Trace intensity is defined as mean total trace length of discontinuities per unit area, which is an important geometric parameter to describe fracture networks. The probability of each trace appearing in the sampling surface is different since discontinuity orientation has a scatter and is probabilistically distributed, so this factor should be taken into account in trace intensity estimation. This paper presents an approach to estimate the two‐dimensional trace intensity by considering unequal appearing probability for discontinuities sampled by rectangular windows. The estimation method requires the number of discontinuities intersecting the window, the appearing probability of discontinuities with both ends observed, one end observed, and both ends censored, and the mean trace length of discontinuities intersecting the window. The new estimator is validated by using discontinuity data from an outcrop in Wenchuan area in China. Similarly, circular windows are used along with Mauldon’s equation to calculate trace intensity using discontinuity trace data of the same outcrop as a contrast. Results indicate that the proposed new method based on rectangular windows shows close accuracy and less variability than that of the method based on circular windows due to the influence of finite sample size and the variability of location of the window and has advantage in application to sampling surfaces longer in one direction than in the other such as tunnel cross sections and curved sampling surfaces such as outcrops that show some curvature.

Analytical solutions serve as primary benchmarks for verifying model test design, provide rapid predictive tools for preliminary design, and offer fundamental physical understanding of complex structure interaction problems of the geological body. It is essential for ensuring the reliability of experimental results. For the study on slope stability under earthquakes, the seismic behavior of key inclined cracks in the slope is a hot topic, which is a crucial issue in rock mechanics and engineering geomechanics. This paper studies the dynamic propagation of the inclined crack under seismic conditions, proposes the analytical solution of fracture mechanics, and conducts a centrifuge shaking table test accordingly for monitoring and validation. The analytical solution results have been validated experimentally by a centrifuge shaking table test on the seismic behavior of an inclined crack. Results indicate that the amplitude of seismic waves significantly affects crack propagation: the greater the amplitude, the faster the propagation rate. Analysis of crack propagation and maximum surface displacement reveals hysteresis and sudden jumps of surface deformation caused by rock mass structure and locked segments, both in indoor tests and in strong earthquake regions. This paper combines a theoretical and experimental monitoring study, providing a good example of integrating analytical solutions and modeling validation for research on earthquake-induced landslide disasters.