Mathematics of crime scene investigation

Summary: Crime scene investigation uses sophisticated mathematical models to determine what events took place at a crime scene, based on the available physical evidence.

Crime scene investigation (CSI) is the rigorous preservation and documentation of physical evidence at a specific location related to a criminal event. Investigators meticulously collect and measure crime-related evidence for scientific and mathematical analysis, reconstruction, and courtroom presentation. Overall, crime scene investigation and reconstruction involve the application of basic mathematical formulas and equations, as well as physics, geometry, and analytical thinking. Applied mathematical procedures based on well-collected data produce accurate results that generate reliable evidence for presentation in a criminal trial. Analysis of bullets, blood patterns, fingerprints, vehicle skid marks, chemical traces, and other data yield quantitative results that are invaluable in finding, arresting, and convicting suspects.

According to mathematician Chris Budd, “Many of the mathematical techniques used by forensic scientists are similar to those used in medical imaging for brain tumors, oil prospecting and remote sensing by satellites.…It is remarkable how often ideas which might be thought of as pure mathematics often find very real and important applications.”

Mathematical Modeling of Projectiles

An automated ballistic identification system (ABIS) is a computer system designed to capture, store, and compare digital images of bullets and cartridge casings. A scanner captures images of bullets and cartridges so that a mathematical algorithm can extract their unique shapes, marks, and striation patterns (signatures), which are compared to a vast database of stored images. Both wavelets and statistical correlation techniques play a role in these analyses. Forensic ballistics involves the study of a projectile in motion, from the time of shooting to the time of impact with the target. Mathematics is used to analyze and describe a projectile’s path though both the air and any obstructions, such as a body, as well as the mechanical characteristics of the weapon that fired the projectile.

Unobstructed projectile motion through air is typically parabolic, but a bullet may trace a complex path if deflected or stopped by an object, which requires more advanced mathematics, such as fractional differential equations, to describe. Blood droplets are another sort of projectile found at crime scenes, and the blood spray patterns are analyzed with geometric and trigonometric methods to determine the point of origin and other crucial characteristics. Along with ballistics and blood spatter, precise wound descriptions, which are closely related to fields like surveying and topography, can be mathematically modeled to suggest the type of weapon or bullet most likely to have made the wound.

Locations and Relationships

The locations and relative relationships among the various pieces of evidence are also important in making sense of a crime scene. Precise measurements allow investigators to place every item of evidence in its original location with some degree of certainty. These may be represented in a two-dimensional diagram, or in a computer reconstruction that uses two- or three-dimensional representation. Newer laser technology can record distance very quickly and precisely, as well as compute height using trigonometry. Mathematical computer algorithms can then combine data from multiple measures of a single object, taken from many angles, to produce three-dimensional models with minimal error. Another example of imaging used to solve a famous ancient “murder mystery” is the case of King Tutankhamen. X-rays from the 1960s, which could only provide two-dimensional images, were inconclusive. However, using CAT scans, which can mathematically construct three-dimensional images, scientists concluded that the king probably died from an infection in a broken leg.

Probability

Though the phrase “innocent until proven guilty” is often heard in connection with criminal investigations, in many cases the available evidence allows only a statement of what probably happened versus absolute certainty. Homicide investigators must logically infer or deduce what transpired at the crime scene by using evidence to reconstruct events and by matching a crime scene’s characteristics to other examples. They may hypothesize a timeline or scenario and then apply scientific analysis to verify or refute the sequence of events to a high degree of probability.

This process requires critical scientific thinking and logical analysis. Investigators may use controlled experimentation, such as firing several bullets from the same weapon to look for variations in the pattern. Increasingly, they can also use computerized reconstructions of crime scenes and data to manipulate critical variables and conduct multiple “what if” simulations to eliminate unlikely scenarios and narrow the set of possible suspects and causes. Probability also comes into play in DNA analysis, where results are given the form of probability matches, and scientific tests such as gunshot residue, which are not 100% accurate and may occasionally result in a false outcome.

Conclusion

In summary, crime scene investigation requires investigators to apply many scientific and mathematical analyses to determine an accurate sequence of events and reconstruct what actually happened at a crime scene. Physical evidence helps investigators focus on a suspect and the manner in which the crime was committed. Successful crime scene investigations, reconstructions, and interpretations are the result of sound hypothesis formulation, experimentation, laboratory examination, and logical analysis. Applied mathematics provides the logic and rational simulations for scientific reasoning and assumptions.

Bibliography

Adam, C. Essential Mathematics and Statistics for Forensic Science. Hoboken, NJ: Wiley, 2010.

Budd, C. “Crime Fighting Maths.” +Plus Magazine 37 (2005).

Devlin, Keith, and Gary Lorden. The Numbers Behind NUMB3RS. New York: Penguin, 2007.