Blast Vibration: Scaled Distance and Peak Particle Velocity

Blast vibration is regulated on peak particle velocity — the fastest the ground itself moves at the point of interest, not the acceleration and not the displacement. The prediction equation is $PPV = K (D/\sqrt{W})^{-\beta}$, where $D/\sqrt{W}$ is the square-root SCALED DISTANCE: the distance to the receiver divided by the square root of the charge fired on a single delay.

The square root comes from dimensional reasoning about a cylindrical charge, and its practical consequence is worth stating on its own. Quadrupling the charge on one delay does not quadruple the vibration. It halves the scaled distance, and at a typical $\beta$ of 1.6 that multiplies the ground velocity by $2^{1.6} \approx 3.0$. Conversely, splitting a round so that only a quarter as much explosive fires on any one delay cuts the vibration to a third. That is the entire economic argument for delay detonators, and it is why a large shot properly delayed shakes less than a small one fired instantaneously. Note that the total explosive on the shot appears nowhere in the equation — only the charge per delay does.

Now the thing that matters more than the arithmetic. $K$ AND $\beta$ ARE SITE-SPECIFIC REGRESSION FITS AND THEY BELONG TO ONE SITE ONLY. They are obtained by putting seismographs at a range of distances, firing a series of shots with known charges per delay, plotting recorded PPV against scaled distance on log-log axes, and fitting a line: $K$ is the intercept, $\beta$ the slope. They absorb the rock, the overburden, the water table, the topography, the direction the shot faces, the initiation timing, the confinement, and how the structure sits on the ground. A prediction made with somebody else's $K$ is not a prediction; it is an assertion with a number attached. This matters more than it does anywhere else on this site, because vibration complaints and damage claims are settled on exactly these two numbers.

If you must borrow a constant, borrow an upper bound and know that is what you have done. The curves published in USBM RI 8507 — Siskind, Stagg, Kopp and Dowding, 1980 — are ENVELOPES drawn above a scatter of hundreds of records, not central estimates. Even at a well-characterised site the scatter about a fitted line is commonly a factor of two or three either way, so a responsible prediction is quoted at a confidence level, using the 95th percentile line rather than the mean line, with a design margin below the limit. And the units are baked into $K$: a constant fitted in inches per second against feet and pounds is a completely different number from one fitted in millimetres per second against metres and kilograms, and no arithmetic can catch the substitution. Many jurisdictions therefore offer a MINIMUM SCALED DISTANCE route instead, which avoids needing $K$ and $\beta$ at all — more restrictive than a proper site regression, and offered precisely for operations that have not done one.

PPV is not the whole story, and RI 8507's central finding was exactly that. FREQUENCY matters as much as amplitude. Low-frequency ground motion, below about 40 Hz — which is what long distances and soft overburden produce — couples into the natural frequencies of houses and does damage at velocities that would be harmless at high frequency. That is why modern criteria are frequency-dependent curves rather than a single number, and why the often-quoted 0.5 in/s (12.7 mm/s) is a conservative threshold for cosmetic cracking in residential construction rather than a universal limit. Airblast is regulated separately and for good reason: most complaints that people describe as vibration are actually the overpressure rattling the windows.

Two last points of proportion. Human perception of ground vibration begins around 0.5 mm/s, roughly a fortieth of any damage threshold, so most of what a blasting operation deals with is perception and annoyance rather than damage — and notifying the neighbours before the shot does more good than shaving the charge does. And this equation says nothing whatever about FLYROCK, which is a completely different mechanism with a completely different distance, and which is what actually sets the exclusion zone.