  By choosing more orthogonality between pre-selection and post-selection
states, one can significantly improve the sensitivity in the general optical
quantum metrology based on the weak-value amplification (WVA) approach.
However, increasing the orthogonality decreases the probability of detecting
photons and makes the weak measurement difficult, especially when the weak
measurement is disturbed by strong noise and the pointer is drowned in noise
with a negative-dB signal-to-noise ratio (SNR). In this article, we investigate
a modified weak measurement protocol with a temporal pointer, namely, the
auto-correlative weak-value amplification (AWVA) approach. Specifically, a
small longitudinal time delay (tiny phase shift) $\tau$ of a Gaussian pulse is
measured by implementing two simultaneous auto-correlative weak measurements
under Gaussian white noise with different SNR. The small quantities $\tau$ are
obtained by measuring the auto-correlation coefficient of the pulses instead of
fitting the shift of the mean value of the probe in the standard WVA technique.
Simulation results show that the AWVA approach outperforms the standard WVA
technique in the time domain with smaller statistical errors, remarkably
increasing the precision of weak measurement under strong noise background.
