Rate-and-state models are widely used physical models for the relation between changes in pore pressure due to fluid injection or gas extraction and the induced seismic hazard in a field. We consider the modification where the pore pressure measurements are affected by noise and provide explicit expressions for the first and second moments of the state variable. We show that when the pressure increases, there is positive correlation. In the case of decreasing pressure, both positive and negative correlation is possible. Using the delta method, approximate first and second moments of the rate variable are derived and compared to empirical moments.

翻译：率-状态模型是广泛应用于流体注入或气体开采引起的孔隙压力变化与场地诱发地震危险性之间关系的物理模型。我们考虑了孔隙压力测量受噪声影响的情形，并给出了状态变量的一阶和二阶矩的显式表达式。结果表明，当压力增加时存在正相关；在压力降低的情况下，正相关和负相关均可能发生。利用delta方法，推导出了率变量的近似一阶和二阶矩，并将其与经验矩进行了比较。