TIME-STATISTICAL RESOLUTION

If we flipped a fair coin 1000 times, it is unlikely that we would get exactly 500 heads and 500 tails. More likely the number of heads would lie somewhere between 400 and 600. Or would it lie in another range? The theoretical value, called the ``mean" or the ``expectation," is 500. The value from our experiment in actually flipping a fair coin is called the ``sample mean.'' How much difference $\Lambda m$ should we expect between the sample mean $\hat m$ and the true mean m? Both the coin flips x and our sample mean $\hat m$are random variables. Our 1000-flip experiment could be repeated many times and would typically give a different $\hat m$ each time. This concept will be formalized in section 11.3.5. as the ``variance of the sample mean,'' which is the expected squared difference between the true mean and the mean of our sample.

The problem of estimating the statistical parameters of a time series, such as its mean, also appears in seismic processing. Effectively, we deal with seismic traces of finite duration, extracted from infinite sequences whose parameters can only be estimated from the finite set of values available in these seismic traces. Since the knowledge of these parameters, such as signal-to-noise ratio, can play an important role during the processing, it can be useful not only to estimate them, but also to have an idea of the error made in this estimation.