An apparatus for determining the orientation of a local coordinate system (208, 209) of the apparatus relative to a geographical coordinate system, the apparatus being stationary on or beneath the Earth's surface, the apparatus comprising: a body providing a reference for the local coordinate system (208, 209); a rotatable sensor package having a rotatable sensor package coordinate system and comprising at least three angular rate sensors that are mutually non-orthogonal and misaligned with the axes of the rotatable sensor package coordinate system, the rotatable sensor package further having a rotational axis about which the rotatable sensor package rotates with respect to the local coordinate system (208, 209), the rotational axis being fixed in the local coordinate system (208, 209); more than one accelerometer, whereby values measured by the accelerometers are used to compute a gravity vector for Earth in the local coordinate system (208, 209); a rotational drive system for rotating the rotatable sensor package both to precise positions relative to the local coordinate system (208, 209) and at a precise rate relative to the local coordinate system (208, 209); and an angular position indicator to measure the angular position of the rotatable sensor package, whereby values measured by the rotatable sensor package are used to compute an unknown direction of the Earth's rotation vector in the local coordinate system (208, 209) based on an overdetermined system of equations in which the mutually non-orthogonal angular rate sensors have an axis of sensitivity that is represented by a unit vector having non-zero components in directions that are parallel to the rotational axis and orthogonal to the rotational axis, and whereby the gravity and rotation vectors for Earth in the local coordinate system (208, 209) are used to compute a rotation quaternion required to rotate the local coordinate system (208, 209) into the geographical coordinate system. a body providing a reference for the local coordinate system (208, 209); a rotatable sensor package having a rotatable sensor package coordinate system and comprising at least three angular rate sensors that are mutually non-orthogonal and misaligned with the axes of the rotatable sensor package coordinate system, the rotatable sensor package further having a rotational axis about which the rotatable sensor package rotates with respect to the local coordinate system (208, 209), the rotational axis being fixed in the local coordinate system (208, 209); more than one accelerometer, whereby values measured by the accelerometers are used to compute a gravity vector for Earth in the local coordinate system (208, 209); a rotational drive system for rotating the rotatable sensor package both to precise positions relative to the local coordinate system (208, 209) and at a precise rate relative to the local coordinate system (208, 209); and an angular position indicator to measure the angular position of the rotatable sensor package, whereby values measured by the rotatable sensor package are used to compute an unknown direction of the Earth's rotation vector in the local coordinate system (208, 209) based on an overdetermined system of equations in which the mutually non-orthogonal angular rate sensors have an axis of sensitivity that is represented by a unit vector having non-zero components in directions that are parallel to the rotational axis and orthogonal to the rotational axis, and whereby the gravity and rotation vectors for Earth in the local coordinate system (208, 209) are used to compute a rotation quaternion required to rotate the local coordinate system (208, 209) into the geographical coordinate system.