The first question often asked about Trolling Angles is “is it accurate?”.
I have very little doubt about the theory of the physics of the method and the math used in the Trolling Angles app.
Years of software development experience tells me not to deny the possibility of errors in the coding implementation. However, we take a professional approach to design, implementation, and testing, to minimize errors, and are committed to fixing bugs as they are discovered. I’m confident that the data analysis algorithm in Trolling Angles is reliable.
However, as we software people say, “garbage in, garbage out”. The biggest unavoidable vulnerability to accuracy in Trolling Angles is probably in the process of measuring line angles. We measure the angle of a line segment at one end, and we project this changing angle out to meet another angle at the end of a long line. A small error in the angle results in a larger error at the end of the line. This is much like aiming a rifle, which can be done accurately only with technique and care. Of course, there are no sights or scopes on a fishing line, and we can’t normally sight down the line in normal trolling, so we can’t expect “minute of angle” accuracy, or even 1 inch at 100 feet. But, can we achieve accuracy of 1 foot of depth with 100 feet of line?
Let’s do some “back of the envelope” math and statistics. Assume that the uncertainty follows a Normal Distribution. Also, to make the math easier, pretend that the fishing line is straight in each interval, the line is horizontal, and that we are using equal line lengths in each measurement. We will still be in the ballpark.
First, test out the uncertainty when taking a single measurement. This will depend on conditions and your own technique. Take a series of measurements of the same (rig, length, and speed) line angle and find the standard deviation of these angles.
A spreadsheet will find the standard deviation easily with the STDDEV function. To calculate a standard deviation manually: Find the mean (average) of the angles. For each measurement, find the square of its difference from the mean, sum these, divide by the number of measurements, and take the square root. The result is the standard deviation of the uncertainty of a single measurement.
I did this myself with separate samples at three different line lengths of a dozen measurements each, using the smartphone sensor. I was being careful but not obsessive and was pleased that the worst measurement was only about 2 degrees from the mean, and standard deviations were 0.97, 0.50, and 0.76.
Let’s say the standard deviation of the line angle measurements is about one degree.
The sine of one degree is 0.0175, so if the measurement of a horizontal line is off by 1 degree, the difference at the end of a 10-foot line interval is 0.175 foot, and so there is 1 standard deviation, or a 68% chance of being less than 0.175 feet from the true value.
With Trolling Angles, we take multiple measurements and piece them together. Some measurements will be smaller than actual, others will be larger, so we don’t just add the uncertainties when we take multiple measurements. For equal lengths, the formula is u times square root of n, where u is the uncertainty and n is the number of measurements.
If we take n=10 measurements with equal line lengths of 10 feet, the total expected uncertainty is 0.175 feet times the square root of 10, or 0.553 feet.
About half a foot, at the end of 100 feet of line, with 68% certainty.
If we take n=5 measurements with equal line lengths of 20 feet, the uncertainty for each measurement is 0.349, and the total uncertainty is 0.780 feet.
Less than a foot error with only 5 measurements.
Doing the same thing with sloppy measurements, with an uncertainty standard deviation of 5 degrees for a single measurement, sin(5) = 0.087, u = 0.87, n=10, 0.87*sqrt(10)=2.75 feet (which the fish probably will notice).
Note that taking more measurements, closely spaced, leads to better final accuracy, for a couple of reasons.
- Using more measurements helps track the real-world changes in drag and lift coefficients as the line angles change at different parts of the line.
- Statistically, the errors at each line length tend to partially cancel each other out. This assumes that there is no systematic bias causing predominately low or high measurements.
Here is a Monte Carlo simulation to demonstrate the range of likely error using the actual Trolling Angles app. We start with a real measured calibration, one of the Demo calibrations provided with Trolling Angles Version 2 (shown in green). In each of 15 simulations, each angle is changed to add or subtract a value from a normal distribution random number generator with a standard deviation of 1.0 degrees. One could say that, given the actual but somewhat imprecise angle measurements which result in the green line, each of the simulated curves is equally likely to be the underwater profile of the actual fishing line.
