Використання методу Монте-Карло при дослідженні похибок геодезичних приладів

Abstract

У роботі розглянуто принципи використання методу Монте-Карло при дослідженні складових похибок геодезичних приладів. Проведено дослідження з урахуванням систематичних і випадкових похибок. Запропоновано використовувати коефіцієнти кореляції Пірсона та кореляційну матрицю на їх основі для оцінки впливу різних складових систематичної похибки. Визначено основні параметри врахування випадкових похибок у відсотковому співвідношенню до систематичних. Запропоновано масштабовану модель для аналітичної та графічної оцінки впливу складових систематичної похибки на точність вимірювань. Проведено розрахунки, що підтверджують коректність застосування масштабованої моделі для дослідження точності вимірювань геодезичних приладів. This study investigates instrumental errors in geodetic instruments through Monte Carlo simulation, a powerful computational method for modelling stochastic processes. Geodetic measurements are inherently affected by both systematic and random errors, necessitating statistical analysis to quantify their impact on accuracy. The paper outlines the application of the Monte Carlo method, detailing its key stages: defining input data domains, generating random inputs, performing deterministic calculations, and synthesizing results. This approach proves particularly effective for modelling systematic errors, as it allows for the individual assessment of each component's influence and, crucially, their intricate interactions affecting overall measurement precision. The research focuses on an electronic total station, demonstrating a realistic approach to modelling error components. Initially, a flawed statistical modelling approach, which disregarded specific geodetic instrument characteristics, is discussed. Subsequently, a more refined methodology is proposed, enabling a more accurate evaluation of various error constituents on measurement precision. The study specifically models common systematic errors, including the collimation error of horizontal angles, the vertical circle zero error, and the additive and multiplicative errors of the EDM (Electronic Distance Measurement) unit. A significant finding emphasizes the critical role of correctly modelling random errors (noise). Excessive random variability can obscure the linear relationships sought through correlation analysis, rendering the assessment of systematic error influence ineffective. The authors present a method to mitigate this “masking effect” by optimizing the magnitude of random errors and, where appropriate, separating them for different measurement types (e.g., horizontal angles and zenith distances). This refined modelling strategy results in error distributions that logically shift from zero and exhibit clearer correlations between specific error types. To enhance the realism and generalizability of the simulations, the study incorporates variations in “true” measured values, such as distances ranging from 10m to 1000m and zenith angles from 30 to 150 degrees. An increased sample size of 5000 iterations further strengthens the statistical robustness of the findings. The analysis employs Pearson correlation coefficients and correlation matrices to evaluate dependencies between measured quantities and errors, confirming the validity of the developed mathematical models. For instance, the absence of correlation between distance values and the multiplicative error component (when its expected value is zero) is discussed, indicating that while the instrument may exhibit scale deviations, these average out over numerous measurements or diverse conditions. In conclusion, this work validates the applicability of the Monte Carlo method for analyzing the impact of systematic and random error components on geodetic measurement accuracy. It underscores the necessity of precise random error modelling to ensure meaningful analysis. The proposed scalable model offers a robust framework for detailed investigation of various systematic and random error components, individually and in combination, contributing to optimized measurement methodologies in geodesy.