### Probe scanning equations

Rhiannon Ad Astra recently asked me for anything I knew about the mathematics of a target being probe scanned. Here are what remains of my research notes. (This used to exist as a forum thread, but the forums get autopurged.) A graph that used to accompany this was lost.

Over on the "Exploration and Wormholes" board (which isn't public), I've been soliciting the help of Unistas in trying to figure out the mathematics of the new probe equations that went into effect in either Incarna or Crucible (I forget which). The research isn't done, yet, but I decided that it might be a good idea to publish results-to-date for the general EVE Online public. Once this has been figured out, players can better evaluate (on a theoretical basis) which probe formations are better than other probe formations. For example, is it always better (for the purposes of signal strength) to have 8 probes in a cube formation rather than 7 probes in a north-south-east-west-up-down-center formation?

That and some of my YouTube subscribers might have been silently wondering why I haven't been making videos, recently. Well, this is what I'm working on, right now.

Step One

For one probe seeing one target and in the limit of low signal strengths (<25%), the new probe signal strength equation seems to be:

Signal strength from single probe = Constant * F * Probe strength * Target Signature Radius / (Target Sensor Strength * Probe Scan Radius)

• "Signal strength from single probe" is here expressed as a decimal. 1 in this equation is a 100% strength, and 0.0435 would be a 4.35% signal strength.
• Constant = 0.00128 AU / meter (If you want to insist upon expressing the signal strength as a percent, then this is instead 0.128 AU % / meter and you should treat "%" as a measurement unit.)
• F is a function of the distance divided by the probe scan radius. F is 1 if the target is right on top of the probe, and 0.4 if the target is exactly at the edge of the scan radius.
• Probe strength is the "Base sensor strength" of the probes as shown to the probe user in SHOW INFO on the probes actually in his launcher. Note that for display purposes, it will cut off decimals, always rounding down. This does take rig bonuses, ship bonuses, Astrometric Rangefinding, and (I presume) implants into account.
• Target Signature Radius is the Signature Radius of whatever the probe user is trying to scan down.
• Target Sensor Strength is the Sensor Strength of whatever the probe user is trying to scan down, but is not allowed to be more than 90% of the signature radius.
• Probe Scan Radius is the current Scan Radius setting of the probe. For a Combat Scanner Probe, this could be anywhere from 64 AU to 0.5 AU.

A lot of this is going to look familiar from my last thread on this subject two years ago (that link will only work for members and alumni), but there are changes. First, the constant has changed from 0.0025 AU/meter to 0.00128 AU/meter. Second, the target's sensor strength in points is capped at 90% of the signature radius in meters. So, if the signature radius is 100 meters, then all sensor strengths above 90 points are treated as equal to 90 points, instead. (This is the limitation that makes any ship probable, now, but you need the full Low-Grade Virtue implant set and everything else, too.) Third, this equation seems to hold when the single-probe signal strength is below 25%. If it's more than 25%, then the result deviates downwards from what this equation predicts; I will describe that next. Fourth, the single-probe case isn't capped at a signal strength of 50%.

Step Two

Let's call the result of the above equation the "Old Signal". This Old Signal is what you get for one probe seeing one target and the result being 25% or less, and this can be predicted given the probe strength, distance to target, scan radius, signature radius of the target, and sensor strength of the target.

Above 25%, however, the actual signal strength deviates from this prediction, resulting in what I'll call "New Signal". I started with the assumption that, if Old Signal is more than 25%, then the game takes Old Signal and puts it through a mathematical function to get New Signal. So, I decided to plot y=Log(New Signal,2) versus x=Log(Old Signal,2), and the curve seems to fit y = -4 / (x+4) reasonably well given the slightly noisy data.

Log(New Signal,2) = -4/(Log(Old Signal,2)+4)
New Signal = 2 ^ (-4/(Log(Old Signal,2)+4))

For clarification, Log(A,B) is the B-base logarithm of A. 2^6 = 64 therefore Log(64,2) = 6

After adding a column to my spreadsheet to compare this new theory to the experimental data, I did find that predictions for "New Signal" were off by anywhere from +0.34 to -1.65 percentage points on the signal strength, but the deviations from this refined theory don't follow a smooth pattern. I don't know if this is due to noisy data, or if the equation is close but not quite right.

Step Three

If two probes can see a target, then the total signal strength is not the sum of the signal strengths of each probe (unless the two probes are directly opposite from each other). Consider the case where each probe, by itself, would give an equal signal strength on the target. (An unusual situation, to be sure, but this simplifies the explanation.) Probe 1 - Target - Probe 2 forms an angle, and as this angle increases, the overall signal strength increases. When the two probes are seeing the target from exactly the same direction (a zero-degree angle), one probe eclipses the other. When the two probes are exactly on opposite sides (a 180-degree angle), then their individual results are added. For angles in-between, the amount of contribution from the second probe follows an overlapping-circles equation:

Probe Crucible Hypothesis.jpg

The general equation for circle-circle intersections may be obtained from MathWorld. If we then take into account that the center-to-center distance between the probes is dictated by the probe-target-probe angle (call it "theta"), then the fraction overlap of one circle with another is:

1 - theta/(pi * 1 radian) - sin(theta) / pi

The fraction of each circle that isn't overlapped is:

theta/(pi * 1 radian) + sin(theta) / pi

(Engineers and students thereof: Please observe appropriate unit conversions, here. The result should be dimensionless.)

For example, if two probes both see a target, and are at right angles to the target, and each probe would be 10.00% signal strength by itself, then the non-overlapped fraction of the second circle is 0.818 and the two probes combined give a signal strength of:

10.00% * (1 + 0.818) = 18.18%

In practice my data has been noisy for reasons I haven't been able to determine. Some angle measurements were done by flying the probe ship around the target on-grid and launching probes, and even those signal strengths vary by amounts that don't have a pattern that I can discern. So, when you try this, it might not be exactly 18.18%, or exactly 1.818 times a solitary probe.

Future work

This is still a work in progress. For two probes with unequal solitary signal strengths, an overlapping circles equation still works, but the ratio of the radii to use for the overlapping calculation is not merely equal to the ratio of the solitary signal strengths; I've only studied this briefly. Also, I haven't figured out exactly how the calculations work for three or more probes seeing a target. I can calculate an overlap factor for each pair of probes, but figuring how out that works into the overlap factor for three or more probes is still puzzling me.