Damage to public infrastructure at or below the ground surface (streets, curbs, and water and gas lines) in southwestern Santa Clara Valley, California, associated with the 1989 Loma Prieta earthquake, is used to support the assertion that the series of photointerpreted lineaments are tectonic in origin and related to long-term reverse faulting along the range front of the Santa Cruz Mountains. We quantitatively analyze whether the photointerpreted lineaments are spatially correlated with earthquake-induced damage by examining whether the damage was located preferentially closer to the mapped lineaments than to a spatially random set of points. The analysis confirms that damage related to the Loma Prieta earthquake is located preferentially close to mapped lineaments. This result supports the assertion that the lineaments have a tectonic origin related to range-front faulting along the Santa Cruz Mountains, and that their presence may be related to primary fault rupture, spatially focused shaking damage, or slip triggered by strong motion induced by nearby faults.
The Santa Clara Valley is located in the southern portion of the San Francisco Bay area, California (Fig. 1), along the active tectonic boundary of the San Andreas fault separating the Pacific and North American plates. The dextral San Andreas fault is the largest and most persistent tectonic feature in the area, but other faults with more oblique and reverse senses of displacement are also present (e.g., Graymer et al., 2006). Active faults bound Santa Clara Valley and may be present within the alluvium of the valley floor itself (Fig. 2). Earthquake slip on any of these faults may pose a threat to those living and working there. Identifying and understanding the damage associated with faults within Santa Clara Valley are critical to understanding the severity of earthquake hazards in the region.
Locating faults that are potentially active but have not ruptured or experienced creep in the recent past is difficult in urbanized areas. Urban growth in the last century in the Santa Clara Valley has covered or modified many of the clues needed to identify such faults. Although geologic mapping that overlaps the study area (Wentworth et al., 1998; Brabb et al., 2000; McLaughlin et al., 2001) depicts surface traces and concealed faults, these faults have been difficult to trace through Quaternary alluvium. Their location is based on geophysical and geomorphic evidence. An important data set for demarcating potential fault locations is a set of lineaments mapped from aerial photographs taken prior to the widespread urbanization. This data set has been interpreted as being associated with bending-fault moments (Hitchcock and Kelson, 1999) and has helped guide the location of several mapped fault traces (Hitchcock and Kelson, 1999; Graymer et al., 2006), and yet direct evidence to support this interpretation from field data is limited.
The 1989 Mw 6.9 Loma Prieta earthquake, located roughly 30 km south of Santa Clara Valley, provided a unique opportunity to identify possible faults within the alluvium of the Santa Clara Valley. The earthquake caused damage to many types of public works, including roads and sidewalks, throughout Santa Clara Valley (Fig. 2; Schmidt et al., 1995, 2014). In total, 1427 observations (previously erroneously reported as 1284 observations in Schmidt et al.  and Langenheim et al. ) combined from extensive field work and the records of governmental and public service agencies were recorded over a 663 km2 area. Schmidt et al. (1995, 2014) reported damage indicators that included pavement breaks in asphalt (roads) and concrete (sidewalks), and subsurface gas and water line ruptures. Damage to above-ground structures was not included because it depends upon building construction, materials, and design, and damage cannot be unequivocally attributed to the Loma Prieta earthquake. Sidewalks, asphalt, and other public infrastructure works are frequently repaired and generally built to uniform specifications and are therefore consistent indicators of ground motion.
The damage data set of Schmidt et al. (1995, 2014) can be used to quantify and test for a possible association between recorded damage and the photointerpretive lineaments that have been used as the basis for subsequent mapping of fault traces in the alluvium outboard of the range front of the Santa Cruz Mountains. A positive spatial association between the two data sets (i.e., they are proximal in space) would reinforce the likelihood that both data sets represent different aspects of fault activity. The association would strengthen ties between earthquakes, such as the Loma Prieta earthquake, that are isolated in time and the long-term processes, such as faulting or distributed deformation associated with blind faults, which leave an imprint on the landscape in the form of lineaments. Either spatially focused shaking due to a change in the material properties across faults (see, for example, Harmsen et al., 2008), or triggered slip along existing fault planes could explain the spatial association of damage with the mapped lineaments.
The Santa Clara Valley is bounded by actively uplifting mountains, on the west by the Santa Cruz Mountains and on the east by the East Bay Hills. The San Andreas fault is the largest and most persistent tectonic feature in the area, exposed southwest of the Santa Clara Valley, whereas the Hayward-Calaveras fault system bounds the range front of the East Bay Hills. The southwestern range front of the Santa Clara Valley, along the base of the Santa Cruz Mountains, contains a series of active southwest-dipping reverse faults approximately parallel to, and northeast of, the San Andreas fault (Fig. 2). These faults, which include the Monte Vista, Berrocal, and Shannon faults, possibly root into the San Andreas fault at depth (McLaughlin and Clark, 2002). The Santa Cruz Mountains are being thrust over the rocks and sediments of the Santa Clara Valley (McLaughlin and Clark, 2002), with an average uplift rate of 0.8 mm/yr (Bürgmann et al., 1994). The existence of a –28 mGal negative gravitational anomaly (Robbins, 1971) centered over the Santa Clara Valley was interpreted by Stanley et al. (2002) as an asymmetrical sedimentary basin of Cenozoic deposits overlying older, denser Franciscan deposits at least 3 km thick. One of the deepest parts of the basin lies between the towns of Saratoga and Los Gatos (Stanley et al., 2002).
The Monte Vista, Berrocal, and Shannon faults are mapped discontinuously along the range front, and several authors have suggested that the faults extend through the alluvium of Santa Clara Valley. For instance, Langenheim et al. (1997) proposed the extension of a northwest-striking reverse fault into the Quaternary alluvium based on two-dimensional gravity and magnetic modeling along a northeast-trending cross section near the town of Saratoga (purple star in Fig. 2). The proposed reverse fault is consistent with the asymmetrical gravitational gradient that extends along the range front, the location of hypocenters at depth, and the direction of motion implied by focal mechanisms of Loma Prieta aftershock events. They visually related concentrations of damage from the data set of Schmidt et al. (1995) to gravity and aeromagnetic gradients. The inferred fault resulting from the Langenheim et al. study connects the Monte Vista fault to the Shannon fault to the southeast.
Geologic mapping in the region over the last three decades has delineated the Monte Vista, Berrocal, and Shannon faults (Sorg and McLaughlin, 1975; McLaughlin et al., 1991; McLaughlin and Clark, 2002). Mapping in the Santa Cruz Mountains has shown that northwest-striking reverse faults are progressively younger to the northeast toward Santa Clara Valley (McLaughlin and Clark, 2002). On the basis of this pattern, one might expect reverse faults to be forming at or near the western edge of the study area in the Santa Clara Valley.
Detailed Quaternary geologic mapping (Hitchcock et al., 1994) along the range front of the Santa Cruz Mountains between the towns of Los Gatos and Saratoga (Fig. 2) revealed potential signs of faulting from offset terraces, stream gradients, and changes in channel sinuosity. Hitchcock et al. (1994) also mapped a series of aerial photographic lineaments, based on topographic scarps, vegetation lineaments, tonal changes, and topographic depressions and ridges over a larger area (the study area in this paper). They proposed to extend and connect the Monte Vista and Shannon faults along the base of the Santa Cruz Mountains and to add a zone of unnamed faults outboard of the range front (Fig. 2), on the basis of the linear trends observed in the map pattern of the lineaments (Hitchcock and Kelson, 1999).
Because the map area containing the lineaments (Hitchcock et al., 1994) is a subset of the study area covered by the Loma Prieta earthquake damage data set (Schmidt et al., 1995), the two data sets are well suited for comparison (Fig. 3). If the mapped lineaments are spatially associated with the damage, one can be confident that the two data sets are related to a common spatial phenomenon, and the best current explanation is faulting. Geomorphic evidence suggests that the lineaments represent long-term surface deformation resulting from reverse-fault activity along and outboard of the range front of the Santa Cruz Mountains. The damage in Santa Clara Valley from the Loma Prieta earthquake could result from triggered slip along these faults or from shaking damage caused by a change in material properties across the faults. In this paper, we test for a spatial association between the damage and the mapped traces of the photointerpretive lineaments.
DATA AND SOFTWARE
The damage location data are from a digital data set (Schmidt et al., 2014) that was also published earlier in map form (Schmidt et al., 1995). These damage data were field-checked, and address locations were plotted by hand. All of the damage locations were digitized from standard U.S. Geological Survey 7.5 min topographic quadrangles. The authors performed quality control by visual inspection of paper plots of the data sets and by inspection of the registration of the damage data to scanned, registered 1:24,000 scale topographic stable-base greenline quadrangles. The damage data were classified according to material type and properties and sense of deformation (e.g., concrete, contractional). Where deformation was clearly observed in the field, it primarily expressed contractional strain. See Schmidt et al. (1995, 2014) for additional details. Visual inspection of the location of the damage relative to the base maps showed that the damage is accurately located to within a few meters.
The lineament database is an unpublished digital rendition of the lineaments shown on plate 2 of Hitchcock et al. (1994). Although specific statements of locational uncertainty of the mapped lineaments are not present in the report, the data were derived from photointerpretation and compiled at 1:24,000 scale, so the locational uncertainty is likely less than a few meters. All of the data sets are either available publicly or are digital versions of publicly available data.
The street map used to restrict the area of statistical analysis was taken from 800 dpi (dots per inch) scans of U.S. Geological Survey 1:100,000 scale topographic maps of the San Jose (1978) and Palo Alto (1982) quadrangles. The scans were merged and used as a single base map.
The software package Arc/Info™ (a commercial geographic information system [GIS] package available from Environmental Systems Research Institute1) was used to perform map analysis: calculating distances to map features, generating two-dimensional density plots, converting vector data to raster data, and visual overlays. Splus™ (a commercial statistical package available from Insightful1) was used to perform statistical analysis: generating and comparing distributions by graphical analysis and statistical testing.
The method of analysis is based on the distance of the Loma Prieta earthquake damage (treated as points) from the mapped lineaments (treated as lines). If damage is close to a lineament, then this suggests the two are related to a common phenomenon, such as surface rupture due to faulting or enhanced shaking from material property contrasts across faults. Throughout the analyses, distance is used as a measure of association. Sources of potential error in the locations of damage or lineaments, such as those stemming from georeferencing errors or the use of different base maps, are assumed to be minor relative to the distances identified as a measure of association.
In order to demonstrate that the damage is spatially associated with the lineaments, rather than being collocated by chance, we compared the observed distances of damage from the nearest lineament with a model of distances from the nearest lineament for a set of randomly located points. If the distance of damage to the nearest lineament was shorter on average than for the random set, then the damage and the lineaments were categorized as spatially associated.
A random set of points has equal probability of being located anywhere in the study area. This means that the probability of a single point being located in a small square within the study area is simply the area of the small square divided by the total study area, i.e., the percentage of the study area covered by that pixel. To obtain the probability of a random point being within a given distance of the nearest lineament, the study area was approximated by pixels, and the distance to the nearest lineament was measured for each pixel. The sum of the area within a given distance of the nearest lineament divided by the area of the whole study area is the probability that a random point will be within that given distance of the nearest lineament (Okabe and Fujii, 1984). For example, to estimate the probability of damage being located between 100 and 200 m from the nearest lineament, we calculated all of the pixels that are between 100 and 200 m from the nearest lineament, summed their areas, and divided this by the total study area. This is equivalent to finding the percentage of the study area that is between 100 and 200 m from the nearest lineament. This calculation was performed at a resolution of 5 m for the entire study area. Approximating the study area with pixels and measuring the distance of each pixel (calculated using the pixel centroid) to the nearest lineament were easily accomplished using GIS tools (Fig. 4). Once the distribution had been approximated using GIS, it was compared with the observed distribution of the distance of the damage to the nearest lineament (also easily calculated using GIS software). The distributions were then compared. If the observed distribution was significantly closer to the lineaments, then the damage was located preferentially closer to the lineaments than a random distribution, and the damage and lineaments showed a certain degree of spatial association.
Caution should be exercised to ensure that the locations used to calculate distances for both the observed, and the approximation of the theoretical random distributions are not taken from the same set of pixel centroids. If both distributions are calculated using the same set of pixel centroids, then the observed distribution is a subset of the approximated theoretical random distribution, and the two samples are not independent. The lack of independence violates the assumptions of many statistical tests, such as the Mann-Whitney test (Conover, 1999). In this analysis, the locations of the observed damage were digitized from topographic maps and were independent of the grid of pixels used to characterize the theoretical random distribution.
The study area is not capable of uniformly containing the type of damage recorded by Schmidt et al. (1995), however, and this complicates the simple model shown in Figure 4. As previously mentioned, damage was recorded for areas associated with streets (pavement and sidewalk damage, and pressurized water and gas lines, which for the most part are associated with urbanized areas). Pavement or pipe breaks can only be produced in areas that contained recordable damage, which in general are paved areas (water and gas lines often largely follow street networks). As a result, the probability of damage is inhomogeneous, because areas where pavement and pipes are not present have a low to zero probability of exhibiting that type of damage. This must be taken into consideration when developing a null hypothesis involving the set of randomly located points. Although detailed maps of areas covered by pavement were not available, a scanned topographic map of streets within the study area served as a reasonable proxy (fig. 7 inPhelps, 2007). The study area was restricted to those portions that were covered by streets, to account for the inhomogeneous probability of damage across the study area.
The analysis was further complicated by the presence of a highly concentrated area, or cluster, of damage within the town of Los Gatos (Fig. 3). Previous authors (Hitchcock and Kelson, 1999; Langenheim et al., 1997; Schmidt et al., 1995) highlighted the linear nature of the damage and suggested a pattern of linear zones. Schmidt et al. (1995) noted, however, that the damage also exhibited spatial clustering. A more detailed quantitative assessment of the damage revealed two patterns in the data set: The first consists of clusters and is illustrated well by a dominant clustering pattern of damage at a locus within the town of Los Gatos, and the second subordinate pattern consists of linear zones of damage that occur largely along the range front of the Santa Cruz Mountains. The most prominent clustering pattern is the roughly circular feature within the town of Los Gatos, the cause of which is unclear; perhaps it is the result of local site conditions of the alluvium, or the result of a focusing of energy due to the local shape of the basin, or perhaps it is related to the same phenomenon that produced the apparently linear pattern of damage. Because its shape suggests the damage may be due to some local effect, an analysis of the spatial association of the damage with the photointerpreted lineaments must account for possible bias introduced by spatial clustering in the data.
Density maps can be used to convert a map of discrete damage locations into a smooth, continuous surface by counting the number of points in a circle, assigning the count to the center of the circle, and then repeating the process at regular intervals across the map (for a discussion of the mathematics of the procedure, see Silverman, 1986). This generates a continuous surface of counts, which can be used to quantitatively demonstrate the dominance of the clustering of damage as an important pattern in the damage data. The choice of the window size (the radius of the circle) is made by the analyst; larger window sizes create increasingly smoothed maps and show only the largest features, while very small window sizes approach a map of the individual damage points.
A density map of the damage locations was generated for the study area (Fig. 5, top). A range of window sizes was examined, and a window with a 400 m radius was chosen as the best compromise between capturing the detail of larger patterns without blurring the pattern excessively. The cluster of points within the town of Los Gatos (C in Fig. 5), in the southern corner of the study area, stands out as an area of exceptional concentration of damage amid smaller, somewhat linear concentrations of damage dispersed across the study area. This damage cluster (C in Fig. 5), with a maximum density of 114 points per 400 m radius at its center, is approximately four times the density of other concentrated areas, which have between 25 and 30 points per 400 m radius (Fig. 5, top). This fact can be hidden if the damage is plotted at a scale where symbols plot on top of one another, making a quantitative assessment difficult. The density map presents the information that is lost when the scale of the map causes symbols to be occluded. It provides a quantitative means by which to compare the concentration of damage across the study area.
The dominance of this cluster of damage (C in Fig. 5), in terms of the amount of damage concentrated in a comparatively small area, could interfere with the quantitative analysis of the linearly trending damage locations observed in previous studies. In this paper, we wish to study the linear patterns in the damage data by examining the distance from the damage to the mapped lineaments. A significant cluster in the data could bias the results positively if by chance the cluster was located near mapped lineaments, or negatively if by chance it was located far from mapped lineaments. In this case the cluster (C in Fig. 5) is located near to the lineaments, so that this cluster introduces a strong positive bias. A statistical test of the distance to the nearest lineament would yield information primarily about the cluster of damage within the town of Los Gatos, and not about the rest of the data. It is the broader distribution of damage we wish to investigate in this paper, so the effect of this cluster must be mitigated.
In order to examine the linear trends in the damage, the data were declustered by modeling the cluster as a circular anomaly and removing the anomalous concentration of damage from the study. A profile of the anomaly was extracted from the density plot, and a Gaussian curve with a standard deviation of 250 m was fit to the values (Fig. 6). The Gaussian curve is a good approximation of the density for roughly two standard deviations. This implies that the cluster of damage can be reasonably modeled as a circular phenomenon with a standard deviation of 250 m. The cluster phenomenon is the main effect within two standard deviations, and it has little effect outside that range.
The area within 500 m (two standard deviations) of the cluster center was removed from the data in subsequent analysis. By removing this area from the analysis, we assume that near the cluster, the damage is due solely to whatever phenomenon caused the cluster; that outside of the cluster, this effect is negligible; and that the remainder of the data are free of other such clustering effects. We additionally assume that the cluster can be modeled as a circular phenomenon. While the cluster is not perfectly circular, the circular model does account for most of the clustering effect. The updated damage data set (subsequently referred to as the declustered damage) has all points within 500 m of the center of the cluster removed, and a comparison of the original density map and the density map resulting from the declustered damage data is shown in Figure 5. The cluster contains about one third of the damage in the study area.
Once the effect of clustering in the data and the inhomogeneities of the study area were taken into account, the study area was modified (subsequently referred to as the restricted study area (Fig. 7). We then developed a model of randomly located points for the restricted study area and compared the distance to the nearest lineament for the random points to that of the observed distances for the declustered damage data.
Note that in this analysis, the mapped lineaments are assumed to be fixed; that is, they are taken as given, and not themselves randomly distributed over the study area. The damage is treated as a process that is either related to these fixed features or not. The question could have been posed conversely: Are the lineaments located more closely to the mapped damage than a random process for generating lineaments? In this case, the lineaments are the more stable features. There are several lineament types, including vegetation changes, changes in photographic tone, geomorphic depressions, and scarps cutting Quaternary units of different ages, indicating a range of time for their formation. If they were tectonic features, then one would expect them to span multiple earthquakes. The damage represented by a single earthquake would not necessarily be associated with every lineament, but with a subset belonging to one or more regions that were active in the Loma Prieta earthquake. Therefore, the question is posed such that the particular earthquake is compared with the general tectonic framework.
The theoretical distribution of the distance to the nearest lineament for randomly located points was approximated within the restricted study area (Fig. 7). The histograms for this theoretical distribution and for the distribution of distances to the nearest lineament for the declustered observed damage are shown in Figure 8. The distributions are clearly different. The theoretical distribution is rather broad, with peaks (has a mode) at ∼100 m, and it has a thick tail. The declustered observed distribution shows a more pronounced peak that is at the very edge of the distribution (the first 25 m bin), and it has a thin tail. The medians differ by ∼100 m, with the median of the random distribution being 252 m and that of the observed distribution being 154 m.
For confirmatory analysis, a one-tailed Mann-Whitney test (a nonparametric test that compares distributions) was performed in order to test whether or not the damage is generally closer to the lineaments than samples from the theoretical random distribution. The damage distances from lineaments were compared to a random sample of 1000 values chosen from the values that approximate the theoretical distribution. The Mann-Whitney test has three requirements: that each sample is an independent sample from its respective distribution, that the samples are mutually independent, and that the measurement scale is at least ordinal (Conover, 1999). Addressing the assumptions, the theoretical distribution was randomly sampled, and the damage is assumed to be a random sample of potential damage. The sampling of each distribution is independent of the other—the distance of damage from the nearest lineament was measured directly, whereas discretizing the study area approximated the distance from the nearest lineament for the theoretical distribution. Finally, both measurements were made on the ratio scale. Using the formula for a one-tailed Mann-Whitney test (Conover, 1999, p. 272–274), the sum of the ranks of the damage distances was less than the estimated first quantile of the random (null) distribution, indicating that the declustered observed data are closer to the lineaments than the sample from the declustered random distribution and are significant at greater than the 99% confidence level. Statistically, the declustered damage is spatially associated with the lineaments.
The effect of the cluster of damage (C on Fig. 5) on the analysis can be seen (Fig. 9) by comparing the distribution of the distance to the nearest lineament for both the declustered damage data set and the original damage data set for the study area. The histograms show a dramatic increase in the amount of damage close to the lineaments when the cluster of damage is included in the data set. The effect of the cluster is to strengthen the spatial relationship of the damage to the lineaments. This demonstrates the significant positive bias caused by the cluster of damage.
These analyses indicate that damage associated with the Loma Prieta earthquake is related to the mapped lineaments of Hitchcock et al. (1994). This supports the idea that the damage and the lineaments are related to a common process. We propose that much of the damage likely was the result of local rupture or shaking due to triggered slip on reverse faults that are demarcated by geomorphic features, mapped as lineaments, along the range front of the Santa Cruz Mountains.
It is also possible that the damage was caused by focused shaking within the Cupertino Basin or from focused shaking along a subsurface discontinuity. Harmsen et al. (2008) modeled multiple earthquake scenarios, including a modeled M 6.7 earthquake on the Monte Vista–Shannon faults, and noted that the Cupertino Basin exhibited strong shaking in many of these scenarios. They noted that contrasting lithologic properties between the rocks and sediments of the Cupertino Basin and the rocks that bound the basin can intensify strong ground motion, and that basin geometry can complicate those motions. As previously noted, one of the deepest parts of the Cupertino Basin lies between the towns of Saratoga and Los Gatos. Such a deep basin could have intensified the strong ground motion caused by the Loma Prieta earthquake.
Support for the link of a common process between the damage and the mapped lineaments can be found by examining the increased concentration of damage close to the lineaments. Note that the amount of observed declustered damage decreases with distance from the lineaments, with the highest percentage of damage located within 25 m to the nearest lineament (Fig. 8). That is, in the observed data set, the probability of damage occurring decreases with distance from the nearest lineament. The theoretical random distribution exhibits a slightly different behavior; it initially increases with distance from the nearest lineament, and then decreases, with the mode of the theoretical random distribution being ∼100 m from the nearest lineament (Fig. 8A). Thus, for randomly located points, the highest probability of occurrence is ∼100 m from the nearest lineament, and then the probability falls off at shorter (and farther) distances. This contrasts with the behavior of the observed damage, which is greatest close to the lineaments and then continues to decrease in a roughly logarithmic decay.
Support for the hypothesis of the damage being indicative of triggered slip can be found by examining the subset of the damage (Figs. 10 and 11) that exhibited significant contractional strain (29% of the entire damage data set, and 33% of the declustered data set). The median of this subset of damage is closer to the mapped lineaments than the rest of the declustered data. The damage, then, is not only close to the mapped lineaments, and not only decays with distance, but additionally it exhibits the most contractional strain close to the lineaments. Contractional strain is consistent with movement on reverse faults along the range front of the Santa Cruz Mountains. Any competing hypothesis must explain all of these observations. Because the cluster contains about one third of the damage in the study area, approximately two thirds of the damage could be due to triggered slip.
The most significant feature, in terms of the concentration of damage, is the cluster of damage (C in Fig. 5) located within the town of Los Gatos. This damage accounts for about one third of the total damage in the area. The cluster’s roughly circular shape and relatively small diameter (1 km) suggest that it is due to a local effect, but the specific cause is unclear.
Using quantitative statistical methods, we determined that damage from the Loma Prieta earthquake in Santa Clara Valley is spatially associated with a set of photointerpreted lineaments mapped along the range front of the Santa Cruz Mountains. Histogram plots that compare the distance to the nearest lineament from randomly located points with the distance to the nearest lineament from the observed damage locations demonstrate that the observed damage is located preferentially closer to the mapped lineaments than randomly located damage. The median distance from the observed damage to the lineaments is ∼100 m less than the expected median distance from randomly located damage to the lineaments, and a Mann-Whitney test confirms that the difference is statistically significant. The observed damage is also greatest closest to the lineaments and falls off in number with distance, suggesting that a common process produced the observed damage and lineaments. Furthermore, damage that exhibited significant contraction is preferentially closer to the mapped lineaments than the rest of the damage, suggesting contractional movement on reverse faults along the range front of the Santa Cruz Mountains caused by triggered slip. An alternative hypothesis is that the observations regarding the damage were caused by focused shaking along the basin margin or along discontinuities, such as senescent or inactive faults, in the subsurface.
Carl Wentworth generously provided time for discussions and feedback, as well as contributing the digital lineament data. Donald Singer and Robert Simpson provided guidance and thoughtful discussions as well. We would also like to acknowledge the reviewers (M. Nathansen, C. Perkins, and V. Langenheim) for their time and efforts. Their comments helped to significantly improve the manuscript. This work would not have been possible without all of their help.