N. R. Goulty writes: Yao & Reddish published an empirical formula for the duration of mining subsidence in months, t, expressed as a linear function of the mining depth in metres, h, and the percentage of hard rock in the overburden, p:
t = 1.5553 + 0.0321h + 0.037p. (1)
They presented a nomograph based on this formula,which they recommended as a tool for civil engineers to calculate the time span after which no precautions need be taken to protect surface buildings from further damage due to residual subsidence.
The observations used to find (1) are shown in Fig. 10 and Table 2 of Yao & Reddish's paper. There were a total of twelve observations of residual subsidence at known depths, but the percentage of hard rock in the overburden was only known in nine cases. They first found the line of regression for t on h, using all twelve observations:
t = 3.1854 + 0.0321h. (2)
Then, in the nine cases for which p was known, they found the line of regression for the residuals, Δt, between observed values of t and those predicted from (2), on p:
Δt = - 1.6301 + 0.037p. (3)
Finally, they combined (2) and (3) to produce (1). This procedure may appear to be attractive in the absence of a complete dataset, but in fact it is not statistically rigorous and results in a prediction function which is not optimum.
Generalized matrix inversion produces the best solution, in the least-squares sense, for the