While propagating through the rocks the different frequencies will decay at different rate \alpha(f) and if noise sensor is located at {\bf r}_0 = \{0, \, 0, \, 0\} then the it will capture:
| (1) | \delta N_S = \int_V A(f) \cdot {\bf u} \cdot \nabla p \cdot \exp[-\alpha(f)r] \cdot \delta V |
The decay decrement \alpha(f) is growing with frequency: \displaystyle \frac{d \alpha}{df} > 0.
There is no universal model but it can be approximated by a linear-quadratic dependance:
| (2) | \alpha(f) = \alpha_1 \cdot f + \alpha_2 \cdot f^2, \quad \alpha_1>0,\, \alpha_2>0 |
with \alpha_1 and \alpha_2 having much slower dependance on frequency than \alpha(f) and in most practical cases can be assumed constant.
See also
Physics / Mechanics / Continuum mechanics