典型优化目标函数下源参数反演性能对比研究

基于外场实验数据，从反演高估率、准确性、稳定性角度系统评估了8种典型目标函数在不同未知源参数反演情形下的反演性能差异.研究发现，不同目标函数反演性能差异显著.仅反演单参数源强时（Q），对数变换目标函数高估率最大（79.4%），偏差平方和目标函数准确性最高（P_ARD<50%=82.3%，ARD=（35.3±9.1）%），目标函数稳定性无明显差异（CV<0.01）.三参数反演（Q，x，y）时，标准化均方根误差目标函数源强高估率最大（98.5%），对数变换目标函数准确性、稳定性最高（P_ARD<50%=91.1%，ARD=（48.4±9.8）%；CV=0.01）；位置方面，偏差平方和目标函数准确性最高（AD=（36.12±11.39） m），对数变换目标函数稳定性最强（CV=0.0018）.四参数反演（Q，x，y，z）时，标准化均方根误差目标函数源强高估率最大（98.5%），对数变换目标函数准确性、稳定性最优（P_ARD<50%=61.7%，ARD=（55.2±16.5）%；CV=0.03）；位置方面，相关系数总体表现最优（AD=（34.37±10.72） m；CV=0.011）.整体上，随反演参数变化，对数变换目标函数源强反演性能最稳定.

Comparative study on source parameters inversion performance of typical cost functions

Based on Prairie grass experiments, the differences in inversion performance from eight typical cost functions were systematically evaluated with metrics of accuracy, robustness and overestimation rate. It was found that substantial differences in the inversion performance presented cross models with different cost functions. In terms of single unknown source parameter inversion (only for source strength (Q)), the cost function based on logarithm transformation scheme had the highest overestimation rate (79.4%), and that associated with the sum of deviation squares scheme exhibited the best accuracy (P_ARD<50%=82.3%, ARD=(35.3±9.1)%); while no significant difference (CV<0.01) was found in the robustness from different cost functions. In terms of three unknown source parameter inversion (source strength (Q) and location (x, y)), the cost function based on normalized root mean square error scheme had the highest overestimation rate (98.5%) for source strength, the logarithm transformation cost function performed the best in the measure of accuracy and robustness (P_ARD<50%=91.1%, ARD=(48.4±9.8)%; CV=0.01); the cost function based on the sum of deviation squares scheme had the highest accuracy for the source location (AD=(36.12±11.39)m), while the logarithm transformation cost function showed the best robustness (CV=0.0018). In terms of four unknown source parameters inversion (source strength (Q) and location (x, y, z)), the cost function based on normalized root mean squared error exhibited the highest overestimation rate (98.5%) for source strength, and the logarithm transformation scheme got the best accuracy and robustness (P_ARD<50%=61.7%, ARD=(55.2±16.5)%; CV=0.03); the correlation coefficient scheme performed the best in accuracy and robustness for source location (AD=(34.37±10.72)m; CV=0.011). In general, the logarithm transformation cost function had the most stable inversion performance with the increase of the unknown source parameters.