Systems and methods for determining tool center, borehole boundary, and/or mud parameter

A method comprising:

• placing a downhole tool into a borehole in a geological formation;
• obtaining a set of measurements using the downhole tool, wherein the set of measurements comprises, for each of a plurality of measurement times:
• a plurality of measurement representative of a distance between a sensor and a borehole boundary;
• and
• a measurement representative of an azimuth of the downhole tool;
• and
• iteratively estimating, using a processor, a set of borehole boundary distances from a point in the borehole and a tool center trajectory in relation to the point in the borehole, at least partly by:
• estimating at an iteration one of the set of the borehole boundary distances and tool center trajectory based on the set of measurements assuming an estimation relating to the other of the set of borehole boundary distances and tool center trajectory at a previous iteration is true;
• estimating at the iteration the other of the set of borehole boundary distances and tool center trajectory based on the set of measurements assuming the estimation relating to the one of the set of borehole boundary distances and tool center trajectory at the iteration is true;
• and
• repeating at least the estimating of the set of borehole boundary distances and the estimating of the tool center trajectory at a further iteration at least until convergence.

A method comprising:

• placing a downhole tool into a borehole in a geological formation;
• obtaining a set of measurements using the downhole tool, wherein the set of measurements comprises, for each of a plurality of measurement times:
• a measurement representative of a distance between a sensor and a borehole boundary;
• a measurement representative of an acceleration of the downhole tool;
• and
• a measurement representative of an azimuth of the downhole tool;
• iteratively estimating, using a processor, a set of borehole boundary distances from a point in the borehole and a tool center trajectory in relation to the point in the borehole, at least partly by:
• estimating at an iteration one of the set of the borehole boundary distances and tool center trajectory based on the set of measurements assuming an estimation relating to the other of the set of borehole boundary distances and tool center trajectory at a previous iteration is true;
• estimating at the iteration the other of the set of borehole boundary distances and tool center trajectory based on the set of measurements assuming the estimation relating to the one of the set of borehole boundary distances and tool center trajectory at the iteration is true;
• and
• repeating at least the estimating of the set of borehole boundary distances and the estimating of the tool center trajectory at a further iteration at least until convergence.

A system comprising:

• at least a sensor disposed in a downhole tool configured obtain a set of measurements representative of a distance between the sensor and a borehole boundary;
• at least a sensor disposed in the downhole tool configured to obtain a set of measurements representative of an azimuth of the tool;
• and
• a processor configured to estimate, a set of borehole boundary distances from a point in the borehole and a tool center trajectory in relation to the point in the borehole, at least partly by iteratively:
• estimating at an iteration one of the set of the borehole boundary distances and tool center trajectory based on the set of measurements assuming an estimation relating to the other of the set of borehole boundary distances and tool center trajectory at a previous iteration is true;
• estimating at the iteration the other of the set of borehole boundary distances and tool center trajectory based on the set of measurements assuming the estimation relating to the one of the set of borehole boundary distances and tool center trajectory at the iteration is true;
• and
• repeating at least the estimating of the set of borehole boundary distances and the estimating of the tool center trajectory at a further iteration at least until convergence.