Assessment of the accuracy of several methods for measuring the spatial attitude of geological bodies using an android smartphone

Journal Pre-proof Assessment of the accuracy of several methods for measuring the spatial attitude of geological bodies using an Android smartphone

Jue Wang, Nengpan Ju, Chaoyang He, Junchao Cai, Da Zheng PII:

S0098-3004(19)30399-1

DOI:

https://doi.org/10.1016/j.cageo.2019.104393

Reference:

CAGEO 104393

To appear in:

Computers and Geosciences

Received Date:

28 April 2019

Accepted Date:

28 November 2019

Please cite this article as: Jue Wang, Nengpan Ju, Chaoyang He, Junchao Cai, Da Zheng, Assessment of the accuracy of several methods for measuring the spatial attitude of geological bodies using an Android smartphone, Computers and Geosciences (2019), https://doi.org/10.1016/j. cageo.2019.104393

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.

Journal Pre-proof

Assessment of the accuracy of several methods for measuring the spatial attitude of geological bodies using an Android smartphone

Jue Wang, Nengpan Ju, Chaoyang He, Junchao Cai, Da Zheng1 State Key Laboratory of Geological Hazard Prevention and Geological Environment Protection, Chengdu University of Technology, Chengdu 610059, Sichuan, China

Abstract: Spatial attitudes are widely used in geological field investigations, to describe the geometry of geological structural surfaces. Smartphones have a variety of sensors that can be used to obtain these measurements. This paper examines several methods for measuring attitudes using smartphones and analyzes the principles and calculation formulas of these methods. We programmed an Android application to test and analyze the accuracy of different calculation methods. We propose the use of an orientation angle to quantitatively describe the position of the smartphone during measurement. We designed and manufactured a test platform that can change three angles separately, thereby greatly increasing the convenience of the test. The results show that the trigonometric function method can deliver an accurate measurement regardless of the orientation of the phone, and this method is convenient and quick to use.

1

Jue Wang, mainly responsible for the writing of the article and most of the experiments and analysis work;

Nengpan Ju, gave guidance on experiments and ideas; Chaoyang He, gave guidance on the programming of Android applications; Junchao Cai, helped in the process of testing and analysis; Da Zheng, helped in the design and manufacture of the test platform.

1

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1 Introduction Smartphones are employed not only to surf the internet, watch videos, and play games but also to find locations and get directions. To perform these functions, smartphones have an array of sensors to determine device orientation and position. These sensors include GPS receivers, accelerometers, gyroscopes, magnetometers, and even barometers. With the aid of these sensors, it is possible to determine device orientation, position, velocity, and both linear and rotational acceleration (Allmendinger et al., 2017). Smartphones can be utilized in the field of geosciences for various purposes due to their multifaceted abilities that combine both hard- and software features (Lee et al., 2018). Many geologists have taken advantage of smartphones in geological work, such as taking photos, positioning, recording, and slope stability analysis (He et al., 2018; Marcai et al., 2015; Tien Bui et al., 2015; Wang et al., 2016; Wolniewicz, 2014). Some researchers have realized that smartphone sensors can also be used to determine the spatial attitude of the phone, and thus, the smartphone can be placed on a geological surface to directly measure its orientation. These researchers have conducted analyses and proposed different algorithms to describe the spatial information provided by smartphone. At present, there are many smartphone apps that can measure spatial attitudes, such as eGEO Compass, Field compass, FieldMove Clino, GeoClino, Structural Compass, et al. Smartphone apps for spatial attitudes measurement currently available on the Android and iOS platform are incompletely summarized in Table 1. Table 1

Name 3D Geocompass Clinometer compass eGEO Compass FieldMove Clino FieldMove Clino GeoClino for Android GeoClino for iPhone Geocompass

List of smartphone apps for spatial attitudes measurement

Developer Edison Wang Yu Saito Marco Foi Midland Vally Exploration Ltd Midland Vally GSI CO., LTD Geological Survey of Japan, AIST Luca Innocentu 2

Platform Android iOS Android Android iOS Android iOS android

Accessibility Free Pay for download Free, pay for full features Free Free Free, pay for full features Free Free

Journal Pre-proof GeoCompass 2 - Geologist’s Compass GeoID Geological Compass Geostation Lambert PocketTransit Rockd Rocklogger Sterenet Mobile Strike and Dip Structural Compass

HUNG HSUN LIN Engineering Geology & GIS Lab., SNU. THSoft Co., Ltd. Geomecanica Apps Peter Apple Thomas Redfield University of Wisconsin-Madison Shared Apps RockGecko Richard Allmendinger Hunt Mountain Software Apps Medion

iOS

Pay for download

iOS

Pay for download

Android Android iOS iOS

Free, pay for full features Free, pay for full features Pay for download Free, pay for full features

iOS

Free

Android iOS iOS Android

Free, pay for full features Free Pay for download Free

Novakova and Pavlis (2017) compared and analyzed the results of different geological compass applications available for the Android platform. They showed that although it is fast and convenient to use the smartphone for field measurements, the accuracy of the different software and hardware is variable. The shortcoming of this study is that tested outcrops are insufficient in number to find a regular error pattern of the measurements at different orientations. Furthermore, we do not know the algorithms of each application, so we cannot optimize and improve algorithms. Many researchers have described how they use their smartphones to measure spatial attitudes. Weng et al. (2012) rotated the phone to a specific angle and then read the values of the orientation sensor as dip direction and dip angle. Lee et al. (2013) calculated the orientation by analyzing the relationship between sensor value vectors and the surface being measured. Hama et al. (2014) use the device attitude class called CMAttitude which is unique to the iOS system to calculate dip direction and dip angle. Allmendinger et al. (2017) developed an iOS software Stereonet Mobile which uses the rotation matrix to calculate the orientation of the phone. Cheng et al. (2017) calculated the rotation matrix of the phone from the initial attitude to measure attitude and further calculate the orientation. In this research, we analyzed the relationship between the values of the orientation sensor and the geological body, using trigonometric functions to calculate the spatial parameters. To compare and analyze the applicability and accuracy of different calculation methods, we programed a test application and designed and manufactured a test 3

Journal Pre-proof platform. We tested these methods under different conditions and analyzed the calculation results.

2 Background Theory Since Android is the most widely used smartphone system in the world, we chose Android for research. In terms of coordinate systems and sensors, iOS and Android systems have similar definitions.

2.1 Android platform coordinate system

Fig. 1. Three-dimensional coordinate system

To accurately describe the values of different sensors, the Android system establishes a three-dimensional coordinate system based on the direction of the phone itself. As in Fig. 1, the 3 axes are orthogonal to each other with the center of the phone as the origin point. The X-axis is parallel to the short side of the smartphone screen and is positive to the right; the Y-axis is parallel to the long side of the smartphone screen, and the direction of the earphone is positive; and the Z-axis is perpendicular to the screen of the smartphone and is positive outward the screen. 4

Journal Pre-proof When the phone rotates, the coordinate system rotates with the phone, and the relationship between the axis and the screen remains unchanged.

2.2 Android platform sensors For Android smartphones, there are 3 sensors can be used to determine device orientation: the acceleration sensor, magnetic field sensor, and orientation sensor. The acceleration sensor and magnetic field sensor are real, physical sensors, while the orientation sensor is a virtual sensor calculated by the previous 2 sensors. Each sensor can obtain 3 values, and the values describe the x, y, and z axes of the device coordinates. The acceleration sensor is a sensor that can obtain the acceleration of the 3 axes, represented as a 3-numbered array containing the acceleration values (unit: m/s2) of the x, y, and z axes. The magnetic field sensor obtains the magnetic field intensity of the 3 axes, represented as a 3-numbered array, which describes the magnetic field intensity (unit: μT) of the x, y, and z axes. The orientation sensor is a sensor that provides the angles of the 3 rotational angles of the phone, namely: the azimuth, the angle between the positive direction of the Y-axis and the north direction; pitch, the angle of the phone rotates around the X-axis; and roll, the angle of the phone rotates around the Y-axis.

3 Acquisition of dip direction and dip data Many researchers have introduced their methods of using smartphones to obtain spatial attitudes. The following are four calculation methods that can be implemented on the Android platform, including: 1. Direct reading method, proposed by Weng et al. (2012); 2. Vector method, proposed by Lee et al. (2013); 3. Rotation matrix method, proposed by Cheng (2017); 4. Trigonometric function method, proposed by the author. 5

Journal Pre-proof The following are the specific calculation processes for these methods.

3.1 Direct Reading Method Weng et al. (2012) introduced an Android phone application, GeoTools, that is adapted to measure strike and dip of any planar feature. When measuring a surface, users need to place the phone in a readable portrait fashion onto the surface so that the lengthwise dimension conforms to the dip direction of the surface, and the short ends of the phone represent the strike direction of the surface if the phone is adjusted to the level of X-Y plane as sensed by the accelerometer. The attainment of such an alignment is indicated when the ‘‘level’’ recorded by the app reads zero. In the meantime, it measures the inclination along the lengthwise dimension of the surface, that is the dip angle. This method is advantageous because it makes direct use of the output from the orientation sensor, eliminating potential ambiguities like sign errors or inverse trigonometric function errors in computer codes.

3.2 Vector Method Lee et al. (2013) calculates the strike and dip by analyze the relationship between the sensor data vectors and the orientation of the smartphone. They use a 3-dimensional vector G (Gx, Gy, Gz) to record the data acquired by the acceleration sensor, and another vector M (Mx, My, Mz) for the magnetic field sensor. The vector normal to the screen is equal to N (0, 0, ±1).

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Fig. 2 Schematic diagram of the vector method (modified after Lee et al., 2013)

The spatial relationship of each vector is shown in Fig. 2, different colors represent different planes. M is toward the magnetic north, usually not on the horizontal plane. G is facing downwards, N is normal to the screen plane, G and N are on the vertical plane. When the screen of the smartphone is facing up, the vector N = (0, 0, -1), and the direction of the N vector is as illustrated in Fig. 2, otherwise, N = (0, 0, 1). The cross product of these vectors, G×M, is a vector directed to the east (90 degrees) on the horizontal plane, G× N, is parallel to the smartphone screen and normal to the direction of gravity, thus equal to the strike of the screen plane, both on the screen plane and the horizontal plane. The calculation of the dip angle is relatively simple, the dip angle d is equal to the angle between the N vector and the G vector (Fig. 2). The relationship between the d, N and G is as follows: 𝑁 ∙ 𝐺 = |𝑁||𝐺|cos 𝑑

(1)

After substitute N= (0, 0, ±1) and rearranging, the formula for calculating the dip angle is as follows: 7

Journal Pre-proof 𝑑 = cos ―1

𝐺𝑧

| | |𝐺|

(2)

Lee et al. (2013) do not have a specific description of the dip direction calculation method. From our perspective, the detailed calculation steps are as follows: As M vector is usually not on the horizontal plane, the direction of strike is not equal to the angle between G × N vector and M vector. We need two vectors both on the horizontal plane to calculate the direction of strike. The two vectors are the strike vector S = G × N, and the east vector E = G × M. The angle between these vectors is therefore the angle between the strike line and magnetic east. We define θ as the angle between the S vector and the E vector, we can get the relationship between θ, S and E as follows: 𝑆 ∙ 𝐸 = |𝑆||𝐸|cos 𝜃

(3)

After rearranging, θ can be calculated by using the following equation. 𝑆∙𝐸

𝜃 = cos ―1 |𝑆||𝐸|

(4)

Since 0° ≤ θ ≤ 180°, the azimuth of the strike (vector S) cannot be directly obtained by the angle value (90°) of the E vector. It is also necessary to determine the positional relationship between the S vector and the E vector. The positional relationship between two two-dimensional vectors can be simply judged by the positive or negative of the result of the cross-multiplication. We found that if the E vector is projected to the screen plane, the positional relationship between the S'= (Sx, Sy) and the E' = (Ex, Ey) is similar to the positional relationship between S and E. At this time, both the S' and E' vectors are in the smartphone coordinate system, eliminating the complicated coordinate system conversion process. Therefore, the positional relationship between the S vector and the E vector can be obtained by calculating the positional relationship between the S' vector and the E' vector.

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Fig. 3 Relationship between the Strike and the East

Specifically, we first calculate the cross-multiplication of the S' vector and the E' vector, 𝑆' × 𝐸' = 𝑆𝑥𝐸𝑦 ― 𝑆𝑦𝐸𝑥. If the result is positive, the direction is in the clockwise direction from east, otherwise it is in the counterclockwise direction. If S is in the clockwise direction of E, as Fig. 3 (c) and (d), then S = 90° + θ. If S is in the counterclockwise direction of E, as Fig. 3 (a) and (b), then S = 90° - θ. When the screen of the smartphone is facing up, dip direction = strike +90°, else, dip direction = strike -90°.

3.3 Rotation Matrix Method According to the measurement principle of the acceleration sensor and the magnetic field sensor, Cheng (2017) lists the linear equations required to solve the three rotation angles φ, θ, and ψ of the smartphone rotating along the three axes when the smartphone is moved from the initial state to the measurement state, thereby obtaining the rotation matrix R. The solved rotation matrix is used to find the geological surface normal vector, and bring it into the corresponding formulas to 9

Journal Pre-proof calculate the geological body orientation. Values of Acceleration Sensor Gx , Gy , Gz

Values of Magnetic Field Sensor Mx , My , Mz

Rotation Angles φ, θ

Rotation Angle

ψ Rotation Matrix R Normal Vector V

Dip Direction

Dip Angle

Fig. 4 Rotational matrix method calculation step

Fig. 5 Transformation of coordinate systems

Using the Android platform coordinate system as the smartphone coordinate system, when the attitude of the smartphone changes, the coordinate system changes from X0, Y0, Z0 to X', Y', Z'. The change of the coordinate system can be regarded as the result of the smartphone rotating the φ, θ, ψ along the X0, Y0 and Z0 axes. These 10

Journal Pre-proof three rotation angles correspond to the following three rotation matrices, which can convert any vector to a new coordinate system.

[

1 𝑅𝑥(𝜑) = 0 0

𝑅𝑦(𝜃) =

[

0 cos 𝜑 ― sin 𝜑

cos 𝜃 0 sin 𝜃

0 1 0

]

(5)

]

(6)

]

(7)

0 sin 𝜑 cos 𝜑 ―sin 𝜃 0 cos 𝜃

[

cos 𝜓 sin 𝜓 0 𝑅𝑧(𝜓) = ― sin 𝜓 cos 𝜓 0 0 0 1

When the smartphone keeps horizontal, the Y0 axis towards the north and the location is in the northern hemisphere. The readings of the acceleration sensor and magnetic field sensor are:

[]

0 𝐺= 0 𝑔

[

(8)

]

0 M = 𝐵 cos 𝛿 ― sin 𝛿

(9)

Where g is the acceleration of gravity and B is the strength of the magnetic field, δ is the angle between the sensor axis and the direction of the earth's magnetic field. After the smartphone attitude changes, the readings of acceleration sensor and magnetic field sensor are:

[] []

𝐺𝑥 0 𝐺 = 𝐺𝑦 = 𝑅𝑥(𝜑)𝑅𝑦(𝜃)𝑅𝑧(𝜓) 0 𝑔 𝐺𝑧

[]

(10)

𝑀𝑥 0 M = 𝑀𝑦 = 𝑅𝑥(𝜑)𝑅𝑦(𝜃)𝑅𝑧(𝜓)𝐵 cos 𝛿 ― sin 𝛿 𝑀𝑧

[

]

(11)

Solve the equations (10) and (11), the three rotation angles can be calculated as follows: tan 𝜑 = tan 𝜃 =

𝐺y 𝐺𝑧 ― 𝐺𝑥

𝐺2𝑦 + 𝐺2𝑧

11

(12) (13)

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tan 𝜓 =

𝑀xcos 𝜃 + 𝑀𝑦sin 𝜃sin 𝜑 + 𝑀𝑧sin 𝜃cos 𝜑

(14)

𝑀𝑦cos 𝜑 ― M𝑧sin 𝜑

It should be noted that when calculating the rotation angle, the result range of the function atan(y/x) is (-π/2, π/2], and the result range of the function atan2(y, x) is (-π, π] , here we need to use the function atan2(y, x) for calculation. After the phone is rotated, in the new coordinate system, the normal vector of the screen is (0, 0, 1), and the following equations can be listed:

[]

0 𝑅𝑥(𝜑)𝑅𝑦(𝜃)𝑅𝑧(𝜓)V = 0 1

(15)

[]

0 Solve the equation 𝑅𝑥(𝜑)𝑅𝑦(𝜃)𝑅𝑧(𝜓)V = 0 (15), the normal vector can be 1 further obtained as follows.

[

sin 𝜑sin 𝜓 + cos 𝜑cos 𝜓sin 𝜃 V = cos 𝜑sin 𝜃sin 𝜓 ― cos 𝜓sin 𝜑 cos 𝜑cos 𝜃

]

(16)

Cheng calculated the angles between the vector V and the Y-axis and Z-axis separately, which are the dip direction and dip angle of the structural plane. The calculation formulas are as follows. Table 2 dip direction calculation formula

𝑉𝑧 ≥ 0

𝑉𝑧 < 0

𝑉𝑥 > 0

𝑉𝑥 < 0

𝑉𝑥 > 0

𝑉𝑥 < 0

𝑉𝑦 𝜋 ― tan ―1 𝑉𝑥 2

𝑉𝑦 3𝜋 ― tan ―1 𝑉𝑥 2

𝑉𝑦 3𝜋 ― tan ―1 𝑉𝑥 2

𝑉𝑦 𝜋 ― tan ―1 𝑉𝑥 2

Specifically, when 𝑉𝑥 = 0, the dip direction calculation is as follows: Table 3 Special cases of dip direction

𝑉𝑥 = 0 𝑉y > 0

𝑉y = 0

𝑉y < 0

0

Horizontal Plane

π

Table 4 dip angle calculation formula

𝑉𝑧 > 0

𝑉𝑧 < 0 12

𝑉𝑧 = 0

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tan ―1

𝑉2𝑥 + 𝑉2𝑦 𝑉𝑧

tan ―1

𝑉2𝑥 + 𝑉2𝑦 ― 𝑉𝑧

𝜋 2

3.4 Trigonometric Function Method We propose a method that can quickly measure the spatial attitude no matter the direction of the phone using the orientation sensor. In theory, the three values azimuth, pitch and roll acquired by orientation sensor should be the angles which the smartphone rotates around three axes. But under this assumption, the calculated result of dip direction and dip angle is always unsatisfactory. Until we designed and manufactured the test platform in Section 5.1, we found that, in our test data, pitch and roll represented in Fig. 4 as γ and β, respectively. Azimuth shows the sum of the angle which the smartphone rotates in the horizontal plane and the angle which the smartphone rotates around the Z axis, that is azimuth = θ + δ. The dip direction of the plane is θ , and the dip angle is α .

Fig. 6. Trigonometric function method schematic

Dip angle can be calculated easily, according to geometric relationship, 13

Journal Pre-proof 𝑂𝐴

(17)

𝑂𝐴

(18)

𝐴𝐵 ∙ 𝐴𝐶

(19)

AB = sin 𝛽 AC = sin 𝛾 AD =

𝐴𝐵2 + 𝐴𝐶2 𝑂𝐴

(20)

sin 𝛼 = 𝐴𝐷

Substituting equations (17), (18), and (19) into equation (20), and then rearranging, we can calculate the dip angle α according to the following equations: (21)

sin 𝛼 = sin2 𝛽 + sin2 𝛾

The dip direction can be obtained by calculating the angle δ which is the angle that the smartphone rotates on the plane. When the smartphone is placed as shown in Fig. 4, we can get: (22)

δ + ε = 180° AC =

OA sin 𝛾

(23)

AD =

OA sin 𝛼

(24)

cos 𝜀 =

AD 𝐴𝐶

sin 𝛾

(25)

= sin 𝛼

Since the roll and pitch angles may be negative when the phone changes direction, it is necessary to adjust the calculation formula according to the position of the phone. When the smartphone is placed in different directions, the relationship between these angles is as shown in Table 5. Different calculation formulas can be selected according to the positive or negative of β (roll): Table 5 Relationship between different angles

δ

β

γ

δ

0~90

>0

>0

ε

90~180

>0

<0

180 - ε

180~270

<0

<0

180 + ε

ε sin 𝛾 sin 𝛼 sin 𝛾 180 ― cos ―1 sin 𝛼 sin 𝛾 180 ― cos ―1 sin 𝛼 cos ―1

14

θ sin 𝛾 sin 𝛼 sin 𝛾 𝜃 = azimuth ― cos ―1 sin 𝛼 sin 𝛾 𝜃 = azimuth + cos ―1 sin 𝛼 𝜃 = azimuth ― cos ―1

Journal Pre-proof 270~360

<0

>0

360 - ε

cos ―1

sin 𝛾 sin 𝛼

𝜃 = azimuth + cos ―1

sin 𝛾 sin 𝛼

4 Development of a smartphone-based Application 4.1 Development environment Currently, there are two popular smartphone operating systems, Android and iOS. Because the Android system is the world’s most widely used system, we decided to develop an Android application to perform the functions above. To develop an Android application, there are many languages and tools to choose from. For most developers, the preferred language is JAVA or Kotlin while the main IDE software programs are eclipse, IntelliJ IDEA, and Android Studio. In recent years, with the popularity of cross-platform development, some new IDE software allows developers to use other languages to develop cross-platform applications. Xamarin is one of the cross-platform IDE programs and is included in the Visual Studio 2017, which offers free use for students and individual developers. Based on the experience of developing small tools using the C# language, we chose Xamarin as the development tool. By opening the Visual Studio installer, we selected Visual Studio Community 2017, which is a full-featured IDE that is free to students, open-source and individual developers.

4.2 Data acquisition To obtain data from different sensors, we need to instantiate a SensorManager and register the listeners for each sensor. After they have been registered, we can access the sensors and obtain the data from each sensor service. We created a corresponding textview widget for each sensor's value, to show the value of the sensors. These sensors receive and process data through the interface function

ISensorEventListener.OnSensorChanged,

which

is

used

to

receive

notifications from the sensor manager when sensor values have changed. The 15

Journal Pre-proof RunOnUiThread function is used to update the values shown in the textview widget, so that changes could be displayed on the screen in real time. When saving, we create and open a csv file, read the value of each textview, and write the values into the csv file at 0.1 second intervals. When suspending the operation, we close the csv file, so as to copy the file to the computer and process the data by Excel or other software.

5 Test and analysis 5.1 Test conditions 5.1.1

Test Platform

Natural exposures typically have a limited range of surface orientations available for measurement, and typically surface roughness generates an error in repeat measurements of the same surface. Thus, we rejected the use of natural exposures to test smartphone precision and accuracy. Instead, we developed a test platform system that can change the dip direction and dip angle independently. With this system, a large number of combinations can be obtained for data analysis. We also consider whether the orientations in which the smartphone is placed on the plane may also affect the measurement results. We propose the concept of orientation angle to indicate the orientation of the smartphone when placed on the plane. The orientation angle ranges from 0 to 360 degrees. We define the positive direction of the Y-axis of the phone at 0° when it is the same as the dip direction, and then increase the degrees in the clockwise direction. When designing the test platform, an orientation angle turntable and dial are added to measure the orientation angle.

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Fig. 7 The test platform

To reduce the influence of the magnetic properties of the test platform itself on the test results, the test platform is mainly made from plastic, and the brackets and supports made from aluminum; all the screws are made of nylon plastic. This test platform is made up by following parts: a foundation, a dip direction turntable, 2 brackets and supports, a test platform and a dip dial. The test platform edge has been cut at an angle to read the value number on the dip dial, and it contains an orientation angle turntable. Before testing, we need to adjust the level of the test platform and align the north direction. There is a square groove in the middle of the dip direction turntable, the same size as the traditional compass, and the test platform is adjusted by the bubble and pointer of the compass. After the adjustment is completed, the compass is removed to avoid the influence of the compass's magnet on the measurement of the smartphone. 17

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Test Device

We chose a Xiaomi Mi6 smartphone for testing. This phone is used daily by the author and comes with MIUI 10, Android 8.0.0, and Qualcomm Snapdragon 835. According to AIDA64, the phone’s acceleration sensor is an ICM20690, and its magnetic field sensor is an AK09916C. Details of the sensors are in the table below. Table 6

Details of the sensors (“AK09916C | AKM - Asahi Kasei Microdevices,” 2016, “ICM-20690 | TDK,” 2016)

Acceleration sensor

Magnetic field sensor

Name

ICM20690

AK09916C

Vendor

InvenSense

Asahi Kasei

Range

±16 g

± 4900 μT

Sensitivity

±1%

0.15 μT

5.2 Stability test First, we tested the feasibility of obtaining sensor data by using the application we developed. We tried to get the data from the acceleration sensor, magnetic field sensor, and the orientation sensor every 0.1 seconds. We placed the Mi 6 smartphone on the test platform to continuously record data for more than 500s and obtained more than 5000 data, then plotted the data as a curve (Fig. 8).

Acceleration (m/s2)

(a)

A[x]

10

A[y]

A[z]

8 6 4 2 0 0

1000

2000

Sequence

18

3000

4000

5000

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M[x]

15

M[y]

M[z]

Magnetic Field (μT)

10 5 0 -5 -10 -15 -20 -25 -30 -35 0

1000

2000

Sequence

3000

(c)

4000

O[x]

O[y]

5000

O[z]

Orientation (°)

30 20 10 0 -10 -20 0

1000

2000

Sequence

3000

4000

5000

Fig. 8 Data obtained from (a) acceleration, (b) magnetic field, (c) orientation sensor

These test results illustrate points made in previous studies that sensors vary strongly among devices. In the test by Novakova and Pavlis (2017), they found that the sensor data of Android devices are not stable enough and have some notable transients that may cause an undetectable error. Different devices use different sensor hardware. In our test, the data obtained from the Mi 6 is stable, just like the performance of iOS device in Allmendinger’s test (Allmendinger et al., 2017). It can be seen from Fig. 8 that although the data acquired by each sensor has a certain degree of fluctuation, the amplitude is limited to a relatively small range, and the overall trend is stable. Among the sensors, the magnetic field sensor has the largest fluctuation, the acceleration sensor is the second most variable, and the orientation sensor has the smallest fluctuations. The value of the sensor can be represented by an average number over a period of time to reduce the errors caused by fluctuation. 19

Journal Pre-proof To determine the quantity of data needed to calculate the average, the first 100 datasets acquired by the sensor are analyzed. It can be seen that the values of the acceleration sensor and the magnetic sensor are discrete and distributed within a certain range, and an average value is needed to indicate the value of the sensor. The value of the orientation sensor is relatively stable and shows little change. In the beginning, due to fluctuations in the data, the average value will also change significantly. When a certain amount of data is reached, the curve of the average tends to stabilize and decreases as the data fluctuates. As seen in Fig. 9, the average curve is relatively stable after 20 repetitions. Therefore, in the subsequent tests, the average of the more than 20 data was used as the value of sensors. (a)

(b)

Acceleration Sensor X Value

9.36

A[z]

1.27

9.35

9.35

2.63

1.26

9.34 20

40

60

80

100

(e) 0

Magnetic Field Sensor X Value

12.0

-13.5

M[y]

-14.0

20

40

60

80

(f) 0

100

Magnetic Field Sensor Y Value

-28.0

11.5

-28.5

11.0

-29.0

20

40

60

80

100

Magnetic Field Sensor Z Value

M[z]

(d) 0

M[x]

Acceleration Sensor Z Value

9.36

2.64

A[y]

A[x]

1.28

(c)

Acceleration Sensor Y Value

10.5

-29.5

-14.5 10.0 -15.0

-30.0

9.5

(g) 0

20

40

60

80

30.0

Orientation Sensor X Value

100

-30.5

(h) 0

20

40

60

80

100

(i) 0

20

7.7

Orientation Sensor Z Value

Orientation Sensor Y Value

-15.6

-15.5

40

60

80

100

7.6

O[z]

O[y]

O[x]

29.9 -15.4

7.5

29.8 7.4

-15.3

29.7

-15.2 0

20

40

60

Sequence

80

100

7.3 0

20

40

60

80

100

Sequence Single Data

0

20

40

60

80

100

Sequence Average Value

Fig. 9 First 100 values of each sensor (a), (b), (c) data from acceleration sensor. (d), (e), (f) data from magnetic field sensor. (g), (h), (i) data from orientation sensor.

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5.3 Accuracy test To measure accuracy, the phone was placed on the test platform and different spatial attitudes were simulated by rotating different parts. The value of the sensors through the smartphone was recorded and exported, using different calculation methods to calculate the dip angle and dip direction. In addition, comparing that with the settings of the test platform, the accuracy of the different calculation methods was analyzed.

5.3.1

Test 1 dip direction

To test the accuracy of the smartphone under different dip directions, we fixed the dip angle at 45° and orientation angle at 0°, and rotated the dip direction turntable to make the dip direction vary through the range of 0° ~ 360°. When the orientation angle is 0°, the roll of the orientation sensor is 0°, and the azimuth and pitch angle readings from the orientation sensor are coincident with the dip direction and dip angle of the geological body, which is the method adopted by Weng (2012). More than 20 data were acquired every 30°, and the average value was calculated. The dip direction and dip angle were calculated following the methods listed in Section 3, and the results are shown in the table below. Usually, geological spatial data values should contain 2 integers. Here, in order to better compare the accuracy of the calculation results of different methods, one decimal place is retained. Table 7 Spatial attitude calculated by different methods when dip direction changes Spatial settings

Vector Method

Rotation Matrix

Trigonometric

Direct Reading

Method

Function Method

Method

Index

DD

DA

OA

DD

DA

DD

DA

DD

DA

DD

DA

1

0

45

0

1.0

45.3

1.0

45.3

0.3

45.6

0.2

45.6

2

30

45

0

32.0

45.3

32.0

45.3

30.2

45.5

30.2

45.5

3

60

45

0

63.0

45.2

63.0

45.2

59.9

45.5

60.0

45.5

4

90

45

0

93.4

45.3

93.4

45.3

89.7

45.6

89.8

45.6

5

120

45

0

122.3

45.3

122.3

45.3

121.5

45.3

121.2

45.3

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150

45

0

151.8

45.4

151.8

45.4

150.9

45.4

150.5

45.4

7

180

45

0

181.0

45.4

181.0

45.4

181.3

45.4

180.9

45.4

8

210

45

0

210.2

45.4

210.2

45.4

211.4

45.6

211.0

45.6

9

240

45

0

239.4

45.5

239.4

45.5

241.6

45.7

241.4

45.7

10

270

45

0

268.6

45.5

268.6

45.5

268.9

45.9

268.4

45.9

11

300

45

0

298.4

45.5

298.4

45.5

299.2

45.5

299.4

45.5

12

330

45

0

329.1

45.4

329.1

45.4

329.5

45.4

329.6

45.4

Note: Due to the width limit, the angle names in the table are abbreviated, where DD means Dip Direction, DA means Dip Angle, and OA means Orientation Angle.

2

2

2

2

0

-2

0

-2

-4 60 120 180 240 300 Dip Direction (°)

0

-2

-4 0

Difference (°)

(d) 4

Difference (°)

(c) 4

Difference (°)

(b) 4

Difference (°)

(a) 4

60 120 180 240 300 Dip Direction (°)

0

-2

-4 0

Dip Direction Dip Angle

-4 0

60 120 180 240 300 Dip Direction (°)

0

60 120 180 240 300 Dip Direction (°)

Fig. 10. Difference of dip direction and dip angle calculated by (a): Vector Method, (b): Rotation Matrix Method, (c): Trigonometric Function Method, (d): Direct Reading Method

Since the range of dip direction changes is large and the errors are small, it is difficult to directly view the error from the scatter diagram. We find the difference between the results calculated by different calculation methods and the test platform settings, and plot the residuals of different calculation methods. The vector method and the rotation matrix method are calculated using the data of the acceleration sensor and the magnetic field sensor, and the same result is obtained. The trigonometric function method and the direct reading method are using the data of the orientation sensor, and the same result is obtained. As we can see from Fig. 10, the dip angle calculated by different methods is relatively accurate, and the error is within 1°. For dip direction, the deviations of the vector method and rotation matrix method are relatively significant, up to 3°. The errors of other methods are within 2°. For the dip direction calculated by the vector method and rotation matrix method, the error changes with the dip direction to a sinusoidal-like pattern. We speculate that this change may be caused by the calculation of the trigonometric function when 22

Journal Pre-proof calculating the component of the geomagnetic field on each axis of the smartphone.

5.3.2

Test 2 dip angle

To test the measurement accuracy of the smartphone with different dip angles, we fixed the dip direction at 0° and orientation angle at 0°, and rotated the dip angle turntable to make the dip direction vary in the range of 0 ~ 90°. This test was carried out at 10° intervals, and the results obtained by various calculation methods are shown in the following table. As above, the following results are calculated using an average of 20 or more datasets. Similarly, when the orientation is 0°, the data of the orientation sensor can be directly read as dip direction and dip angle. Table 8 Spatial attitude calculated by different methods when dip angle changes Spatial Settings

Vector Method

Rotation Matrix

Trigonometric

Direct Reading

Method

Function Method

Method

Index

DD

DA

OA

DD

DA

DD

DA

DD

DA

DD

DA

1

0

0

0

294.0

0.1

294.0

0.1

57.8

0.0

0.5

0.0

2

0

10

0

0.2

10.1

0.2

10.1

0.6

10.2

0.6

10.2

3

0

20

0

0.5

20.3

0.5

20.3

0.3

20.3

0.6

20.3

4

0

30

0

0.8

30.0

0.8

30.0

0.1

30.0

0.0

30.0

5

0

40

0

1.0

40.4

1.0

40.4

0.1

40.4

0.1

40.4

6

0

50

0

1.0

50.4

1.0

50.4

0.0

50.4

0.0

50.4

7

0

60

0

1.0

60.1

1.0

60.1

0.1

60.8

0.0

60.8

8

0

70

0

1.0

70.0

1.0

70.0

0.1

71.7

0.1

71.7

9

0

80

0

0.7

79.9

0.7

79.9

1.5

79.9

1.6

79.9

10

0

90

0

359.4

89.4

180.6

89.4

0.4

89.4

0.2

90.6

Note: Due to the width limit, the angle names in the table are abbreviated, where DD means Dip Direction, DA means Dip Angle, and OA means Orientation Angle.

2

2

2

2

0

-2

0

-2

-4 30 60 Dip Angle (°)

90

0

-2

-4 0

Difference (°)

(d) 4

Difference (°)

(c) 4

Difference (°)

(b) 4

Difference (°)

(a) 4

30 60 Dip Angle (°)

90

0

-2

-4 0

Dip Direction Dip Angle

-4 0

30 60 Dip Angle (°)

90

0

30 60 Dip Angle (°)

90

Fig. 11 Difference of dip direction and dip angle calculated by (a) Vector Method, (b) Rotation 23

Journal Pre-proof Matrix Method, (c) Trigonometric Function Method, (d) Direct Reading Method. Dip direction difference of Rotation Matrix Method is excessive when dip angle at 90°, data point is not shown in the figure.

Since the dip direction adopts the range of 0 to 360°, when the measurement error is negative, the measurement result of 0° dip direction may in fact be 359.9°. To make the error calculation accurate, we change the data to a negative number manually, that is, -0.1 ° prior to calculation. When the dip angle is 0°, the plane is horizontal, and there is no dip direction. If we continue to use the previous calculation methods, we will get results with no reference value Therefore, when the dip angle is 0°, the data will not be added to calculate the dip direction error. When the dip angle is not 0°, all the methods can obtain relatively accurate results in general. The calculation results of the vector method and rotation matrix method have the smallest error, all within 1°. The calculation results of the other 2 methods are similar, most of the errors are within 1°, and the overall error is within 2°. For dip direction, when the dip angle is 90°, the strike is usually used to describe the direction of the geological structure. The dip direction here is the orientation of the smartphone screen, that is, the orientation of the free face. Except for the rotation matrix method, the result is reversed when the dip angle is 90°, and the dip direction obtained by other methods is relatively accurate, with an error of 1~2°. We also noticed that the pitch angle of the smartphone is 90.6° at this time, and the theoretical dip direction is indeed 180°. From this point of view, it can also be said that the dip direction calculated by rotation matrix method is the only correct one. In terms of dip angle, the results obtained by the vector method and rotation matrix method are relatively more accurate, and the accuracy of results from other methods is slightly worse.

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Test 3 orientation angle

The orientation in which the smartphone is placed is not explained when other researchers use a smartphone to make measurements. Therefore, the impact of different orientations of smartphones on the accuracy of measurement results is unknown. In this study, we quantify the orientation angle to test the accuracy of the smartphone's measurement in different directions. We fixed the dip direction at 0° and dip angle at 45° and rotated the orientation angle turntable to make the orientation angle vary between 0 ~ 360°. When the orientation angle changes, the Y-axis direction of the smartphone is no longer the same as the dip direction. Hence, the values of the orientation sensor cannot be directly read as the spatial attitude, and the result of the direct reading method is no longer counted. Table 9 Spatial attitude calculated by different methods when orientation angle changes Spatial Settings

Vector Method

Rotation Matrix

Trigonometric

Method

Function Method

Index

DD

DA

OA

DD

DA

DD

DA

DD

DA

1

0

45

0

0.7

45.2

0.7

45.2

0.8

45.2

2

0

45

30

1.3

45.1

1.3

45.1

359.8

45.1

3

0

45

60

2.2

45.2

2.2

45.2

358.9

45.3

4

0

45

90

2.7

45.3

2.7

45.3

0.1

45.3

5

0

45

120

2.5

45.6

2.5

45.6

1.1

45.6

6

0

45

150

2.5

45.6

2.5

45.6

2.0

45.5

7

0

45

180

1.7

45.6

1.7

45.6

1.2

45.6

8

0

45

210

1.0

45.6

1.0

45.6

2.4

45.6

9

0

45

240

0.2

45.6

0.2

45.6

2.3

45.6

10

0

45

270

359.9

45.5

359.9

45.5

1.6

45.4

11

0

45

300

0.8

45.4

0.8

45.4

1.2

45.4

12

0

45

330

0.4

45.2

0.4

45.2

0.4

45.2

Note: Due to the width limit, the angle names in the table are abbreviated, where DD means Dip Direction, DA means Dip Angle, and OA means Orientation Angle.

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2 0 -2 -4

(c)

4 2

Difference (°)

(b)

4

Difference (°)

Difference (°)

(a)

0 -2 -4

0

60 120 180 240 300 Orientation Angle (°)

Dip Direction Dip Angle

4 2 0 -2 -4

0

60 120 180 240 300 Orientation Angle (°)

0

60 120 180 240 300 Orientation Angle (°)

Fig. 12 Difference of dip direction and dip angle calculated by (a): Vector Method, (b): Rotation Matrix Method, (c): Trigonometric Function Method

The dip angle calculations of these methods are relatively accurate, all the errors are within 1°. In terms of dip direction, all the methods maintain good accuracy with an error of 0~3°. This result shows that all 3 methods can calculate the dip direction and dip angle when the smartphone is placed in any direction.

5.3.4

Test 4 multi-angle combination

The dip direction turntable, the dip angle turntable and the orientation angle turntable are rotated to test the accuracy of the measurement results of these calculation methods under different conditions. To cover as many orientations as possible while reducing the number of test groups, the tendency and orientation were tested by rotating the turntable at 45° intervals, and the dip angle was tested at intervals of 30°. Since the dip direction measurement does not make sense when the dip angle is 0°, the dip angle change does not start from 0°, but 15°, 45°, and 75° are selected to represent a gentle tilt, medium tilt, and steep slope. The test obtained a total of 192 sets of data, the specific data of which are shown in Table 11.

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(b) 10

8

8

8

6

6

6

4

4

4

2

2

2

0

0

0

-2

-2

-2

-4

-4

-4

-6

-6

-6

-8

-8

-8

Difference

(a) 10

-10

10

-10 0

90

180

270

Orientation Angle

Dip Angle 15° 45° 75°

Dip Direction 0° 45° 90° 135° 180° 225° 270° 315°

-10 0

90

180

270

Orientation Angle

0

90

180

270

Orientation Angle

Fig. 13 Difference of dip direction calculated by (a): Vector Method, (b): Rotation Matrix Method, (c): Trigonometric Function Method

The errors in dip direction are as shown in

Fig. 13. It can be seen that the dip

direction errors obtained by the vector method and the trigonometric function method are concentrated approximately 0°, and the error of the rotation matrix method is concentrated in different regions depending on the orientation angle. For the vector method and rotation matrix method, the error of dip direction calculation result is up to 8°. In general, when the dip angle is shallow, the error is small, and the error is large when the dip angle is steep. Under the same dip direction, the change of the difference of the dip direction does not show a correlation with the change of the orientation angle, so it is unable to reduce the error of the dip direction measurement by rotating the smartphone. The regulation of the trigonometric function method error has not been found to be obvious, and the error is mainly within 4°, which is more stable and concentrated than other methods.

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(c) 2

1

1

1

0

0

0

-1

-1

-1

Difference

(a) 2

-2

-2 0

90

180

270

Dip Direction 0° 45° 90° 135° 180° 225° 270° 315°

-2 0

Orientation Angle

Dip Angle 15° 45° 75°

90

180

270

0

Orientation Angle

90

180

270

Orientation Angle

Fig. 14 Difference of dip angle calculated by (a): Vector Method, (b): Rotation Matrix Method, (c): Trigonometric Function Method

The errors of the dip angle are shown in Fig. 14. It can be seen that all 3 methods can get accurate results, the errors of vector method and rotation matrix method are within 1°, the errors of trigonometric function method are mostly less than 1°, and the individual results are slightly larger than 1°. In general, for these 3 methods, when the orientation angle is 0°, the error of the dip angle is the smallest, and when the orientation angle is 180°, the error of the dip angle is the largest. Relatively speaking, the error of the vector method is relatively stable, and the trigonometric function method is slightly larger.

5.3.5

Correlation analysis

The resulting data from the above tests are selected for correlation analysis, and the Spearman correlation coefficient is selected for statistical analysis. The Spearman correlation and significance level (P-value) of the data obtained by different calculation methods are shown in the following table. It should be noted that since the 28

Journal Pre-proof dip direction used in geological work is in the range of 0 to 360°, the actual dip direction may be 0°, but the smartphone may obtain 359°. In fact, the deviation is only 1°, but the deviation of 359° is calculated in the correlation analysis. Therefore, for similar data, we changed the 359° to -1° manually before performing the correlation analysis. Table 10 Correlation analysis of spatial attitude Dip Direction Spearman

P

correlation

Dip Angle Spearman

P

correlation

Vector Method

0.99123

5.25297×10-198

0.94749

6.93952×10-113

Rotation Matrix Method

0.98452

1.1739×10-170

0.94749

6.93952×10-113

0.99126

3.53039×10-198

0.94716

1.37068×10-112

Trigonometric Function Method

It can be seen from the above table that all the 3 methods can obtain the dip direction and dip angle data well, and the correlation coefficients are quite high. In our test data, the results obtained by the rotation matrix method is basically the same as the results obtained by the vector method. When the dip angle is 90°, the dip direction calculated by the rotation matrix method is reversed. Therefore, the two methods have the same correlation coefficient of the dip angle, and the dip direction correlation of the rotation matrix method is lower than the vector method. Among the three methods, the dip directions obtained by the trigonometric function method are slightly better, and the dip angles obtained by vector method and rotation matrix method are slightly better. Although the calculation results of the dip direction have comparatively larger fluctuation than the dip angle, the correlation coefficient is higher because the error is the same as the positive and negative deviation. The dip angle calculation results are smaller, but in general, the positive error is more than the negative value, so the correlation coefficient is slightly lower. This error in the dip angle may be caused by test platform errors and further research is needed.

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6 Conclusion Different Android smartphones have different hardware and software systems. For some Android smartphones (in our test, that is Xiaomi Mi 6), the stability of sensor readings has increased, and the fluctuations are no longer as great as Novakova's tests (Novakova and Pavlis, 2017, 2019). However, the magnetic field sensor and the acceleration sensor still have a certain degree of fluctuation, and it is necessary to continuously record multiple sets of data to calculate a stable average. Android smartphones can be used to quickly acquire geological spatial attitude information from outcrops. Different researchers analyzed from different perspectives and proposed different methods for calculating geological spatial attitude using the sensors of a smartphone. This paper introduces and tests these methods, supplementing some of the details that need to be noted during the calculation process. This paper proposes the use of an orientation angle to describe the orientation of the smartphone placed on the plane. These methods can obtain a relatively accurate dip direction and dip angle when the smartphone is placed at any orientation angle. Specifically, for the dip direction, the change of the error is not obvious with the change of the orientation angle, and the error is quite different from the change of dip direction. For the dip angle, the overall trend is that the error is minimal when the orientation angle is 0°, and the error is greatest when the orientation angle is 180°. On the whole, we recommend using the trigonometric function method to obtain geological spatial data for outcrops. The smartphone can be placed at any angle to obtain a relatively accurate dip direction and dip angle. At the same time, the orientation sensor data are relatively stable, and it is not necessary to obtain 20 sets of data for averaging, making this method most efficient and convenient.

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Acknowledgments Funding: This work was supported by the National Key R&D Program of China [2018YFC1504905]; and the National Natural Science Foundation of China [41772317]. The authors are also grateful to the anonymous referees for their useful comments and careful review of the manuscript.

Computer Code Availability Code name: SensorDataGetter. Developer: Jue Wang. Address: State Key Laboratory of Geological Hazard Prevention and Geological Environment Protection, Chengdu University of Technology, Chengdu 610059, Sichuan, China. Telephone: +86 15882143472. E-mail: [email protected]. Year first available: 2019. Hardware required: The phone should contain an acceleration sensor and a magnetic field sensor. Software required: Android 8.0. Program language: C#. Program size: 6.62MB. Access to the source code: https://github.com/Kingjue2/SensorDataGetter.

References AK09916C | AKM - Asahi Kasei Microdevices [WWW Document], 2016. URL https://www.akm.com/akm/en/aboutus/news/20160318AK09916_001/ 31

Journal Pre-proof (accessed 3.4.19). Allmendinger, R.W., Siron, C.R., Scott, C.P., 2017. Structural data collection with mobile devices: Accuracy, redundancy, and best practices. J. Struct. Geol. 102, 98–112. https://doi.org/10.1016/j.jsg.2017.07.011 Cheng, L., 2017. Algorithms Study on Strike and Dip of Structural Planes Using Three- dimensional Electronic Compass. Chinese J. Undergr. Sp. Eng. 13, 592–597. Hama, L., Ruddle, R.A., Paton, D., 2014. Geological Orientation Measurements using an iPad : Method Comparison. EG UK Comput. Graph. Vis. Comput. https://doi.org/10.2312/cgvc.20141207 He, C., Ju, N., Xu, Q., Li, Y., Zhao, J., 2018. Designing an Android-Based Application for Geohazard Reduction Using Citizen-Based Crowdsourcing Data. Mob. Inf. Syst. 2018, 1–11. https://doi.org/10.1155/2018/1328098 ICM-20690 | TDK [WWW Document], 2016. URL https://www.invensense.com/products/motion-tracking/6-axis/icm-20690/ (accessed 3.4.19). Lee, S., Suh, J., Choi, Y., 2018. Review of smartphone applications for geoscience: current status, limitations, and future perspectives. Earth Sci. Informatics 11, 463–486. https://doi.org/10.1007/s12145-018-0343-9 Lee, S., Suh, J., Park, H. dong, 2013. Smart Compass-Clinometer: A smartphone application for easy and rapid geological site investigation. Comput. Geosci. 61, 32–42. https://doi.org/10.1016/j.cageo.2013.07.014 Marcai, E., Viana, W., Andrade, R.M.C., Rodrigues, D., 2015. A mobile learning system to enhance field trips in geology. Proc. - Front. Educ. Conf. FIE 2015–February. https://doi.org/10.1109/FIE.2014.7044030 Novakova, L., Pavlis, T.L., 2017. Assessment of the precision of smart phones and tablets for measurement of planar orientations: A case study. J. Struct. Geol. 97, 93–103. https://doi.org/10.1016/j.jsg.2017.02.015 32

Journal Pre-proof Novakova, L., Pavlis, T.L., 2019. Modern methods in structural geology of twenty-first century: Digital mapping and digital devices for the field geology, Springer Geology. Springer Singapore. https://doi.org/10.1007/978-981-13-2781-0_3 Tien Bui, D., Tran, C.T., Pradhan, B., Revhaug, I., Seidu, R., 2015. iGeoTrans – a novel iOS application for GPS positioning in geosciences. Geocarto Int. 30, 202–217. https://doi.org/10.1080/10106049.2014.902114 Wang, J., Ju, N., Huang, J., He, C., Fan, Q., 2016. Smart Phone-based Method of Rapid Rock Slope Stability Analysis. Geol. Sci. Technol. Inf. 35, 172–176. Weng, Y.H., Sun, F.S., Grigsby, J.D., 2012. GeoTools: An android phone application in geology. Comput. Geosci. 44, 24–30. https://doi.org/10.1016/j.cageo.2012.02.027 Wolniewicz, P., 2014. SedMob: A mobile application for creating sedimentary logs in the field. Comput. Geosci. 66, 211–218. https://doi.org/10.1016/j.cageo.2014.02.004

Attached Table Table 11 Spatial attitude calculated by different methods Spatial Settings

Vector Method

Rotation Matrix

Trigonometric

Method

Function Method

Index

DD

DA

OA

DD

DA

DD

DA

DD

DA

1

0

15

0

2.8

15.2

2.8

15.2

0.5

15.3

2

45

15

0

46.8

15.2

46.8

15.2

45.4

15.7

3

90

15

0

89.6

15.2

89.6

15.2

89.6

15.6

4

135

15

0

132.5

15.3

132.5

15.3

134.3

15.3

5

180

15

0

177.3

15.3

177.3

15.3

180.3

15.3

6

225

15

0

225.0

15.3

225.0

15.3

226.2

15.3

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270

15

0

271.4

15.3

271.4

15.3

270.1

15.2

8

315

15

0

318.3

15.3

318.3

15.3

315.6

15.2

9

0

45

0

3.8

45.1

3.8

45.1

0.7

45.1

10

45

45

0

48.1

45.1

48.1

45.1

46.2

45.1

11

90

45

0

90.6

45.2

90.6

45.2

91.1

45.2

12

135

45

0

132.6

45.3

132.6

45.3

135.0

45.4

13

180

45

0

176.6

45.4

176.6

45.4

179.7

45.4

14

225

45

0

223.4

45.3

223.4

45.3

225.8

45.4

15

270

45

0

270.6

45.3

270.6

45.3

270.8

45.3

16

315

45

0

318.0

45.2

318.0

45.2

315.1

45.2

17

0

75

0

3.4

75.0

3.4

75.0

0.4

75.0

18

45

75

0

51.3

75.0

51.3

75.0

45.4

75.2

19

90

75

0

95.3

75.1

95.3

75.1

90.5

75.3

20

135

75

0

136.6

75.2

136.6

75.2

135.1

75.4

21

180

75

0

177.7

75.3

177.7

75.3

180.1

75.3

22

225

75

0

220.4

75.2

220.4

75.2

224.6

75.3

23

270

75

0

263.8

75.2

263.8

75.2

269.2

75.3

24

315

75

0

313.0

75.1

313.0

75.1

314.6

75.1

25

0

15

45

1.4

15.1

1.4

15.1

359.8

15.0

26

45

15

45

44.6

15.1

44.6

15.1

45.8

14.8

27

90

15

45

87.8

15.2

87.8

15.2

90.5

15.0

28

135

15

45

132.5

15.3

132.5

15.3

135.8

14.9

29

180

15

45

178.7

15.3

178.7

15.3

175.7

15.2

30

225

15

45

226.6

15.4

226.6

15.4

225.4

15.2

31

270

15

45

273.5

15.3

273.5

15.3

269.8

15.1

32

315

15

45

317.6

15.2

317.6

15.2

314.5

15.0

33

0

45

45

1.2

45.1

1.2

45.1

0.2

45.1

34

45

45

45

45.7

45.0

45.7

45.0

45.0

45.0

34

Journal Pre-proof 35

90

45

45

89.5

45.2

89.5

45.2

90.3

45.0

36

135

45

45

133.7

45.3

133.7

45.3

135.2

45.0

37

180

45

45

179.7

45.3

179.7

45.3

180.6

45.3

38

225

45

45

225.7

45.4

225.7

45.4

226.3

45.4

39

270

45

45

271.6

45.3

271.6

45.3

270.0

45.3

40

315

45

45

316.7

45.2

316.7

45.2

314.3

45.3

41

0

75

45

1.5

75.0

1.5

75.0

0.6

74.8

42

45

75

45

50.2

75.0

50.2

75.0

45.7

74.8

43

90

75

45

94.7

75.1

94.7

75.1

90.6

75.1

44

135

75

45

137.4

75.2

137.4

75.2

134.8

75.2

45

180

75

45

179.0

75.3

179.0

75.3

181.2

75.2

46

225

75

45

221.7

75.2

221.7

75.2

225.6

75.4

47

270

75

45

264.8

75.2

264.8

75.2

270.4

75.1

48

315

75

45

311.8

75.1

311.8

75.1

315.4

74.9

49

0

15

90

358.5

15.0

358.5

15.0

0.1

15.0

50

45

15

90

48.6

15.1

48.6

15.1

45.4

14.9

51

90

15

90

87.4

15.1

87.4

15.1

92.9

15.0

52

135

15

90

135.3

15.2

135.3

15.2

135.7

15.1

53

180

15

90

182.9

15.2

182.9

15.2

181.5

15.2

54

225

15

90

228.5

15.3

228.5

15.3

225.6

15.3

55

270

15

90

272.1

15.2

272.1

15.2

269.2

15.3

56

315

15

90

315.0

15.1

315.0

15.1

314.8

15.2

57

0

45

90

358.6

45.2

358.6

45.2

0.4

44.8

58

45

45

90

43.1

45.2

43.1

45.2

46.0

45.0

59

90

45

90

89.5

45.3

89.5

45.3

90.6

45.3

60

135

45

90

136.4

45.4

136.4

45.4

136.1

45.5

61

180

45

90

182.6

45.8

182.6

45.8

181.9

45.8

62

225

45

90

227.1

45.5

227.1

45.5

225.7

45.5

35

Journal Pre-proof 63

270

45

90

270.8

45.5

270.8

45.5

270.9

45.3

64

315

45

90

314.4

45.3

314.4

45.3

315.4

45.8

65

0

75

90

359.3

75.2

359.3

75.2

0.7

75.4

66

45

75

90

48.6

75.3

48.6

75.3

44.7

74.6

67

90

75

90

94.9

75.4

94.9

75.4

90.8

74.7

68

135

75

90

138.5

75.5

138.5

75.5

135.4

75.5

69

180

75

90

180.8

75.4

180.8

75.4

179.8

75.5

70

225

75

90

221.9

75.4

221.9

75.4

225.1

75.8

71

270

75

90

263.7

75.3

263.7

75.3

269.7

75.3

72

315

75

90

309.8

75.2

309.8

75.2

314.9

75.2

73

0

15

135

356.7

15.3

356.7

15.3

359.1

15.3

74

45

15

135

42.1

15.3

42.1

15.3

43.5

15.1

75

90

15

135

90.3

15.3

90.3

15.3

91.1

15.3

76

135

15

135

138.1

15.4

138.1

15.4

135.3

15.3

77

180

15

135

183.6

15.5

183.6

15.5

179.9

15.1

78

225

15

135

227.3

15.5

227.3

15.5

225.0

15.4

79

270

15

135

269.8

15.4

269.8

15.4

270.4

15.4

80

315

15

135

312.6

15.4

312.6

15.4

314.6

15.4

81

0

45

135

356.6

45.5

356.6

45.5

0.1

45.5

82

45

45

135

43.1

45.5

43.1

45.5

45.4

46.0

83

90

45

135

91.6

45.6

91.6

45.6

90.8

45.4

84

135

45

135

138.8

45.7

138.8

45.7

135.3

45.4

85

180

45

135

183.3

45.8

183.3

45.8

180.3

45.5

86

225

45

135

226.3

45.8

226.3

45.8

224.5

45.7

87

270

45

135

269.2

45.7

269.2

45.7

268.4

45.7

88

315

45

135

312.0

45.7

312.0

45.7

315.6

45.4

89

0

75

135

356.9

75.5

356.9

75.5

0.8

75.5

90

45

75

135

48.1

75.5

48.1

75.5

45.2

76.1

36

Journal Pre-proof 91

90

75

135

97.0

75.6

97.0

75.6

90.6

75.5

92

135

75

135

140.6

75.7

140.6

75.7

135.2

75.5

93

180

75

135

181.3

75.8

181.3

75.8

180.8

75.8

94

225

75

135

222.4

75.8

222.4

75.8

225.2

75.7

95

270

75

135

264.3

75.7

264.3

75.7

269.9

75.8

96

315

75

135

308.7

75.6

308.7

75.6

314.8

75.6

97

0

15

180

356.7

15.3

356.7

15.3

2.8

15.4

98

45

15

180

45.5

15.3

45.5

15.3

45.5

15.3

99

90

15

180

94.0

15.3

94.0

15.3

91.1

15.3

100

135

15

180

139.8

15.6

139.8

15.6

135.0

15.3

101

180

15

180

182.6

15.7

182.6

15.7

180.4

15.5

102

225

15

180

225.5

15.5

225.5

15.5

224.8

15.4

103

270

15

180

267.5

15.5

267.5

15.5

269.0

15.5

104

315

15

180

311.1

15.4

311.1

15.4

313.8

15.3

105

0

45

180

358.0

45.6

358.0

45.6

0.4

45.6

106

45

45

180

48.9

45.6

48.9

45.6

46.2

45.8

107

90

45

180

97.2

45.6

97.2

45.6

91.5

45.6

108

135

45

180

140.6

45.7

140.6

45.7

135.5

45.8

109

180

45

180

181.5

45.8

181.5

45.8

180.0

45.8

110

225

45

180

222.9

45.9

222.9

45.9

225.3

45.8

111

270

45

180

264.8

45.8

264.8

45.8

270.2

45.7

112

315

45

180

309.6

45.7

309.6

45.7

315.4

45.4

113

0

75

180

357.8

75.6

357.8

75.6

359.8

75.4

114

45

75

180

49.3

75.6

49.3

75.6

46.0

75.5

115

90

75

180

97.2

75.7

97.2

75.7

91.0

75.7

116

135

75

180

140.1

75.9

140.1

75.9

133.5

75.9

117

180

75

180

180.9

75.9

180.9

75.9

179.5

75.9

118

225

75

180

221.8

75.9

221.8

75.9

225.3

75.6

37

Journal Pre-proof 119

270

75

180

263.3

75.9

263.3

75.9

271.0

75.9

120

315

75

180

308.7

75.8

308.7

75.8

316.3

75.8

121

0

15

225

359.5

15.2

359.5

15.2

359.6

15.2

122

45

15

225

48.6

15.3

48.6

15.3

46.3

15.2

123

90

15

225

95.4

15.3

95.4

15.3

91.8

15.2

124

135

15

225

138.5

15.4

138.5

15.4

136.0

15.3

125

180

15

225

180.9

15.5

180.9

15.5

180.8

15.5

126

225

15

225

222.9

15.5

222.9

15.5

224.5

15.5

127

270

15

225

265.9

15.5

265.9

15.5

268.7

15.3

128

315

15

225

311.3

15.3

311.3

15.3

314.2

15.2

129

0

45

225

0.3

45.1

0.3

45.1

1.2

44.8

130

45

45

225

49.7

45.1

49.7

45.1

44.7

45.3

131

90

45

225

95.8

45.1

95.8

45.1

89.2

44.7

132

135

45

225

138.7

45.2

138.7

45.2

134.3

45.1

133

180

45

225

180.2

45.3

180.2

45.3

177.6

45.3

134

225

45

225

222.5

45.2

222.5

45.2

225.8

45.9

135

270

45

225

265.4

45.2

265.4

45.2

270.1

45.5

136

315

45

225

310.8

45.1

310.8

45.1

315.8

45.1

137

0

75

225

359.7

75.6

359.7

75.6

0.7

75.8

138

45

75

225

50.6

75.6

50.6

75.6

45.8

75.6

139

90

75

225

97.7

75.7

97.7

75.7

91.1

75.8

140

135

75

225

139.5

75.9

139.5

75.9

135.4

75.9

141

180

75

225

179.7

75.9

179.7

75.9

180.1

75.8

142

225

75

225

221.2

75.9

221.2

75.9

223.9

75.9

143

270

75

225

263.7

75.8

263.7

75.8

269.8

75.6

144

315

75

225

309.8

75.8

309.8

75.8

315.5

75.8

145

0

15

270

3.7

15.2

3.7

15.2

359.5

15.8

146

45

15

270

50.6

15.1

50.6

15.1

45.9

15.2

38

Journal Pre-proof 147

90

15

270

94.6

15.2

94.6

15.2

92.1

15.1

148

135

15

270

136.8

15.4

136.8

15.4

136.3

15.4

149

180

15

270

178.8

15.4

178.8

15.4

181.1

15.4

150

225

15

270

221.3

15.4

221.3

15.4

224.4

15.4

151

270

15

270

267.0

15.4

267.0

15.4

268.4

15.4

152

315

15

270

314.3

15.2

314.3

15.2

314.3

15.2

153

0

45

270

2.3

44.8

2.3

44.8

0.7

44.6

154

45

45

270

50.8

44.8

50.8

44.8

46.7

44.1

155

90

45

270

96.0

44.9

96.0

44.9

89.9

45.0

156

135

45

270

138.0

45.2

138.0

45.2

135.5

44.9

157

180

45

270

179.1

45.2

179.1

45.2

179.6

45.4

158

225

45

270

221.1

45.2

221.1

45.2

225.2

45.2

159

270

45

270

264.9

45.1

264.9

45.1

270.6

45.3

160

315

45

270

311.6

45.0

311.6

45.0

315.5

45.0

161

0

75

270

2.7

75.5

2.7

75.5

0.5

75.5

162

45

75

270

52.6

75.5

52.6

75.5

45.6

75.6

163

90

75

270

97.0

75.6

97.0

75.6

90.3

75.7

164

135

75

270

138.0

75.8

138.0

75.8

135.5

75.9

165

180

75

270

178.2

75.8

178.2

75.8

180.5

75.9

166

225

75

270

220.0

75.8

220.0

75.8

226.2

75.8

167

270

75

270

263.6

75.6

263.6

75.6

269.8

75.7

168

315

75

270

311.3

75.5

311.3

75.5

314.4

75.6

169

0

15

315

5.0

15.1

5.0

15.1

1.2

15.4

170

45

15

315

49.3

15.1

49.3

15.1

45.9

15.0

171

90

15

315

92.3

15.2

92.3

15.2

93.0

15.3

172

135

15

315

134.8

15.3

134.8

15.3

136.8

15.3

173

180

15

315

176.9

15.3

176.9

15.3

180.2

15.4

174

225

15

315

221.9

15.3

221.9

15.3

224.6

15.3

39

Journal Pre-proof 175

270

15

315

269.2

15.2

269.2

15.2

270.0

15.3

176

315

15

315

317.7

15.1

317.7

15.1

316.2

15.1

177

0

45

315

3.2

44.7

3.2

44.7

0.2

45.0

178

45

45

315

50.9

44.7

50.9

44.7

45.6

44.7

179

90

45

315

95.0

44.8

95.0

44.8

90.7

44.9

180

135

45

315

137.5

44.9

137.5

44.9

135.8

44.8

181

180

45

315

178.5

44.9

178.5

44.9

180.6

45.2

182

225

45

315

221.2

44.9

221.2

44.9

226.2

44.8

183

270

45

315

265.8

45.2

265.8

45.2

270.8

45.1

184

315

45

315

313.8

45.1

313.8

45.1

315.1

44.1

185

0

75

315

4.7

75.1

4.7

75.1

0.2

74.7

186

45

75

315

53.3

75.1

53.3

75.1

45.2

74.8

187

90

75

315

96.5

75.2

96.5

75.2

89.9

75.2

188

135

75

315

136.7

75.3

136.7

75.3

136.1

75.3

189

180

75

315

177.4

75.4

177.4

75.4

181.8

75.4

190

225

75

315

218.5

75.4

218.5

75.4

225.4

75.3

191

270

75

315

263.4

75.3

263.4

75.3

269.4

75.1

192

315

75

315

312.6

75.2

312.6

75.2

315.2

75.2

Note: Due to the width limit, the angle names in the table are abbreviated, where DD means Dip Direction, DA means Dip Angle, and OA means Orientation Angle.

40

Journal Pre-proof Highlights

1. Introduced a variety of complete calculation methods of measuring the spatial attitude; 2. A trigonometric function method using the orientation sensor is proposed, and the results are relatively accurate; 3. Proposed the orientation angle to quantitatively describe the posture of the mobile phone during measurement; 4. Designed and manufactured a test bench to simulate multiple angle combinations.