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Amidst recent advances in terahertz scanning tunneling microscopy (THz-STM), having already demonstrated simultaneous atomic spatial resolution and sub-picosecond time resolution, the THz-driven bias resulting from coupling THz pulses to the STM junction is not completely understood. Here, we use finite-element simulations and equivalent circuit models to define and quantify the THz-transient bias that determines the THz-induced electron tunneling signal in the STM junction. Waveforms sampled using THz-STM on an optically excited GaAs sample exhibit features predicted by the simulations.