The Practical Application of Finite Difference Analysis in Accident Reconstruction

The effective application of a Finite Difference Analysis routine in accident reconstruction is demonstrated. Such analysis finds not only the central value but the reliability side-bands; not only the mean but the probable extremes of the analysis, given the probable investigative reliability.

Furthermore, because the controlling transfer functions depend only on the physics of the event, the side-band spans found by one means of reconstruction apply also to the central values found by some other means. Such ubiquity allows the practitioner to add to pre-existing central results the probable bounds of those results due to the practical limitations of investigative accuracy.

Finite Difference Analysis is the variation of the inputs to an analytical solution each in turn to the extent of the statistically likely bounds of error in that measurement, and the probabilistic summation of the consequences. The result is the likely effect on the solution had all the inputs varied simultaneously but randomly from the original set, as in Monte Carlo analysis. The reliability side-bands flanking the central value then portray the confidence with which the solution is presented. The principles are reviewed in Appendix A.

In forensic accident reconstruction there is a perennial choice between prospective and retrospective analysis — time-forward from postulated causes to their likely effects, or time-reversed, from postulated effects to their likely causes — offering a tradeoff between the rich detail of dynamics-and-statics simulation⁷ and the faithfulness to field data of momentum-and-energy reconstruction (reconstruction per se^3-5,8-10). However, the present procedure^2,6 is applicable only when one final effect at a time can be considered to have varied from its as-measured value. This restricts our present scope to time-reversed reconstruction.

Finite Difference Analysis could be accomplished using any time-reversed algorithm, given sufficient analyst time. The reconstruction would be repeatedly perturbed and rerun, with each consequent change evaluated, stored, and eventually root-sum-squared. But for more expeditious execution those procedures can be provided algorithmically, as in, and to date only in, the author’s CRASHEX^3-5 program.

A representative accident reconstruction will be shown for the fairly common case of impact with an oncoming vehicle during a left turn.

An engineering opinion was requested as to movements and speeds involved in and avoidability of a particular collision. Figure 1, drawn on an available site survey which has been rotated to a conventional azimuth, shows the reported approach paths 1 (Southbound) and 2 (Northbound) and the probable positions and yaw attitudes of the vehicles at impact and at rest according to the applicable police report.

Vehicle 1 proceeded upgrade to a point 250' South of the crest of a hill AND turned left across the path of Vehicle 2. Phantom positions of Vehicle 1 are shown along its path immediately before blocking and immediately after clearing the path of the oncoming vehicle.

Vehicle 2 had antiskid brakes and left no detectable tire marks, but its pre-impact tire noise was reported by the driver of an uninvolved Southbound vehicle (the unshaded outline in Figure 1), and pedal response denoting full antiskid brake actuation prior to impact was reported by Driver 2. This testimony inferred pre-impact hazard sighting and consequent avoidance braking of Vehicle 2.

Then the front of Vehicle 2 impacted the right side of Vehicle 1, impelling it Northeast a measured 43' as it rotated clockwise to rest off-pavement facing West. Meanwhile, Vehicle 2 traveled Northeast a measured 34' 3" while rotating clockwise to rest facing slightly South of East with its front wheels on the East shoulder. The post-impact tire marks and rest positions were shown in police photographs.

Figures 2 through 6 show the numerical inputs to and the numerical and graphical outputs from a CRASHEX reconstruction of this event.

Shown in Figure 2 are data descriptive of the vehicles and their trajectories, with Vehicles 1 and 2 in left and right columns respectively. Zero values denote the absence in this instance of an initial yaw rate, a final runout, and a final speed. The nonzero entries define the approach paths and heading, the vehicle locations and headings at impact and at rest, and the curvature of the intervening spin path. These 16 inputs when combined with the 10 vehicle parameters are sufficient for a minimal momentum solution.

An independent energy-based solution⁴ was found from the probable structural properties and crush profiles of the vehicles (Figure 3). As side benefits this gave also the duration of impact and hence, in the momentum solution, the tire forces and the consequent loss of momentum to earth during impact³.

Figure 4 lists the resulting output data of the reconstruction. In five pairs of rows, with Vehicles 1 and 2 in respective rows of each pair, 68 items of data describe the event in terms of

This array must closely match the results of any prior study of the base case, or if none exists it is used in developing the base case directly from the field data. In the fifth set each entry is an alternative (primed) value of a prior (starred) output in a previous row, providing checks on the internal consistency of the reconstruction. Further reconciliation by iterative reduction of discrepancies so found would impose final constraints sometimes initially imposed in other programs. The reconciliation of all constraints within any reconstruction should improve that reconstruction.

While not used during FDA, and accordingly shown only to reduced scale in Figure 5, the Site Plot will closely match the results of any prior study, absent which it is used in base-case development and presentation. The position of each vehicle (Vehicle 2 gray) is shown before impact, at impact, at separation, at half-second intervals after impact, and at rest, with corresponding inferred skidding (double) or rolling (single) tire marks. During operation the entire event can be animated from start to finish in real time, with drag-and-drop revision of inputs, while the pop-up table at the top summarizes a few pivotal numeric inputs and outputs from Figures 2 through 4.

While again not used during FDA, the Vector Plot shown in Figure 6 also is of use in prior-study matching or in base-case development. The vehicle perimeters, crush contours, and impact force vectors are shown (left), with Vehicle 2 gray. The momentum vector summation (right) is the primary solution for the unknown speeds and speed changes. Superposed broken lines denote the independent ΔV's due to energy. When these independent plots barely differ (as here) the two independent speed change solutions have been adequately reconciled.

On command all of this data and graphics are displayed on one screen and optionally printed on one sheet, much as seen here. As will be seen, the same sheet less graphics plays a vital role in FDA.

In the subject case the best estimates of the two approach (impact) speeds, which in this instance as in many are the numerical outputs of primary interest, are

Heretofore this has been the end point of the reconstruction, leaving the reconstructionist with nothing substantive to say about the quality or reliability of such a "best estimate." Now, however, thanks to recent SAE tests and papers, by the method of Finite Difference Analysis the boundaries of reasonable confidence in such best estimates can be numerically evaluated and recited.

Finite Difference Analysis commences after such a central or best-case reconstruction has been achieved. In the first stage of FDA, user-specified changes, all statistically equally likely, are applied in turn to each of the inputs to the established base case. In the second stage the results are statistically summed.

• for each element respectively in each resulting output array, subtraction of the previously saved base-case output;

• tabular presentation of both the single input difference and the array of output differences as a printed report; and

• digital saving of all such arrays for use in the second stage of the analysis.

In Step 1 each input could be a unit quantity (e.g. 1 meter or 1 foot or 1 pound); then each output array would constitute a printable array of outputs per unit input. These are sensitivities or transfer coefficients which would be useful when tuning not only reconstruction but also simulation input changes to effect desired outputs.

However, in FDA each such input is set to equal the span of probable uncertainty or deviation in the associated measurement; then each such output array lists the consequent and equally probable deviation of each output.

Figure 7 shows the characteristic “bell curve" for deviations which are normally distributed. Presumption of 2 SD deviations of measurement as the inputs will include 95.44 % of the probable input and (hence) output deviations, excluding at each end of the spectrum 2.28% (1 part in 44). (While some small difference always exists as between positive and negative perturbations, in practice the difference is negligible.)

For the exemplar case, using as inputs twice the "Medium" standard deviations of measurement found in juried and other studies (see Appendix B), and with the initial position of Vehicle 1 taken as a reference, 43 input differences were used consecutively in 43 runs of the differencing routine. For each run a single physical sheet listed the change in one of the inputs and the consequent changes in all of the outputs shown in Figures 2 through 4.

These 43 separate sheets then could be hand sorted to order-rank the inputs in terms of their effect on any one output of interest — and then sorted again in terms of any other one output. This procedure sequences for each output "the significant few" influences among "the trivial many" (see Appendix A). We will show below the results of such an order-ranking for the two impact speeds, the primary parameters of the exemplar case.

In the second and final stage of Finite Difference Analysis, for each output the root of the sum of the squares of all previously calculated output differences is found. As shown in Appendix A, by basic statistical theory this is the probable effect of the random occurrence of all the input deviations. Vectorially, as shown in Figure A1 it is a progressive summation of successive normals to each preceding sum. Such summation avoids the excessively conservative linear summation of the absolute values of all effects, yet preserves a due effect for every influence.

Given 2 SD input deviations (having doubled the 1 SD values listed in Appendix B), this sum is the 2 SD reliability of each output, that is, the "95% reliability side-band" values which exclude at each end all but one part in 44 of the effects of all probable deviations of measurement.

For the exemplar reconstruction, Figure 8 enumerates in the format of Figures 2, 3 and 4 all of the 43 difference inputs to the final summing routine and the root-sum-square values for all 68 output differences. Since there were 68 responses to each of 43 perturbations, the effects of over 2900 responses are summarized in Figure 8.

Each "output" item in Figure 8 defines the width from center of the band which encompasses 95% of the likely deviations of some reconstructed parameter from the value found in the base case.

The reconstruction variations often of greatest interest are the two impact speeds, seen in the third pair of rows in the first column of Figure 4 for the base case, and of Figure 8 for the deviations.

In Figure 9 these results are summarized for the exemplar case. Notably, whereas the speed of Vehicle 1 has a span of uncertainty which at 5.5 mph is 45% of its 12.1 mph central value, at 3.9 mph the span of uncertainty for Vehicle 2 is a mere 8% of its 48.4 mph central value. The relative uncertainty (output deviation due to measurement uncertainty) for Vehicle 2 is barely over 1/6th that for Vehicle 1. Although inherent, such an asymmetric result might hardly have been guessed; but it issues automatically from Finite Difference Analysis.

For the 12.1 mph impact speed of Vehicle 1, the four most influential deviations of measurement shown in the FDA output sheets were, in order of magnitude,

These four components, combined vectorially, contributed 5.1 mph of the eventual 5.5 mph vector sum. As all the other deviations combined contributed only the remaining 0.4 mph, evidently they were trivial by comparison.

In hindsight we can see that all these effects are due to the sensitivity of the magnitude of the approach momentum vector of Vehicle 1 to the directions of all the vectors. In the momentum vector diagram (see Figure 6), where the shared vector is the mutual momentum change during impact, each solitary vertex angle is the change in path direction of a vehicle during impact. The length of the leftmost vector (O~I) is evidently sensitive to the directions of all the other vectors, accounting for all four of the major deviations recited.

For the 48.4 mph impact speed of Vehicle 2, the four most influential contributors were, in order of magnitude,

which combine to account for 3.4 mph of the 3.9 mph total, the remaining 0.5 mph being trivial by comparison. Of these four only one, the opposing vehicle's approach direction, appeared in the previous list. The other three are all magnifiers of the size of the vector diagram for one vehicle or the other.

Had the field investigators known that these 8 parameters would be found to be the significant few among the trivial many in the reconstruction, during initial site examination extra attention could have been given to those 8 measurements, and the overall reliability of the reconstruction could have been improved. With on-site skid tests of replicates the liberal allowances of +0.14 for the tire-road friction of both vehicles might well have been halved, as even more readily might the +10° and +130 pounds of uncertainty for Vehicle 1's direction and weight at impact; together halving the span of all the uncertainties of speed for Vehicle 2,

Such foreknowledge of which measurements would be most significant could result from a preliminary reconstruction and FDA analysis of the same event (time permitting), or (more likely) from experience with FDA results in some prior similar reconstruction.

Conversely, as to other measurements — the trivial many — even larger uncertainties than we have considered here would not much have altered the reconstructed speeds. Field investigation could thus be performed not only more accurately but perhaps in less time by knowing from FDA where accuracy will matter most or least.

By re-sorting the FDA output sheets a similar ranking of components could have been found for any other outputs.

FDA results are not peculiar to the particular reconstruction algorithm in use; rather, they embody transfer functions which are (as just demonstrated) determined by the physics of the particular impact configuration. The reliability bandwidth of the finding is due only the deviations of the original measurements and the physics of the event, not the specific means of reconstruction of the event.

From this follows the transferability of the span of the "bell curve" (the normal distribution) in Figure 7 from the central value found with one algorithm to the central value found with another reconstruction algorithm. It would be equally transferrable to the central value found by near-duplicate simulation or, for that matter, re-enactment. This greatly broadens the useful scope of FDA analysis performed with some suitably structured reconstruction algorithm.

The prudent reconstructionist would pre-correct the central values found by any means for known errors of treatment. This could well include iterative trimming of a reconstruction by reference to a trusted simulation of the same case. Such iterative refinement would fulfill the original design philosophy behind SMAC⁷ and CRASH⁸. In the present study such correction was not needed because prior study⁵ had established the relatively negligible errors of treatment of CRASHEX with respect to SMAC for collision configurations similar to the one involved here.

When accident avoidance is of interest any reconstruction can be extrapolated back to the time when the hazard might have been recognized and avoided by a participant. We will consider such back-extrapolation first without and then with Finite Difference Analysis.

In the exemplar case, as in most left-turn impacts, during the last moments of their mutual approach evidently each driver took the other by surprise. Their speeds are critical as to deciding who had the last clear chance of avoidance of impact. At initiation of a left turn a sufficiently distant oncoming vehicle traveling at ordinary speed would have presented no evident hazard — or as in the present case might have been hidden from view. But an imminently approaching vehicle presents a clear and present hazard, avoidable by stopping before blocking its path. The apparent responsibility for avoidance thus switches from one driver to the other depending on the reconstructed speed of the oncoming vehicle.

Given the reconstruction of a probable 48.4 mph speed of Vehicle 2 at impact, a prior speed in excess of the 55 mph speed limit at the site seems likely but remains to be established. While no pre-impact tire marks had been observed, its driver had reported activation of the anti-lock brakes before impact, confirming the report of the Southbound witness that she heard tire squeal, turned her head to the left about 90 degrees, and saw the impact occur. Preliminarily this suggested a witness response time of one second.

As shown in Figure 10 that assumption was combined with available driving-simulator time histories of experimental foot and brake pedal position and force and Vehicle 2's probable tire-road downhill deceleration. Considering the use of anti-skid braking the friction coefficient was taken to be more than the Dry Slide value of 0.69 according to Appendix A of Bartlett 2003², yet somewhat less than the accompanying 0.90 value for Dry Peak; hence, 0.85.

The figure shows the probable braking time history from hazard detection through pedal response to brake application, for a rise in 1 second to a steady-state value of 0.823 (downhill) and the start of tire noise, maintained long enough for the witness to respond and see the impact occur.

By integration from the probable final speed of 48.4 mph, the probable initial speed was 70.8 mph with travel distance of 187 feet from hazard recognition to impact. The speed before braking thus was 15.8 mph in excess of the 55 mph legal limit. The point of recognition was 250 - 187 = 63 feet and 0.6 seconds past the crest of the hill, which allows enough time for Driver 2 to respond.

But for the speed violation, the speed at hazard recognition would have been 55 mph. Given the same braking the time and distance for a full stop would have been 3.8 seconds and 184 feet. As this is 3 feet short of the path of Vehicle 1, no impact would have occurred.

Furthermore, with Vehicle 1 moving at about 18 feet per second, given an extra seven-tenths of a second — 2.7 seconds after having been sighted (broken vertical line) — it would have fully cleared the Northbound lane. Vehicle 2 avoidance braking could have been terminated by then, when Vehicle 2 had traveled only 167 feet and had slowed to 20 mph, at 187 - 167 = 20 feet from the departing right rear of Vehicle 1. Each vehicle could then have gone its way unaffected.

As decisive as this might seem, it would be fairly questioned. The opposing counsel, but before that the presenting expert, would be concerned as to its reliability. What if something had been different? What, indeed, if everything had been different — as is statistically certain, since no measurement can be made exactly? How reliable is the finding?

This is exactly the question which FDA can answer. Every measurement made can be presumed to have been as much in error as the high or the low extreme of the range experimentally shown to be likely — not, of course, all at the same time. Except that the same variability of tire-road adhesion persists, all of these deviations can be assumed to have occurred in a random manner, and their equally likely combined effect can be found.

That is Finite Difference Analysis, and as already noted in Figures 8 and 9 its result in the case of 2 SD variation of the reconstructed impact speed of Vehicle 2 is +3.9 mph. That vertical span is shown centrally in Figure 11. Shown to the right of center is the equally likely +0.140 g span of deceleration upon brake application.

So, by integration from these final values to find speed and distance, flanking each of the mean values are now probability sidebands denoting the probable extremes, or the reliability, of the reconstruction. Note that the distance at which the hazard was visible is being inferred not from the visibility of the approach skid but from its audibility. The time then is constant while the distance is variable with the tire-road friction.

In a more fully litigated case, in due course a human factors expert would have been retained for a more confident, clinically based evaluation of the response-and-braking time of Driver 2 and the hear-turn-and-see response time of the witness, and their standard deviations. We show here the preliminary analysis which was sufficient for settlement in the exemplar case

With 1 chance in 44, as shown by the flanking fine lines in Figure 11, for the case of improbably high friction of 0.963 before impact at 52.3 mph, Vehicle 2 actually traveled

While indicative of prompt observance of Vehicle 1 by Driver 2 while sighting over the crest of the hill, this scenario cannot be ruled out. However, as will soon be seen this is not the pivotal case.

It is equally likely that, as shown by the sidebands with symbols, for the case of improbably low deceleration of 0.683 before impact at 44.5 mph, Vehicle 2 traveled

In either case impact ensued; but the question is, would it have ensued absent the speed excess?

In each instance the impact occurred at the same time, 2.0 seconds after the hazard was observed. Noting the near-constancy of the ending time of lane blockage at 0.7 seconds thereafter, if in the alternative the pre-response approach speed of Vehicle 2 had been 55 mph, then for the higher-friction case (fine lines) a full stop in

• 168 feet (solid horizontal line), the right front bumper of Vehicle 2 would have impacted at

Thus, within the reasonable bounds of the evidence the avoidance of impact cannot be guaranteed. This is the adverse “what-if” contingency for which the reconstructionist should have been prepared, and about which he or she would properly have been cross-examined.

Had it been crucial, as previously mentioned on-site skid tests of replicates of both vehicles might have been made and some extra care might have been taken in site investigation. Chances are good that this would have eliminated all apparent chance of impact even in the low-friction case.

But even with the evidence as known, Vehicle 1 would have cleared the path of Vehicle 2 (whatever its speed) at time t = 2.7 seconds (broken vertical line). By careful interpolation on Figure 11 (other broken lines), this circumstance occurs for a case which is 22% toward the mean from the low-friction extreme. This has a probability of (1-0.22)*2 SD = 1.56 SD, occurring with 1 chance in 17, for

The arriving right front corner of Vehicle 2 then would just miss the departing right rear corner of Vehicle 1, and each vehicle would go its way unaffected.

We thus find that the chances of no impact whatsoever but for the speed violation were 16 out of 17. We now know the reliability of that finding.

Even in the remaining one chance in 17, which occurs at the probable low limit of tire-road friction, only minimal impact would have occurred; and, with correspondingly minimal occupant impact.

Not only would the Vehicle 2 speed at the time of such impact have been about half as great (in the low 20's rather than the mid 40's), but due to Vehicle 1 chassis rotation about the instant center, located ahead of the front axle for such aft impact, the fraction of that speed inflicted on a centrally seated occupant would have been halved again. By reducing the speed of occupant impact in Vehicle 1 by approximately three-quarters — from 31.4 mph per Figure 4, third row pair, second column, to about 8 mph — this would have reduced the associated energy of occupant impact to one-sixteenth as much, or, by about 94%.

Thus, even in the worst case both the intervehicular impact and the occupant-vehicle impact would have been very greatly reduced. But what is much more likely is that absent the speed violation neither the vehicles nor their occupants would have been impacted at all.

On occasion the merely preliminary and verbal recital of such a significant finding by the client attorney will be sufficient for settlement. As this was so in the subject case we cannot recite from the exemplar case the much more extensive preparation necessary in more protracted litigation. Well-qualified experts would be needed for independent assessments of the human factors values and the injury reduction, while demonstrative exhibits could have included detailed simulations and animations of the event for not only the means but for the extremes as reviewed here.

This paper has illustrated the application to an exemplar case of a time-reversed collision reconstruction program incorporating an automated Finite Difference Analysis routine. This combination was used to accurately establish both a most likely case and the credible departures from that case due to uncertainties of measurement.

Absent Finite Difference Analysis only the central case could have been presented. Although that would have indicated impact avoidance but for a speed violation, analytic testimony would have been lacking as to the reliability of that finding and for disputation of a contrary finding within the range of the known evidence.

Without the presentation of reliability sidebands the susceptibility of any bare finding to inevitable but unknown errors of measurement can never be denied. Only by means of Finite Difference Analysis can the jury be spared the necessity of sheer speculation, their own and that of the testifying experts, in that regard. Only by means of Finite Difference Analysis can any forensic expert using any method or means of reconstruction offer more than a supposedly educated guess, a sheer expression of personal confidence lacking in analytic foundation, as to the magnitude of such susceptibility. Finite Difference Analysis thus is able to contribute critically in practice to the forensic utility of accident reconstruction.

As was further noted, the transfer functions which determine the consequences of small deviations of measurement depend on the physics of the event, not on its analysis. Therefore, the reliability sidebands, the FDA deviations from the mean value of a parameter of interest, established by any one means of accident reconstruction are applicable to the mean value of the same parameter found for the same event by some other means of reconstruction. Specifically, in Figure 11 the 48.4 mph mean probable speed of impact together with its +3.9 mph sidebands could be shifted up or down to suit the findings by the alternative means of reconstruction, with corresponding effects on the speeds at recognition and the distances traversed prior to impact.

This transferability applies even to time-forward re-enactment of a surveyed subject incident (the staging of a hopefully duplicate collision between replicate vehicles) or (what is far more likely) duplicative mathematical simulation, both of which as initial-state methods are resistant to analysis of statistically known deviations of measurement of the end effects of the actual collision.

Whatever the treatment of choice, then, and whatever its constraints, this approach brings to accident reconstruction in general a salutary level of sophistication. The practitioner not only can present as usual the treatment and the reconstruction which is in his or her opinion the most reliable, but now can by brief further use of any FDA-enabled time-reversed reconstruction recite also the probable extremes between which his or her opinion falls given the practical limitations of investigative accuracy. All "what-if" questions issuing from the inherent uncertainty of physical measurement are thus answered before they can be asked.

Although the possibility of Finite Difference Analysis had been foreseen and the necessary routines had been from the outset included among the programmed extensions of the CRASH program on which this paper has relied, their presence was academic until by the foresight of and under the direction of Dr Raymond M Brach of the University of Notre Dame, and with the cooperation of the organizers of and the many host organizations of and attendees at the WREX-2000 Conference at Texas A&M, the necessary clinical studies of measurement reliability were performed and in due course summarized and published. Without these pivotal contributions the present paper could not have been offered.

1. 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 2002-01-0546

2. Bartlett, W.D., and Fonda, A.G., Evaluating Uncertainty in Accident Reconstruction with Finite Differences, SAE 2003-01-0469

5. Fonda, A. G., Partially-Braked Impact and Trajectory Benchmarks, and Their Application to CRASH3 and CRASHEX, SAE 2000-01-1315

6. Fonda, A. G., The Effects of Measurement Uncertainty On the Reconstruction of Various Vehicular Collisions, SAE 2004-01-418

7. Jones, I. S., & Baum, A. S., Research Input for Computer Simulation of Automobile Collisions, Volume IV. Staged Collision Reconstructions, Dec 1978. DOT HS 805 040.

8. Marquard, E., Progress in the Calculations of Vehicle Collisions, ATZ 68/3, pp 74-80, 1966

9. McHenry, R R, A Comparison of Results Obtained with Different Analytical Techniques for Reconstruction of Highway Accidents, SAE 750983, Oct 1975

10. Smith, R.A., and Noga, J.T., Accuracy and Sensitivity of CRASH, DOT HS 806-152, March 1982.

Mr. Fonda can be reached at support@crashex.com and at 649 S Henderson Rd (C307), King of Prussia, Pa. 19406. The accident reconstruction and finite difference analysis program CRASHEX is accessible at www.crashex.com in a form suitable for abbreviated usage in time-share style at a minimal fee with simple pass-through of per-case expense to the end user.

Given the assumption that two-body momentum is conserved, from the planar directions of approach of vehicles to and directions and speeds of departure from oblique collision it is a simple matter of vector summation (as in Figure 6) to find the speeds of two vehicles at impact. In case of post-impact spin, however, the post-impact speeds cannot be found straightforwardly because the tire-to-ground forces then vary cyclically. Recognizing that this is a problem in accident reconstruction in general, in a 1966 publication⁸ Marquard developed a parameterized treatment of pre-calculated alternate periods of predominantly angular and predominantly translational deceleration.

Thereafter Raymond McHenry and others at Cornell Aeronautical Laboratory (later CALSPAN) developed⁹ a unique algorithm based on Marquard’s paper. This became NHTSA’s “CRASH” program, an acronym for “Cornell Reconstruction of Accident Speeds on the Highway.” In instances of axial impact, when the two momentum equations coalesce, a second equation in the two unknowns was found by means of energy.

As this met NHTSA’s need for consistent if modestly inexact treatment of large numbers of collisions, further federal development of the algorithm was terminated in 1982. That left the algorithm in the public domain and available for refinement by others. Accordingly the present author developed further refinements of the algorithm for greater convenience and accuracy in single-event accident reconstruction, calling the result CRASHEX (Computerized Reconstruction of Approach Speeds on the Highway, EXtended), and published a series of papers based thereon.

Over the years, while the option of reversion to a CRASH solution using the same set of inputs was maintained the various refinements and extensions have included improved graphics, issuance rather than suppression of the redundancies of the treatment, issuance of the yaw rate of each vehicle prior to spin and the coefficient of restitution of the impact, and various refinements of the energy treatment.

Rather than neglect force applied to the vehicle by the tires during impact, the production of tire forces during impact has since 1995 been modeled in CRASHEX using the nonlinear, friction-limited, brake-sensitive tire model of SMAC, applied for the quarter-period of the harmonic oscillation of the two effective impacting masses when jointly compressing the two effective structural springs.

Although Finite Difference Analysis, under the name of Deviation Analysis, had been incorporated in CRASHEX almost from its inception, it became far more useful in 2002 with the publication of a paper by Bartlett, Brach, et al¹ which reported experimental studies performed at the WREX-2000 Conference. Attendees at that conference, all experienced field investigators, were assigned tasks of measurement typical of those of field investigation. The standard deviations found for these juried measurements were reported more concisely by Bartlett and Fonda in 2003³, together with supplemental data obtained from the literature. For convenience this data, essential for Finite Difference Analysis in accident reconstruction by any means, is provided again as Appendix B.

Finite Difference Analysis consists of perturbation of the inputs to some treatment one at a time, each assigned as if it were a single uncertainty of known (e.g. 2 SD) probability. The result,

reduces for small perturbations to Δr = Δx * ∂r/∂x. (Monte Carlo analysis depends on the same relationships, with massive application of a multitude of statistically chosen perturbations to substantially the same effect.)

Then, for each reconstruction result (r) of interest, all the individual changes for each measurement of interest (x) are statistically combined by summation of their squares, giving

The basis in statistics is that the variance is the square of the deviation and the variance of the sum is the sum of the variances. This is equivalently a series of vector summations with each vector added along the normal to the preceding sum, which when taken in order of magnitude forms a vector spiral, Figure A1. Obviously the smaller vectors may be many yet their effect will be small. This allows strategic isolation of “the significant few” from “the trivial many” (see www.juran.com) measurement uncertainties.

Figure A1 also shows the difference between the two summations for two involved vehicles, and the improvement which can be made by successive improvement of field measurement accuracy from Casual (black) to Normal (gray) (applicable in the present exemplar case) to Meticulous (white) — that is, reduction of uncertainty from High to Medium to Low in Appendix B.

In the absence of such treatment of such information, under no circumstances can a reasonably confident and statistically substantiated opinion be rendered regarding either the likely bounds of the values reconstructed or the magnitude and consequent ranking of the various contributions to those bounds.

From Bartlett and Fonda, “Evaluating Uncertainty in Accident Reconstruction with Finite Differences,” SAE 2003-01-0469², Appendix A, using data from Bartlett, Brach, et al, “Quantifying The Uncertainty in Various Measurement Tasks Common to Accident Reconstruction,” SAE 2002-01-0546¹ and Smith and Noga, “Accuracy and Sensitivity of CRASH,” DOT HS 806-152, March 1982¹⁰. Some representative findings from both prior papers are summarized in the 2003-01-0469 Appendix; the summary table follows.

	Values for one standard deviation (1 SD)
Measurement	Low Uncertainty	Medium Uncertainty	High Uncertainty	Units
Range¹	0	0.07	0.19	%
X (from Y axis²)	0	0.17	1.4	% of Y
Y (from X axis²)	0	0.17	1.4	% of X
Chord Rise³	0.04 (0.01)	0.14 (0.04)	0.7 (0.2)	foot (meter)
Angle⁴	0	0.6	3	degree
Weight⁵	10 (44)	65 (286)	400 (1778)	pound (newton)
Mass Dispersion	0	0.02	0.2	unitless
Wheelbase	0	1 (2.5)	3 (7.5)	inch (cm)
Tire-Road dry-slide μ⁶	0.03	0.07	0.12	g
Crush Depth⁷	0.5 inch (1.2 cm)	20%	40%
Crush Width⁷	0.5 inch (1.2 cm)	6%	16%
Crush Direction⁸	0	10	20	degree
Crush Properties⁹	10	20	40	%
Skidmark Measurement	1 (0.3)	2 (0.6)	5 (1.5)	foot (meter)
Typical Values for Uncertainty Measurements 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 Bests of the same or replicate vehicle; ‘Medium’ if based on vehicle size; ‘High' denotes a casual estimate.