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  GPS and SONAR Sensor Location

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Author Topic:   GPS and SONAR Sensor Location
jimh posted 05-19-2012 11:00 AM ET (US)   Profile for jimh   Send Email to jimh  
In this brief analysis we look at the accuracy of depth soundings which are located by a GPS receiver and examine the possible errors. If we want to know the depth of water at a particular position, we first must locate the position.

A typical marine GPS receiver with a precision fix augmentation from WAAS can have an accuracy of about 10-feet. If a GPS receiver says we are at a particular location, we can be somewhere within a circular area of radius 10-feet of that location with good probability. We can express the uncertainty in terms of the size of the area. With an error of 10-feet we have an area of 62.8-square-feet of uncertainty, that is, a circle of radius 10-feet.

Now we take a depth sounding. If we take the sounding precisely at the location of the GPS sensor, we do not add any further error, but, as often happens, if the sounding is taken at another position, we introduce further error. Suppose the sounding is taken by a SONAR that is located 10-feet away from the GPS sensor. We do not record the precise heading when the sounding is taken, so we don't know where the two sensor were when the depth and position were recorded. If you imagine the SONAR sensor being in a fixed location, the boat could rotate around the SONAR sensor. This puts the GPS sensor at any location on a circle of radius 10-feet. Now we add the 10-feet error of the GPS measurement. We now have an uncertainty of position that is a circle or 20-feet radius, or an area of 125.6-square-feet.

The depth sounding from SONAR also involves some imprecision. The SONAR beam is a cone, and the bottom echo that is measured could come from a reflection that occurs anywhere in the area of the cone. Let us assume the cone angle is 30-degrees and the water depth is about 38-feet. The SONAR cone at that depth will have a radius of 10-feet. That is, the echo we see as indicating the depth could be from a reflector that is as far as 10-feet away from directly below the SONAR sensor. This adds another 10-feet to our uncertainty circle. We now have a combined uncertainty of position and depth that suggests the sounding and position could be located in a circle with a radius of 30-feet. This is an area of 188.5-square feet.

Now we compare the uncertainty between the SONAR and GPS sensor locations. If they are co-located, we have no offset. This reduces the error to just the 10-feet of GPS position error and the 10-feet of SONAR sounding error: a 20-foot radius. If the sensors are not co-located and are 10-feet apart, we have a larger uncertainty: a 30-foot radius. The increase in the uncertainty is proportional to the two areas, 188.5 and 125.6. These are in a ratio of 1.5:1.

We can then say that the co-location of the SONAR sensor with the GPS sensor increases the accuracy by 50-percent in our example using 10-foot error of position, 10-foot offset between sensors. (The error of the SONAR cone does not affect the difference in accuracy due to offset locations.)

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