Despite the prevalence of elastic anisotropy in rocks, most models used routinely for wave propagation, imaging, and rock physics assume an isotropic Earth. The main reason for using isotropic approximations is that we seldom measure enough parameters to characterize the stiffness tensor of a rock completely. Fluid substitution is an important example: because of an incomplete knowledge of the stiffness tensor, often we choose isotropic equations over their anisotropic form. Assuming weak anisotropy, we derive an approximate form of the anisotropic fluid-substitution equation for seismic waves propagating along the symmetry axis of a transversely isotropic medium. The approximation takes the form of the usual isotropic calculation, with a simple first-order correction proportional to the Thomsen anisotropic parameter δ, thus requiring only three elastic constants. Because δ can be either greater than or less than zero in a VTI medium, we can explain why isotropic fluid substitution sometimes overpredicts or underpredicts the full anisotropic result. Numerical simulations show that the approximate equation is valid for anisotropic medium with absolute value of δ as high as 0.3.

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