Research on Maximum Likelihood b Value and Confidence Limits Estimation in Doubly Truncated Apparent Frequency–Amplitude Distribution in Rock Acoustic Emission Tests
Abstract
:1. Introduction
2. Derivation of MLE b Value and Confidence Limits Estimation Scheme in DTFAD for Rock AE Tests
3. Performance Verification of the Derived MLE b Value and Confidence Limits Estimation Scheme in DTFAD Based on Synthetic Data
4. Applicability of the MLE b Value and Confidence Limits Estimation Scheme in Specifically Designed Rock Deformation Test
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Data Volume | bT | bA | bF | Confidence Limits of bF | Segment Determined by FGS | |||
---|---|---|---|---|---|---|---|---|
Lower | Upper | Left Endpoint | Right Endpoint | Span/20 | ||||
300 | 2.1715 | 3.2272 | 2.0804 | 0.7956 | 4.0562 | 40.71 | 48.55 | 0.3920 |
1.0666 | 2.4161 | 1.0753 | 0.2272 | 2.3090 | 41.96 | 53.46 | 0.5750 | |
0.6245 | 1.7274 | 0.5716 | 0.1478 | 1.4862 | 43.92 | 62.64 | 0.9360 | |
500 | 2.1715 | 3.2211 | 2.1098 | 1.0917 | 3.5840 | 40.50 | 48.24 | 0.3870 |
1.0666 | 2.1184 | 1.0090 | 0.4725 | 1.7834 | 41.61 | 53.22 | 0.5805 | |
0.6245 | 1.9165 | 0.6119 | 0.2102 | 1.1016 | 43.68 | 58.36 | 0.7340 | |
800 | 2.1715 | 3.0030 | 2.1350 | 1.3169 | 3.2732 | 40.42 | 48.83 | 0.4205 |
1.0666 | 2.0981 | 1.0634 | 0.5701 | 1.6672 | 42.15 | 54.49 | 0.6170 | |
0.6245 | 1.6907 | 0.5981 | 0.2518 | 1.0192 | 43.03 | 58.34 | 0.7655 | |
1000 | 2.1715 | 2.9978 | 2.1307 | 1.3936 | 3.0845 | 40.37 | 48.75 | 0.4190 |
1.0666 | 2.0112 | 1.0649 | 0.5634 | 1.6356 | 42.05 | 54.70 | 0.6325 | |
0.6245 | 1.4890 | 0.6068 | 0.3329 | 0.9525 | 43.74 | 61.26 | 0.8760 | |
2000 | 2.1715 | 2.7451 | 2.1751 | 1.6421 | 2.4591 | 40.22 | 50.31 | 0.5045 |
1.0666 | 1.8940 | 1.0618 | 0.7085 | 1.4579 | 41.73 | 54.83 | 0.6550 | |
0.6245 | 1.1968 | 0.6191 | 0.4270 | 0.8486 | 44.11 | 66.35 | 1.1120 | |
3000 | 2.1715 | 2.6223 | 2.1502 | 1.7768 | 2.5957 | 40.21 | 51.60 | 0.5695 |
1.0666 | 1.6856 | 1.0676 | 0.8410 | 1.3162 | 40.96 | 57.22 | 0.8130 | |
0.6245 | 1.0594 | 0.6190 | 0.4856 | 0.7686 | 42.56 | 67.91 | 1.2675 | |
5000 | 2.1715 | 2.4491 | 2.1387 | 1.8641 | 2.4501 | 40.16 | 53.52 | 0.6680 |
1.0666 | 1.4626 | 1.0627 | 0.9261 | 1.2111 | 40.68 | 60.44 | 0.9880 | |
0.6245 | 0.9476 | 0.6224 | 0.5268 | 0.7226 | 41.94 | 71.78 | 1.4920 | |
8000 | 2.1715 | 2.3873 | 2.1558 | 1.9541 | 2.3842 | 40.10 | 55.16 | 0.7530 |
1.0666 | 1.2874 | 1.0645 | 0.9721 | 1.1652 | 40.28 | 64.87 | 1.2295 | |
0.6245 | 0.8366 | 0.6209 | 0.5616 | 0.6840 | 41.67 | 75.96 | 1.7145 | |
10,000 | 2.1715 | 2.3371 | 2.1548 | 1.9770 | 2.3495 | 40.02 | 55.52 | 0.7750 |
1.0666 | 1.2592 | 1.0612 | 0.9782 | 1.1495 | 40.26 | 65.82 | 1.2780 | |
0.6245 | 0.8086 | 0.6226 | 0.5694 | 0.6784 | 41.27 | 77.59 | 1.8160 | |
50,000 | 2.1715 | 2.1990 | 2.1558 | 2.0857 | 2.2296 | 40.14 | 62.15 | 1.1005 |
1.0666 | 1.1207 | 1.0653 | 1.0356 | 1.0961 | 40.04 | 77.32 | 1.8640 | |
0.6245 | 0.7092 | 0.6245 | 0.6058 | 0.6436 | 40.14 | 86.99 | 2.3425 | |
100,000 | 2.1715 | 2.1709 | 2.1608 | 2.1146 | 2.2087 | 40.00 | 65.19 | 1.2595 |
1.0666 | 1.1011 | 1.0655 | 1.0454 | 1.0861 | 40.00 | 81.30 | 2.0650 | |
0.6245 | 0.6962 | 0.6242 | 0.6115 | 0.6372 | 40.04 | 89.26 | 2.4610 |
Rock | Channels | b Value | Confidence Limits | Standard Deviation | Data Volume | ||
---|---|---|---|---|---|---|---|
Lower | Upper | Width = Upper − Lower | |||||
Red sandstone | 1 | 1.5105 | 1.4016 | 1.6250 | 0.2234 | 0.0469 | 6368 |
2 | 1.6539 | 1.4712 | 1.8541 | 0.3829 | 0.0750 | 2434 | |
3 | 1.8771 | 1.6670 | 2.1121 | 0.4452 | 0.0781 | 2030 | |
4 | 1.6674 | 1.5356 | 1.8081 | 0.2725 | 0.0532 | 4677 | |
5 | 1.7288 | 1.5892 | 1.8786 | 0.2893 | 0.0548 | 4317 | |
6 | 1.6653 | 1.5180 | 1.8238 | 0.3058 | 0.0597 | 3742 | |
Marble | 1 | 1.4081 | 1.3278 | 1.4914 | 0.1637 | 0.0360 | 11,376 |
2 | 1.3400 | 1.2553 | 1.4278 | 0.1725 | 0.0391 | 10,026 | |
3 | 1.3614 | 1.2897 | 1.4353 | 0.1456 | 0.0327 | 14,139 | |
4 | 1.3050 | 1.2541 | 1.3571 | 0.1030 | 0.0237 | 27,372 | |
5 | 1.3358 | 1.2686 | 1.4049 | 0.1364 | 0.0310 | 15,875 | |
6 | 1.3399 | 1.2558 | 1.4272 | 0.1714 | 0.0389 | 10,155 | |
Granite | 1 | 1.1973 | 1.1443 | 1.2514 | 0.1071 | 0.0257 | 24,435 |
2 | 1.1897 | 1.1392 | 1.2412 | 0.1020 | 0.0246 | 27,154 | |
3 | 1.1575 | 1.1074 | 1.2086 | 0.1011 | 0.0247 | 27,340 | |
4 | 1.1566 | 1.1098 | 1.2042 | 0.0944 | 0.0231 | 31,333 | |
5 | 1.1602 | 1.1130 | 1.2083 | 0.0953 | 0.0232 | 30,764 | |
6 | 1.1669 | 1.1143 | 1.2206 | 0.1063 | 0.0259 | 24,673 | |
Limestone | 1 | 1.0767 | 0.8617 | 1.3100 | 0.4483 | 0.1123 | 1592 |
2 | 1.0980 | 0.8963 | 1.3159 | 0.4197 | 0.1044 | 1812 | |
3 | 1.0212 | 0.7795 | 1.2852 | 0.5057 | 0.1291 | 1244 | |
4 | 1.1382 | 0.9565 | 1.3335 | 0.3770 | 0.0924 | 2178 | |
5 | 1.0485 | 0.8060 | 1.3140 | 0.5080 | 0.1284 | 1247 | |
6 | 1.1170 | 0.9017 | 1.3511 | 0.4494 | 0.1109 | 1582 |
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Xia, C.; Chen, D.; He, W.; Liu, H.; Liu, X. Research on Maximum Likelihood b Value and Confidence Limits Estimation in Doubly Truncated Apparent Frequency–Amplitude Distribution in Rock Acoustic Emission Tests. Mathematics 2022, 10, 2409. https://doi.org/10.3390/math10142409
Xia C, Chen D, He W, Liu H, Liu X. Research on Maximum Likelihood b Value and Confidence Limits Estimation in Doubly Truncated Apparent Frequency–Amplitude Distribution in Rock Acoustic Emission Tests. Mathematics. 2022; 10(14):2409. https://doi.org/10.3390/math10142409
Chicago/Turabian StyleXia, Changgen, Daolong Chen, Wei He, Huini Liu, and Xiling Liu. 2022. "Research on Maximum Likelihood b Value and Confidence Limits Estimation in Doubly Truncated Apparent Frequency–Amplitude Distribution in Rock Acoustic Emission Tests" Mathematics 10, no. 14: 2409. https://doi.org/10.3390/math10142409