0.98 is adopted as the representative attenuation model for the Mexicali Valley, as it was derived from the joint inversion of multiple acceleration spectra from well-recorded seismic events. This model suggests a slightly higher quality factor across all frequencies compared to that of Singh et al. (1982), indicating lower attenuation, whereas the model by Domínguez et al. (1997) indicates stronger attenuation at frequencies above 15 Hz.
5. Discussion and conclusions
The two main objectives of this study were to analyze the shear-wave spectral decay parameter (κ) in northern Baja California and to estimate the quality factor Qβ in the Mexicali Valley. For this purpose, a high-quality database of strong motion recordings from both sediment and rock sites in northern Baja California was used. A linear increase of κ with distance was observed in both geological settings. The zero-distance intercept, κ0, was estimated for each site in the two provinces. Averaging the site-specific estimates within each region yielded mean κ₀ values of 0.0200 s for rock sites and 0.0367 s for sediment sites. These values are consistent with those reported in other seismic regions worldwide. The slopes of the κ–distance trends for rock and sediment sites differ by only 0.00009 s/km, suggesting that seismic energy attenuation in these provinces is governed by slightly different mechanisms.
Rebollar (1990) estimated κ using data recorded on granitic outcrops within the San Miguel fault zone in northwestern Baja California. Based on digital recordings of several microearthquakes (M = 1.7–3.0) at epicentral distances of 7–30 km, he obtained average κ0 values of 0.0328 s and 0.0252 s at the two sites indicated by white asterisks in Figure 1. These estimates are higher than the κ0 value of 0.020 s determined in this study, which is based on a larger and higher-quality dataset. Furthermore, Rebollar (1990) concluded that κ was independent of distance. A similar finding was reported by Anderson and Hough (1984), who analyzed recordings from five Hollister earthquakes (magnitudes 5.0 to 5.8) at distances of less than 40 km.
In this study, κ was estimated using data recorded over a broader range of distances (up to 140 km) than in the works of Rebollar (1990) and Anderson and Hough (1984). Notably, several stations used here—RAC, RSL, HDI, TRH, and VTR—are located within the same seismic zone analyzed by Rebollar (1990). When κ values from these stations are plotted for distances up to 50 km, they exhibit a clear increasing trend, with a slope of κR = 0.00014 s/km and an intercept of κ0 = 0.0200 s. These parameters are nearly identical to those obtained using the full distance range, indicating that κ increases with distance even at short ranges. This trend is likewise evident in κ values computed at sedimentary sites within 50 km, as illustrated in Figure 7 (see also Ktenidou et al., 2013). Regarding the κ0 values estimated at similar finding was reported by Anderson and sediment sites, the highest were observed at CHI (κ0 = 0.0464 s), TAM (κ0 = 0.0407 s), and SAL (κ0 = 0.0415 s), all located in the northeastern part of the study area along the Imperial fault zone. The average of these three κ0 values is 0.0429 s, which is consistent with the t* = 0.047 s estimated by Singh et al. (1982) for the uppermost 4 km of crust beneath the Imperial fault zone. In that study, Singh et al. modeled the displacement spectra of several aftershocks from the 1979 Imperial Valley earthquake (Mw 6.5). Figure 3 of their paper presents logarithmically averaged displacement spectra for four distance intervals. Here, the high-frequency portions of those spectra were converted to acceleration, and approximate κ and κ0 values were derived by analyzing the slopes of the resulting linear trends. From the average spectra of groups A, B, and C, κ values of 0.0483 s, 0.0541 s, and 0.0543 s, respectively, were obtained. A linear regression of the three κ values against their corresponding average distance intervals of 8, 23, and 32 km (not shown for brevity) yielded a slope of κR = 0.00025 s/km and an intercept of κ0 = 0.0469 s. Remarkably, despite the approximate nature of this analysis, the estimated κ0 is in good agreement with both the t* value of Singh et al. (1982) and the average of the highest κ₀ values obtained at CHI, TAM, and SAL. Lower κ0 values were observed at sites farther south, associated with the Cerro Prieto and Indiviso fault systems.
Another topic of interest concerning the decay parameter κ is its potential dependence on earthquake magnitude. The relationship between κ and magnitude remains unclear and continues to be an active area of research. While some studies suggest that κ may vary with earthquake size, others have found no significant correlation between these parameters (Atkinson and Silva, 1997; Purvance and Anderson, 2003; Fernández et al., 2010; Van Houtte et al., 2011; Yadav et al., 2018; Kumar et al., 2020; Biro et al., 2020, among others). Given that many of the earthquakes analyzed in these studies have magnitudes comparable to or smaller than those in the present work, we constructed Figure 15 to explore whether a dependence between κ and magnitude exists. The Figure presents κ estimates grouped into four narrow magnitude intervals, covering events with magnitudes between 4.0 and 5.8. Within this range, neither the slope κR nor the intercept κ0 shows substantial variation across magnitude bins, suggesting no clear dependence on earthquake size. The minor variations observed are likely due to data scatter within each interval rather than a systematic trend.
Values of κ and κ0 were also calculated using data from three regional earthquakes with magnitudes 6.3, 6.5, and 7.2. The results show that the slope κR and intercept κ0 differ by only 0.00008 s/km and 0.0054 s, respectively, from the average values obtained for smaller-magnitude events. Small earthquakes typically have simple rupture processes and cause limited environmental effects. In contrast, larger earthquakes tend to involve more complex source processes, potential nonlinear soil behavior, and additional attenuation mechanisms that may influence spectral decay (e.g., Biro et al., 2020; Biro, 2024). Therefore, it is reasonable to expect κ to be independent of magnitude for small earthquakes, as observed in this study, but to exhibit some dependence on magnitude for larger events. However, the limited dataset available for the larger earthquakes in this study prevents drawing definitive conclusions about the κ–magnitude relationship in such cases. A more robust understanding will require additional data with broader coverage in both distance and magnitude for the northern Baja California region.
The second main objective of this study was to estimate the frequency-dependent quality factor Q β for sediments in the Mexicali Valley. This was accomplished by first inverting the data from each of the eight events individually and then jointly inverting the spectra from all events. Both approaches yielded consistent results; however, the exponential model Q β = 24.1 f 0.98, derived from the joint inversion of all available data, was ultimately adopted as the most representative. This model indicates that Q β increases nearly linearly with frequency, like the findings of Singh et al. (1982) for the Imperial fault zone, but yields slightly higher values, implying weaker intrinsic and scattering attenuation. The coda-Q model proposed by Domínguez et al. (1997) for a small area within the Cerro Prieto geothermal field indicates weaker attenuation at frequencies below 15 Hz and stronger attenuation at higher frequencies.
The agreement between the final model and previous regional attenuation models supports the reliability of the inversion results and indicates that attenuation in the study area is generally consistent with earlier findings, aside from moderate local variations. These variations may reflect spatial heterogeneities in the crustal structure, differences in site conditions, or methodological and dataset-related factors. Overall, the new model provides an updated, region-specific attenuation relationship that may enhance seismic hazard assessments in the Mexicali Valley area. Finally, the consistency of κ and Qβ estimates across various sites and events underscores the reliability and robustness of our findings.
Contributions of authors
All aspects of the work, including conceptualization, methodology, data analysis, and manuscript preparation, were carried out by the sole author.
Funding
This work was carried out with the financial support of the Center for Scientific Research and Higher Education of Ensenada, Baja California (CICESE).
Acknowledgments
I gratefully acknowledge the long-term financial support provided by the Center for Scientific Research and Higher Education of Ensenada, Baja California (CICESE). I also thank Manuel Luna Munguía for his valuable assistance with data processing and Sergio Arregui Ojeda for his technical support. Many others contributed to this study in various ways, and I extend my sincere appreciation to all of them for their support. Finally, I gratefully acknowledge the valuable comments of two reviewers, which significantly improved this manuscript.
Conflicts of interest
The author declares no conflicts of interest.
Handling editor
Raúl Castro Escamilla.
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