Noise robust algorithm for measuring gravitational tool-face

A system, comprising:

• a drilling rig;
• a pipe string attached to the drilling rig;
• a bottom hole assembly attached to the pipe string, wherein the bottom hole assembly comprises at least one sensor;
• a drill bit, wherein the at least one sensor measures a revolutions-per-minute (RPM) of the drill bit;
• and
• a computing subsystem connected to the at least one sensor and configured to:
• determine a revolutions-per-minute (RPM) measurement based on a zero crossing of sinusoidal magnetometer data;
• determine a gyroscope bias from gyroscope data using the determined RPM measurement;
• determine bias compensated gyroscope data by subtracting the determined gyroscope bias from the gyroscope data;
• determine a gyro-based angular displacement measurement using the bias compensated gyroscope data;
• compute a first tool-face measurement using the determined RPM measurement;
• compute a second tool-face measurement using accelerometer data;
• determine a selected tool-face measurement based on a selection between the first tool-face measurement and the second tool-face measurement using a predetermined selection criterion;
• compute a weighted sum of the selected tool-face measurement and the gyro-based tool-face measurement;
• and
• determine a gravitational tool-face measurement from the computed weighted sum.