The one-dimensional translational dynamic time histories were simulated using sinusoidal waves of various frequencies and real ground motion records (including the Northridge, Loma Gilroy, and Tangshan waves) as input loads. This was done to determine the suitable image acquisition frame rates required for the image deformation measurement technique to accurately capture displacement and acceleration time histories under these different excitations. The Northridge, Loma Gilroy, and Tangshan waves are primarily characterized by high, medium, and medium-to-low frequency components, respectively. Together, they encompass a broad frequency spectrum, thus facilitating the investigation into the effects of load characteristics and frequency content on the performance of the image-based technique in measuring dynamic soil displacement and acceleration time histories.

The dynamic time history combining translation and shear was simulated through superposition of translational motion and shear deformation formulations. Let ( x 0 n , y 0 n ) denote the initial coordinates of the n-th particle in the image, where n = 1 , ⋯ , N ; and let ( x t n , y t n ) represent the position of the n-th particle at time t. The dynamic translation functions along the orthogonal x- and y-directions are defined as u(t) and v(t), respectively. Each particle is assumed to be located at the deformation center of a soil unit with length ∆L and height ∆h Denote the horizontal motion function of the soil unit as p ( t , y 0 n ) and the shear strain function as γ ( t , y 0 n ) .The total horizontal displacement is then given by: γ ( t , y 0 n ) Δ h 2 + p ( x 0 n , y 0 n ) + u ( t ) , This expression accounts for both the horizontal movement of the soil unit and the relative motion of the particle within the unit. Thus, the position of each particle at time tcan be described as follows:

A simulated image of artificial sand, generated using Equation (1) to represent combined translation and uniform shear, is shown in Figure 1. The figure clearly demonstrates the deformation effects under superimposed translational and shear loading, illustrating the overall morphologic evolution of the sand subjected to this composite motion.

Figure 1. Simulation image of uniform shear deformation of sand.

The realization of non-uniform shear deformation artificial sand simulation image is defined by trigonometric function, as shown in Equation (2). The simulated images of artificial sand generated by translation and shear superposition according to Equation (2) are shown in Figure 2.

Figure 2. Simulation image of non-uniform shear deformation of sand.

Among them, MAX_t is the maximum deformation of the simulated deformation time history at time t, h is the height of the image, and y is the longitudinal coordinate value of the current particle.

We first investigated the influence of window size on computational error under uniformly distributed strain. Uniform shear strains of 0.1%, 1%, and 10% were applied to the simulated images. The displacement fields were computed through image deformation analysis. Missing data, erroneous vectors, and noise were then filtered from the obtained displacement fields. Strain was subsequently calculated using different window sizes, and the computational errors are summarized in Table 1.

Table 1. Errors of different strain sizes and calculation windows.

The results under uniform strain conditions indicate that: 1) the mean error, coefficient of variation, and maximum error(maximum value of absolute errors) all decrease with increasing window size, demonstrating that larger windows significantly enhance computational accuracy; 2) a small calculation window results in a high coefficient of variation, indicating higher dispersion in the strain field, as smaller windows are insufficient to smooth the noise present in the displacement field; 3) a strong correlation exists between the required window size and the strain magnitude. Smaller strain values are associated with greater relative errors, thus necessitating larger windows for effective filtering and smoothing.

In summary, strain calculation error is subject to the coupling effect between the calculation window size and the strain magnitude. Assuming a mean error of 20% as the acceptable threshold, the minimum window sizes for strains of 0.1%, 1%, and 10% should be no less than 17 × 17, 13 × 13, and 5 × 5, respectively.

For non-homogeneous strain distributions, a sinusoidal function was applied to simulated images to introduce nonlinear shear strain variations from 0% to 10%. The displacement fields were derived using image deformation analysis methods and then smoothed. Strain fields were computed using different computation window sizes, as shown in Figure 4. The corresponding mean and maximum errors for each window size are presented in Figure 5. Figure 4 and Figure 5 reveal that under non-homogeneous strain conditions, when a small computation window is used, the strain field exhibits high variability. In contrast, a larger window reduces variability and yields a smoother strain field. However, an excessively large window leads to significant deviations from the theoretical strain values, particularly an underestimation of the maximum strain. This occurs because a large window covers numerous regions with different strain levels, and the strain computation process averages the values over the entire window. The greater the strain variation within the window, the greater the deviation of the computed result from the true value. These findings highlight the crucial importance of selecting an appropriate computation window size in strain analysis under non-uniform deformation conditions. A well-chosen window size ensures both smoothness of the strain field and accuracy in capturing local variations.

Figure 4. Calculation results of different strain calculation windows. (a) image of sand simulation, (b) original displacement, (c) smoothed displacement, (d) strain by 9 × 9 window, (e) strain by 17 × 17 window, (f) strain by 25 × 25 window.

Figure 5. Variation of calculation error of non-uniform strain with window size.

Accurate measurement of dynamic displacement, acceleration, and strain in soil under cyclic or seismic loading is essential in geotechnical and earthquake engineering research. Recent advances in image-based measurement techniques provide non-contact solutions for tracking soil deformation, offering spatial and temporal resolution that traditional sensors may lack. However, the accuracy of displacement derived from image sequences is influenced by several factors, including image acquisition frame rate, loading frequency, and the signal-to-noise ratio of the imaging system.

In this study, simulated image sequences of an 8 g, 50 Hz sinusoidal excitation were analyzed. The dynamic displacement at a selected point—obtained by applying a 3 × 3 spatial averaging filter around the measurement point to mitigate noise—was measured under various image acquisition frame rates. As shown in Figure 6 and Table 2, the errors in peak displacement values were systematically evaluated for sinusoidal waves with a fixed amplitude (8 g) but varying frequencies and frame rates. These errors are of particular concern as they can be amplified during subsequent calculations of acceleration and strain. Thus, the following thresholds were adopted: a peak displacement error of 10% and peak acceleration and strain errors of 20% were considered acceptable for engineering applications. The results indicate that when the image acquisition frame rate is at least six times the input frequency, displacement can be tracked with high accuracy, with the mean error of the measured peak displacements consistently below 10%. Moreover, the close agreement among the peak displacement values measured at different frame rates demonstrates the robustness of the image-based computational approach.

Table 2. Average peak error of sine wave displacement.

Figure 6. The displacement time history of 8 g 50 Hz sine wave.

Subsequently, acceleration time histories were derived by applying a low-pass filter and double differentiation to the displacement data. The resulting acceleration measurements under different frame rates are presented in Figure 7. Table 3 summarizes the errors in the calculated peak acceleration across different input frequencies and frame rates. The results confirm that when the frame rate is at least six times the loading frequency, the acceleration time history is accurately captured, with mean peak errors remaining below 20%.

Figure 7. Acceleration time history of a sine wave of 8 g 50 Hz.

Table 3. Average peak error of sine wave acceleration.

To further evaluate the reliability of the method, simulated image sequences of three types of seismic waves were generated and analyzed at a fixed image acquisition frame rate of 100 Hz. For each wave, displacement time histories at three arbitrarily selected points were compared, as shown in Figure 8. The excellent agreement among these results underscores the exceptional stability of the computational method. Additional analyses were conducted using the Tangshan wave at varying frame rates—100 Hz, 10 Hz, 4 Hz, and 2 Hz (Figure 9). The results show a clear trend: measurement error decreases as frame rate increases. As summarized in Table 4, when the frame rate is ≥4 Hz, the average errors (arithmetic mean of absolute errors) in peak displacement are ≤5.5%, ≤7.0%, and ≤7.4% for the three seismic waves, respectively, confirming high accuracy under these conditions.

Figure 8. Comparison of displacement time history of 3 points selected from three seismic wave calculation results.

Figure 9. Comparison of seismic wave displacement time history.

Table 4. Average peak error of seismic wave displacement.

Acceleration time histories under seismic loading were obtained by processing the displacement records via low-pass filtering and double differentiation. For the Tangshan wave, comparisons at frame rates of 100 Hz, 10 Hz, 4 Hz, and 2 Hz (Figure 10) reveal that accurately reproducing acceleration requires higher frame rates than displacement. Both the peak values and frequency content of the calculated acceleration degrade noticeably as the frame rate decreases. It is also noted that excessively high frame rates (e.g., 100 Hz in Figure 10) can introduce high-frequency glitches, indicating that beyond a certain threshold, further increases in frame rate may not improve accuracy and can even introduce noise. Table 5 presents the mean peak acceleration errors for the three seismic waves at frame rates of 20 Hz, 10 Hz, 6 Hz, 4 Hz, and 2 Hz. The results demonstrate that higher-frequency ground motions require higher frame rates to maintain precision. The mean peak acceleration errors are ≤17.7%, ≤8.9%, and ≤10.5% for frame rates ≥ 20 Hz, ≥10 Hz, and ≥6 Hz, respectively—all within the acceptable limit of 20%, confirming that image-based measurements are suitable for dynamic soil response analysis under seismic conditions.

Figure 10. Comparison of seismic wave acceleration time history.

Table 5. Average peak error of seismic wave acceleration.

Simulated image sequences were subjected to sinusoidal loading with maximum deformation amplitudes of 0.01 mm, 0.1 mm, 1 mm, and 10 mm to simulate one-dimensional translational motion. The displacement fields were computed using two image deformation analysis methods: RG-DIC and PIVlab (a particle image velocimetry toolbox based on MATLAB). After filtering out missing data, anomalous vectors, and noise from the displacement fields, a point was randomly selected for displacement time history analysis. The resulting errors are shown in Figure 11, which demonstrates that under one-dimensional translation conditions, the analysis errors of both RG-DIC and PIVlab decrease with increasing vibration amplitude. When the vibration amplitude is ≥0.1 mm, the errors from both methods are negligible and nearly identical. However, when the amplitude is reduced to ≤0.01 mm, the error for RG-DIC can reach 10%, whereas PIVlab exhibits a superior performance with an error of only approximately 3%.

Figure 11. Comparison of the analysis results of different amplitude sine waves.

Using the same approach, the simulated images were subjected to EL Centro seismic loading with maximum amplitudes of 0.01 mm, 0.05 mm, 0.5 mm, 5 mm, and 50 mm. A comparison between the simulated and computed displacement time histories is presented in Figure 12. The results indicate that RG-DIC yields closer agreement with the simulated input than PIVlab. The analysis error of both methods decreases with increasing displacement amplitude. However, at the amplitude of 50 mm, the results from PIVlab exhibit severe deviation from the prescribed values. This is primarily due to the propagation and accumulation of outliers during the cumulative displacement calculation in PIVlab.

Figure 12. Comparison of displacement analysis results of EL Centro waves with different amplitudes.

Acceleration time histories were obtained by applying a low-pass filter and double differentiation to the displacement data, as shown in Figure 13. In contrast to the displacement time histories, the acceleration profiles are largely unaffected by outliers from the cumulative displacement process. As the vibration amplitude decreases, the error in the computed acceleration increases, accompanied by high-frequency fluctuations or “glitches”. For instance, at a low acceleration level of 0.02 m/s², the acceleration time history from PIVlab shows significant oscillation (Figure 13). This occurs because the displacement time history obtained by PIVlab is noisier than that from RG-DIC, and these minor discrepancies are greatly amplified during differentiation. While RG-DIC accurately recovers the prescribed seismic motion, PIVlab only produces consistent results when the maximum displacement reaches ≥5 mm.

Under large deformation conditions, PIVlab obtains the instantaneous displacement field through particle matching between adjacent image frames, and then derives the velocity field by performing differentiation on the displacement field. This process carries significant risk of error accumulation: in large deformation scenarios, when the pixel displacement between adjacent frames exceeds the preset search window range of PIVlab, particle matching is prone to mismatching, which introduces local errors into the displacement field; subsequent displacement calculations rely on this displacement field, and the local displacement errors are continuously amplified during the differentiation process, eventually leading to obvious deviations.

Figure 13. Comparison of acceleration analysis results of EL Centro waves with different amplitudes.