Skip to content

Latest commit

 

History

History
136 lines (109 loc) · 5.55 KB

Adjusted.md

File metadata and controls

136 lines (109 loc) · 5.55 KB
Raw

Geomagnetic Adjusted Data

Abram Claycomb <[email protected]>

Background and Motivation

The magnetic field measured at an observatory of the USGS is measured by a three-axis fluxgate sensor roughly aligned with the magnetic field. The three axes are:

  • h - Horizontal 'leading' ahead of the local magnetic declination (magnetic north) at the time of installation, so that the local magnetic vector would eventually cross the h axis
  • e - Horizontal, nominally orthogonal to h (roughly magnetic east)
  • z - Nominally vertical, downward, and nominally orthogonal to both h and e. Vertical at installation on a balancing device with the intended purpose of staying level if the pier on which it is mounted tilts under the sensor, all enclosed under a glass dome to keep air movements from convecting heat directly to the sensor from the room, or pushing the balanced system

Simultaneously, the field is measured by an Overhauser-effect scalar magnetometer (non-directional). This is called the total field:

  • F - Total field at the Overhauser pier

A third magnetometer, called a declination-inclination magnetometer (DIM) is used to manually find direction of the field for the purpose of calibrating the three-axis sensor mentioned above, and converting the coordinate system to that of a geographic north, east, and downward set of axes:

  • X - Geographically North component of the magnetic field, based on a survey of the absolute pier, and the azimuth mark, at the time of installation, and periodically on a time scale of a few years
  • Y - Geographically East component of the magnetic field, again based on the survey mentioned in X above
  • Z - Vertical component of the magnetic field, downward, based on leveling the theodolite at each absolute measurement session

The declination and inclination measured by the DIM are:

  • D - declination
  • I - inclination

The measurements with the DIM are called absolutes and measured on a timescale on the order of 1 week. Four sets of four measurements each are recorded on four orientations of the DIM sensor and these measurements are averaged, to account for errors in the sensor and its alignment to the optical axis of the theodolite to which it is mounted.

The real-time measurements (to the nearest second) of h, e, z and F are used to compute what are known as baselines, or the differences in the pseudo-vector cylindrical coordinate representation. The equations relating these quantities, with some definitions, are found below:

  • F_pier_correction - measured on the order of once or twice per year, by a second Overhauser recording for a few hours at the absolute pier location (in place of the absolute DIM theodolite)
  • F_corrected = F + F_pier_correction
  • X = F_corrected*cos(I)*cos(D)
  • Y = F_corrected*cos(I)*sin(D)
  • Z = F_corrected*sin(I)
  • H_absolute = sqrt(X**2 + Y**2) = F_corrected*cos(I)
  • D_absolute = arctan2(Y,X) = D
  • Z_absolute = F_corrected*sin(I)
  • H_ordinate = sqrt(h**2 + e**2) - were the angles small, this may have been historically approximated as h
  • D_ordinate = arctan2(e,h) - were the angles small, this may have been historically approximated as e/h
  • Z_ordinate = z
  • H_baseline = H_absolute - H_ordinate
  • D_baseline = D_absolute - D_ordinate
  • Z_baseline = Z_absolute - Z_ordinate

Calibration to Reduce Errors and Transform Coordinates

The purpose of making the manual absolute measurements is to account for errors in the vector magnetometer, and transform the recorded h, e, z data into X, Y, Z coordinates. There are several types of errors:

  • non-orthogonal sensor error, which can be corrected by a transformation matrix as a linear operator
  • scale error: measurement by one unit in one sensor not being equal to one unit of the field, which can again be corrected by a different kind of transformation matrix; for fluxgate (and DIM) magnetic sensors, this is known to be temperature-dependent
  • offset error: measurement with no field applied gives a non-zero sensor output; can be corrected by adding a vector, which can be re-cast as a matrix transformation and linear operator by an affine transformation; for fluxgate (and DIM) magnetic sensors, this is known to be temperature-dependent
  • local magnetic disturbances - usually minimized by site selection and disciplined operations during maintenance and measurement.

The combined effect for the above mentioned errors, as well as a final rotation to transform coordinates to X,Y,Z can be found using a least-squares algorithm and baseline calculator data.
This is phase one of the Adjusted Data project.

Example

Usage for this algorithm is shown in this Adjusted Usage example.

Example calculations of affine transformation matrices for USGS observatories are shown in this Adjusted Example IPython notebook.

There's a Generation Tool notebook, for generating the tranformation matrices in an automated fashion, with tools for adjusting manually and previewing the effect on delta F, etc.

References