# Source Parameter Estimation Method for Assessment of Structural Resiliencies

Authors: Zijun Wang; Boming Zhao
Journal: Shock and Vibration
 (2017)

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## Abstract

Assessing structural integrity and sustainability during natural hazards, for instance, strong earthquakes, is an effective way to reduce or even avoid large losses of life and property damage. With the established vulnerability relationships between source parameter and building damage, the seismic resilience of structures can be obtained after the source parameter is estimated in the early stage of an earthquake. For this purpose, we propose a method that employs theP wave displacement parameter to estimate earthquake magnitude in real time to quick assess the structural resiliencies. By selecting period and amplitude parameter as comparisons, the magnitude estimation formulas are derived, respectively, where the proposed P wave displacement parameter method is of the highest precision. Through the evolutionary estimation of the P wave displacement parameter as a function of the time window used, we show that the existing regression relationships can be extended to the large earthquake. Therefore, this paper provides a quick earthquake magnitude estimation method for the establishment of a more reasonable and accurate resilience assessment system for structures.


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## Body

## 1. Introduction

Assessing the structural integrity and sustainability during natural hazards, for instance, strong earthquakes, is an effective way to reduce or even avoid large losses of life and property damage. With the established vulnerability relationships between source parameter and building damage, if we can estimate the earthquake’s magnitude before the high-intensity shaking occurs, the seismic resilience of structures can be obtained. Earthquake early warning systems [1–4] can provide alerts of impending ground motions within a few seconds to tens of seconds after an earthquake occurs so that appropriate measures can be immediately taken to mitigate seismic risks.Strong earthquakes occur frequently in Mainland China, where more than 500 earthquakes with magnitude above 6.0 have occurred within 28 provinces from 1900 to the present day. Moreover, the region of earthquake occurrence has increased in size, and events occurred with greater intensity and frequency, meaning that the period of active earthquakes has arrived. These earthquakes destroy buildings and cause severe losses of life and property. For example, more than eighty thousand deaths or disappearances occurred in the 2008Ms 8.0 Wenchuan earthquake, which was a major tragedy for the Chinese people. Therefore, it is of great significance to seek an approach through scientific research to mitigate the potential threats associated with earthquake hazards.In this paper, the P wave displacement parameter(MId) method, which uses data from a single station, is then proposed. We additionally select the representative period (τc) and the amplitude (Pd) parameters, and we compare the relationships between each of these 3 parameters and the magnitude of the aftershocks of the 2008 Ms 8.0 Wenchuan earthquake. In addition, magnitude estimation formulas are derived and the accuracy of these formulas is explored by comparing the estimated magnitude with the catalog magnitude. Furthermore, we investigate to what degree the initial parameters indicate the Wenchuan mainshock magnitude and analyze the relationships among the expanding time window, the proposed MId parameter, and the estimated magnitude. With the established vulnerability relationships between magnitude and building damage, by quickly predicting the early structural antiseismic capacity from the estimated magnitude, this paper can provide a scientific basis for the establishment of a more reasonable and accurate resilience assessment system for structures.


## 2. Dataset

A robust input database is essential in identifying reliable regression functions for magnitude estimation using a statistical approach. According to the China Strong Motion Net Centre (CSMNC), after theMs 8.0 Wenchuan earthquake on May 12, 2008, a total of 383 aftershocks were recorded by the end of 30 September 2008. These events were generated over a rupture length of approximately 300 km with focal depths ranging from 2 to 20 km, and the records were obtained by strong motion seismographs with a dynamic range of ±2 g, which were mainly installed at free-field sites. The sampling rate was 200 sps.In this study, we select the mainshock and 43 aftershocks of the 2008Ms 8.0 Wenchuan earthquake. As criteria, we specify that the selected earthquakes should have magnitude greater than Ms=4.0 and hypocentral distances less than 150 km. Furthermore, each selected event is required to have at least three records to ensure good station coverage and avoid the path effects. Thus, we use a total of 306 acceleration waveforms from the selected events, which demonstrate a large range of focal depths and mechanisms varying from thrust to strike slip.Since correct picking of P arrivals and ensuring the exclusion of S waves from the analysis are prerequisites for accurately calculating the characteristic parameters, we use the three-step P phase detection method proposed by Wang and Zhao [5], and we double-check the arrival time via manual inspection for each waveform. After performing baseline error correction for the acceleration records, the signals are integrated to velocity records. Moreover, the velocity records are integrated to displacement records, which are required to calculate the parameter values. Then, for real-time applications, a high-pass recursive Butterworth filter with a cutoff frequency of 0.075 Hz is applied to the vertical components to remove the long-period drift that occurs after integration [6].


## 3. Source Parameter Estimation

### 3.1. Magnitude Estimation Method

The proposed P wave displacement parameter(MId) is defined as the integral of the squared high-pass filtered displacement of the vertical component ground motion, which ranges from 0 to 3 seconds of the initial P wave. The definition of MId shows that it is an integral of the squared displacement within the selected time window, which can reflect information with different periods carried by an advancing rupture on a fault plane. As a magnitude estimator, MId is a physically fundamental and source-dependent property. Nielsen [7] concluded that the flow rate of elastic energy controls earthquake fracture development and propagation. Fractures with higher initial energy are more likely to continue propagating over long distances and grow into earthquakes with large magnitude.To verify the validity of the proposed method, we select the two characteristic parameters that have so far proved to be the most robust, namely,τc and Pd [8, 9], for comparison. To correct the calculated MId and Pd values for the effects of distance, we normalize them to a reference distance of 80 km, which is the average of the hypocentral distances from the analyzed data set, as the way followed by Zollo et al. [10] and Festa et al. [11]. The final values of MId and Pd are referred to as MIdRef and PdRef, and they can be obtained after preliminary locations are available. Such locations can be determined using real-time procedures and data from a single station; for example, see the methods proposed by Odaka et al. [12] and Horiuchi et al. [13]. We average the characteristic parameters MIdRef, τc, and PdRef from the multiple observation records associated with each event and assume a linear regression model between the catalog magnitude Ms and the averaged parameters as log⁡parameter=A+B×Ms, whereA andB are constants that are to be determined from the regression analysis.


### 3.2. Relationships between Magnitude andMId, τc, and Pd

Using the current analytical form, the resulting best-fitting regression relationship betweenMIdRef and the magnitude is given by(1)log⁡MIdRef=-11.175+1.218×Ms±0.399.The individual data points and the average values oflog⁡(MIdRef) as a function of magnitude Ms are shown in Figure 1(a), where SDV means the standard deviation andR stands for the correlation coefficient (similarly hereinafter). The logarithm of the P wave displacement parameter shows a striking linear correlation with earthquake magnitude within the magnitude range considered (4≤Ms≤6.5).Figure 1
(a)–(c) showMIdRef, τc, and PdRef calculated from the first 3 seconds of P wave data (grey squares) and the averaged values (red squares). The magnitude regression relationships determined by this study are shown as solid lines, and the standard deviations are shown as dashed lines.
(a)
(b)
(c)The average period of ground motionτc is a measurement of the average period of the P wave within the first few seconds, which is frequently used to estimate magnitude [1]. According to the dataset, the regression relationship between τc and magnitude Ms is described by(2)log⁡τc=-1.362+0.246×Ms±0.148.Sinceτc approximately represents the P wave pulse width, there is a good relationship between the average period of ground motion and the final earthquake magnitude in Figure 1(b).As for the peak displacement amplitudePd method, the linear regression relationship between the logarithm of PdRef and earthquake magnitude Ms is established as follows:(3)log⁡PdRef=-5.076+0.548×Ms±0.275.The regression curve in Figure1(c) shows that PdRef is well correlated with earthquake magnitude.


### 3.3. Estimated Magnitude Comparison

We can invert the regression results in (1)–(3) to obtain equations for estimating earthquake magnitude from the above 3 characteristic parameters as follows:(4)MMIdRef=9.175+0.821×log⁡MIdRef±0.328Mτc=5.537+4.065×log⁡τc±0.603MPdRef=9.262+1.824×log⁡PdRef±0.502.We call these 3 estimated magnitude valuesMe the P wave displacement parameter magnitude MMIdRef, the average period of ground motion magnitude Mτc, and the peak displacement amplitude magnitude MPdRef. Figure 2 shows that the Me values calculated from (4) display approximately 1 : 1 relationships with the catalog magnitude Ms using the selected data.Figure 2
Magnitude determined using the 3 characteristic parameters versus the catalog magnitudeMs. Solid lines show the least squares fits, whereas the two dashed lines show the range of one standard deviation.
(a)
(b)
(c)On the 45° line,Me=Ms, and the dashed lines illustrate the one-standard-deviation envelopes. Most of the estimated magnitude values fall within one standard deviation of the regression line. The proposed MIdRef is associated with the smallest deviation, that is, 0.328, whereas the corresponding values for Mτc and MPdRef are 0.603 and 0.502, respectively.Based on the prediction equations (4), we calculate the differences Δ between the estimated and catalog magnitude values for each record (i.e., Δ=Me-Ms), which obey normal distributions. Figure 3 shows histograms that reflect the corresponding frequency distributions. According to the computed probability density curves, the standard deviation associated with the proposed parameter MIdRef is the lowest within the effective estimated range, demonstrating that the MId method displays the highest reliability and has apparent advantages over the τc and Pd methods.Figure 3
Frequency distribution histograms and the corresponding probability density curves of the estimated magnitude differences using each of the 3 characteristic parameters.
(a)
(b)
(c)


## 3.1. Magnitude Estimation Method

The proposed P wave displacement parameter(MId) is defined as the integral of the squared high-pass filtered displacement of the vertical component ground motion, which ranges from 0 to 3 seconds of the initial P wave. The definition of MId shows that it is an integral of the squared displacement within the selected time window, which can reflect information with different periods carried by an advancing rupture on a fault plane. As a magnitude estimator, MId is a physically fundamental and source-dependent property. Nielsen [7] concluded that the flow rate of elastic energy controls earthquake fracture development and propagation. Fractures with higher initial energy are more likely to continue propagating over long distances and grow into earthquakes with large magnitude.To verify the validity of the proposed method, we select the two characteristic parameters that have so far proved to be the most robust, namely,τc and Pd [8, 9], for comparison. To correct the calculated MId and Pd values for the effects of distance, we normalize them to a reference distance of 80 km, which is the average of the hypocentral distances from the analyzed data set, as the way followed by Zollo et al. [10] and Festa et al. [11]. The final values of MId and Pd are referred to as MIdRef and PdRef, and they can be obtained after preliminary locations are available. Such locations can be determined using real-time procedures and data from a single station; for example, see the methods proposed by Odaka et al. [12] and Horiuchi et al. [13]. We average the characteristic parameters MIdRef, τc, and PdRef from the multiple observation records associated with each event and assume a linear regression model between the catalog magnitude Ms and the averaged parameters as log⁡parameter=A+B×Ms, whereA andB are constants that are to be determined from the regression analysis.


## 3.2. Relationships between Magnitude andMId, τc, and Pd

Using the current analytical form, the resulting best-fitting regression relationship betweenMIdRef and the magnitude is given by(1)log⁡MIdRef=-11.175+1.218×Ms±0.399.The individual data points and the average values oflog⁡(MIdRef) as a function of magnitude Ms are shown in Figure 1(a), where SDV means the standard deviation andR stands for the correlation coefficient (similarly hereinafter). The logarithm of the P wave displacement parameter shows a striking linear correlation with earthquake magnitude within the magnitude range considered (4≤Ms≤6.5).Figure 1
(a)–(c) showMIdRef, τc, and PdRef calculated from the first 3 seconds of P wave data (grey squares) and the averaged values (red squares). The magnitude regression relationships determined by this study are shown as solid lines, and the standard deviations are shown as dashed lines.
(a)
(b)
(c)The average period of ground motionτc is a measurement of the average period of the P wave within the first few seconds, which is frequently used to estimate magnitude [1]. According to the dataset, the regression relationship between τc and magnitude Ms is described by(2)log⁡τc=-1.362+0.246×Ms±0.148.Sinceτc approximately represents the P wave pulse width, there is a good relationship between the average period of ground motion and the final earthquake magnitude in Figure 1(b).As for the peak displacement amplitudePd method, the linear regression relationship between the logarithm of PdRef and earthquake magnitude Ms is established as follows:(3)log⁡PdRef=-5.076+0.548×Ms±0.275.The regression curve in Figure1(c) shows that PdRef is well correlated with earthquake magnitude.


## 3.3. Estimated Magnitude Comparison

We can invert the regression results in (1)–(3) to obtain equations for estimating earthquake magnitude from the above 3 characteristic parameters as follows:(4)MMIdRef=9.175+0.821×log⁡MIdRef±0.328Mτc=5.537+4.065×log⁡τc±0.603MPdRef=9.262+1.824×log⁡PdRef±0.502.We call these 3 estimated magnitude valuesMe the P wave displacement parameter magnitude MMIdRef, the average period of ground motion magnitude Mτc, and the peak displacement amplitude magnitude MPdRef. Figure 2 shows that the Me values calculated from (4) display approximately 1 : 1 relationships with the catalog magnitude Ms using the selected data.Figure 2
Magnitude determined using the 3 characteristic parameters versus the catalog magnitudeMs. Solid lines show the least squares fits, whereas the two dashed lines show the range of one standard deviation.
(a)
(b)
(c)On the 45° line,Me=Ms, and the dashed lines illustrate the one-standard-deviation envelopes. Most of the estimated magnitude values fall within one standard deviation of the regression line. The proposed MIdRef is associated with the smallest deviation, that is, 0.328, whereas the corresponding values for Mτc and MPdRef are 0.603 and 0.502, respectively.Based on the prediction equations (4), we calculate the differences Δ between the estimated and catalog magnitude values for each record (i.e., Δ=Me-Ms), which obey normal distributions. Figure 3 shows histograms that reflect the corresponding frequency distributions. According to the computed probability density curves, the standard deviation associated with the proposed parameter MIdRef is the lowest within the effective estimated range, demonstrating that the MId method displays the highest reliability and has apparent advantages over the τc and Pd methods.Figure 3
Frequency distribution histograms and the corresponding probability density curves of the estimated magnitude differences using each of the 3 characteristic parameters.
(a)
(b)
(c)


## 4. Magnitude Estimation for Large Earthquakes

The results above imply that the parameters determined from the first 3-second P wave agree well with the catalog magnitude in the considered range(4≤Ms≤6.5); thus, the regression equations are practicable to estimate the magnitude for earthquakes under 6.5 reliably. However, for large earthquakes, the use of a limited portion of signals allows capturing only the slip contributions from a limited fault portion, which may result in the underestimation of final earthquake magnitude (i.e., the saturation problem) [11, 14].For the 2008Ms 8.0 Wenchuan mainshock, there were 32 records available within the criteria (hypocentral distance less than 150 km), which provided us with a unique opportunity to investigate the methodologies up to great earthquakes. We first check the extent to which these characteristic parameters could link the mainshock magnitude using the initial 3-second P wave. Then, we analyze the relationships among the increased time window, proposed MId, and predicted magnitude.Together with the mainshock and aftershock records, we establish the correlations betweenMIdRef, τc, PdRef, and magnitude, respectively (see Figure 4), using the procedure discussed above. According to the regressions, the characteristic parameters derived for the mainshock using the 3-second P wave were all located without the one standard deviation and the anticipated magnitude values were calculated as MMIdRef=7.18, Mτc=7.05, and MPdRef=6.91, respectively. Although these three parameters all exhibited a saturation problem, the proposed parameter MId was relatively more accurate and of the lowest standard deviation, meaning that it could be a reliable and robust magnitude estimator in the earthquake’s early stage.Figure 4
Reestablished regressions of catalog magnitudeMs with the 3 characteristic parameters (based on 3 seconds of P wave data) and considering the Wenchuan mainshock (red stars) concurrently. Solid lines show the least squares fits, whereas the two dashed lines show the one-standard-deviation envelopes.
(a)
(b)
(c)For the saturation problem in predicting large earthquakes, theoretically, a trade-off strategy between time and accuracy can be considered where better estimations could be obtained by enlarging the observation time window to update the characteristic parameters [15–17]. Since our proposed MId had the most accurate estimation for the mainshock with 3-second P wave compared with τc and Pd, we further investigate MId using different time windows ranging from 3 to 10 seconds for all of the available records.To avoid the S wave contamination while increasing the time window, we compute the theoretical S wave arrival times and exclude from our analysis all the stations for which the estimated S wave arrival occurred within the considered time window. Figure5 shows that the MIdRef magnitude for the Wenchuan mainshock increases with the length of the P wave records used in determining MIdRef. The MIdRef growth pattern implies that MIdRef determined from the first 6-7 seconds of P wave data may be sufficient to estimate the magnitude of earthquakes of magnitude 8.0. As the window length continues to grow, the magnitude converges at around 8.2. Since the magnitude is estimated only by the initial P wave, the results above illustrate that the parameter MId may be an accurate and rapid magnitude estimator for earthquakes with magnitude under 6.5. Moreover, it can also provide stable and accurate estimates for large earthquakes if an appropriately increased time window is used.Figure 5
Growth in theMIdRef parameter (averaged values) for the Wenchuan mainshock versus the lengths of the initial P wave records used. The corresponding magnitude scale is also represented; this scale is derived using the equation shown in Figure 4(a).


## 5. Discussion and Conclusion

We proposed a method in terms of the P wave displacement parameter(MId) to anticipate the magnitude in real time in order to quick assess the seismic resilience of structures. The MId, τc, and Pd values for the 43 aftershocks of the 2008 Wenchuan earthquake were calculated and the magnitude estimation formulas from these 3 parameters were derived, respectively. Our result showed that the magnitude obtained from MId of 3-second initial P wave agreed well with the catalog magnitude, where the smallest uncertainty was acquired compared with those of τc or Pd. With the established vulnerability relationships between magnitude and building damage, by quickly predicting the early structural antiseismic capacity from the estimated magnitude, this paper can provide a scientific basis for the establishment of a more reasonable and accurate resilience assessment system for structures.Using only a small portion of the initial P wave signals may cause saturation problems in predicting large earthquakes [18]. In the case of the Wenchuan Ms 8.0 mainshock, using the first 3-second P wave may underestimate the magnitude by about 0.8 to 1.0 unit with the investigated parameters (MMIdRef=7.18, Mτc=7.05, and MPdRef=6.91); however, the proposed MId was of the best performance. To mitigate this problem, several methodologies use a longer time window of P wave to update magnitude estimates [17, 19]. The evolutionary estimation of MId as a function of the time window shows that the existing methodologies and regression relationships can be extended to large earthquakes, and the saturation effect can be removed through the use of time windows of approximately 6-7 seconds with the investigated MId parameter.



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*Source: 5830124-2017-10-30.xml*