Regional model of peak ground motion in Southwestern Germany

Abstract

Ground motion prediction equations (GMPEs) and the effects of site amplifications are substantial for the assessment of seismic hazard. To investigate the regional earthquake ground motion in southwestern Germany, we fit ground motion models to observed horizontal peak ground acceleration from earthquakes with $0.9 \le M_{\text {L}} \le 4$ using the earthquake catalogue of the joint federal seismological services of Baden-Württemberg and Rhineland-Palatinate (Erdbebendienst Südwest), Germany. We use GMPEs that consider first-order geometrical spreading, first-order magnitude-scaling, and apparent anelastic attenuation. Due to indications from the data residuals, we additionally introduce a heuristically defined expression to consider Mohorovičić reflection phases, and a second-order geometrical decay term that is derived to approximate the decay of a general moment-tensor source. While the expression for the Mohorovičić reflection phases improved the data fit, the second-order decay term is hardly changing the resulting model. Averaged site deviations from the median model are incorporated to account for site effects. Depending on the local geological conditions, these deviations show a strong variability within individual seismogeographical regions.

Appendix A

1.1 A.1 List of stations and operators

Table 4 Stations used in this study, the corresponding operators or related project (abbreviations stated in Section 9), locations (WGS-84 coordinates in decimal degree and affiliations to seismogeographical regions), number $N_\text {obs}$ of the used PGA-values at each station, the determined station corrections $s_s$ for $\text {GMPE}^\text {Moho,IS}$, and the standard deviation $\sigma _\text {s}$ (in log$_{10}($m/$s^2)$) of the observed log(PGA)-values for each station

1.2 A.2 Description of optimised station corrections

Following, we describe the optimised staton corrections of $\text {GMPE}^{\text {Moho,IS}}$ considering geological aspects.

Northern Limestone Alps and the Alpine Foreland

The three stations DAVA, RETA, and OBER (in BY and eastern GV; station group a) of the Northern Limestone Alps have strong negative station corrections between about −0.7 and −0.6. Only MEM and several subsurface stations (BFO, FREU, ROMAN, SWS, IMS, LAGB, BIW: $-0.4$ to $-0.6$) reach similar negative values. The corrections of the stations A103D, UBR, TETT, KONZ, STEIN, SISB, and WALHA (station group b) in the vicinity of the Lake Constance (BO and western BM) are mostly negative ($-0.5$ to 0.1). They are typically placed on a Quarternary layer above Tertiary molasse rocks or on molasse rocks.

Swabian Jura and Eastern Württemberg

The station corrections of the regions SA and EW vary strongly. Stations MSS, ERPF, BHBD, GUT, BUCH, ZWI, HDH, and DEGG (station group c) which are placed on Jurassic limestone show moderate corrections between $-0.3$ and 0.2. In contrast, other stations (station group h: JUNG, BALG, SAUL, EBIN, TUEB, MSGN, REUL, A108A) of SA have essentially higher values of 0.3 to 0.7. These stations are placed on unconsolidated sediments (mostly Quarternary) with an estimated thickness of about 10 to 50 m which overlays rocks of mostly Jurassic origin. One exception is A104C with a moderate correction (0.0) although placed on a unconsolidated layer above Triassic consolidated rocks. ONST and EMING have station corrections of about 0.1 and 0.4, for which a thin Quarternary layer is estimated (thickness < 10 m or less). Some stations of SA and EW (A100A, URBA, ROTE) including WOER of FA and SIND of NF are placed on Triassic rocks and show moderate corrections $-0.1$ to 0.1 with exception of A117A (0.3).

URG with Eastern and Western Graben Shoulders

The surficial stations of the URG (including graben shoulders) mostly have positive corrections ranging from 0.0 to 0.5. Thereby, most of the surficial stations (station group e: WBA, WBB, KTD, ECH, PEB, VOEL, METMA, ENDD, FELD, BERGE, OPP) with relatively small corrections (-0.1 to 0.2) are placed on metamorphic or magmatic rocks of the graben shoulders.

Also, other stations (station group f: LBG, BRET, GALG, A124A in NW; SLE in SW; WBG, FLIN, A115A, MILB in NF) on the graben shoulders mostly have moderate corrections of $-0.2$ to 0.2 and are situated on different consolidated sedimentary rocks (limestone, siltstone, sandstone) of Triassic origin. However, slightly increased values are observed at A115A (0.4) and MILB (0.4), whereby at A115A a thin surficial layer of clay and silt is suspected.

The remaining surficial stations (station group g: FBB, A123A, WLS, BREM, LOES, BREI, OFFE, FREI, STAU) are placed within the graben and have tendentially larger corrections of about 0.0 to 0.5. These are mostly placed on a layer of Quarternary rocks over Tertiary sediments. VOGT (0.0) is one exception and is located on volcanic rocks of the Kaiserstuhl.

Central area of the Rhenish Massif

The stations in MR, HU, EI, and RS are commonly placed on Devonian claystone or siltstone. The station corrections of the surface stations on Devonian rock are in the range between $-0.3$ and 0.3 (station group d: AHRW, BEUR, DEP08, RIVT, ABH, FSH, GWBD, OCHT, GWBC, GLOK/DEP12, DEP14, TDN, TNS, BHE). Other surface stations (GWBE: 0.0, DEP02: 0.1) of MR are placed on different volcanic rocks (meta-igneous or unconsolidated rocks). The two stations WLF ($-0.3$) and WMG (0.2) are located more apart from the others and are placed on Jurassic respective Triassic rocks. This is in accordance with the corrections of the stations on Jurassic and Triassic rocks in the regions of SA, EW, and FA.

Subsurface stations

The subsurface stations commonly show clear negative station corrections. The subsurface stations (FACH, NICK, LAGB, BIW) in the central area of the Rhenish Massif, FREU in SA, BFO in NW, KIZ in SW, IMS in PS, and ROMAN in BO are in the range from $-0.2$ to $-0.6$. Still, the subsurface stations within SR and NR (SWS, BODE, BABA, WALT, NEEW, ROTT, LDE, LDO, ILLF, HOHE) show a wide range of corrections between $-0.6$ and 0.3. We think the existance of more positive corrections compared to subsurface stations at other regions is reasonable since probably not all stations reach the depth of the bedrock.

1.3 A.3 Coefficients for subregions

Table 5 Final coefficients of $\text {GMPE}^\text {Moho}$ and the unweighted cost $C_{w_i=1}$ from optimising regional data subsets

1.4 A.4 Statistics of bootstrap analysis

Table 6 Standard deviation $\sigma _\text {bootstrap}$ as well as median, 16th and 84th percentile ($Q_\text {50}$, $Q_\text {16}$, $Q_\text {84}$) of the model parameters determined from 250 bootstrap replications for $\text {GMPE}^\text {Moho}$

1.5 A.5 Approximate distance-decay considering the intermediate wavefield

Several authors have reported an increased attenuation slope at small distances for shallow earthquakes (Chang et al. 2001; Cotton et al. 2008; Atkinson 2015). To model this effect, Cotton et al. (2008) recommended to include a term that is dependent on focal depth of the earthquake as done by Chang et al. (2001). Cotton et al. (2008) further associate the dependency on depth with an effect investigated by Frankel et al. (1990). Frankel et al. (1990) showed that a steep decay proportional to $r^{-1.5}$ between 15 and 90 km can be explained by reflections at the bottom side of the layer interfaces above the sources. For this purpose, they compared a velocity model with two layers above the source with a model without discontinuities above. Without discontinuities above the source, a decay of $r^{-1}$ was observed.

In this study, we observe a steep decay of amplitude residuals within the distance up to 30 km which could not be explained from the decay term $f_{\text {geom}}(r_{e,s}) =a \, \log _{10}(r_{e,s})$ with the fitted regression coefficients of about $a \approx -1.5$. In the following, we consider the wavefield from a general moment-tensor source in an unbounded medium and allow several simplifications. The approximated near-/intermediate-field terms are used to test, if near-/intermediate-fields can reproduce the increased slope near the source.

1.5.1 A.5.1 Wavefield terms of a general moment-tensor source

The various components of the seismic wavefield from a general moment-tensor source are stated by Lokmer and Bean (2010). First, we rearrange their equation

$$\begin{aligned} \varvec{\Psi }_{\text {N}}= & {} \mathbf {R}^{\text {N}}(\theta ) \, \frac{M(\omega )}{4 \pi \rho \alpha ^{2} r^{2}} \Bigg \{ \left[ \frac{i}{2 \pi n_{\lambda }} - \left( \frac{1}{2 \pi n_{\lambda }} \right) ^{2}\right] \cdot e^{2 i \pi n_{\lambda }} \nonumber \\&- \left[ \frac{\alpha }{\beta } \frac{i}{2 \pi n_{\lambda }} - \left( \frac{1}{2 \pi n_{\lambda }} \right) ^{2} \right] \cdot e^{2 i \frac{\alpha }{\beta } \pi n_{\lambda }} \Bigg \} \end{aligned}$$

(15)

for the near field to

$$\begin{aligned} \varvec{\Psi }_{\text {N}}= \varvec{\Psi }_{\text {N1}} + \varvec{\Psi }_{\text {N2}} \end{aligned}$$

(16)

with

$$\begin{aligned} \varvec{\Psi }_{\text {N1}} = \mathbf {R}^{\text {N}}(\theta ) \, \frac{M(\omega )}{4 \pi \rho \alpha ^{2} r^{2}} \frac{1}{4 \pi ^{2} n_{\lambda }^{2}} \left( e^{i 2 \pi n_{\lambda } } - e^{i 2 \frac{\alpha }{\beta } \pi n_{\lambda } } \right) \end{aligned}$$

(17)

and

$$\begin{aligned} \varvec{\Psi }_{\text {N2}}= & {} \mathbf {R}^{\text {N}}(\theta ) \, \frac{M(\omega )}{4 \pi \rho \alpha ^{2} r^{2}} \frac{1}{2 \pi n_{\lambda }} \nonumber \\&\cdot \left( e^{i \left( 2 \pi n_{\lambda } + \pi /2 \right) } - \frac{\alpha }{\beta } e^{i \left( 2 \frac{\alpha }{\beta } \pi n_{\lambda } + \pi /2 \right) } \right) . \end{aligned}$$

(18)

The corresponding intermediate wavefields $\varvec{\Psi }_{\text {IP}}$ and $\varvec{\Psi }_{\text {IS}}$ as well as the far wavefield $\varvec{\Psi }_{\text {FP}}$ and $\varvec{\Psi }_{\text {FS}}$ for P- and S-waves read as (Lokmer and Bean 2010)

$$\begin{aligned} \varvec{\Psi }_{\text {IP}}= & {} \mathbf {R}^{\text {IP}}(\theta ) \, \frac{M(\omega )}{4 \pi \rho \alpha ^{2} r^{2}} e^{i 2 \pi n_{\lambda } } \end{aligned}$$

(19)

$$\begin{aligned} \varvec{\Psi }_{\text {IS}}= & {} \mathbf {R}^{\text {IS}}(\theta ) \, \frac{M(\omega )}{4 \pi \rho \beta ^{2} r^{2}} e^{i 2 \frac{\alpha }{\beta } \pi n_{\lambda }} \end{aligned}$$

(20)

$$\begin{aligned} \varvec{\Psi }_{\text {FP}}= & {} \mathbf {R}^{\text {FP}}(\theta ) \, \frac{M(\omega )}{4 \pi \rho \alpha ^{2} r^{2}} 2 \pi n_{\lambda } e^{i ( 2 \pi n_{\lambda } + \pi /2 )}\end{aligned}$$

(21)

$$\begin{aligned} \varvec{\Psi }_{\text {FS}}= & {} \mathbf {R}^{\text {FS}}(\theta ) \, \frac{M(\omega )}{4 \pi \rho \beta ^{2} r^{2}} 2 \pi \frac{\alpha }{\beta } n_{\lambda } e^{i ( 2 \frac{\alpha }{\beta } \pi n_{\lambda } + \pi /2)} \end{aligned}$$

(22)

with P- and S-wave velocity $\alpha$ and $\beta$, the radiation patterns $\mathbf {R}^{\text {N}}(\theta ) \,$ , $\mathbf {R}^{\text {IP}}(\theta ) \,$, $\mathbf {R}^{\text {IS}}(\theta ) \,$, $\mathbf {R}^{\text {FP}}(\theta ) \,$, $\mathbf {R}^{\text {FS}}(\theta ) \,$ of the wavefield parts (near, intermediate P-wavefield, intermediate S-wavefield, far P-wavefield, far S-wavefield), density $\rho$, the Fourier spectrum $M(\omega )$ of the source depending on angular frequency $\omega$, the distance r to the source, and the distance ($n_{\lambda } = \frac{r \omega }{ 2 \pi \alpha }$) to the source measured in P-wavelengths.

The dependencies of the wavefield parts on the distance are

$$\begin{aligned} \varvec{\Psi }_{\text {N1}}\propto & {} \frac{1}{r^{2}} \cdot \frac{1}{n_{\lambda }^{2}}\end{aligned}$$

(23)

$$\begin{aligned} \varvec{\Psi }_{\text {N2}}\propto & {} \frac{1}{r^{2}} \cdot \frac{1}{n_{\lambda }} \end{aligned}$$

(24)

$$\begin{aligned} \varvec{\Psi }_{\text {IP}}\propto & {} \frac{1}{r^{2}} \end{aligned}$$

(25)

$$\begin{aligned} \varvec{\Psi }_{\text {IS}}\propto & {} \frac{1}{r^{2}} \end{aligned}$$

(26)

$$\begin{aligned} \varvec{\Psi }_{\text {FP}}\propto & {} \frac{1}{r^{2}} \cdot n_{\lambda }\end{aligned}$$

(27)

$$\begin{aligned} \varvec{\Psi }_{\text {FS}}\propto & {} \frac{1}{r^{2}} \cdot n_{\lambda }, \end{aligned}$$

(28)

disregarding the constants of the medium, the source coefficients and the spatial periodicities in Eqs. 17 to 22. We see that the distance-decay of the far (intermediate) P-wavefield behaves as the distance-decay of far (intermediate) S-wavefield. Further on, the intermediate field decay is one order higher than the far field decay with respect to $n_{\lambda }$. The part $\varvec{\Psi }_{\text {N1}}$ ($\varvec{\Psi }_{\text {N2}}$) of the near field decay is two (one) order higher than the intermediate fields with repect to $n_{\lambda }$.

1.5.2 A.5.2 Simplifications for an approximated PGA-decay

Since the spatial decay of the far (intermediate) P-wavefield behaves as the far (intermediate) S-wavefield behaviour (Appendix A.5.1), and since PGA on the horizontal components is rather composed by the S-phases than by P-phases, we omit the P-wavefields to derive a simple form for an approximated overall decay. We also neglect the near wavefield, since its order of decay is still higher than the intermediate wavefield decays. It remains the intermediate S-wavefield and the far S-wavefield:

$$\begin{aligned} \varvec{\Psi }_{\text {IS}}(\omega )= & {} \mathbf {R}^{\text {IS}}(\theta ) \, \frac{M(\omega )}{4 \pi \rho \, \beta ^{2} \, r^{2}} \, e^{i 2 \frac{\alpha }{\beta } \pi n_{\lambda } } \end{aligned}$$

(29)

$$\begin{aligned} \varvec{\Psi }_{\text {FS}}(\omega )= & {} \mathbf {R}^{\text {FS}}(\theta ) \, \frac{M(\omega ) \, \alpha \, n_{\lambda }}{2 \rho \, \beta ^{3} \, r^{2}} \, e^{i ( 2 \frac{\alpha }{\beta } \pi n_{\lambda } + \pi /2)} . \end{aligned}$$

(30)

In the following, we consider the decay of Eqs. 29 and 30 only at the dominant period $T_{0}$ of the spectral wavefield so that we can write for the number of wavelengths for the dominant angular frequency $\omega _{0}$ or dominant period $T_{0}$:

$$\begin{aligned} n_{\lambda 0} = \frac{ \omega _{0} \, r}{ 2 \pi \alpha } = \frac{r}{T_{0} \, \alpha } \,. \end{aligned}$$

(31)

We further neglect the azimuthal dependency of the radiation patterns ($\mathbf {R}^{\text {IS}}=\mathbf {R}^{\text {FS}}=1$), the spatial periodicity ($e^{i ( 2 \frac{\alpha }{\beta } \pi n_{\lambda } + \pi /2)}=1$), and introduce the factors $z^{\text {IS}}=\frac{M(\omega ) }{4 \pi \rho \beta ^{2}}$ and $z^{\text {FS}}=\frac{M(\omega ) \,\alpha }{2 \rho \beta ^{3}}$. With Eq. 31 and the described simplification, we can write for the considered wavefields (Eqs. 29 and 30):

$${\mathbf\Psi}_{\mathrm{IS}}\;\vert_{T=T_0}\;= z^{\mathrm{IS}}\cdot\frac1{r^2}$$

(32)

$${\mathbf\Psi}_{\mathbf{FS}}\vert_{T=T_0}=\frac{z^{\mathrm{FS}}}{\mathrm\alpha}\cdot\frac1{r\cdot T_0}\cdot$$

(33)

Consequently, we can write for their ratio:

$$\begin{aligned} \frac{ \varvec{\Psi }_{\text {IS}}}{ \varvec{\Psi }_{\text {FS}}} \Bigg |_{T=T_{0}} = \frac{z^{\text {IS}} \,\cdot \,\alpha }{z^{\text {FS}}} \cdot \frac{T_{0}}{r} . \end{aligned}$$

(34)

We sum up the wavefields $\varvec{\Psi }_{\text {FS}}$ and $\varvec{\Psi }_{\text {IS}}$ and use Eqs. 33 and 34:

$${\mathbf\Psi}_{\mathrm{FS}}\vert_{T=T_0}\;+\;{\mathbf\Psi}_{\mathrm{IS}}\vert_{T=T_0}\;\;\;={\mathbf\Psi}_{\mathrm{FS}}\cdot{\left.\left(1+\frac{{\mathbf\Psi}_{\mathrm{IS}}}{{\mathbf\Psi}_{\mathrm{FS}}}\right)\;\right|}_{\mathit\;T=T_0}\mathit\;\mathit\;\mathit\;=\frac{\mathrm z^{\mathrm{FS}}}{\mathrm\alpha}\cdot\frac1{T_0}\cdot\;r^{-1}\cdot\left(1+\frac{\alpha\cdot z^{\mathrm{IS}}}{z^{\mathrm{FS}}}\cdot T_0\cdot r^{-1}\right)\cdot$$

(35)

On the lines of the summation of the wavefield parts, we test a summation of the PGA parts, consider the logarithmic space, and introduce the exponents a and p to allow deviations from the theoretical decay ($p=-1$ and $a=-1$):

$$\begin{aligned}&\log _{10} \left( \text {PGA}^{\text {FS}} + \text {PGA}^{\text {IS}} \right) \nonumber \\= & {} \log _{10} \left( \frac{z^{\text {FS}}}{\alpha \,\cdot \, T_{0}} \,\cdot \, r^{a} \cdot \left( 1 + \frac{z^{\text {IS}} \cdot \alpha }{z^{\text {FS}}} \, \, T_{0} \,\cdot \, r^{p} \right) \right) \nonumber \\= & {} \log _{10} \left( \frac{z^{\text {FS}} }{ \alpha \,\cdot \, 10^{m\cdot M + c^{\prime }} \, } \right) \,+ \, a \, \log _{10}(r) \nonumber \\&+ \, \log _{10}\left( 1 + \frac{z^{\text {IS}} \cdot \alpha }{z^{\text {FS}}} \cdot 10^{m\cdot M + c^{\prime }} \,\cdot \, r^{p} \right) . \end{aligned}$$

(36)

The expression $10^{m\cdot M + c^{\prime }}$ estimates the predominant period (in s) from the event magnitude according to empirical scaling relations. Sato (1979) states several empirical relations of predominant period $T_{0}$ (in s) with P- and S-waves from studies by Kasahara (1957); Terashima (1968); Furuya (1969), and Yamaguchi et al. (1978). The relations have the form of $\log _{10}(T_{0}) = m \cdot M + c^{\prime }$ with a constant $c^{\prime }$ and a coefficient m that ranges from 0.4 to 0.58 depending on the study.

The transfer from wavefield summation to PGA summation is strictly speaking not valid, since the various wavefield parts do not necessarily superimpose to maximum values. However, we believe that the approach is still useful. The part $\text {PGA}^{\text {IS}}$ of the intermediate S-wavefield vanishes at large hypocentral distances. At small hypocentral distances, the PGA part of the far field is small in comparison to the near field part. Hence, we expect that Eq. 36 can be used as a first-order approximation including the decay of $\text {PGA}^{\text {IS}}$.

We get an alternative definition of the geometrical decay from Eq. 6 by considering the terms of Eq. 36 which are dependent on r and by defining $z=\frac{z^{\text {IS}} \cdot \alpha }{z^{\text {FS}}} \cdot 10^{c^{\prime }} = \frac{\beta }{2 \, \pi } \cdot 10^{c^{\prime }}$ ($\alpha$ and $\beta$ in km/s):

$$\begin{aligned} f_{\text {geom}}(r,M) = a \, \log _{10}(r) + \log _{10}\left( 1 + z \cdot 10^{m \cdot M } \cdot r^{p} \right) \,. \end{aligned}$$

(37)

Hereby, the term $a \, \log _{10}(r)$ corresponds to the geometrical decay of $\text {GMPE}^{\text {basic}}$. The omitted expression $\log _{10} \left( \frac{z^{\text {FS}} }{ \alpha \cdot 10^{m\cdot M + c^{\prime }} } \right) = \log _{10} \left( \frac{z^{\text {FS}} }{ \alpha } \right) - m M - c^{\prime }$ of Eq. 36 is independent from r and will be projected into $f_{\text {M}(M)} = d \cdot M$ and into the constant c (cf. Eqs. 5 and 8).

The coefficient p controls the steepness of the PGA-decay near the source relative to the far field decay. The coefficients z and m scale the PGA-values in relation to the far field PGA-values. Whereas z is independent from the event magnitude, the product $m \cdot M$ accounts for an amplitude scaling of the intermediate field with magnitude that differs from the scaling of the far field.