Of our software packages, the only one that can produce a depth estimate from magnetic or gradient profile data is MagPick (https://www.geometrics.com/software/). This program is free to download and use, but since it is free there are definitely some limitations to its functionality and list of features. What most geophysical firms use is Oasis Montaj by Seequent which has many more depth of burial calculation techniques with very sophisticated calculations behind them. With our system we typically estimate that the error in burial is about 20% of the total distance between the sensor and the target. So if the sensor is 1m above the ground and the object is estimated at 9m deep (10m total) it would be +/- 2m in vertical estimation.
As for how you would know if something is shallow or deep you just need to look at the width of the magnetic profile of the feature. The wider the feature the deeper it is. This is explained in some detail in the Applications Manual for Portable Magnetometers (see attached).