posted 11-23-2014 05:03 PM ET (US)

Let us say we are boating and looking for a particular target on a SONAR. (The target is stationary and is on the sea bottom.) The water depth is 100-feet. We are using a SONAR transducer with a cone angle of 30-degrees of flat response.While boating we observe a target on SONAR. We immediately mark our position on our chart plotter as determined by our GNSS receiver. Sometime later we try to return to the position marked and can't find the target. What caused the errors?

The first error in this process occurs in the SONAR. The target reflection is occurring from some point on the bottom that is in the cone of the SONAR signal. We assumed a cone angle of 30-degrees. If the depth is 100-feet, then the radius of the circular area on the bottom that intersects our SONAR cone is defined by

circle radius = tan(coneAngle/2) x depth

--or--

circle radius = tan(15) x 100 = 26.8-feet

The actual target causing the reflection we see on the SONAR screen could be as far as 26.8-feet away from the SONAR transducer's position directly overhead on the sea bottom.

The second error in the process occurs when the chart plotter marks the position. The position recorded is the position of the GNSS sensor, not the SONAR transducer. On a small boat it would be common that the SONAR transducer was mounted on the transom while the GNSS antenna was mounted at the helm. These two locations could be 10-feet apart. This is the GNSS antenna offset error

The third error occurs when the GNSS receiver deduces the position of its antenna. The deduced position is subject to many influences for accuracy. A nominal accuracy for an autonomous GPS without any assistance is plus or minus 5-meters in the horizontal plane. That is about 15-feet of error. This is the GNSS position error.

If we are particularly unlucky, all of these errors will lay on a straight line, and thus they will be additive. This gives us a maximum error for the waypoint position that is supposed to mark the SONAR target of

26.8-feet + 10-feet + 15-feet = or about 50-feet of error.

To get back over this target at some time in the future, we navigate to the recorded position. Our navigation is again subject to the error of the GPS position, or 15-feet. We can position the GPS sensor at the same position as the recorded waypoint, but we can only count on being within a circle of 15-foot radius. If we are again unlucky, the GPS position error will be in the opposite direction that it was when the waypoint was recorded. So not only are we 15-feet away from where the target is located by the error in the initial position, we are another 15-feet away due to the second GPS position error made on the return. Ouch, now we are 30-feet away from the real position of the target.

But things are not done getting worse. If the orientation of the boat is also very unlucky for us, then the GPS tells us we are back to the original position plus 15-feet of error, the boat orientation will be aligned in the worst way, putting our SONAR transducer another 10-feet away from where it should be to find the target. Total error in re-locating to the waypoint is 40-feet.

Now we have to consider the SONAR cone problem, but this time it actually helps us. Even though we might be out of position by 40-feet, we know that the SONAR cone is going to cover a circular area on the bottom of 25-foot radius. This means we have a good chance of seeing the target, even though we have positioned the SONAR transducer 40-feet away from where we though should be.

In actual use, the errors do not tend to align for the worse outcome. The errors are likely to be uncorrelated. We repeat the analysis with the consideration that the errors will add as the root sum of squares:

Error = [ ( (SONARconeRadius)^2 + (GNSS antenna offset)^2 + (GNSS position error)^2 ]^0.5

Error = (26.8^2 + 15^2 10^2)^0.5

Error = 32-feet

Our initial position is now only likely to be 32-feet in error at most from where the actual target is located on the sea bottom.

When we return to our stored waypoint, our GNSS position error and GNSS sensor offset also will be uncorrelated. The likely error will be

Error = [ (GNSS antenna offset)^2 + (GNSS position error)^2 ]^0.5

Error = (15^2 + 10^2)^0.5

Error = 18-feet

This makes the chances of finding the original target better. We should be back to within 18-feet of the actual boat position that found the first target, and we will be searching with a SONAR that paints the bottom with a 26.8-foot radius. We should see something from that target on our display.