Laser speech measurement is a new sound capture technology based on Laser Doppler Vibrometry (LDV). It avoids the need for contact, is easily concealed and is ideal for remote speech acquisition, which has led to its wide-scale adoption for military and security applications. However, lasers are easily affected by complex detection environments. Thus, speckle noise often appears in the measured speech, seriously affecting its quality and intelligibility. This paper examines all of the characteristics of impulsive noise in laser measured speech and proposes a novel automatic impulsive noise detection and removal method. This method first foregrounds noise using decorrelation based on a linear prediction (LP) model that improves the noise-to-signal ratio (NSR) of the measured signal. This makes it possible to detect the position of noise through a combination of the average short-time energy and kurtosis. The method not only precisely locates small clicks (with a duration of just a few samples), but also finds the location of longer bursts and scratches (with a duration of up to a hundred samples). The located samples can then be replaced by more appropriate samples whose coding is based on the LP model. This strategy avoids unnecessary processing and obviates the need to compromise the quality of the relatively large fraction of samples that are unaffected by speckle noise. Experimental results show that the proposed automatic speckle noise detection and removal method outperforms other related methods across a wide range of degraded audio signals.