Region-specific linear site amplification model for peaty organic soil sites in Hokkaido,Japan

Abstract

We develop a site amplification model for a regional geologic condition that includes surficial deposits of peaty organic soils in Hokkaido, Japan. We use ground motion data from national Japanese networks along with local data from the study region. We apply a non-reference site approach to infer site effects from misfits of reference ground motion models to data from 10 subduction zone earthquakes. Application of this approach requires removal of source-specific biases and careful consideration of source-to-site path effects. These considerations are essential to avoid mapping source- or path-related model misfits into estimates of site response. We consider three subduction ground motion models as reference models. By paying special attention to the conditions for which the path models are effective, and making adjustments for between-island path misfits (Hokkaido to Honshu and vice versa), we identify regional site effects that are insensitive to the ground motion models used in their derivation. Observed site responses are characterized by strong resonances at first-mode site frequencies, and as a result the regional model is conditioned on peak frequency from horizontal-to-vertical spectral ratios.

Keywords

Site response peaty organic soils subduction zones ground motion

Introduction

The application of ergodic site terms in ground motion models (GMMs) to regions having very soft, peaty organic soils is a process that carries large epistemic uncertainty. One substantial driver of this uncertainty is that the ground conditions at such sites can have very slow time-averaged shear wave velocities in the upper 30 m (V_S₃₀), perhaps 100 m/s or less. Figure 1 shows V_S₃₀ histograms from site databases in the NGA-West2 and NGA-Subduction projects. The lowest V_S₃₀ values in these histograms are in the range of 100–200 m/s. As a result, the application of ergodic models to soft peat sites represents an extrapolation beyond the data range—the ergodic mean site response is simply unknown.

Figure 1.

Histograms of total and measured V_S₃₀ values for sites in NGA-West2 database (left) and NGA-subduction database (right). From Seyhan et al. (2014) and Ahdi et al. (2020), respectively.

A second driver of large uncertainty can be anticipated because soft peats usually overlie relatively firm, non-organic soils at depth. As a result, there likely exists within the site profile steep velocity gradients that can give rise to more pronounced impedance and resonance effects than would be typical at non-peat sites. Such effects produce site response transfer functions with strong peaks at one or more site frequencies. These features of site response cannot be fully captured by V_S₃₀-based models as used in typical GMMs, although they could potentially be captured by a model that combines V_S₃₀-scaling terms with site resonance terms that take the peak frequency ( $f_{peak}$ ) as a site parameter (e.g. Harmon et al., 2019; Hassani and Atkinson, 2018a, 2018b; Kwak et al., 2017).

As a result of these issues, the ergodic site terms in current GMMs for subduction earthquakes in Japan, whether based on V_S₃₀ (Abrahamson et al., 2018; Parker et al., 2022; denoted Aea18 and Pea22) or site classes defined from V_S₃₀ and site period (four such classes are used in Zhao et al., 2016a, 2016b; denoted Zea16) are expected to have bias and large uncertainty when applied to peaty organic soil sites with very low V_S₃₀ such as those in Hokkaido. In this study, we develop region-specific site amplification models based on features of local site response derived from non-ergodic analyses. The developed site amplification model is specific to the Hokkaido region, but the procedure used in its development is fully general and as such could be applied to other regions with ground motion recordings.

The approach taken in this article is to develop a linear site amplification model using relatively low-amplitude recordings, for which significant nonlinear effects are not expected. We then examine nonlinear effects through residuals analysis using data from one event that produced stronger shaking.

Data sources

The region-specific analysis performed in this study applies to the portion of the Tokachi River in Hokkaido, Japan passing through peaty organic soil layers that extend roughly from the river mouth to 40 km upstream of the river mouth, as shown in Figure 2. This region contains seven strong motion stations operated by Obihiro Development and Construction Department (ODCD), for which we have obtained recordings from ten earthquakes in the NGA-subduction database. This section discusses the data compiled for analysis of non-ergodic site responses at these seven stations.

Figure 2.

Map of northern Japan showing event locations considered in this study and location of Tokachi River study region. Detail map of study region shows locations of local ground motion stations.

Ground motions and related metadata

Table 1 provides metadata for the seven ODCD stations in the study region. The station locations were provided by ODCD and the basis for the site information (V_S₃₀ and classification) is provided in the following section. As shown in Figure 2, the station locations are roughly evenly distributed across the study region.

Table 1.

Metadata of seven stations owned and operated by Obihiro Development and Construction Department (ODCD)

Station name	Longitude	Latitude	V_S ₃₀ (m/s)	f_peak (Hz)	Site class (Zea 16)
Ushishubetsu	143.4622	42.7921	130	1.14	IV
Inaho	143.6063	42.7876	212	1.60	III
Horooka	143.5489	42.7841	118	1.42	IV
Reisakubetsu	143.4744	42.8359	181	1.62	IV
Tabikorai	143.5642	42.7556	149	1.14	IV
Toitokki	143.6043	42.7281	148	1.60	IV
Tonai	143.4250	42.8917	181	1.50	IV

The stations in Table 1 have recorded 10 earthquakes. Figure 2 shows the epicenter locations of these events, each of which has ground motion data that is incorporated into the NGA-subduction database (Kishida et al., 2020). Metadata for the 10 events is provided in Table 2 along with the numbers of processed recordings produced by each event in the NGA-subduction database. The numbers of records per event range from 13 to 1293; there is a dramatic increase in station density and data quality since the 1995 Kobe earthquake.

Table 2.

Summary of event metadata and numbers of recordings for the 10 considered earthquakes

Event	Date/time	Mag.	Strike/dip/rake	Fault type	Event type	Longitude, latitude	Hypocenterdepth (km)	No. of recordingsin NGA-subduction	Range of R_maxfor NGA-subductionrecordings	Max R_rupof ODCDstations (km)	No. ofselectedNGA-subductionRecordings	No. ofrecordingsODCD array
1	1994/12/28, 12:19	7.76	180/17.3/90	Reverse	Subduction interface	143.7558, 40.4365	10.6	31	(148, 218)	220	10	3
2	1995/01/06, 22:37	6.98	179/21/69	Reverse	Subduction interface	142.3055, 40.2233	47.8	13	(113, 114)	300	0	3
3	2004/11/28, 18:32	7.01	211/24/81	Reverse	Subduction interface	145.2529, 42.9533	51.0	378	(262, 450)	150	84	7
4	2004/12/06, 14:15	6.77	222/26/90	Reverse	Subduction interface	145.3371, 42.8463	46.0	204	(189, 324)	160	57	7
5	2008/09/11, 00:21	6.8	228/21/108	Reverse	Subduction interface	144.1493, 41.7792	30.7	407	(155, 269)	130	64	6
6	2011/03/11, 05:46	9.12	200/12.3/88	Reverse	Subduction interface	142.823, 38.1165	17.5	1293	(157, 610)	360	682	7
7	2011/04/07, 14:32	7.15	24/37/87	Reverse	Subduction slab	141.9645, 38.253	66.4	799	(252, 600)	515	428	7
8	2011/11/24, 10:25	6.19	223/16/103	Reverse	Subduction interface	142.8422, 41.7457	43.0	177	(135, 230)	136	57	6
9	2012/12/07, 08:18	7.23	188/44/-100	Normal	Subduction slab	143.8381, 38.0216	21.0	866	(230, 409)	520	192	7
10	2003/09/25,19:50	8.29	230/20/90	Reverse	Subduction interface	144.0786, 41.7796	25.0	645	(149, 538)	70	302	6

ODCD: Obihiro development and construction department.

We apply data screening for each event in which recordings beyond a limiting distance, R_max, are removed owing to concerns of incomplete data sampling (i.e. an appreciable fraction of recordings may not appear in the data as a result of having amplitudes below the trigger/detection threshold). The NGA-subduction project developed R_max criteria that depend on earthquake magnitude and instrument type (Contreras et al., 2022). When those criteria are applied to the 10 events considered in this study, the range of R_max are as given in Table 2. As also shown in Table 2, the maximum rupture distance (R_rup) of ODCD stations is within or beyond the R_max range for 6 of the 10 events, thus strict application of the NGA-Subduction criteria would severely restrict the dataset. Because those R_max criteria are intentionally strict (conservative), they were relaxed for the present application to allow the inclusion of recordings having modestly larger distance than R_max (by no more than 40%) when doing so extends the distance range for an event sufficiently to encompass R_rup for ODCD stations. Backarc stations are not used because the Obihiro sites of interest in this study are located in forearc regions, and our principal interest is source-site wave paths within the forearc region. Moreover, by excluding backarc stations, we also avoid complex backarc path effects (Ghofrani and Atkinson, 2011; Skarlatoudis and Papazachos, 2012; and Cramer and Jambo, 2020). Event 2 data were excluded in our analyses because of the 13 recordings in the NGA-subduction database, 11 do not meet either the weaker R_max criteria or are backarc stations, and the remaining two have rupture distances much less than those for ODCD stations (113 km vs 300 km). Despite not using Event 2, we do not renumber other events (i.e. we use Events 1 and 3–10).

The NGA-subduction data for the 10 events did not include recordings at the ODCD stations. We were provided with raw (digital but unprocessed) recordings from the seven ODCD stations by T. Sato (last update 10 November 2017, personal communication). The data were processed following standard procedures as given in Kishida et al. (2020), which include instrument correction, application of both high- and low-pass acausal filters at operator-determined corner frequencies, and baseline correction of the processed signals. Following the application of this processing, we computed median-component intensity measures RotD50 (Boore, 2010) for peak acceleration, peak velocity, and 5% damped pseudo-spectral accelerations (PSA) using the R package by Wang et al. (2017).

As shown in Figure 3, Events 1–9 produced relatively weak motion recordings at the ODCD stations, which are useful for developing the linear component of a regional site amplification model. In contrast, Event 10 (2003 Tokachi-Oki Earthquake) produces appreciably stronger shaking. Accordingly, our approach for model development is to develop a linear model using data from Events 1–9, and then to perform residuals analysis using data from Event 10 to investigate potential nonlinearity effects.

Figure 3.

Histogram of median-component peak accelerations at ODCD sites.

Site conditions

Each of the seven ODCD sites in the study region is underlain by peaty organic soils in the Tokachi River basin (Hokkaido River Disaster Prevention Research Center (HRDPRC), 2004; Tsai, 2018). The geomorphic terrain class for six of the seven sites per the Japanese Engineering Geomorphologic Classification scheme (Wakamatsu and Matsuoka, 2013) is “back marsh” (the seventh, in the furthest downstream area, is “delta and coastal lowland”). Based on boring logs at strong motion accelerograph sites and other areas from HRDPRC (2004), summarized and translated to English by Tsai (2018), the peaty organic soils extend to depths of 2–6 m, and are underlain by alluvial deposits of sandy, silty and clayey sediments. Under the levees, the peat layer is typically 0.5–1.0 m thinner. One typical geotechnical and V_S profile at a site adjacent to the Tabikorai ODCD station is shown in Figure 4. The classification of materials as peats in these logs is based on visual assessments and/or lab tests for organic content. Penetration resistance measured during cone penetration tests and standard penetration tests, as well as shear wave velocities, are low in the area. Relatively firm material, likely Pleistocene in age, is located at greater depths generally around 35–40 m. Because the peaty organic soils are of modest thickness at the ODCD sites, the wave propagation path through the sediments occurs mainly in non-peat materials. As such, while the site response evaluated from these sites may be only modestly impacted by the peaty soils themselves, it is controlled by the depositional environment of the region, which produced surficial peats and low seismic velocities. The site class as given by Zhao et al. (2016b) is mainly IV (Table 1), which is described as soft soils with V_S₃₀ < 200 m/s and site period T ≥ 0.6 s. The site class for US codes (Building Seismic Safety Council (BSSC), 2020) would be F for most of the study region, based on peat thickness >3 m.

Figure 4.

Geologic log at a site adjacent to Tabikorai ODCD station.

We performed geophysical testing at or in the vicinity of the Toitokki, Horooka, Tabikorai, Reisakubtsu, and Ushishubetsu stations using spectral analysis of surface wave (SASW) and ambient noise techniques (full results in Tsai, 2018). Rayleigh wave dispersion curves and horizontal-to-vertical spectral ratio versus frequency plots (HVSR plots) were obtained from these tests. The dispersion curves are provided in Electronic Supplement Figure S1 in the Appendix and the HVSR plots are provided in Figure 5. We use Geopsy (Wathelet et al., 2020) and a joint inversion approach (García-Jerez et al., 2016; Griffiths et al., 2016), which fits the lowest-frequency peak of the Raleigh wave ellipticity curve to the peak of HVSR, to derive the V_S profiles and calculate V_S₃₀ for the five sites.

Figure 5.

HVSR versus frequency plots for ODCD sites as established from microtremor or pre-event noise data, and fitted pulse with identified peak frequencies.

Sites Tonai and Inaho were not part of the site exploration program. For these sites, HVSR plots were developed using pre-event noise signals by first estimating the p-wave arrival time, and then taking preceding portions of the signals for use in analysis. Prior comparisons of HVSR from ground motion recording stations (using time intervals without earthquake shaking) and temporarily deployed sensors (as used in site characterization) have shown similar results (Wang et al., 2021), which supports the use of non-seismic signals for the Tonai and Inaho sites. V_S₃₀ values were estimated from a geomorphology and slope proxy-based model (Wakamatsu and Matsuoka, 2013).

The HVSR data was processed as described in Wang et al. (2022), which is revised somewhat from earlier protocols in Site Effects Assessment Using Ambient Excitations (SESAME, 2004). These procedures involve windowing and low-cut filtering the data, removing windows with anomalous features, smoothing the resulting spectral ratios, and plotting the data for the selected windows along with the arithmetic mean and mean ± one standard deviation.

Figure 5 plots HVSR versus frequency for the seven Obihiro stations. Gaussian pulses were fitted to prominent peaks in the HVSR (Ghofrani and Atkinson, 2014; Kwak et al., 2017) and the resulting $f_{peak}$ values are given in the figure, which will be used for site amplification model development. Sites Toitokki and Tabikorai have two HVSR peaks, which raises the question of how to select $f_{peak}$ for such cases. We identify the most fundamental peaks for the ODCD sites as follows: (1) if the peaks are of comparable amplitude but distinct in frequency (ratio of the peak frequencies is greater than about 3–5), adopt as $f_{peak}$ the value for the lower frequency peak; (2) if the peaks are of significantly different amplitude (>factor of two), use the peak with the larger amplitude (this is usually the lower frequency peak); and (3) if the peaks are of comparable amplitude and the frequencies are similar (ratio of the peak frequencies is less than about 3–5), refit the Gaussian function to encompass both peaks together. Case 1 applies to the Tabikorai site, Case 2 applies to the Ushishubetsu site (the peak at ∼0.25 Hz is not selected), and Case 3 applies to the Toitokki site. The larger uncertainty in HVSR that is shown for sites Tonai and Inaho is associated with the use of pre-event noise data from relatively few records of generally short duration. Table 1 summarizes the V_S₃₀ and $f_{peak}$ values obtained from these measurements for all of the sites.

Ground motion data analysis

This section presents our analyses of ground motion recordings to support model development. We first describe the analysis of residuals and the main data trends. Next we evaluate event-specific biases and regional (between-island) complexities in path. The intent of those analyses is to provide region-adjusted, within-event residuals for each event and recording at the seven ODCD sites. Finally, we evaluate non-ergodic site responses for each site.

Residuals analysis and trends

Our approach to data analysis operates on residuals, which are the difference between the natural log of an observation and its prediction from a GMM. “Observation” in this case is a computed RotD50 intensity measure from a ground motion recording. As shown in Table 2, we consider all recordings that meet data selection criteria (not just those in the study region). The GMMs applied in these analyses are Zea16, which is specifically intended for use with Japan subduction earthquakes, Aea18 applied with Japan-regionalized path correction terms, and the Pea22 NGA-Subduction ground motion model. The three GMMs are used to investigate sensitivity of derived site terms to the alternate reference GMMs. The maximum rupture distances recommended for use are 300 km for Zea16 and Aea18 and 1000 km for Pea22. We consider data at distances beyond 300 km, although measures are taken (limiting the maximum distances for event terms, using only forearc stations, and adjusting models for between-island effects) to minimize path biases at larger distances.

Total residuals are computed using the reference GMMs as

R_{ij} = \ln (Y_{ij}) - {(μ_{lnY})}_{ij}

(1)

where Y_ij is an observed RotD50 intensity measure for site j from event i, and ${(μ_{\ln Y})}_{ij}$ is the natural log mean prediction from the alternate GMMs given event magnitude, site-source distance, site condition, and other parameters used in the models. Nonzero residuals have many potentially causative factors. An event can produce ground motions systematically low or high relative to the GMM. Likewise, a source-to-site path may be associated with attenuation rates that are relatively fast or slow in comparison to the model. Our goal ultimately is for the residuals to be used to analyze site response. As a result, if the residuals include systematic effects related to source or path, our ability to infer site response is compromised. This problem is addressed in the analysis by removing, to the extent justified by the data, systematic (repeatable) source and path effects from total residuals. The remainder of this section describes how these adjustments were made.

Figure 6 plots total residuals vs distance for four events and at four oscillator periods (0.005, 0.08, 0.8, and 5.0 s) from the Pea22 GMM (qualitatively similar results are obtained for other events and from the Aea18 and Zea16 GMMs; results provided in Electronic Supplement A Figure S2). The figure is organized by event location; Part (a) applies to events off the coast of Hokkaido, Part (b) to Honshu. Important attributes of the data to consider in Figure 6 are (1) the slope of the data relative to distance and (2) offsets in residuals between islands or different attenuation rates for ground motions recorded on the two islands. Based on Figure 6 and Electronic Supplemental A Figure S2, we find the following:

A flat slope of residuals for a particular island indicates an unbiased GMM path term, whereas a gradient indicates path term bias. Events 6 and 7 exhibit downward gradients in short-period (0.005 and 0.08 s) residuals from recordings on the event-adjacent island. Events 3 and 4 lack gradient features for most periods, with the exception of upward gradients at 0.8 s. for Hokkaido stations. The gradients observed in Figure 6 are mainly associated with large-distance attenuation. Data for other events (Electronic Supplemental A Figure S2) lack gradient features.

The data show diverging trends for Hokkaido and Honshu stations beyond about 300 km for events recorded on both islands (Events 3, 4, 6, and 7). This suggests the presence of between-island regional attenuation effects not captured by the GMMs.

Figure 6.

Variation of Pea22 total residuals with distance for four events at periods 0.005, 0.08, 0.8, and 5 s: (a) events off coast of Hokkaido and (b) events off coast of Honshu.

These issues motivated the application of distance limits for the analysis of event terms, as described in the following section.

Event terms

The vertical offset from zero of the total residuals in Figure 6 is an event attribute referred to as an event term. Event terms are computed during GMM development when mixed effects regression methods are applied (Bates et al., 2015). Event terms computed in this manner reflect model-to-data misfits resulting from all model components, but mainly the source and path components. For the present application, we require an event term estimate that reflects source model misfit to the extent practical, because path errors are addressed separately. For this reason, we do not use directly the event terms from Pea22, but instead take the event term for a given event i as the average of the residuals

η_{E, i} = \frac{1}{n_{i}} \sum_{j = 1}^{n_{i}} R_{ij}

(2)

where $n_{i}$ is the number of recordings for event i that meet certain data screening criteria. We recognize that estimating random effects such as event terms with average residuals (as in Equation 2) can introduce errors (e.g. Abrahamson and Youngs, 1992; Stafford, 2014) when $n_{i}$ is small. Fortunately, the error is small for most of the subject events (Events 3–10; Table 2) due to ample recordings. Note that the use of Equation 2 implies the lack of an overall bias (across all events) in the GMM, which if present would be a constant term added to the left side of Equation 2.

As described in the previous section, several events exhibit gradients in total residuals with distance, indicating problems with the GMM path model. To avoid mapping path errors into event terms, we limit the maximum considered distances for the calculation of event terms with Equation 2. From examination of Figures 6 and S2 in the Electronic Supplement, considering results from all three GMMs, we used maximum distances of 200 km (Events 3 and 4) and 300 km (Events 6 and 7). Within these distances, residuals for event-adjacent islands are reasonably flat in most cases. For the other considered events where residual gradients did not occur, no data screening was applied beyond that used for all data, which involves (1) removal of stations in the backarc and (2) removal of stations with $R_{rup} < x R_{\max}$ , where x ranges from 1.0 to 1.4 (to encompass distances to ODCD stations).

Event terms obtained from Equation 2 are shown in Figure 7. Positive and negative event terms indicate under- and overprediction, respectively, and both conditions are encountered. The event terms for Event 9 are very large—values above 1.5 for periods less than about 0.3 s.

Figure 7.

Event terms (PSA) for the eight events recorded in the ODCD array from (a) Zea16 GMM, (b) Aea18 GMM, and (c) Pea22 GMM.

Regional (between-island) terms

Japan is known to have strong regional variations in anelastic attenuation (e.g. Ghofrani and Atkinson, 2011). These effects, if not included in a GMM, can introduce large differences in residuals for recordings on opposite sides of the volcanic front (i.e. forearc vs backarc sites), and potentially for recordings on different islands. The effects of paths crossing the volcanic front were addressed in our analysis by screening out backarc data. Additional regional complexities in the forearc, where encountered (Events 3, 4, and 7) were addressed with application of distance cutoffs. The remaining issue considered here is potential effects of source-to-site paths passing between islands, specifically Honshu to Hokkaido and vice versa. To investigate this, we first compute within-event residuals as

δ W_{ij} = R_{ij} - η_{Ei}

(3)

Next, we plot $δ W_{ij}$ with binned means and ±one standard derivations versus distance for Honshu and Hokkaido stations using data from all events. As shown in Figure 8, there are divergences of within-event residuals from the Pea22 GMM for these data groups for distances beyond about 350 km at short periods (similar features were seen in the other GMMs shown in Electronic Supplement A Figure S3). Because these divergences are a path issue, they should be corrected to the extent practical prior the analysis of site terms. Accordingly, we separate sources as Hokkaido-adjacent (denoted North; latitude >39°) and Honshu-adjacent (denoted South; latitude <39°), and sites by location (Hokkaido or Honshu). The North events are 1, 3, 4, 5, and 8, and the South events are 6, 7, and 9. This categorization of events and sites facilitates analysis of between-island terms that correct ground motions when an earthquake occurs in one region (e.g. South) and a subset of the recordings are in the other region (e.g. North).

Figure 8.

Event-corrected within-event residuals for eight events, separated by region, derived using Pea22 GMM.

Region terms are computed for each combination of event and station regions as the average of within-event residuals

η_{l, k}^{reg} = \frac{1}{n_{l, k}} \sum_{\forall i \in l, \forall j \in k} δ W_{ij}

(4)

where the ∀ symbol in combination with ∈ (e.g. $\forall i \in l$ ) indicates “for any value of $i$ among the set specified by array $l$ ,” which is used to parse the data into source-site groups. As before, $i$ is the event index and $j$ the site index. We take $l \in {0, 1}$ (i.e. $l$ can be 0 or 1) to segregate event regions (0 refers to South Events and 1 refers to North Events) and $k \in {0, 1}$ to segregate station regions (0 refers to Honshu and 1 refers to Hokkaido). Figure 9 plots region terms over the period range 0.08–5 s with their 95% confidence intervals for each source-site regional combination.

Figure 9.

Region terms for Hokkaido and Honshu stations from South and North regional events, as derived from (a) Zea16 GMM, (b) Aea18 GMM, and (c) Pea22 GMM.

The results in Figure 9 show that for common source-station regions, such as South events and Honshu stations (index $l = 0$ and $k = 0$ ), region terms of both models are relatively small (less than 0.2) and have little trend with period. However, when the indices differ, there are large biases. The large positive region terms at short periods (< ∼1 s) for South events and Hokkaido stations are consistent with the corresponding upward (positive) trends of residuals in Figure 8. There is a large negative bias at long periods for North events and Honshu stations from all GMMs, whereas the Aea18 and Pea22 GMMs also have a large positive bias at short periods for this combination. We have tested the statistical significance of these between-island effects using F tests, the results of which are presented in Electronic Supplement B. The testing confirms that the separation of regions for each event group is justified for most periods using each of the GMMs.

To incorporate the between-island regional terms into residuals analyses, we adjust the within-event residuals from Equation 4 as follows

δ W_{ijlk}^{reg} = δ W_{ij} - η_{l, k}^{reg}

(5)

where $δ W_{ijlk}^{reg}$ is the region-adjusted within-event residual. Figure 10 shows the distance variation of $δ W_{ijlk}^{reg}$ (with binned means and ±one standard derivation bars) for all data (Figure 10a), North events only (Figure 10b) and South events only (Figure 10c). Figure 10 applies to the Pea22 model; similar plots are provided in Electronic Supplement A Figures S4 and S5 for the Aea18 and Zea16 models. For each figure, results are shown separately for South (Honshu) stations in blue and North (Hokkaido) stations in red. For the present study, the most critical results are those for Hokkaido stations and both event types, particularly over the distance range applicable to the ODCD study region. The two vertical lines drawn in Figure 10b and c indicate the distance range of ODCD stations across all events in the respective groups (∼100–220 km for North events, ∼320–520 km for South events).

Figure 10.

Trend of Pea22 region-corrected within-event residuals with closest distance at periods of 0.08, 0.8, and 5 s for (a) all data, (b) North events only, and (c) South events only. Distance range of ODCD stations marked. Similar results for Aea18 and Zea16 GMM appear in the electronic supplement.

The results in Figure 10a can be compared with those in Figure 8 to see the effect of the regional terms on residuals trends—those trends are slightly reduced but not eliminated. While the residual trends for North events are generally flat (Figure 10b), the much better-recorded South events produce mixed results (Figure 10c). The Pea22 residuals are reasonably flat for Hokkaido stations in the distance range of interest. The Aea18 GMMs (Electronic Supplement A Figure S4c) produces an upward trend for some periods (0.08 s) and a trends for other periods that is generally flat within the distance range, but trends upward at the upper limit of the range (approximate 500 km or greater). To address this, we apply a 450-km cutoff to the Aea18 model for the case of South events and Hokkaido recordings. No further adjustments to the three path models are applied for the analysis of site terms.

Site terms

By adjusting residuals for event biases, $η_{E, i}$ and regional biases, $η_{lk}^{reg}$ , the remaining region-adjusted within-event residual, $δ W_{ijlk}^{reg}$ , represents errors in the prediction of observed intensity measures from the GMM that can be attributed to the combination of systematic site effects at each station and relatively random, event-to-event path errors. If the path errors are indeed random, they would average to zero when summed over many observations. With this in mind, we estimate the effect of site response, also called the site term, $η_{S, j}$ , at site j as follows

η_{S, j} = \frac{1}{n_{j}} \sum_{i = 1}^{n_{j}} δ W_{ijlk}^{reg}

(6)

where $n_{j}$ is the number of recordings for station j. As with the event term computation, this represents a frequentist interpretation of the problem statistics.

The site response model assumed to apply for a given intensity measure at a given site is taken as (Stewart et al., 2017)

F_{S} = f_{1} + f_{2} \ln (\frac{x_{IMr} + f_{3}}{f_{3}})

(7)

where F_S is site amplification in natural log units, $x_{IMr}$ represents the amplitude of shaking for a reference site condition (generally rock) for a particular earthquake at a particular site (expressed as an intensity measure), f₁ is the coefficient representing linear site response, f₂ represents the slope (generally negative) in amplification- $\ln (x_{IMr})$ space for $x_{IMr} >> f_{3}$ , and f₃ represents a transitional value of the reference site intensity measure below which the site response is nearly linear, and above which the trend of amplification with $x_{IMr}$ is nearly linear in log–log space.

The site term in Equation 6 represents the misfit between the observed site response and the site response predicted by the ergodic model in the GMM. If the observations are primarily from weak ground motions that do not introduce a significant nonlinear response, the only site response coefficient from Equation 7 that can be evaluated empirically is f₁, which is taken as

{(f_{1})}_{j} = η_{S, j} + F_{S, j}

(8)

where $F_{S, j}$ is the ergodic site response for site j as evaluated from the selected model, and f₁ is site-specific amplification relative to the GMM reference condition (V_s₃₀ = 760 m/s for Zea16 and Pea22; 1000 m/s for Aea18).

Figure 11 shows for each of the seven sites the region-adjusted within-event residuals $δ W_{ijlk}^{reg}$ and site terms (top) and the total site response (bottom) computed as in Equation 8, using the Pea22 GMM. Each of the sites exhibits a peak at a specific natural period. For example, at the Tonai site, the peak site response occurs at a period of about 0.7 s, and the amplification at that period is approximately $e^{2.3} \approx 10$ . This very high site amplification is likely associated with resonance of the upper soft peat and clay layers (with very low $V_{S}$ ) relative to deeper, stiffer sediments (with higher $V_{S}$ ). A somewhat surprising feature of this dataset is the similarity of the observed site responses across the seven sites. While some details change, for each site the amplification peaks between 0.5–0.9 s, has a peak width of about one log cycle of period, a minimum amplification near 0.1 s, and a gradual increase in amplification toward PGA. As such, the Tokachi River floodplain provides a nearly ideal setting for the development of a regional site response model.

Figure 11.

Region-adjusted within-event residuals (a) and estimated site responses (b) for the seven Obihiro stations as derived from Pea22 GMM. Similar results obtained for other GMMs as shown in the electronic supplement (a) δW^reg and (b) linear amplification.

Figure 12 compares linear site response terms ( $f_{1}$ ) for each site as derived from the three reference GMMs. This comparison shows that despite the different source and path models in the reference GMMs, the derived site responses are quite similar. Because the site responses derived from the three models are consistent, any one of them could be used for subsequent model development. We utilize results from the Pea22 GMM for this purpose.

Figure 12.

Comparison of site-specific linear site responses as derived using the Zea16, Aea18, and Pea22 GMMs.

Site amplification model development

We develop a site amplification function that takes site frequency as derived from microtremor-based HVSR ( $f_{peak}$ ) as input to capture the observed site amplification in an average sense across the seven sites and which presumably would have general applicability across the study region shown in Figure 2. We also develop an alternative model in which the only site information is that it is located on peat in the general study region, but $f_{peak}$ is unknown.

Site parameters

Because the very soft sites (with peaty organic soils and clays) in the study region exhibit site amplification effects suggestive of a resonant ground response effect at a site frequency, we sought to use HVSR to estimate the frequency of the peak response ( $f_{peak}$ ). The concept is to use $f_{peak}$ as a site parameter to be used in the regional model. It is desirable that the process used to estimate $f_{peak}$ be independent of the ground motions, which can be best accomplished by measuring $f_{peak}$ from microtremors. Before going forward, a point of clarification on notation—frequency $f_{p}$ as used in the site amplification model (next subsection) is the fitted oscillator frequency of the peak site response, which is not an independent variable (it is derived from ground motions, hence it is not independent of those motions). There are two reasons why $f_{p}$ and $f_{peak}$ may not match: (1) one corresponds to oscillator frequency ( $f_{p}$ ) and the other to frequency as used in Fourier analysis ( $f_{peak}$ ) and (2) even when Fourier representations of site response are used, site response peaks may not match HVSR peaks. Regarding point (1), some prior models have taken the oscillator peak as 80% of the HVSR peak (Harmon et al., 2019). Point (2) has been investigated in prior research, which has shown that $f_{peak}$ is consistent with the frequencies of peak site response for many, but not all, soil sites (Bonilla et al., 2002; Cadet et al., 2012; Ghofrani et al., 2013; Hassani et al., 2019; Kawase et al., 2011; Lachet et al., 1996; Lermo and Chávez-García, 1993; Theodulidis et al., 1996; Wang et al., 2021). In this study, $f_{p}$ is allowed to deviate from $f_{peak}$ ; differences are set by the data, as described below.

Mean amplification

In order to capture the peaked shape of site amplification observed at the ODCD sites, we selected a Mexican hat wavelet function (adapted from Ricker Wavelet function; Ryan, 1994). This function is intended to capture site resonance effects that dominate amplification shapes at short to intermediate periods (T < 2 s). A linear decay function is used at longer periods (T > 2 s). The recommended site amplification function for linear conditions is as follows

f_{1} (T, t_{0}) = {\begin{matrix} c_{0} + \frac{2 c_{1}}{\sqrt{3 c_{2}} π^{1 / 4}} (1 - {(\frac{\ln (T f_{p})}{c_{2}})}^{2}) e^{- \frac{1}{2} {[\frac{\ln (T f_{p})}{c_{2}}]}^{2}} & for T \leq T_{tr} \\ c_{3} \ln (\frac{T}{T_{tr}}) + f_{1} (T_{tr}, f_{p}) & for T > T_{tr} \end{matrix}

(9)

where c₀ controls the overall level of site amplification, c₁ scales the amplitude of the hat function, c₂ describes the width of the PSA peak in natural log period space, T_tr = 2 s is the transition period between the Mexican hat and linear functions, and c₃ describes the linear decay of amplification with log period beyond T_tr. Frequency $f_{p}$ is the frequency of the peak in the Mexican hat fitting function. A similar function was used by Harmon et al. (2019).

We performed a least squares fit of Equation 9 to the observed amplification for each of the seven sites, with the results in Figure 13 and Table 3. Potential realizations of the coefficients are assumed to be normally distributed; p-values in Table 3 are estimated using estimated mean values and standard errors. The p values indicate that all coefficients are statistically significant except for $c_{3}$ , which is not statistically distinct from zero. Nonetheless, we retain nonzero $c_{3}$ to improve fits at long periods.

Figure 13.

Fit of model to observed amplification when model coefficients are taken from site-specific optimization.

Table 3.

Coefficients for Mexican hat model parameters for each site

Station name	c ₀	c ₁	c ₂	c ₃	f_p (Hz)
	Standard error	Standard error	Standard error	Standard error	Standard error
	p value	p value	p value	p value	p value
Toitokki	1.416	1.145	1.161	−0.469	1.271
	0.163	0.360	0.264	0.448	0.302
	0	0.001	0	0.295	0
Tabikorai	0.929	1.435	1.178	−0.552	1.063
	0.162	0.365	0.248	0.458	0.215
	0	0	0	0.228	0
Horooka	1.350	1.270	1.144	−0.729	1.209
	0.163	0.361	0.252	0.450	0.253
	0	0	0	0.105	0
Reisakubetsu	1.475	0.962	0.948	−0.610	1.727
	0.166	0.346	0.264	0.396	0.368
	0	0.005	0	0.123	0
Tonai	1.344	1.224	1.114	−0.671	1.242
	0.164	0.359	0.250	0.447	0.258
	0	0.001	0	0.133	0
Ushishubetsu	1.210	1.457	1.113	−0.704	1.126
	0.163	0.361	0.219	0.450	0.198
	0	0	0	0.118	0
Inaho	1.228	1.009	0.975	−0.587	1.770
	0.167	0.347	0.259	0.403	0.378
	0	0.004	0	0.145	0

To develop the model for sites other than the seven stations in the study region, we examined the relationship between $f_{p}$ and $f_{peak}$ as shown in Figure 14 ( $f_{p} = a f_{peak}$ , where a = 0.948). Confidence intervals on the best fit line (shown in the figure) encompass the 1:1 line. Our interpretation is that the limited data do not reject the null hypothesis of $f_{p} = f_{peak}$ , which we adopt to estimate $f_{p}$ for use in the Mexican hat function. All other coefficients are taken as constant across all sites. The other coefficients were obtained by minimizing the sum of square of errors after specifying $f_{p}$ as above, with the resulting values in Table 4 in the row labeled “Model using $f_{p}$ from $f_{peak}$ .” The resulting model predictions are compared with data and the site-specific fits in Figure 13. The results are generally good, although some loss of fidelity occurs relative to the site specific (non-ergodic) models.

Figure 14.

Relationship between HVSR peak frequency ( $f_{peak}$ ) and peak in Mexican hat fitting function of site response ( $f_{p}$ ). Note that $f_{p}$ corresponds to a peak frequency for oscillator response.

Table 4.

Coefficients for Mexican hat model parameters

Model	c ₀	c ₁	c ₂	c ₃
Model using f_p from f_peak	1.28 ± 0.06	1.17 ± 0.13	1.10 ± 0.09	−0.53 ± 0.14
Model using f_p = 1.30 ± 0.10 Hz (population mean)	1.28 ± 0.06	1.18 ± 0.14	1.11 ± 0.09	−0.61 ± 0.17

We also develop a model for the case in which $f_{peak}$ is unknown. In this case, we regress Equation 9 to the combined data set for all sites to obtain a new set of coefficients as shown in Table 4 in the row labeled “Model using $f_{p}$ = 1.30 ± 0.10 Hz (population mean)”; this value of f_p was regressed as part of the model development, and represents a regional average site frequency. By combining all sites, the pulse width and linear decay rate at long periods stay about the same. The regional average curve is plotted relative to the observed amplification levels at all Obihiro sites in Figure 15. This model is intended for application to sites within the study region (Tokachi River flood plain deposits) for situations in which $f_{peak}$ is not available. Where $f_{peak}$ is available, its use for site amplification prediction is preferred.

Figure 15.

Fit of model to observed amplification when model coefficients are taken from regional average model.

Aleatory variability model and model bias

The standard deviation terms to use with the proposed site amplification model are $τ$ for between-event variability and $ϕ$ for within-event variability. The $τ$ model is assumed to be unaffected by the site amplification model described here, and can be taken from GMMs. The $ϕ$ model can be taken from the standard deviation of the within-event residuals obtained through the use of the GMMs in combination with the proposed site amplification models. To develop this within-event standard deviation model, we compute residuals as in Equation 2, but now using the region-specific site amplification model in lieu of the Pea22 GMM’s site terms. After subtracting event terms (Equation 3) and regional terms (Equation 4), we then partition the within-event residuals as

δ W_{ijlk}^{reg} = c + η_{S, j}^{'} + ε_{ij}

(10)

where $c$ is the model bias, $η_{S, j}^{'}$ is the site term at site $j$ obtained through the use of the modified GMM (the prime ′ differentiates this site term from that in Equation 6) and $ε_{ij}$ is the remaining residual. The model bias indicates the overall model misfit relative to the data (equivalent to the average of all $δ W_{ijlk}^{reg}$ ). The standard deviation of $η_{S, j}^{'}$ is denoted the site-to-site dispersion ( $ϕ_{S 2 S}$ ) while the standard deviation of $ε_{ij}$ is the single-station within-event dispersion ( $ϕ_{SS}$ ).

These standard deviations are shown in Figure 16 for both the model employing regional average parameters and site-specific $f_{peak}$ values. Figure 16a shows that $ϕ_{S 2 S}$ is approximately 0.2–0.3 for both of the proposed models, which is significantly below average values at short periods (<1 s) for the entirety of Japan (Al Atik, 2015). This is expected because of the relatively uniform geotechnical conditions in the study region. Figure 16b shows single-station standard deviations ( $ϕ_{ss}$ ), which are slightly lower than the Japan average values obtained previously from KiK-net data by Rodriguez-Marek et al. (2011).

Figure 16.

Comparison of within-event standard deviation terms from Obihiro stations (this study) and Japan averages: (a) site-to-site standard deviations as compared with Japan-average from Al Atik (2015) and (b) single-station standard deviation as compared with Japan-average from Rodriguez-Marek et al. (2011).

Figure 17 compares the model bias for the regional average and site-specific models, both of which are effectively zero. Also shown for comparative purposes is the bias obtained using the site term in the Pea22 GMM, which is very large (indicating underprediction). The substantial bias of the GMM for the Obihiro sites demonstrates the need for site-specific site factors for these very soft sites with strong impedance contrasts.

Figure 17.

Comparison of model bias for the ergodic model of Pea22 GMM and the Pea22 GMM combined with the two proposed region-specific site amplification models.

Nonlinearity

The previously developed models for mean site response (Equation 9, Figure 13) are based on relatively weak ground motions from 8 events, and hence provide linear site amplification. To investigate nonlinear effects, we examine model misfits relative to the data from Event 10 (the 2003 Tokachi-Oki Earthquake), which produces significantly stronger ground motions than the other considered events at the ODCD stations (Figure 3).

The analysis of Event 10 follows the process used for other events. Equation 1 is used to compute total residuals, and Equation 2 is used to compute the event terms ( $η_{E, 10}$ ). Figure 18 compares the event terms for Event 10 to the other events considered. In the case of ODCD stations, the GMM used for residuals calculation is modified from the published version

{(μ_{lnY})}_{ij} = μ_{lnY}^{r} + f_{1}

(11)

where $μ_{lnY}^{r}$ is the mean ground motion prediction for reference rock from Pea22 and $f_{1}$ is from Equation 9.

Figure 18.

Event term by Pea22 for Event 10 compared with those for Events 1–9.

Region-adjusted within event residuals are computed using Equation 5. Figure 19 plots $δ W^{reg}$ for Event 10 along with those for the other events, using ODCD stations only. The residuals are plotted as a function of $PG A^{r}$ , which is the median peak acceleration for the reference site condition from the Pea22 GMM. The within event residuals of $δ W^{reg}$ with respect to $PG A^{r}$ exhibit no trend for $PG A^{r} < 0.03 g$ , which is the weak shaking condition associated with Events 1–9. This confirms our initial assumption of linearity for those events. The Event 10 data, however, indicate $PG A^{r}$ dependencies that are downward at short periods and upward at 5.0 s.

Figure 19.

Region adjusted within event-residuals for data recorded at ODCD stations versus reference rock site PGA^r.

The trends of the results in Figure 19 are fit by regression using the relation in Equation 7 with $f_{1} = 0$ (the setting of $f_{1}$ to zero is because of its inclusion in the model used for residuals analysis; Equation 9) and $f_{3} = 0.1 g$ (a typical value). As a result, only parameter $f_{2}$ is set by regression. Nonlinearity is evident from curvature in the fit line, and is quantified by $f_{2} \neq 0$ . The downward curvature at short periods is expected, and results from increased damping in sediments as strains increase. The upward trend at long periods is also fairly common. This typically occurs because nonlinearity softens the soil, increasing its fundamental period. Because the elastic (small strain) period is in the range of 1–2 s, this softening will bring the soil deposits to resonance at longer periods, which would be reflected by increased long period PSA as indicated by the trend line. Figure 20 plots $f_{2}$ with period for application in Obihiro, along with recommended values for this parameter from Seyhan and Stewart (2014) (a semi-empirical model for active tectonic regions, exercised using the regional geometric mean V_S₃₀ of 150 m/s). The Obihiro data has generally the same pattern with period as the ergodic model, but with reduced nonlinearity, and positive values of $f_{2}$ at long periods. The $f_{2}$ values in Figure 20 are based on very limited data, and would ideally be further investigated using ground response simulations using region-specific profiles prior to using the nonlinear component of the model in engineering applications. In particular, difficulties might be anticipated for very strong shaking conditions that might introduce soil failure, which is not reflected in the current estimates of parameters $f_{2}$ or $f_{3}$ .

Figure 20.

Nonlinear term f₂ as function of period as regressed from Obihiro data and from ergodic model for active regions (applied for V_S₃₀ = 150 m/s).

Summary and recommendations

GMMs generally include site terms intended for application over a range of V_S₃₀≈ 200–1500 m/s. These limits naturally lead to questions of how to model site response for sites with conditions that fall outside of this range. We address this question for a very soft soil region (Ohibiro) in Hokkaido Japan underlain by peaty organic soils and other, non-organic, recent sediments. The range of V_S₃₀ in the study region is 100–210 m/s (geometric mean 150 m/s).

We evaluate non-ergodic site responses at seven Obihiro sites using data from nine earthquakes. This analysis required removing regional biases in path effects identified for conditions when ground motions propagate between islands (e.g. event is off the coast of Honshu and ground motions are recorded on Hokkaido). After correcting for these effects, and applying event term adjustments tailored to account for source misfits, site responses for the seven sites are found to have common characteristics, which include peaks at periods of 0.5–0.9 s that are associated with resonances and impedance contrasts of the soft and weak surficial soils (e.g. peats and clays) in the region relative to stiffer underlying soils.

The recommended site response model consists of a modified Mexican Hat wavelet function (Equation 9), the most important parameter of which is a peak frequency $f_{p}$ that is estimated from peaks of microtremor-based HVSR ( $f_{peak}$ ) as $f_{p} = f_{peak}$ . Although less desirable, the model could be applied when site-specific measurements of $f_{peak}$ are unavailable by using a regional average of $f_{p} = 1.3 Hz$ . Using data for an event that produces strong shaking, we investigate approximate nonlinear site response effects, which reveal patterns that are broadly similar to characteristics of an ergodic model for active regions. Application of the regional model substantially reduces site-to-site aleatory variability from a general model from Japan (e.g. 0.48 to 0.20 for PGA).

The regional model for mean site response and within-event standard deviation is applicable to the Tokachi River flood plain in the Obihiro region of Japan, as shown in Figure 2. The model is based on data from seven sites with relatively similar $V_{S}$ profiles and soil conditions, and it could be in error for sites in the study region if site conditions are appreciably different (e.g. sites without peat). The model does not apply to sites not located on the flood plain geologic unit, as identified on regional maps (HRDPRC, 2004). While the model is not considered applicable to other regions containing very soft peaty organic soils in the absence of validation, the modeling approach outlined herein can be applied to other regions with unusual geologic conditions that may substantially impact site response.

Supplemental Material

sj-docx-1-eqs-10.1177_87552930221082965 – Supplemental material for Region-specific linear site amplification model for peaty organic soil sites in Hokkaido, Japan

Supplemental material, sj-docx-1-eqs-10.1177_87552930221082965 for Region-specific linear site amplification model for peaty organic soil sites in Hokkaido, Japan by Pengfei Wang, Yi Tyan Tsai, Jonathan P Stewart, Atsushi Mikami and Scott J Brandenberg in Earthquake Spectra

Supplemental Material

sj-docx-2-eqs-10.1177_87552930221082965 – Supplemental material for Region-specific linear site amplification model for peaty organic soil sites in Hokkaido, Japan

Supplemental material, sj-docx-2-eqs-10.1177_87552930221082965 for Region-specific linear site amplification model for peaty organic soil sites in Hokkaido, Japan by Pengfei Wang, Yi Tyan Tsai, Jonathan P Stewart, Atsushi Mikami and Scott J Brandenberg in Earthquake Spectra

Footnotes

Acknowledgements

The authors thank Takashi Sato at Civil Engineering Research Institute for Cold Region, PWRI for providing access to the ground motion data files, OCDC for providing access to the sites for field measurements, and Tadahiro Kishida for providing advance access to NGA-subduction ground motion data in support of this work. Robert Kayen of USGS was of great assistance with the geophysical testing, both in the field and in guiding subsequent data analysis.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by California Department of Water Resources (Contract No. 4600010406). Partial support was also provided by the UCLA Civil & Environmental Engineering Department. This support is gratefully acknowledged.

ORCID iDs

Pengfei Wang

Jonathan P Stewart

Scott J Brandenberg

Supplemental material

Supplemental material for this article is available online.

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