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The
(Un)Reliability Of Scene Measurement And Accident Reconstruction
Albert
G. Fonda
Fonda Engineering Associates
Email: afonda@crashex.com
Web: www.crashex.com
PRINTABLE
VERSION
 “So?
Who’s perfect?” was Joe E Brown’s classic closing
line to the equally classic movie “Some Like It Hot.”
The same concession necessarily applies to collision scene measurement
and reconstruction. Nothing can be measured perfectly, and such
uncertainty renders the result somewhat uncertain whatever the
treatment. This is mainly a discussion of the effects of uncertainties
of measurement alone, mentioning only briefly the accompanying
but independent uncertainties of treatment.
We’ve
all known about such uncertainty and we’ve all lived with
it. But it can be awkward to concede the imperfection when we
do not really know its consequences. We have not known how to
persuasively say yes, there is uncertainty, but only so much and
no more, and it does not affect the opinion being offered. Or
conversely, on rebuttal we may have not been able to persuasively
say that no useful opinion can be reached. That situation is changing.
It is changing to your benefit—since, when guided by the
type of analysis described here, the scene investigation will
be more focused and sometimes less expensive, and the resulting
reconstruction will be more complete and will better support your
opinion when rendered.
LANDMARKS
The first of two landmark events was WREX2000: the World Reconstruction
Exposition, September 2000, at College Station, Texas, and a published
summary (BA2002) of a set of “juried” Measurement Tasks
performed there. These statistical findings together with others
established for the first time the repeatability, or the reliability,
of all the usual measurements of scene and vehicle on which reconstructions
are usually based.
The second landmark event, if I am not too biased as one of the
authors, was a recent SAE (Society of Automotive Engineers) paper
(BA2003) reporting the use of the outputs from the WREX2000 studies
as inputs to an exemplary accident reconstruction, to show how
to state their significance in terms of a useful finding. To give
away the result before we start, for the selected, fairly typical
collision the 5% likely variations in the approach speeds reconstructed
by instantaneous conservation of momentum due to only the errors
of measurement was as low as 0.6 mph (1.5%) with extremely accurate
site examination, rising to about 3.0 mph (7.5%) assuming more
ordinary efforts, and slightly exceeded 5 mph for one vehicle
and 11 mph for the other if the scene investigation was rather
casual.
In various instances a low level of uncertainty might be necessary,
or a high level might be adequate, to decide the issue or issues
at hand. In a different type of event a higher or lower level
of uncertainty might be obtained; a family of like studies remains
to be performed.
BENEFITS
The availability of the measurement uncertainty data and the method
of analysis has two major effects: (1) prior to the study, the
pending scene investigation effort can be more fully optimized,
and (2) following the study, the likely accuracy of the reconstruction
as affected by the measurements can be numerically reported rather
than vaguely approximated.
FEASIBILITY
In a few minutes (with practice) to a few days, and depending
on the suitability of the chosen method of reconstruction, a set
of informative calculations may be performed. The statistical
approach can be used with any available method of reconstruction,
if readily repeated with variations of residual scene data; an
option which allows the investigator to use familiar methods and
software. In the alternative, the repetitions and calculations
required can be accomplished much more rapidly if they are dedicated
routines provided by the reconstruction algorithm. In the author’s
program CRASHEX (Computerized Reconstruction of Accident Speeds
on the Highway, Extended) such automation has been incorporated
since before 1990 (FO1995, FO2000), although while there were
some benefits from the beginning, the capability was not fully
useable until recently for lack of proper input data. In the referenced
2003 paper several of the alternative methods are briefly reported;
here only findings using CRASHEX will be recited.
MEASUREMENT
UNCERTAINTIES
At WREX2000, as reported in the 2002 SAE paper "Quantifying
The Uncertainty in Various Measurement Tasks Common to Accident
Reconstruction" by Bartlett, Baxter, Brach, and others (BA2002),
volunteers made measurements of selected distances, tire marks,
crush damage, and drag factors. When statistically analyzed these
tests gave, in many instances for the first time, the variability
of such typical field measurements. For instance,
Using a fiberglass tape, a mean length of 39 feet was measured
with a standard deviation (SD) of 0.025 feet (0.065%). A mean
length of 92 feet was measured with SD of 0.061 feet (0.067%).
Using a single-wheel roller tape, a mean length of 35 feet was
measured with SD of 0.08 feet (0.21%), and a mean of 91 feet with
SD of 0.12 feet (0.13%).
Using a dual small-wheel roller tape, a mean length of 38 feet
was measured with SD of 0.08 feet (0.21%), and a mean of 90 feet
was measured with SD of 0.16 feet (0.17%).
Thus, the standard deviation of errors in measurement of distances
exceeding about 30 feet or 10 meters is typically well under one-fifth
of 1% using a wheel and under one-thirtieth of 1% using a tape.
The question is, how do these and other likely field errors, and
the likely errors of estimation of any quantities not measured
in the field, combine to affect the reconstruction?
Considerably more uncertainty in the location of a point results
from the use of a tape or a wheel if it is initially used to establish
a right triangle whose sides are presumed to define perpendicular
reference axes. Likewise there will be further difficulties in
measuring angles and in identifying the items to be measured.
All such sources of error should be considered in any uncertainty
analysis.
These experimental results and others were summarized by Bartlett
and Fonda (the present author) in Table 1, attached, from the
SAE paper “Evaluating Uncertainty in Accident Reconstruction
with Finite Differences” (BA2003).
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TABLE
1.
Typical values of measurement uncertainty common to accident
reconstruction
NOTES:
Low' error by laser or digital sensor is trivial for all
distances and angles. (1) Range 30 to 90 feet, for points
near the major axis of an elongated cluster. (2) Cartesian
distance, about equally far from each axis. (3) 'Low' for
scribed mark, 'Medium' for tire mark, 'High' for casual
estimate. (4) 'Medium' by dubious jury, 'High' if object
uncertain. (5) 'Low' established by on-site test, typical
scale resolution; 'Medium' per Smith (1982), 'High' for
casual estimate; In cases where the number of passengers
or their weights are unknown, or contents of the vehicle
may have exited the vehicle during collision, the high-uncertainty
level may be much higher, and will have to assessed on a
case-by-case basis. (6) 'Low' established by on-site test
with actual or replicate vehicle and accurate equipment,
'Medium' for less accurate on-site tests or Ebert's generic
values, 'High" for casual estimates. (7) 'Low' is based
on uncertainty in vehicle dimensional data for very well
defined features from WREX-2000, 'Medium' per Smith's NHTSA
investigator(s), 'High' based on WREX-2000 results. (8)
'Medium' per Smith and Bartlett's results from WREX; 'High'
denotes a casual estimate. (9) 'Low' if from tests of the
same or replicate vehicle; 'Medium' if based on vehicle
size; 'High' denotes a casual estimate.
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The first column of Table 1 lists “Low” 1SD levels of
measurement uncertainty, largely assuming the use of laser instruments.
Laser errors by WREX2000 test were vanishingly small (shown as
zero) for purposes of accident reconstruction. The second column
lists “Medium” levels of measurement uncertainty, assuming
use of distance triangulation (not scene monuments) for the establishment
of perpendicular axes, and no on-site friction testing. The third
column lists “High” or liberal levels of measurement
uncertainty, suitable for preliminary assessment of a reconstruction.
Specific assumptions are listed in the footnotes of the table.
The data in the table is merely exemplary, and is subject to confirmation
and revision.
FINITE-DIFFERENCE
ANALYSIS
The methods of analysis reviewed in the 2002 SAE paper were only
the well-known methods of (a) worst-case analysis, (b) algebraic
partial differentiation with statistical analysis, and (c) Monte
Carlo repetitive solution with statistical analysis. In the 2003
paper a fourth method, (d) finite-difference repetitive solution
with statistical analysis, previously introduced, was for the
first time demonstrated for such use.
This “new” method is rather obvious from basics and
has long been known in the field of statistics but was virtually
unknown in accident reconstruction, even though it was briefly
mentioned in several of my earlier SAE papers and the necessary
routines had long been written into my software. From the start
it could be used for sensitivity analysis (Equation (1) below,
with  x=1), and if the statistically probable inputs were
known it could be used for finite-difference analyses; but, at
first the inputs could only be roughly approximated. The WREX2000
data plus earlier data as reported in the 2002 and 2003 papers
provided the necessary spectrum of input data, enabling the practical
use of the method for the first time in this field.
The method depends on disturbance of the base case of the reconstruction
by one amount at a time, all of which are equally probable errors
of measurement, giving changes from the base case, for any selected
output r,
This result
is, if the influence coefficients  r/x
for all x’s are invariant and independent, numerically the
same as that which would have been obtained from algebraic partial
differentiation. For present purposes this requires, however,
that the x’s be site or vehicle measurements which are inputs
to a time-reversed computation (a true reconstruction), not the
speed inputs to a time-forward simulation such as SMAC, EDSMAC,
or PC-CRASH (which develop the trajectories as outputs).
For each reconstruction result (r) of interest, these individual
effects, one for each measurement of interest (x), are combined
by summation of their squares, giving
The basis
in statistical gobbledegook is that the variance is the square
of the deviation and the variance of the sum is the sum of the
variances.
However, this operation can be accomplished by means of a progressive
summation of vector normals, or a series of evaluations of c in
a²+b²=c² where each hypotenuse provides one of
the sides of a further right triangle for another summation. A
practical example of this (and a pretty good common-sense derivation
of the result) is that a series of trips of known length in randomly
differing directions have a probable sum which is the same as
if each trip were normal to the previous sum, favoring neither
minimization nor maximization of the sum. The “worst case”
would give the maximum sum, as if all the trips had been made
in the same direction. The Monte Carlo method would give the sum
of many, many trips of lengths as well as directions varying randomly,
but with known probability as to the lengths.
Obviously the worst-case method is too adverse, while the algebraic
method is at best difficult and at worst well-nigh impossible.
The Monte Carlo method requires special programming and massively
repetitive solutions (in the tens of thousands), which if done
in a spread sheet requires, compared with CRASHEX, unnecessarily
simple equations of reconstruction, introducing avoidable errors
of treatment.
This treatment gives the uncertainty range around the nominal
result to the selected confidence level. If the relationships
are linear and independent, which is true enough if the distinctions
are not close, the results are the same as with algebraic partial
differentiation with statistical analysis as in Equation (2),
or Monte Carlo repetitive solution with standard statistical analysis.
In principle, any method of reconstruction my be used, even “hand”
(handheld calculator) calculations. The user “merely”
need add the likely errors of measurement to the base case one
at a time (finding a particular x+x), run the solution,
and from the resulting value f(x+x) of each variable of
interest subtract the unperturbed result f(x) as indicated in
Equation (1). Then, square all these differences (r) and
find the square root of their sum, as indicated in Equation (2).
If this can be done economically, the finite-difference method
will give qualitatively the same results regardless of the method
of reconstruction, though sometimes with quantitatively slightly
different results.
In the case of CRASHEX the procedure is exactly the same except
that it is automated. As I reported in my 1995 SAE paper (FO1995),
the procedure begins with a return from the base-case output screen
to a data-less input screen; as soon as each desired perturbation
(x) is entered the solution runs and the array of r’s
are reported on an output screen. Then a print may be made and/or
another input perturbed, or each r may be replaced by the
corresponding  r, the root of the sum of the squares
so far. Each print also shows the relevant input perturbation(s),
and rank ordering for the effect on any one output is done by
hand sort of the printouts.
Regardless of treatment there will be some errors of measurement
which have comparatively little effect, while others are more
influential. An order ranking of the contributions (ranking the
r’s within r) will identify “the
vital few among the trivial many,” separating the sheep from
the goats; a known statistical task.
Usually, as we are about to see, the three most influential errors
will dominate the sum. This means that up to about six field measurements
might need to be documented as having been made with the care
designated; but no other measurements, even if considerably less
accurate than claimed, could have significantly degraded the existing
reconstruction.
A TYPICAL
INTERSECTION IMPACT
The case initially selected is a right-angle intersection impact
with both vehicles traveling at 40 mph, Vehicle 2 having been
Eastbound when impacted on the rear of its right side by the Northbound,
smaller Vehicle 1, giving (due to mutual offsets) the spinning
trajectories shown in Figure 1.
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| FIGURE
1.
Typical intersection impact |
This usefully
complex but more or less typical collision configuration was used
by Severy in staged impacts in the 1950's, McHenry in the 1970's
in the development of both SMAC and CRASH, and Fonda in the 1980's
in the development of CRASHEX; and it was the most-severe case within
the family of EDSMAC simulations provided as an accident reconstruction
baseline by Kinney and Woolley (1994). An EDSMAC simulation of RICSAC
10 with both vehicles at exactly 40 mph (and with the struck vehicle
reversed) established the particular scene and vehicle data to be
reconstructed.
FINITE-DIFFERENCE
RESULTS
Taking a momentum-only, instantaneous-impact (CRASH3) type of
solution as a base case, all variable uncertainties were set to
twice the 1SD values outlined in Table 1 for Low, Medium, and
High uncertainties. These input errors are 5% likely to occur;
so more error than this, while possible, is not at all likely.
Then in each instance, by ranking in order of magnitude the three
largest resulting uncertainties in the two impact speeds, which
are for the present the vital few we seek, are found to have the
values shown in Table 2. The same results are shown graphically
in Figure 2, a progressive summation of vectors, building up from
the largest to the smallest in a spiral manner, for each of the
three levels of measurement uncertainty.
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| TABLE
2.
Ranked uncertainties of reconstruction due to measurement
uncertainties |
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| FIGURE
2.
Vectorially summed effects of measurement error |
Table
2 and Figure 2 show that for all levels of error the calculation
was found to be most sensitive to errors of measurement of the
tire-road friction or vehicle drag (which were varied jointly),
with the exception of a greater sensitivity to a High (6 degree)
error in the relative directions of approach (and heading) of
the vehicles. Otherwise there was secondary or tertiary sensitivity
to path curvature error, and to error in the weight of the striking
vehicle if all errors were Low or Medium.
Results
of Casual Investigation
The High levels of error (11.3 and 5.5 mph) might correspond to
a quite tentative reconstruction; yet such results could be quite
adequate when a gross approximation would settle the issue at
hand. Otherwise all of the data should be collected with greater
care, especially (if nothing else) with regard to the angle of
convergence between the vehicles (noting that the paths at impact
might be difficult to discern and may have included last-chance
avoidance attempts).
Results
of Ordinary Investigation
The Medium levels of error (3.0 and 2.8 mph) require at least
the better friction look-up values of Ebert (EB1989), an allowance
for likely occupant and cargo loading of the generic vehicles,
and careful treatment of well-documented tire tracks; yet if initially
well investigated by others a further site visit might not be
necessary. At this level of accuracy the results could be quite
acceptable when exact approach speeds are not crucial to the issue
at hand.
Significant improvements in the end result could be made, however,
by on-site friction measurements (see Bartlett 2003), on-site
studies to refine the vehicle paths, and adoption of a method
of treatment not presuming instantaneous impact (as do CRASH3
and the like).
As I noted in my 2000 SAE paper, referring to errors of measurement
not yet documented, "If all relevant inputs have been perturbed
and each relevant root-sum is at least twice the likely error
of treatment as indicated by the present study or others, the
treatment is as good as it needs to be, and there need be no specific
allowance for errors of treatment." In an intersection impact
such as described here, with Medium errors of measurement this
would apply, as shown in FO2000, if CRASHEX were used; but not
if any “hand” calculation (simplistic, two-body conservation
of momentum), an equivalent spread sheet, or CRASH3, EDCRASH,
RECTEC, or the like were used.
Results
of Meticulous Investigation
The Low levels of error (both 0.6 mph) presume the use of laser
instruments applied to precisely defined targets along tire marks
still visible or recreated at the site, development of the corresponding
mass center paths, careful estimation of specific occupant and
cargo loading, tare weights, and coasting drag of the involved
vehicles or replicates thereof, and onsite friction measurements.
Note that while the use of computer-aided electronic instrumentation
always may be justifiable for its speed, convenience, credibility,
and avoidance of inadvertently gross error, in terms of accuracy
it is overkill unless it is combined with equally careful measurement
techniques in all other respects and with a superior method of
treatment; not one degraded by the simplifying approximation of
instantaneous impact. That is, travel first-class all the way,
or else don’t bother.
ADOPTION
OF THE APPROACH
The results recited here are merely examples, and apply neither
to collisions which differ greatly from the assumed intersection
impact, nor even, with any great precision, to collisions rather
similar to that of Figure 1. For example, as can be seen in Figure
1, Vehicle 2 rolled less than 4 feet after impact, with the result
that any error in measurement of the rise of that short chord
would have an unusually large effect on the path direction at
the end of spin and the yaw change during spin. Accordingly, with
Normal errors, error in that chord rise was found to be of “secondary”
importance. Given more rollout the chord-rise error ranking would
subside to tertiary or less.
Furthermore, for the sake of simplicity and to avoid any perception
that the findings were unique to CRASHEX, only the CRASH3 treatment
(which is the initial case with which CRASHEX seeds its iteration
for the forces and motions during impact) was performed, and only
the approach speeds resulting from idealized conservation of momentum
were studied. This omitted all the input data for the damage to
the vehicles; that is, we ignored the energy solution, also the
influence of damage data on the more sophisticated momentum solution.
We also chose to ignore the speed changes (the V’s)
(which are related to occupant causation, as opposed to collision
causation), and any changes in all these relationships which might
be unique to CRASHEX. This better served the purpose of a mere
demonstration by example.
Thus it will be up to you to adopt and adapt the approach when
the opportunity arises. If enough is known about the event prior
to a field trip, it would be best to rough out a pilot reconstruction
and finite difference analysis, assuming Medium Uncertainty; that
is, reading from the middle column of Table 1 for each likely
error of measurement, considered one at a time. When these are
ranked by magnitude in terms of the output(s) of greatest interest
(and with no need as yet to do a root-sum-square), the most influential
variables can be spotted. These vital few measurements can then
be made with more care, while less care need be given to the trivial
many. For instance, a laser survey might not really be needed;
and indeed in some cases a site visit by the reconstructionist,
reassuring as it may be, might not even be needed.
Later, you can revise the base-case inputs from the pilot array
on the basis of the completed field work, and develop your most
probable reconstruction. Then you can repeat the finite difference
analysis, assuming whatever uncertainties of measurement are judged
to have actually pertained, that is, reading from whatever column(s)
of Table 1 (interpolating as needed) seem appropriate for each
likely error of measurement. You can again rank the results in
terms of effect on each of the outputs of greatest interest, and
combine them into the corresponding root-sum-square. If also the
likely errors of treatment are included (not a part of the present
discussion), the root-sum indicates the confidence with which
the results can be stated, or, the likely upper and lower bounds
flanking the most probable value, and the ranking progresses from
the most to the least influential measurements involved.
Your presentation, if well founded on the evidence and not unduly
broadened by avoidable errors of treatment, becomes virtually
unassailable. When the very appropriate question is asked, “Did
you consider the possibility of (such-and-such) a difference in
(you name it)?”, you then can answer quite confidently, if
you have done your homework: “Yes, and it did not affect
my opinion, which allowed for such a difference and for many others.”
Or, as the case may be, “No, because such a large difference
would contradict the available evidence, and I decline to speculate;”
or, sometimes, “Yes, I did, and this is exactly why I am
saying that the issue cannot scientifically be decided, given
the evidence.”
As things stand you might soon meet a forensic opponent whose
presentation is this well substantiated; so it will behoove you
to do so first, or suffer the consequences. The bar is being raised.
SUMMARY
With the recent establishment of quantitative, juried data as
to likely errors of measurement in accident investigation, a long-known
but little-used tool, Finite Difference Analysis, has become useable
in accident reconstruction. This procedure is useful in identifying
the vital few measurements among the trivial many, and their combined
effects, by quantifying the effects of input uncertainty on calculated
results in accident reconstruction.
The Finite Difference Method provides an immediate uncertainty
evaluation whether implemented by means of hand calculations or
more complicated accident reconstruction algorithms. When combined
with an existing algorithm for accident reconstruction it adds
to the sophistication of that specialized procedure the benefits
of both a sensitivity analysis and statistical information on
the accuracy of the outputs of the reconstruction.
For a typical intersection impact the numerical results in the
present paper suggest the proper allocation of resources for the
field investigation and the general levels of accuracy to be expected
according to the care taken in data collection. In future Finite
Difference studies similar results can be developed not only for
dependent variables other than approach speed, given the same
impact configuration, but also for a variety of other impact configurations.
In the long-range future of the art and science of automotive
collision reconstruction, by application of Finite Difference
Analysis during preliminary analysis the funds and manpower available
for full scene investigation will be more judiciously allocated,
and by re-application of the same method following the completion
of field work the reliability of the analysis and the resulting
opinion will be responsibly reported to those most concerned with
the findings.
PRINTABLE
VERSION
REFERENCES
BA2002. Bartlett,
W.D., Wright, W., Masory, O., Brach, R., Baxter, A., Schmidt,
B., Navin, F., Stanard, T., Quantifying The Uncertainty in Various
Measurement Tasks Common to Accident Reconstruction, SAE Paper
2002-01-0546
BA2003. Bartlett,
W. D., and Fonda, A. G, “Evaluating Uncertainty in Accident
Reconstruction with Finite Differences,” SAE 2003-01-0489
EB1989. Ebert,
N., Tire Braking Traction Survey Comparison of Public Highways
and Test Surfaces, SAE Paper 890638
FO1995. Fonda,
A.G., Nonconservation of Momentum During Impact, SAE 950355
FO2000. Fonda,
A.G., Partially-Braked Impact and Trajectory Benchmarks, and Their
Application to CRASH3 and CRASHEX, SAE Paper 2000-01-1315
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