# Battle Strategy and Terrain Complexity (TERCOM)

Citing Wikipedia: “An advantage of terrain occurs when military personnel gain an advantage over an enemy utilizing, or simply in spite of, the terrain around them. The term does not exclusively apply to battles, and can be used more generally regarding entire campaigns or theaters of war. Mountains, for example, can block off certain areas, making it unnecessary to station troops within the inaccessible area. This deployment strategy can be applied with other formidable environmental features as well, such as forests and mountains. In the former instance, vegetation can provide concealment for tactical movements such as setting up an ambush. In the latter, the elevation can provide an advantage to soldiers using projectile weapons. Elevation itself is perhaps the most well-known example of terrain advantage, with gravity working to the advantage of the more elevated party.

This short blog has the intent of illustrating how with QCM technology it is possible to measure the complexity of terrain and its morphology. Such information can be useful when planning campaigns, strategies, selecting equipment for deployment or trajectories. Clearly, a complex terrain will require a different approach to a less involved one. Terrain complexity, in other words, will dictate what equipment is used and how.

Until recently, QCM technology has been used to process data, such as real-time CAN bus data, stock market data, real-time patient data in ICUs, or air traffic radar output. In  other words, scalar multi-channel data sampled with frequencies ranging from a few Hertz to thousands of Hertz.

With the development of a Multi-thread version of the QCM algorithm, we can now use Supercomputers to process arrays representing terrain. In particular we are able to measure the complexity of 2D domains in which elevation, z, is provided in each point of the domain, i.e. z=f(x, y).

The images which follow are examples of functions of the type z=f(x, y) and of the corresponding Complexity Landscapes, i.e. c=g(x, y), where c is complexity. No analysis or comments shall be provided as the images are shown for illustrative purposes only. These are in fact the first ever images of Complexity Landscapes. Their synthesis, is expensive from a computational standpoint. Some landscapes illustrated below require more than one day of computer runtime.

The following are example of Complexity Landscapes of gravity along a plane passing through the source, i.e. F=-GMm/r^2, where r=sqrt(x^2 + y^2).

Finally, the examples below are Complexity Landscapes of regions of actual terrains described via elevation, z=f(x, y).

The bottom line: More complex terrain means potentially more effort, more energy expenditure, risk, more sophisticated equipment, and a more articulate strategy. Today it is possible to measure terrain complexity (TERCOM), moving from a qualitative checklist approach to a quantitative assessment of terrain morphology-specific difficulties in the context of a given battle strategy. At the end of the day one can select any of the following strategies or routes:

• shortest
• fastest
• least complex

This last solution can be thought of as variational, brachistochrone-type of problem, which can, of course, be solved using conventional optimization tools. Until this is attempted on a real case, it is not at all clear if the least complex path lies in between the shortest and fastest routes.

With the approach we have developed it is also possible to process 3D data, representing any digital landscape or field, such as electromagnetic, in the form V=f(x, y, z). In terms of computational resources an order of magnitude more with respect to 2D domains will of course be necessary. Consider that in order to process an averaged sized area as the ones shown above, it is necessary to partition it into 500.000 to 1.000.000 overlapping domains. Using even the accelerated QCM algorithm still requires a day or more of a high-end CPU. Evidently, running on a multi-thread architecture with a few thousand cores brings down the runtime by 3 or 4 orders of magnitude, i.e. to minutes or even seconds.