# 6.5. Definitive Data Calculation Based on Most Common Orientations

Fluxgate variometers usually have 3 orthogonal fluxgate sensors. These sensors can be oriented in various ways. Most variometers are designed such that two sensors are horizontal and a third sensor is vertical. If the two horizontal sensors are oriented towards geographic north and east, respectively, then the orientation is often referred to as XYZ. If they are approximately oriented towards magnetic north (i.e. approximately along the local H component) and magnetic east, then the orientation is called HDZ. There exist other orientations, but these are less frequently used. One example is to mount two horizontal sensors approximately 45° on either side of geomagnetic north and one sensor along Z. Another example is to align one sensor approximately along the field vector, one sensor horizontal towards east, and one sensor perpendicular to the other two sensors, in which case the three fluxgates will sense magnetic field variations that approximately reflect changes in field strength F, declination D and inclination I.

There are two types of fluxgate sensors. One type is quasi-absolute and measures the full field component (up to 70000 nT) or more. As every fluxgate, it has an unknown sensor offset which typically is small (a few nT). The other type is a variation-type fluxgate with a limited measurement range (typically 6000 nT to 20000 nT). The variation-type requires bias fields to be applied along the fluxgate in order to compensate for geomagnetic field components that exceed the measurement range. The bias field can be very large (up to 70000 nT or more) and adds to the small sensor offset. Both bias field and sensor offset are not known accurately, but are assumed to be stable between absolute measurements.

Here, we first describe the calculation of definitive data based on the HDZ orientation and later for the XYZ orientation, as this is one of the most commonly used orientations. In particular, we consider a variometer-type instrument with selectable bias fields for the individual channels. The HDZ orientation was also the typical orientation for classical instruments with suspended magnets and photographic recordings. With the classical instruments, often simple, linear formulas were used to calculate absolute data from variations and baselines. For this procedure to be sufficiently accurate, the classical instruments required reorientation with changing declination (typically every few years). With the non-linear formulas given here, it is not necessary to reorient the fluxgate variometer with changing declination (as long as the magnetic field stays within the measuring range of the fluxgates, which is typically the case for decades).

The horizontal sensor directions do not exactly correspond to the magnetic field components H and D. In order not to confuse the sensor directions with the magnetic field components HDZ, we label the sensors N for horizontally north, E for horizontally east and V for vertically down. We denote \(N_{var}\), \(E_{var}\) and \(V_{var}\) the fluxgate output in nT, positive towards north, east and down, respectively. We denote \(N_0\), \(E_0\) and \(V_0\) the sum of the (stable) bias fields necessary to compensate field components, the sensor offsets and the pillar differences (between the main absolute pillar and the variometer pillar, see below) for the sensors N, E and V, respectively. The angle between geographic north and sensor N is labelled \(D_0\).

We begin with reviewing the assumptions that are made for the formulas that we use here to be valid:

The three fluxgates N, E, V have to be orthogonal.

The fluxgates N and E have to be horizontal, the fluxgate V has to be vertical.

Fluxgate E points roughly towards magnetic east. It requires a negligibly small sensor offset and a zero bias field, i.e., \(E_0\) can be assumed to be zero.

Fluxgate N points roughly towards magnetic north, i. e., \(D_0\) is similar to declination D.

We assume that the magnetic north direction (declination D) at the variometer is identical to the magnetic north direction at the main pillar of the observatory, where declination is measured.

Fig. 6.2 depicts this situation for horizontal fluxgate sensors E and N and their output in nT, \(N_{var}\) and \(E_{var}\). \(D_0\) is now the baseline of declination, its physical meaning is the orientation of the fluxgate variometer, expressed as the angle from geographic (true) north to the N sensor. \(N_0\) is the baseline of the H component, its physical meaning is the bias plus offset of the fluxgate sensor N. Likewise, \(V_0\) is the baseline of the vertical component and its physical meaning is the bias and offset of that sensor.

While the relationship between the absolute values Z, baselines \(V_0\) and variations \(V_{var}\) for Z component is simply linear (1a), the relationship for H (1b) and D (1c) can be derived from Fig. 6.2:

with \(N_0\) being the baseline of the horizontal component and \(D_0\) being the baseline of declination. \(N_{var}\) and \(E_{var}\) are variations in nT. Note that the definitive values at the absolute pillar can be calculated for any time t, for which both final adopted baselines and checked and cleaned variation recordings are available. Note also, that if either \(E_{var}\) or \(N_{var}\) is missing, then neither H nor D can be calculated.

For completeness, we give the formulas that are necessary to calculate baselines from absolute measurements that have been performed at a point in time \(t_1\). These can easily be derived from (1a) to (1c):

Note that in the case of the absolute values \(H(t_1)\), \(D(t_1)\) and \(Z(t_1)\) all appropriate pillar differences have to be included such that the absolute values represent the magnetic field at the observatory’s main pillar. Since the observed baseline values \(N_0(t_1)\), \(D_0(t_1)\) and \(V_0(t_1)\) are determined as a difference between measurements on two pillars, they also incorporate the pillar difference between the main pillar and the variometer pillar. From the resulting observed baseline values \(N_0(t_1)\), \(D_0(t_1)\) and \(V_0(t_1)\) the continuous final baselines \(N_0(t)\), \(D_0(t)\) and \(V_0(t)\) are adopted and used for the calculation of definitive data as described above in formula (1a) to (1c).

If the XYZ-orientation is used, we label the fluxgates X, Y and Z.

Note

To avoid confusion, that these are the identical symbols as used for the geomagnetic components X, Y and Z, and that you have to determine from the context whether the sensors or the geomagnetic components are being referenced. We also refer to the vertical sensor as V before.

We assume the sensors X, Y and Z are aligned exactly along the magnetic components X (North), Y (East) and Z (vertical). It is thus not trivial to physically set up a magnetometer exactly in this direction. Then, all formula become linear: \(X(t) = X_0(t) + X_{var}(t)\) with \(X_0\) being the baseline (bias plus offset in nT) and \(X_{var}(t)\) being the variometer output of the X sensor; \(Y(t) = Y_0(t) + Y_{var}(t)\) with \(Y_0\) being the baseline (bias plus offset in nT) and \(Y_{var}(t)\) being the variometer output of the Y sensor; \(Z(t) = Z_0(t) + Z_{var}(t)\) with \(Z_0\) being the baseline (bias plus offset in nT) and \(Z_{var}(t)\) being the variometer output of the Z sensor.