Here's my assessment of Chander's analysis.
I've broken it down into three specific errors that Chandler has made. Plus several "technique" improvements that Chandler could have performed to improve his results.
Outright errors in Chandler's analysis.
1. Not using position vs. time data. Use of central difference approximation to estimate velocity.
2. Not performing a competent error analysis.
3. Using the 1.75 second data point. This is a clear outlier.
Techniques that would have improved his results.
1. Use of high-res, uncompressed video
2. Adaptive data sampling instead of fixed interval (higher rate at areas of greater interest / higher velocity).
3. Use of virtual data point to obtain better curve fit.
4. Choose data points with consistent pixel intensity (e.g., 50% between pre & post intensity values).
5. Incorporate video raster scan timing correction.
6. Lots of video processing techniques. (interlace, contrast enhancement, edge detection, etc.)
I'll discuss only the errors.
1. Use of central difference approximation to estimate instantaneous velocity.
The principle error was to NOT use the position vs. time data to generate an empirical equation for the position vs. time, then differentiating that function for velocity vs. time, and differentiating again for acceleration vs. time. This approach would have conditioned the data (i.e., reduced errors) before beginning to manipulate it.
The use of the central difference approximation put large amounts of error into Chandler's raw data. There is no reason to do this. Look at the quality of the curve fit of the position vs. time data in Figure 12-76 (NCSTAR1-9 vol 2, pg. 602 pdf 264). Compare this with the quality of the curve fit of the velocity vs. time curve of figure 12-77 on the next page.
The result is that there is no comparison. The position vs. time data is far superior to the velocity vs. time data. The reason is simple.
The average velocity between two points is given by Vavg = C*(p2 - p1)/(t2 - t1).
where t1 & t2 are the times of the two frames.
p1 & p2 are the pixel numbers of the two points
C = pixel scaling constant (feet / pixel).
t1 & t2 can be determined quite accurately from the frame numbers, so they aren't the problem. But the z values can have moderate errors to them. When you subtract two numbers that are close to each other, but have moderate errors, the answer will have significantly larger precent errors.
By taking this approach, Chandler introduces large errors into his starting data. The proper technique is to smooth the data as early as possible. In signal processing, it is good practice to implement signal conditioning (i.e., cleaning up the signal) PRIOR to amplifying the signal. That way, you don't amplify the noise along with the signal. This is, in essence, what Chandler has done: he's manipulating data that he's introduced a lot of granularity. Differentiating a granular signal in essence dramatically amplifies the step functions in the input data. The empirical equations are much, MUCH better behaved.
In this particular case, the empirical equation that NIST generated is given in Fig 12-76. It is:
Differentiating this expression with respect to time gives the empirical equation for velocity vs. time, which is exactly the equation of the solid line of Fig 12-77.
Differentiating this equation once more gives the acceleration vs. time.
(Thank you, Mathematica.)
Here is a table of the drop distance (ft), velocity (ft/sec), acceleration (ft/sec^2) and acceleration (%G) as a function of time for the first 5 seconds. This is taken directly from the video data. It's the same data that Chandler folded, spindled & mutilated.
...Time... | ...Drop... | ...Velocity... | ...Accel... | ....%G....
(sec) | (feet) | (feet/sec) | (feet/sec2) |
0.00 | 0 | 0 | 0 | 0
0.25 | 0.0 | 0.1 | 1.1 | 3%
0.50 | 0.1 | 0.6 | 3.2 | 10%
0.75 | 0.4 | 1.7 | 5.8 | 18%
1.00 | 1.0 | 3.6 | 9.0 | 28%
1.25 | 2.2 | 6.3 | 12.5 | 39%
1.50 | 4.2 | 9.9 | 16.2 | 50%
1.75 | 7.2 | 14.4 | 20.1 | 62%
2.00 | 11.5 | 19.9 | 23.9 | 74%
2.25 | 17.3 | 26.3 | 27.5 | 85%
2.50 | 24.8 | 33.6 | 30.6 | 95%
2.75 | 34.1 | 41.6 | 33.0 | 102%
3.00 | 45.6 | 50.0 | 34.4 | 107%
3.25 | 59.2 | 58.7 | 34.6 | 108%
3.50 | 74.9 | 67.2 | 33.4 | 104%
3.75 | 92.7 | 75.3 | 30.7 | 95%
4.00 | 112.0 | 82.4 | 26.3 | 82%
4.25 | 134.0 | 88.3 | 20.5 | 64%
4.50 | 156.0 | 92.6 | 13.3 | 41%
4.75 | 180.0 | 94.9 | 5.1 | 16%
5.00 | 204.0 | 95.1 | -3.6 | -11%
5.25 | 227.0 | 93.1 | -12.3 | -38%
Let's plot the results.
Here is the drop distance (blue line), Velocity (red) and acceleration (yellow) vs time for the first 5 seconds.
Now, let's plot the acceleration (as a percent of g) vs time. This gives an ACCURATE accounting of how the acceleration changes as a function of time.
The blue line is the behavior of WTC7's roof top. The red line is the TRUE waveform for an object in free fall (ignoring air resistance).
One can see IMMEDIATELY that Chandler's claim that, over the interval of 1.75 to 4.00 seconds, the acceleration is a constant that is equal to G, as well as NIST's statement that it is "a constant approximately equal to G" is nonsense. The acceleration isn't close to being constant.
And this graph, right here, is why I said that the acceleration looked nothing like free fall. The RED LINE looks like free fall. The blue line could not possibly look more different.
Last comment. Figuring out the BEST ESTIMATE for the integrated average acceleration over the period from 1.75 seconds to 4.00 seconds, calls simply for solving the definite integral of the acceleration as a function of time between those integration limits and then dividing by (4.00 - 1.75 = ) 2.25 seconds.
The answer for that is 30.24 ft/sec^2. Which is equal to 0.94G. This is a number that I can believe. And everything about this analysis makes perfect sense.
Note that Chandler (or whomever) STILL needs to perform a complete error analysis in order to provide accuracy bounds on his answer.
tom
