臺北盆地淺層三維速度構造模型與驗證 Modelling and Validation of 3-D Shallow Velocity Structure in the Taipei Basin

國立臺灣師範大學

地理學系第三屆空間資訊碩士在職專班論文

指導教授：王聖鐸博士

共同指導：林哲民博士

研究生：呂學敏

中華民國一○八年七月

Abstract

The purposes of this study is to test and propose a method combining different existing geophysical survey data and build a three-dimensional S-wave velocity model for ground motion simulation. The inverse distance weighting (IDW) and the ordinary kriging methods were applied to interpolate the data obtained from the suspension P-S logging, the microtremor array, the horizontal-to-vertical spectral ratio (H/V) and the receiver function approaches.

In regard to the ordinary kriging method, the test based on two types of setting, the first setting was to consider only the horizontal semivariogram to interpolate model.

Conversely, the second was to consider both the horizontal and vertical semivariogram.

Moreover, on the basis of the two major setting, the proposed confidence weight factor method was applied. Two set of the confidence weight factors calculated by the seismic

H/V simulation result was tested, the first set based on the types of geophysical approaches. The second set based on considering the different approaches at each site.

According to the spatial distribution of data, three depth scales were divided in this study. The parameters were tested by cross-validation and the fitness of models were verified by the seismic H/V simulation. The result showed that the simulation of

IDW method do not fit the real geological condition well. The value of the ordinary kriging considering both the horizontal and vertical semivariogram varied stronger than the other. And the effect of confidence weight factors was stronger as well. Despite the cross-validation result was worse than the method only considering horizontal semivariogram. The fitness of simulation was higher than the latter.

Keywords: Taipei Basin, S-wave Velocity, Ordinary Kriging

Chinese Abstract

摘要

本研究目的在測試並提出結合既有的地球物理探勘資料的方法，並建立可用

於地動模擬的臺北盆地三維速度構造模型。本研究結合了鑽探井測、微地動陣列、

微地動單站頻譜比法以及接收函數法等資料，利用反距離權重法與普通克利金法

分別進行測試。

在普通克利金法之中，測試了僅考量水平半變異函數的方法以及同時考量垂

直與水平半變異函數兩種主要設定，並分別對該兩種設定的普通克利金法分別加

入了所提出之信心權重因子方法進行測試結果。本研究測試了兩種信心權重因子，

其中，第一種權重因子以方法類別為基礎，利用各方法之強震站模擬之平均值作

為權重因子，第二種信心權重因子則以在相異場址之下，不同的方法模擬的個別

結果作為依據。

本研究依照資料的空間分布將內插模型分為三個部分進行內插，以交叉驗證

法驗證參數的結果，並以強震站單站頻譜法的模擬結果驗證模型用於模擬的適配

性。其結果顯示反距離權重法得到的速度不符合實際地質情況，普通克利金法僅

考量水平半變異函數使得內插數值變化較為緩慢，其交叉驗證的誤差略小於加入

垂直向半變異函數的方法，加入信心權重因子對於結果變化並不明顯。相反地，

考量垂直半變異數的方法對數值的影響變化較大，加入權重信心因子亦造成較大

的數值變化。對兩項空間內插模型進行強地動單站頻譜法模擬，其結果顯示後者

得到較好的適配性。

關鍵字：臺北盆地、剪力波速、普通克利金法

Acknowledgement

First of all, I would like to express my sincere gratitude to my advisor Dr. Sendo

Wang and co-advisor Dr. Che-Min Lin. They showed me a responsible, rigorous, organized scholar model. I learned not merely the knowledge and skills to solve academic problems but also the critical thinking to find problems. Besides, I appreciated the oral examination committee members Dr. Tzai-Hung Wen and Dr.

Chun-Hsiang Kuo. They indicated many insufficient points in this dissertation and gave me some useful suggestions to improve it. Furthermore, I am really grateful to Dr. Kuo-

Liang Wen, Dr. Jyun-Yan Huang, Dr. Zhe-Ming Chen, Dr. Teruo Hatakeyama and Dr.

Yuuzi Tatuoka for leading me to the concerned fields. Finally, I have to thank the people who supported me during the two years. My family, friends, classmates and the seniors in research room 304, 315, 502 and 602. Thanks for your support.

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Index

Abstract ...... i

Chinese Abstract ...... ii

Acknowledgement ...... iii

Index ...... iv

List of Tables ...... vii

List of Figures ...... viii

Chapter 1 Introduction ...... 1

1.1 Motivation ...... 1

1.2 Literature Review...... 3

1.2.1 Geophysical Survey Data ...... 3

1.2.2 Geostatistical Interpolation Method ...... 8

Chapter 2 Study Area and Data ...... 12

2.1 Area of Interest ...... 12

2.2 S-wave Velocity Data ...... 14

Chapter 3 Methodology ...... 18 iv

3.1 Inverse Distance Weighting ...... 18

3.2 Ordinary Kriging ...... 21

3.2.1 Regionalized Variable Theory ...... 22

3.2.2 Stationary ...... 22

3.2.3 Semivariogram ...... 24

3.2.4 Effects in the semivariogram ...... 33

3.2.5 Ordinary Kriging ...... 43

3.3 Confidence Weight factor ...... 48

3.4 Validation ...... 50

3.4.1 Cross-Validation ...... 50

3.4.2 Seismic H/V Simulation ...... 51

Chapter 4 Result and Discussion ...... 53

4.1 Study Process ...... 53

4.1 Result of the IDW Method...... 56

4.2 Confidence Weight Factor Calculation ...... 56

4.3 Semivariogram Analysis ...... 57

4.4 Cross-Validation ...... 61

4.5 Simulation ...... 65

4.6 Discussion ...... 67

Chapter 5 Conclusion and Suggestion ...... 72

5.1 Conclusion ...... 72

5.2 Suggestion ...... 73

References ...... 74

Appendix A ...... 81

Appendix B ...... 92

Appendix C ...... 95

Appendix D ...... 98

Appendix E ...... 101

Appendix F...... 128

List of Tables

Table 2.1 Data format in the study...... 15

Table 3.1 Example of ordinary Kriging...... 49

Table 3.2 Example of proposed method...... 49

Table 4.1 data and methods of models...... 61

Table 4.2 ME, MAE and MAPE result of model A...... 62

Table 4.3 ME, MAE and MAPE result of model B...... 62

Table 4.4 ME, MAE and MAPE result of model C...... 63

Table 4.5 ME, MAE and MAPE result of model D...... 63

Table 4.6 ME, MAE and MAPE result of model E...... 64

Table 4.7 ME, MAE and MAPE result of model F...... 64

vii

List of Figures

Figure 1.1 Borehole logging measurement ...... 7

Figure 1.2 Process of Micretremor Array measurement...... 8

Figure 2.1 The geographic position of the Taipei Basin...... 13

Figure 2.2 Profile of the Taipei Basin...... 13

Figure 2.3 Distribution of different geological measurement approaches ... 16

Figure 2.4 EGDT data process...... 17

Figure 2.5 data integration at the same location...... 17

Figure 3.1 Power function by different P value...... 19

Figure 3.2 Example of search radius...... 20

Figure 3.3 Example of K-Nearest neighbor algorithm...... 21

Figure 3.4 Semivariance scatter plot ...... 28

Figure 3.5 The theoretical relation between covariance and Semivariance. 29

Figure 3.6 Averaged Semivariance scatter plot...... 30

Figure 3.7 Empirical Semivariogram...... 31

viii

Figure 3.8 the relation between Sill, Range and Nugget...... 32

Figure 3.9 Theoretical fitting Models ...... 33

Figure 3.10 Shape effect...... 36

Figure 3.11 The nugget effect...... 37

Figure 3.12 The range effect ...... 38

Figure 3.13 The geometic anisotropy ...... 39

Figure 3.14 The zonal anisotropy ...... 40

Figure 3.15 Geometric anisotropy weight contours ...... 41

Figure 3.16 Zonal anisotropy weight contours ...... 42

Figure 3.17 Mixed anisotropy model ...... 43

Figure 4.1 the study process of IDW method...... 54

Figure 4.2 the study process of ordinary Kriging and proposed method. .... 55

Figure 4.3 The Semivariogram of upper part in x-y direction...... 58

Figure 4.4 The Semivariogram of upper part in z direction...... 58

Figure 4.5 The Semivariogram of middle part in x-y direction...... 59

Figure 4.6 The Semivariogram of middle part in z direction...... 59

Figure 4.7 The Semivariogram of lower part in x-y direction...... 60

Figure 4.8 The Semivariogram of lower part in z direction...... 60

Figure 4.9 the mean of simulated results...... 66

Figure 4.10 the difference of simulated results...... 67

Figure 4.11 The S-wave velocity profile built by model D...... 71

Figure 4.12 The Profile line of figure 4.11...... 71

Chapter 1 Introduction

1.1 Motivation

People living in Taiwan are strongly impacted by earthquake disasters. Over the past few decades, earthquakes have caused numerous human deaths and injuries, such as 1986 Hualien offshore, 1999 Chi-Chi, and 2016 Meinong earthquakes. These earthquakes resulted in not only significant economic damage but also some lasting social problems. Prevention of damage due to earthquake disasters is an important issue for the government and people in Taiwan.

Taiwan is located on the border of the Philippine Sea Plate and the Eurasian Plate.

Several types of landforms have been formed due to the unique natural environment.

The sedimentary basin and the alluvial plain have become the most densely inhabited area during its long historical development, gradually resulting in serval cities with high population density, such as Taipei, Taichung, and the Kaohsiung area. In northern

Taiwan, the Taipei area, located in a triangular sedimentary basin called the Taipei Basin, is the most populous. It is both the economic and political center of Taiwan. The majority of the important government agencies are situated in this area.

When the Chi-Chi earthquake (1999) occurred, the Taipei Basin suffered severe damage. One of the reasons for the serious damage is the “seismic site-effect”. This refers to the effect of the site condition on earthquake intensity. The geological condition inside the Taipei Basin amplified the seismic wave and extended it in the

Taipei Basin. Significant damage was caused by the extended shaking duration and higher earthquake intensity in this event. To prevent such a severe earthquake disaster from occuring again in such a high population density area, it is necessary to understand the geological structure of the Taipei Basin.

Wang et al. (2004) indicated that Taiwan is covered with thick sedimentary layers.

The seismic site effect would amplify seismic waves and cause great damages. A three- dimensional velocity model, when combined with different geophysical survey approaches, can benefit the simulation and prediction of ground motions. Simulation is an effective way to understand the seismic site effect on the basin. However, there is still no a robust model that can simulate the seismic motion in the Taipei Basin. The accuracy of the initial model affects the quality of the results.

An investigation into the S-wave velocity was started by government agencies and academic institutions. However, each investigation approach has its benefits and drawbacks. A method that combines different results is needed. Consequently, the purpose of this study is to combine different types of dataset and build an acceptable model for ground motion simulation. In the study, an interpolated S-wave velocity model of the Taipei Basin was built by using four types of data collected by different geophysical approaches. The model evaluated the site-effect by comparing the predicated result with results obtained from a strong motion station.

1.2 Literature Review

1.2.1 Geophysical Survey Data

Experts and scholars commonly use geophysical approaches to survey underground S-wave velocity. Investigating underground S-wave velocities can help experts describe the characteristics of ground motions. The S-wave velocity is an important parameter for the improvement of the Seismic Design Specifications of

Buildings on different sites (Lin et al., 2011; 陳俊德, 2013). In this study, the data collected from borehole logging measurement, microtremor array measurement, horizontal-to-vertical (H/V) spectral ratio analysis and a receiver function are used.

The National Center for Research on Earthquake Engineering (NCREE) and the

Central Weather Bureau (CWB) executed a project to collect data at seismic stations in the Taiwan Strong Motion Instrumentation Program (TSMIP). So far more than 483 locations have been completed. Most of the recorded depths for the data range from 30 to 40 m. Few data points were recorded at depths greater than 100 m. This approach uses the suspension P–S logging technique to measure S-wave and P-wave velocities at different depths at the seismic station locations. The probe source generated waves at certain depths and the upper and lower receivers received the wave, which passed through the strata. Then the velocity between upper and lower receivers can be calculated (Figure 1.1). Kuo et al. (2012) introduced the origin and technique of of the

Engineering Geological Databases for TSMIP (EGDT). In addition, the EGDT data were applied to classify the site for the Seismic Design Specifications of Buildings.

This study shows that the EGDT data are useful for geological and civil engineering

because of the local accuracy in the shallow subsurface. In contrast, the EGDT data are limited by the depth of the boreholes. Most EGDT boreholes are limited to approximately 30 m deep and the spatial distribution of the EGDT data are typically constrained by the locations of the boreholes. These constraints make EGDT data only useful in the shallow part of the subsurface although this method is certain highly accurate.

The microtremor measurement is one of the most common site survey approaches.

The term “microtremor” refers to seismic background noise. Different from seismic signals, the source is from natural or artificial noise, such as a sea wave or building construction. As compared with seismic waves, microtremor signals can be received at any time. This means that the measurement is quicker and easier than watching for signals; furthermore, it does not require a large budget to maintain stations. Two approaches that apply the microtremor signals are used in this study. The first one is the microtremor array measurement. The basic process of this approach is to distribute over a site a set of portable seismometers and read background noise records. A velocity profile of the studied site is found by fitting the Rayleigh wave dispersion curve (Figure

1.2). 黃有志 (2003) set microtremor arrays on the Yilan Plain to collect the microtremor signals. The Rayleigh wave dispersion curves were analyzed by the frequency-wavenumber method. Then, a genetic algorithm (GA) was used to obtain the optimized one-dimensional S-wave velocity profiles. 郭俊翔 (2004) used the same method to calculate the velocity at some sites in the Taipei Basin, the Yilan Plain and the Coastal Range and evaluated the diversity between the different landform. 林哲民

(2009) applied the microtremor array measurement with the high-resolution frequency 4

wa venumber method on the western plain in Taiwan. The geological interface could be distinguished clearly as referred to the surveyed geological layer depth profile.

Another microtremor measurement approach used in this study is the horizontal- to-vertical spectral ratio (HVSR, H/V). The traditional approach is to determine the two-station spectral ratio, which requires comparison of two stations. However, in some places, such as in wide plains, it is difficult to find a reference station reaching the basement surface. H/V is an approach that uses a single station and a simplified parameter by an empirical equation to eliminate the ground-motion source and path effects. This approach estimates the amplification ratio and the dominant frequency at a site by calculating the ratio of horizontal and vertical spectra (Nakamura, 1989). Then, referring to site characteristics, the H/V ratio can be used to estimate the S-wave velocity by theoretical simulations. 黃雋彥(2009) integrated previously obtained regional H/V survey results and calculated the dominant frequency and the amplification ratio for plains, basins and bajadas sites across Taiwan . The results were compared with EGDT data to evaluate the potential of liquefaction. Lin et al. (2014) evaluated more than 400 locations in the Taipei Basin by the H/V approach. On the

Basis of the measured results, the S-wave velocity structure was modeled by the theoretical SH-wave transfer functions which derived from the GA-Haskell method.

The result can be well verified using previous geological studies in the Taipei Basin.

The high-frequency receiver function approach analyzes seismic acceleration records to retrieve waveforms with converted phases. The S-wave velocity profile at a seismic station can be estimated by modeling its receiver function waveform. 林哲民

(2003) used receiver function approach to analyzed the S-wave velocity structure of the

Yilan Plain. The receiver function waveforms were processed by high-frequency seismic records collected between 1992 and 1998. The S-wave velocity results were estimated by fitting a theoretical model to the waveforms. Moreover, the optimal solutions of the S-wave velocity were searched for by a genetic algorithm. The result depicted the trend with depth in the Yilan Plain and corresponded well to previous studies. Lin et al. (2011) applied the same approach to the Taipei Basin. The results showed that the receiver function approach can clearly detect the velocity discontinuity in strata. Srinivas et al. (2013) applied the receiver function approach with a neighborhood algorithm on the Indo-Gangetic plain. The estimated profile found that the sediment of the northern plain area is thicker than that of the southern. Wu and

Huang (2016) estimated the depth of the basement surface in the Taipei Basin by use of a receiver function. The result reveals that the sediment of the Taipei Basin gradually thickens from the southeast to the northwest. The result also corresponds to a previous study (Wang et al., 2004) that used a reflection survey.

Figure 1.1 Borehole logging measurement (郭俊翔 et al., 2011)

Figure 1.2 Microtremor array measurement process. (Kuo et al., 2016)

1.2.2 Geostatistical Interpolation Method

Interpolation is a compromising method that estimates values with limited data.

Due to budgetary or technique limitations, high-density measurements are difficult to obtain in many cases. The interpolation accuracy of usually affects the study results.

Thus, the importance of interpolation is clear and more and more methods have been proposed and gradually developed to address this.

In this study, the Inverse Distance Weighting (IDW) method and the ordinary kriging method were used. IDW is one of the most popular interpolation techniques. It was modified by Donald Shepard (Shepard, 1968). The IDW method has been applied in various fields such as digital terrain modeling (Garnero et al., 2013), temperature modeling (Ozelkan et al., 2013), river contamination (Madhloom et al., 2017) and 8

p hotovoltaic soiling (Micheli et al., 2018). In regards to seismic S-wave velocities,

Riaño et al, (2017) built a three-dimensional S-wave velocity model of Bogotá,

Colombia by the IDW method. Sixty-four seismic well records collected from 15 wells and the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER)

Global Digital Elevation Model were used. The interpolated result corresponded well with the soft deposits of sediments.

Ordinary kriging method is also a popular interpolation method. This method was developed by Matheron (1963). The kriging method is name after Danie Krige for his contribution to geology. The applications of the kriging interpolation process are mainly in geoscience. Different from the traditional interpolation method, the spatial structure and the spatial relation are considered by the semivariance. The development of ordinary kriging thus represents the start of geostatistics.

Currently, there exists various derivatives of kriging methods, such as simple kriging, ordinary kriging, and universal kriging. Cressie (1990) introduced the origin and some principles of the kriging method. Goovaerts (1997) and Isaaks and Srivastava

(1989) discussed the key concepts of the kriging methods and several factors that could greatly affect the method’s result. Moreover, to slove the three-dimensional problem, the geometric and the zonal anisotropy solutions are proposed. Gringarten and Deutsch

(1999) established a geologically interpretable variogram model to describe the characterization of reservoirs. This semivariogram was analyzed by three scales. The three dimensional reservoir model was estimated by the semivariogram. Ghazi et al.

(2014) used ordinary kriging to interpolate the S-wave velocity in Mashhad, Iran. The

300, 500, 700 and 950 m/s interface contour maps were built by 243 measured S-wave velocity profiles. For the geological condition in Mashhad, a greater from the southeast to the northeast was observed in all contour maps.

Most studies in Taiwan that have applied the kriging method are concerned with the two-dimensional part. 田璦菁 (2003) analyzed the spatial variability of typhoons that affected the Shihmen Reservoir from 1986 to 2001. The arithmetical averaging method, Thiessen polygons method, and ordinary kriging were used and compared. The result showed that increasing duration of rainfall, decreased the error of the interpolated result. Therefore, the short duration rainfall data had larger variation. The ordinary kriging method is not suitable to interpolate data such as that. 陳柏宏(2007) improved the theoretical semivariogram method of the kriging method. The semivariogram determines the relation between weight and distance in the kriging method. The proposed weighted method calculated the error of the traditional model as a weighted factor. The weighted factors were used to correct the semivariogram. As compared with the traditional model, the weighted model obtained better results than the traditional method.

1.3 Content

The aim of this study is to build a three-dimensional S-wave velocity model of the

Taipei Basin. In this study, data collected from different geophysical approaches were first integrated. In addition, different confidence level for each type of data were calculated. Then, the IDW method and the ordinary kriging method were applied for

the building of the S-wave velocity model. The IDW, a simplified interpolation method, differs from the ordinary kriging method as the latter analyzes the spatial structures in the data before interpolation by using three different parts that are divided by depth.

Finally, the fitness of the obtained models was verified by theoretical simulation. The organization of this manuscript is as follows.

Chapter 1 introduces the motivation for the study and reviews several previous studies. The literature review, first discusses the data. The origin of the data and its application are mentioned. The second part of the review focuses on interpolation methods. Some case studies of IDW and the kriging method are described.

Chapter 2 describes the study area and introduces the data used in the study.

Chapter 3 explains the principle of IDW, and the basics of ordinary kriging and the proposed kriging method. Furthermore, the validation and simulation method are introduced.

Chapter 4 illustrates the study process and details the interpolated result. Moreover, the quality of the interpolated models is discussed.

Chapter 5 summarizes and the conclusion and provides the study conclusions.

Chapter 2 Study Area and Data

2.1 Area of Interest

The study focused on the Taipei Basin, which can be divided into Taipei City and

New Taipei City. It is the largest metropolitan area in Taiwan, with a population of more than 6 million people. From the aspect of geomorphology, Tamsui River, Keelung River and Xindian Creek supply enough materials to form a triangular alluvium (Figure 2.1).

The northern, eastern, and western are in the Tamsui, Nangang-Neihu, and Shulin districts, respectively. According to the measurement by 石再添 et al.(1989), the area of the plain and the mountain in the basin is approximately 240 km2 and 220 km2 respectively.

With a geological investigation of the basin, the understanding of its subsurface structure gradually becomes clearer. The Taipei Basin was a highland in the early

Pleistocene. The main fault mechanisms gradually transformed from compression to extension in the Middle Pleistocene. One half-graben landform was formed in the northwestern part of the basin. By the Late Pleistocene, layers of sandstone and conglomerate were alternatively and periodically deposited. In order of time, there are the Banqiao, Wugu, Jingmei and Songshan layers under the subsurface in the Taipei

Basin (Figure 2.2) (Wang et al., 2004 ; 鄧屬予, 2006). Furthermore, the Guandu layer is distributed in the northern area consisting of pyroclastic debris deposited by eruptions of the Tatun volcano Group.

Figure 2.1 Geographic position of the Taipei Basin.

Figure 2.2 Profile of the Taipei Basin. (Wang et al., 2004)

2.2 S-wave Velocity Data

As stated above, the EGDT, the microtremor array, the H/V and the receiver function were used in this study. The data recorded includes the longitudes, latitude, depth, S-wave velocity, code of the measured point, and measurement approach. Table

2.1 shows a sample of the data.

The EGDT data were collected every 0.5 m depth. There are two reasons that high density distribution is not suitable for interpolation. First, selecting similar data for interpolation could not determine the true spatial structure. Second, some data received from small material are considered as noise data. If the distance to noise data is very short, a larger error could occur in the estimation. Therefore, the data were averaged every 10 m in this study. There are seventy-three surveyed locations of the EGDT in the Taipei Basin and the depth of most of the EGDT data is not greater than 40 m.

The microtremor array measurement was only collected at twelve sites in the

Taipei Basin. The number of data records decreases with increasing depth. Twelve points at shallow depths can be detected, but only three points at the depth of 900 m were detected. The H/V approach used only single station to measure microtremor signals with short time. There are four hundred and one measurement positions in the basin. For the same reason as the microtremor array approach, the number of received data recoeds decreased with the depth increasing.

Compared to the microtremotor measurement, the receiver function used seismic signals to analyze the S-wave velocity. The data quality high at deep locations and the number of data records would not decrease with increasing depth. One hundred and 14

thirteen sites were used for the interpolation in this study.

As mentioned above, four types of data were used to interpolate the S-wave velocity in the Taipei basin. The records from the four data type were filtered and integrated.

First, the range of data was selected by longitude and latitude which were defined by the EPSG: 3826. The longitude was limited between 280000 and 318000. The latitude was limited between 2757000 and 2786500. Furthermore, some locations isolated on different landforms were filtered. Data at 113 positions were collected and only the data points that were shallower than 1000 m were used in the study (see Figure 2.3). After filtering the data, the data were integrated. The EGDT data were averaged by every 10 m (see Figure 2.4), and data records were averaged if they were at the same location.

In this step, new data type names were created (see Figure 2.5).

Table 2.1 Data format for the study.

Figure 2.3 Distribution of different geological measurement approaches

Figure 2.4 EGDT data process.

Figure 2.5 Integration of data records taken at the same location.

Chapter 3 Methodology

3.1 Inverse Distance Weighting

IDW is one of the most popular interpolation techniques. This method corresponds to Tobler’s First Law of Geography: “Everything is related to everything else, but near things are more related than distant things” (Tobler, 1970). The basic idea of this method is similar to the inverse-square law in physics; the weight decreases by a power function when distance increases. IDW is typically a deterministic method because its result is only determined by parameters rather than a possibility or initial condition. The only variable in IDW is the distance. This simple equation makes this method quick and widely applied. Equation 3.1 presents IDW equation.

n Zi i1p di (3.1) Z p  ··········································· n 1 i1p di

Here Z p is an estimated value, Z i is the ith observed value, di is the distance

between Z p and Z i , and p is a constant for power function.

The most effective factor of IDW is the value, which decides the relation between distance and weight. Larger values make the weight decline quicker (see

Figure 3.1). Two is the most common default setting in software. Another important factor in IDW is the search strategy. Search radius and k-nearest neighbor are two main search strategies. The search radius defines a geometry shape. If an observed point is within the defined geometry, it is selected (see Figure 3.2). The k-nearest neighbor selects the nearest observed point by distance ordering until selected points up to a defined number (see Figure 3.3). Both strategies can be implemented simultaneously.

Figure 3.1 Power functions under different p values.

Figure 3.2 Example of search radius.

Figure 3.3 Example of the k-nearest neighbor algorithm.

3.2 Ordinary Kriging

Geostatistics has been in development since the 1960’s. Spatial interpolation is one of the most important topics in geostatistics. Ordinary kriging is a representative method that interpolates data by analyzing its spatial structure. The regionalized variable theory is the basis of ordinary kriging. Ordinary kriging uses the semivariogram, which analyzes local or global spatial structure.

3.2.1 Regionalized Variable Theory

A sample space is defined as a set of all possible values. The random variable is a function that assigns all possible values in a sample space. According to whether a value is countable or not, the random variable can be classified as discrete or continuous. In the spatial statistics, the regionalized variable is a random variable in a real specified space. It implies random, error, and uncertainty in real phenomenon. Although a regionalized variable cannot be expressed in a mathematical equation, it still can present a spatial structure or relation in a specified space.

3.2.2 Stationary

It is necessary to estimate unknown values by some assumptions because spatial interpolation must refer to an initial spatial setting. Stationary is a type of spatial setting that assumes that the overall spatial structure has the same statistical properties. In the second-order stationary hypothesis used in the ordinary kriging method, the mean and variance are assumed to be constant between samples (See Equation 3.2) and independent of location. Furthermore, the covariance only depends on distance and is also independent of location (See Equation 3.3. In real-world phenomena, it is difficult to satisfy the second-order stationary hypothesis. The intrinsic hypothesis, a weaker hypothesis, is an alternative. This hypothesis assumes that given the same direction, the variance of the difference only depends on the distance between any two locations.

Namely, the relation between distance and variance can be defined by a function.

Finally, the semivariance is equal to half of the variance (See Equation 3.4). 22

EZx  EZxhconstant  μ ...... (3.2)

CovarianceZx,  Zxh  

E?ZxZxh      

E? ZxZxh    μZx  μZx h μ2 

2 EZxZxh     μ  Ch  ...... (3.3)

2 (3.4) Var Z xZhh xE Z xZ  x2     γh .....

Here E is the expected value, Zx  is the random variable at location x, Zxh   is a random variable at a location that is h meters away from x, h is a distance constant,μ is the mean value, Ch  is the covariance of two points separated by h, and γh is the semivariance of two points separated by h.

3.2.3 Semivariogram

The semivariogram is the core of all the kriging methods. It is a graph that depicts the spatial relation of the semivariance. According to the graph, the spatial continuity and the spatial variability can be analyzed. The equation is defined as

1 2 γh  ZxZxh    ...... (3.5) 2Nh  i

Where N(h) = the number of pairs of data points separated by h.

According to different h values, different γh values can be calculated. Figure 3.4 shows the scatter plot of semivariance. It depicts a trend that the larger the relative distance, the smaller the relative weight. When the relative distance h equals zero, the covariance equals the variance (Per Equation 3.6). Moreover, Equation 3.7 can derived from Equation 3.4. It presents the relation between the covariance and the semivariance

(see Figure 3.5).

C(0) = C[푍(푥), 푍(푥)] = Var(x) ...... (3.6)

1 γhE  [Zx]  hZx   2 2

1122  EZx  hEZxZx    hEZx    22 

112 EZxhEZxhZxEZx  222       22 111 VarZxhVarZxCh      222  (3.7) σ2 ChC0C    h ......

In Equation 3.7, Var is the variance .

The semivariogram based on real data is called the empirical semivariogram.

Serval steps are required to draw an empirical semivariogram. First, calculate the semivariance and draw a two-dimensional graph (see Figure 3.4). Second, find and delete extreme values. Third, groups are divided by lag value and the mean value in each group is taken as its representative value (see Figure 3.6). The final step is to connect representative values (see Figure 3.7). The lag value is the most important parameter in the former steps, because smaller lag values result in over fitting. In contrast, larger lag values simplify the spatial structure. Neither case reasonably represents the spatial structure. After the completion of an empirical semivariogram, three important parameters that significantly affect the result can be found namely, nugget, sill and range.

(1) Nugget : If the distance h is equal to zero, in theory, γ(0) should also be zero .

However, it is possible that the observed values include measurement errors or the discontinued value is detected in an extremely short distance. The phenomenon of spatial noise is called the Nugget effect.

(2) Sill: The semivariance increases with increasing distance. However, it approaches a certain value if the stationary assumption is followed. This value is called the sill.

(3) Range: As distance increases, the semivariance approaches the sill. The distance between the original point and the sill is called the range, which is the maximum effective range. If the value is out of the range, it can be considered as spatially independent.

After obtaining the nugget, sill, and range from the empirical semivariogram (see

Figure 3.8), a theoretical model can be fit. There are five commonly used models (see

Equation 3.8 to 3.12 and Figure 3.9.

(1) Gaussian model :

 h2  h cc0 1exp,0 2 h  r ...... (3.8)    00 

(2) Linear model

 h  h c0  c,0  h  a a  (3.9)   h c0  c , h  a ......   00    

(3) Circular model 26

 2 hh2  h ccha 1cos1,0 1   0  2   aa    h ccha0 , ......   00     (3.10) 

(4) Exponential model

  h  h cc0 1exp,0 h  r ...... (3.11)    00 

(5) Spherical model

 31hh3  h ccha0 (),0 22aa  (3.12)   h cc0 ,h a ......   00    

Here c0 is the nugget, c is the sill, and a is the range.

Figure 3.4 Semivariance scatter plot.

Figure 3.5 Theoretical relation between covariance and semivariance.

Figure 3.6 Averaged semivariance scatter plot.

Figure 3.7 Empirical semivariogram.

Figure 3.8 Relation between the sill, range and nugget.

Figure 3.9 Theoretical fitting models

3.2.4 Effects in the semivariogram

The semivariogram depicts a regional spatial relation. It controls how the relative weights decline. The parameters of the semivariogram plays an important role in it.

Only proper parameters can reflect the true spatial relation and estimate uncertainty well. Some effects are present in a semivariogram. For instance, shape, nugget, range and anisotropy. The following is a short description of these effects.

(1) The shape effect: the shape effect refers to the shape of a semivariogram. If a shape of model is sharp at initiation, if a shape of model is sharp at initiation, there is a higher relative weight between points with shorter distances. Conversely, points with a 33

long er distance receive less or even negative weights. This phenomenon is called the screen effect. Figure ( can be taken as an example. For the blue line, P1 receives almost all the weight because P2 and P3 obtain little relative effect. For the red line, in contrast,

P4, P5 and P6 correspond to the same distance with P1, P2, and P3, respectively.

Although P4 receive a larger weight, P5 and P6 still receive weights with certain effect.

(2) The nugget effect: the nugget effect refers to when the start of the semivariogram does not equal zero. It represents the discontinuance variation in a space.

The nugget effect results from measurement errors or large variances in a short distance.

A larger nugget effect results in the weight close to the average data. Figure 3.11 shows the nugget effect. The relative weight difference between P1 and P2 is larger than that between P3 and P4.

(3) The range effect: the range effect refers to a changing range value. This effect is similar to the shape effect. It controls the decline of the relative weight with increasing distance. The larger range slows the decline. Figure 3.12 shows the range effect. The relative weight difference between P1 and P2 is larger than that between P3 and P4.

(4) Anisotropy: anisotropy is an important effect for the ordinary kriging. It is difficult to find cases without anisotropy, a phenomenon without directional difference in a real space. For instance, when interpolating geological data, directional dependency may result from the directions of the geological structure. Thus, considering anisotropy may improve the estimation. The two basic methods to consider the anisotropy as geometric and zonal anisotropy. Geometric anisotropy can be used when the range 34

value changes with different directional searches but the sill values are same in different directions (See Figure 3.13). By contrast, zonal anisotropy can be used when the sill value changes yet the range values remain constant in different directions (See Figure

3.14). Geometric anisotropy can be transformed to an isotropy model by adjusting its directional semivariances. Figure 3.15 depicts the weight contour difference between isotropy and geometric anisotropy. The directional adjustment relies on the ratio between the sills (a1 and a2). The zonal anisotropy focus on the part where a sill value of a semivariance is greater than the others (the yellow part in Figure 3.14). The effect only depends on one direction whose sill value is higher. Figure 3.16 depicts a zonal anisotropy weight contour example. Furthermore, it is difficult to analyze real data with pure geometric or zonal anisotropy. Figure 3.17 shows a mixed anisotropy model. This model can be divided by a nugget and two structures. The green part can be regarded as a geometric anisotropy model. The yellow part can be analyzed by zonal anisotropy.

The reduced distance is calculated by Equation 3.13. The semivariance of the mixed model is calculated by Equation (3.14).

22    hh12 h g   aa  12 (3.13)  ......  2 h1  h z   a  1

(3.14) γh c2 γ 2h gz  c 1  c 2  γ 1 h  ......

In Equation 3.13 and Equation 3.14, hhg , z are the reduced distance at the geometric and zonal structures. 35

Figure 3.10 Shape effect.

Figure 3.11 Nugget effect.

Figure 3.12 Range effect

Figure 3.13 Geometic anisotropy

Figure 3.14 Zonal anisotropy

Figure 3.15 Geometric anisotropy with weight contours

Figure 3.16 Zonal anisotropy with weight contours

Figure 3.17 Mixed anisotropy model

3.2.5 Ordinary Kriging

Based on the semivariogram, the ordinary kriging only depends on distance to estimate values in a fixed direction. This method is considered as the Best Linear

Unbiased Estimator (BLUE). Linear means that the estimated value is a weighted linear combination (See Equation 3.15). The sum of the weights is one (See Equation

3.16(3.16)). Unbiased means that the estimation error is zero on average (See

Equation3.17). Best means that it minimizes the variance error (See Equation 3.18). 43

N ˆ (3.15) ZZooii  ...... i1

N (3.16) oi 1 ...... i1

E0ZZˆ  ...... (3.17) oo

2 minimizeEZZ ˆ  ...... (3.18)  oo

ˆ Here Zo is the estimated value at location o, Zo is the observed value at location 표,

Z i is the observed value at 푖푡ℎ point, and oi is the relative weight between location o and the observed 푖푡ℎ point.

After substituting Equation 3.16 into Equation 3.17, Equation 3.19 is derived.

2 2 N E ZZEZZˆ   oooiio  i1

2 NN EZZoiioio ii11

2 N

EZZoiio  i1

NN (3.19)  EZZZZ ......  oiojoioj    iJ11

The variance formula can be transferred to Equation 3.20 by using the linearity of the expected value.

2 2 E ZZEZZZZ  ijioj    0 

22 E[()]2 ZZEioioojoj [()()][()] ZZZZE ZZ ...... (3.20)

Equation 3.21 can be derived by transposition of the terms.

E ZZZZ  oioj  

1 222  EZZEZZEZZ ojoiij      ...... (3.21) 2  

After substituting Equation 3.21 into Equation 3.19, Equation 3.22 is obtained

NN EZZZZ  oiojoioj    iJ11

NN 1 222  EZZEZZEZZ   oiojojoiij       iJ11 2  NNNN 1 2 2   EZZEZZ  oiojojoiojoi     iJiJ1111 2

NN 1 2  EZZ  oiojij   iJ11 2 NNN N 111 222 E   ZZE ZZE ZZ  ojojoioioi ojij      Jii111 J 222 1 NN N 1122 2  EZZEZZ ...... oiojoi ojij    iiJ111 22 (3.22)

To find the minimum, partial derivatives can generally be taken. However, the lagrange multiplier is introduced with the constraint of Equation3.23.

NN EZZL[(2 )]  2 (1  ) ii11oi i o oi

NN NN1122 22L 1 E Z ZE Z Z oio joi oji joi    ii11 Ji  1122 ...... (3.23)

W here L is the Lagrange multiplier.

Then, Equation 3.24 can be solved by taking partial derivatives of the Lagrange

multiplier and λoi and setting them to zero. Ordinary kriging minimizes variance to obtain the optimal result.

NN  1122 EZZEZZkN kiiko     1,2,   ii1122 ...... (3.24)  N   1   oi  i1

Equation 3.24 can be rewritten as the following matrix:

1112 1N 1 oo11   1  2122 2N oo22    ...... (3.25)  NNNN22 1 oNNo 1110  1

Where  ij is the semivariance between location i and j, and oi is the relative weight at

the ith point.

3.3 Confidence Weight factor

In this study, the data were collected via several different approaches. Each approach has its advantages and disadvantages. The quality of data and expected cost have to be considered before measuring. Thus, to deal with the problem in which the quality of data is unequal, the proposed method, based on the theory of ordinary kriging, is to give the results from different geological investigation approaches corresponding confidence values. The main idea of this method is to change the

solved weight 휆표𝑖 to the confidence weight C표𝑖 (See Equation 3.26). The confidence weight also follows the principle that the sum of the weight is one (See

Equation 3.27).To ensure that, each C표𝑖 should divide by the sum of C표𝑖. Finally, same as the kriging, An estimated value is a weighted linear combination. Equation

3.15 is rewritten as Equation 3.28. Table 3.1 and Table 3.2 show the difference between the ordinary kriging method and the proposed method.

Coioii   AC ...... (3.26)

N C1oi  ...... i1 (3.27)

ˆ Zooii C Z ...... (3.28)

Here Coi is the relative confidence weight between location o and the ith observation point. ACi is the confidence of approach at location i. oi is the relative weight between location o and the ith observation point.

Table 3.1 Example of ordinary kriging.

Zo oi oi o Z Point A 5 0.3 1.5 Point B 7 0.4 2.8 Point C 3 0.3 0.9

Zˆ o 1.5 + 2.8 + 0.9 = 5.2

Table 3.2 Example of the proposed method.

ACi Coio Z A 5 0.3 3 0.272 1.3635 B 7 0.4 3 0.363 2.5452 C 3 0.3 4 0.363 1.0908

Zˆ o 1.3635 + 2.5452 + 1.0908 = 4.9995

3.4 Validation

3.4.1 Cross-Validation

Cross-validation is conducted to check how fit the parameters are, and if the process works for each observed point and estimates the locations without recorded data. In this study, the mean error (ME), mean absolute error (MAE), and mean absolute percentage error (MAPE) were found via Equation 3.29, Equation 3.30, and

Equation 3.31, respectively.

N 1 ˆ (3.29) ME?ZZoo ...... N i1

N 1 ˆ (3.30) MAE? ZZoo ...... N i1

N ˆ 1 ZZoo MAPE   ...... NZi1 o (3.31)

ˆ Here Zo is the estimated value at the location o and Zo is the recorded value at the location o.

The ME is the average of all the errors. If the value of the ME is higher or lower than zero, then in general, the estimated values are higher or lower than the observed value. The optimum ME value should be close to zero as it is assumed that the average error equals zero in Equation 3.17. The MAE calculates the average of the absolute difference. The ideal MAE value is close to zero. However, it should be noted that the ordinary kriging method does not guarantee the smallest MAE. The MAE measures the average absolute error, but the quality of the total model cannot be understood by the

MAE alone. The MAPE provides the sum of the ratio between the errors and the recorded values. That is, the MAPE indicates the total percent error.

3.4.2 Seismic H/V Simulation

The simulation in the study is based on the H/V approach. The simulation analyzed seismic records from only one station. Same as the microtremor H/V approach, the

Fourier spectra were calculated from each seismic record. Then, the horizontal direction 51

spectra were divided by the vertical spectra. The result depicts the site amplification at different frequencies. Namely, it presents the amplification characteristics at a specific location or site. The results were averaged at each seismic station to determine the observed seismic site effect at a site. To compare with the interpolated models, the theoretical transfer function, based on the Haskell matrix (Haskell, 1960), was adopted to transfer the one-dimensional velocity model to spectra. In this study, the historical seismic records collected between 1992 and 2013 were calculated and averaged. The one-dimensional velocity data at 80 seismic station locations were captured and compared with the historical data via the fitness function. Equation 3.32 presents the linear correlation coefficient. The fitness function is defined by Equation 3.33. F ranges between zero and one.

xxyy   r  i ii ...... (3.32) ()()xxyy iiii

r 1 F  ...... (3.33) 2

Here r is the linear correlation coefficient, xyii , are the ith pairs of the quantities,

xy, are the mean of x and y , and F is the fitness function.

Chapter 4 Result and Discussion

4.1 Study Process

To model and verify the three-dimensional S-wave velocity in the Taipei Basin, the processes of the IDW and the ordinary kriging methods were separated in this study.

Figure 4.1 shows the process of the IDW method. The interpolated results were checked and verified. If the results were acceptable the simulation was implemented. Figure 4.2 depicts the study process of the ordinary kriging and proposed methods. First, four type of data were filtered and integrated. The filtered data (D1) were prepared. Second, this was combined with the confidence factors calculated by Method A and Method B. Data with confidence weight factors from Method A (D2) and Method B (D3) were prepared.

Each of these data records was divided into three parts by depth to analyze the semivariograms. Then, the ordinary kriging method was used under four different scenarios to interpolate the model of the Taipei Basin. The first scenario only used the horizontal semivariogram (OK2D) and did not consider the vertical semivariogram.

Namely, it only extended the two-dimensional semivariogram to three-dimensions. The second scenario considered both the horizontal and vertical semivariograms (OK3D).

The third (OK2DC) and fourth (OK3DC) scenarios were the extensions of OK2D and

OK3D using the confidence weight factors in the calculations (see Section 3.3). Model

A, B, C, D, E, and F were built by three types of data and four scenarios. Finally, the cross-validation and H/V simulation were used to verify the results.

Data Process

Inverse distance Weighting

Data Check and Cross-Validation

H/V Simulation

Figure 4.1 IDW method used in this study.

Confidence Factor Data Process Calculation

Filtered Data (D1) Data with Confidence Data with Confidence Weight Factor by Method Weight Factor by Method B (Ordinary Kriging) A (D2) (D3)

Semivariogram Analysis

OK2D OK3D OK2DC OK3DC OK2DC OK3DC

(Ordinary Kriging) (Confidence Weight (Confidence Weight (Confidence Weight (Confidence Weight (Ordinary Kriging)

Kriging) Kriging) Kriging) Kriging)

Model A Model D Model B Model E Model C Model F

Data Check and Cross-Validation

H/V Simulation

Figure 4.2 Ordinary kriging and proposed method processes used in this study.

4.1 Result of the IDW Method.

The IDW method has fewer parameters than the ordinary kriging method. The power value and search strategy was set in the study. The power value p was set to two. Both the KNN and the search radius were adopted. The maximum number of points was five and the radius of the sphere was 5000 m whose vertical direction was flattened to 50 m. The results are shown in Appendix A. The velocity results show that many unreasonable values exist. Taking TAP072 as an example, the velocity changed rapidly between of 400 m and 600 m depth. Most of the IDW results present the same phenomenon. This phenomenon does not correspond to the layer distribution and observed results. That is, the case result is not corresponded to desired model. the simulation quality of the case IDW result is not high enough to use as a simulation in this study.

4.2 Confidence Weight Factor Calculation

The confidence factors were calculated from the H/V simulation result. The velocity of the microtremor array, HV, and RF was simulated at 80 seismic stations. In this study, two types of confidence factor calculations was used.

1. Method A: Method A averaged the simulation results of each method to calculate the representative value of the method. This gave the data confidence factor by the type of approach. In addition, the EGDT data cannot be used because the spatial distribution is near to the surface, or less than 30 m. In fact, there was little impact to the result for a 1-km scale simulation. Thus, a weight of one was given to all the EGDT data. The confidence factor results are displayed in Appendix B. 56

2. Method B: Method A used one value to represent the overall confidence to an approach. The basic idea of Method B is each different location should get its own confidence factor. Thus, the confidence factor values were found via the simulation results of the microtremor array, HV, and RF at different locations. Factors for different data combinations were created by averaging any two or all confidence factors. The mean of an approach was used to replace the factor if there was no data.

The confidence factor results are displayed in Appendix C.

4.3 Semivariogram Analysis

To fit the data well, the model was divided into three depth scales. The upper part is between 0 and 30 m. Due to the high density of distribution of the EGDT data, this depth range was selected. The middle part of this model is between 30 and 600 m.

Most of the data intervals in this part are approximately 50 m. Furthermore, the deepest observed location is between 600 and 700 m in the Taipei Basin. The lower part is between 600 and 1000 m. The data interval is 100 m. It was assumed that most data had already touched or exceeded the basement depth for this study.

To deal with anisotropy, horizontal (x-y) and vertical (z) directions were separated.

The vertical direction semivariogram has a trend that the deeper locations always receive the higher value. In contrast, the horizontal direction was flatter than the vertical direction. This corresponds to the geological phenomenon in which the horizontal points are more similar than the vertical ones. Figure 4.3 to Figure 4.8 show the semivariogram of the three depth scales in the x-y direction and z direction.

Figure 4.3 Semivariogram of the upper depth scale in the x-y direction.

Figure 4.4 Semivariogram of the upper depth scale in the z direction. 58

Figure 4.5 Semivariogram of the middle depth scale in the x-y direction.

Figure 4.6 Semivariogram of middle depth scale in the z direction. 59

Figure 4.7 Semivariogram of lower depth scale in the x-y direction.

Figure 4.8 Semivariogram of lower depth scale in the z direction. 60

4.4 Cross-Validation

As mention above, the models consisted of three parts. The cross-validation of the three parts was tested by ME, MAE, and MAPE. Table 4.1 shows the data and the ordinary kriging scenario of the models. Table 4.2 to Table 4.7 list the cross-validation results. Model A, B, and C were built by OK2D and OK2DC. In contrast, Model D, E, and F were built by OK3D and OK3DC. Comparing Model A, B, and C, the results of these three are almost the same; however, the ME of B and C were higher than A.

Similarly, the ME of E and F were higher than D. The reason for this is that the ordinary kriging assumed that the averaged error is zero; however, the proposed confidence method changed the value calculated by ordinary kriging. Furthermore, the overall errors of Model A, B, and C are less than Model D, E, and F. This demonstrates a phenomenon that the vertical change is smooth. Finally, the MAPE in the upper part is higher than those in the middle and lower parts. The EGDT data was sensitive to some noise. For this reason, the noise made the cross-validation result worse than the other parts.

Table 4.1 Data and methods of the models.

Model Model A Model B Model C Model D Model E Model F

Component

Data D1 D2 D3 D1 D2 D3

Scenario OK2D OK2DC OK2DC OK3D OK3DC OK3DC

Table 4.2 ME, MAE, and MAPE results of Model A.

Part Upper part Middle part lower part

method (0–30 m) (30–600m) (600–1000m)

ME 1.7964 -3.9349 -7.4432

MAE 84.8629 66.9893 116.2197

MAPE 0.2618 0.0918 0.0812

Table 4.3 ME, MAE, and MAPE results of Model B.