At geomagnetic observatories the absolute measurements are needed to determine the calibration parameters of the continuously recording vector magnetometer (variometer). Absolute measurements are indispensable for determining the vector of the geomagnetic field over long periods of time. A standard DI (declination, inclination) measuring scheme for absolute measurements establishes routines in magnetic observatories. The traditional measuring schema uses a fixed number of eight orientations (Jankowski et al., 1996).

We present a numerical method, allowing for the evaluation of an arbitrary
number (minimum of five as there are five independent parameters) of telescope
orientations. Our method provides

A general approach has significant advantages. Additional measurements may be seamlessly incorporated for higher accuracy. Individual erroneous readings are identified and can be discarded without invalidating the entire data set. A priori information can be incorporated. We expect the general method to also ease requirements for automated DI-flux measurements. The method can reveal certain properties of the DI theodolite which are not captured by the conventional method.

Based on the alternative evaluation method, a new faster and less error-prone measuring schema is presented. It avoids needing to calculate the magnetic meridian prior to the inclination measurements.

Measurements in the vicinity of the magnetic equator are possible with theodolites and without a zenith ocular.

The implementation of the method in MATLAB is available as source code at the
GFZ Data Center

Absolute measurements of the magnetic declination

The alternative method for DI measurements was motivated by a very practical
reason. Zenith oculars are very seldom available. Without them, inclination
measurements are not possible around the magnetic equator where the magnetic
field is almost horizontal. We wanted to enable measurements with a DI-flux
theodolite and without a zenith ocular. The need to evaluate DI measurements at
arbitrary orientations was felt before we started our work on a
viable solution. Peter Crosthwaite

We present a numerical method, allowing for the evaluation of an arbitrary
number (minimum of five as there are five independent parameters) of telescope
orientations. No prescribed fixed measuring scheme is needed. We implement
an instrument model to calculate the fluxgate reading

We took advantage of methods known from geophysical inversion theory such as assessing the residuals, using a priori information and objectively assessing the accuracy of the results (Schmucker, 1975; Tarantola, 1982). Exploitation of the method in practice also showed benefits for the routine work at observatories. We show, for example, that at a given rate of erroneous readings, a higher percentage of successful absolute measurements can be achieved. We apply our method to various data sets. We show the benefit to accuracy and reliability of routine absolute measurements at the Niemegk Observatory and the resulting base values.

For the

This conventional DI scheme has two major advantages:

It allows for a simple numeric evaluation by just averaging four readings.

The influence of the instrument parameters like sensor offset and the misalignment angles between the fluxgate sensor and the telescope cancel out.

In this paper we refer to the angle

In the conventional scheme the calculations of declination

Declination and inclination usually vary during the measuring process. Let

The

Using this declination value

Eight correct measurements are needed to apply Eqs. (

Figure

Graphic presentation of possible telescope
attitudes of a DI flux theodolite.

The new scheme allows us to take DI measurements using a simplified and less error-prone method. Advantages with respect to the conventional methods
are as follows:

No need to calculate the magnetic meridian.

The method tolerates imperfect levelling of the telescope.

The measurement scheme facilitates adjusting the theodolite. Throughout the entire measurement procedure only the adjustment wheels for the horizontal circle are used to adjust the magnetometer reading to a small value.

There is no need for telescope orientations along the magnetic meridian. This avoids orientations requiring a zenith eye piece at observatories close to the magnetic equator.

We feel that, after some practice, the new method (including the optional measurements) needs slightly less time than the conventional, but even without gaining time, the resulting six measured orientations lead to a statistically firmer result than the four conventional ones.

Repeating the new method with different tilt angles

Set vertical circle to a certain value

For the given

The first four orientations of the new scheme are identical to conventional

Set

Invert the telescope, set

Turn telescope to east, set

Invert the telescope, set

The next measurements are taken with the telescope tilted by a certain angle

Set

Invert telescope, set

Optional: invert telescope, set

Optional: invert telescope, set

Set

Invert telescope, set

Optional: invert telescope, set

Optional: invert telescope, set

In steps 1 to 4 there is no need to set

The right panel of Fig. 1 shows the distribution of data points of the new scheme.

As explained above, the eight orientations of the standard DI scheme are used
due to practical reasons. There is no principal reason to take the measurements
at exactly the orientations as in the conventional scheme. Theoretically only
five measurements are needed for the five unknown quantities: declination

The general method described below allows measurements at arbitrary telescope orientations to be evaluated. The number of measurements are arbitrary. Usually it should be at least five. Even fewer than five measurement can be sufficient if a priori information about the instrument parameters is available. The method has a lot of additional advantages. The quality of each single measurement can be assessed and outliers can be identified and discarded. As already mentioned, the measuring set of the eight orientations for the conventional scheme is usually repeated several times, so that 16 or even 24 orientations are available. Using all of them in one joint evaluation leads to a statistically firmer result and better allows us to identify outliers. Even though actual DI measurements at the Niemegk Observatory are known to be at a high accuracy level, we show that its accuracy and reliability could still be improved using the new method. Additionally the new method gives the observer more insight into sources of problems with magnetic cleanliness.

The DI-flux instrument model allows the magnetometer reading

We used the latter, because it facilitated the calculation of the partial derivatives for

For each measurement at time

Each measurement with the DI-flux
theodolite consists of measuring the angles

In order to converge towards the proper solution, the Gauss–Newton method needs
a good first estimate. Instrument parameters

The orientation vectors

The Gauss–Newton method is a straightforward extension of the well-known
Newton method from one to more dimensions. It is also referred to as the
Newton–Raphson method

We are looking for the improvement

With

or by writing the Jacobian matrix

The function

In our case Eq. (

In Eq. (

The linear relation between

Absolute measurements are needed to determine the vector magnetometer offsets and their long-term drift. The dynamic range of the instrument is often adapted to the natural variations of the field, which is far smaller than the constant part. Hence the vector magnetometer is referred to as a “variometer”. A slight drift due to mechanic and electronic instabilities can never be excluded. A very small rotational movement, e.g. in the instrument or its pillar due to temperature, can result in a measurable effect. The variometer orientation in the geographical reference frame must be taken into account. Absolute measurements are indispensable for determining the vector of the geomagnetic field in the geographic reference frame over long periods of time. The wording “absolute” traces back to Gauss and signifies that absolute measurements are inherently adjusting for first-order instrument inexactness (e.g. misalignment of telescope and magnetometer sensor).

The offsets are called “base values”. The stability of the base values is indicative for the stability of the variometer, for the accuracy of the absolute measurements, and for unwanted small scale local magnetic field changes. Thus they are a measure of the quality of an observatory.

Baseline formulas: traditionally baselines refer to

We use the fact that scalar values

The new method data errors, otherwise undetected, can be identified and
discarded, because data from each measured orientation can be assessed. Figure

Panel

Panel

Calculated error bars allow the reliability of each DI measurement to be assessed.
An uncertainty of

A systematic error of a specific theodolite (Zeiss Theo 020, 817992) is
revealed in Fig.

In areas of the globe with a small magnetic inclination, i.e. typically
within 2000 km to the north and south of the magnetic equator, the
conventional DI-flux procedure involves vertical circle readings at steep
telescope orientations. This is not possible without zenith oculars mounted
on the theodolite. The new scheme circumvents this problem if the tilt angle

We have been testing the new method for more than 1 year at the Niemegk
Observatory. A first test was applying the new numeric evaluation to data
produced with the conventional scheme. In case of faultless data, we got
exactly the same result but this time with error bars. We also evaluated
partly corrupted data, which could not be treated conventionally. The method
can still be used if one or even two out of eight measurements have to be
discarded. Investigation of the residuals often allowed us to identify and
correct typos in the data sheet. A data set with the vertical reading set to
90 Gon instead of 100 Gon, produced by a person who was used to a degree scale,
could be evaluated without problems. Furthermore we found that the major
advantage in observatory routine is that several data sets measured on the
same day can be evaluated at once as a single data set. At observatories it
is good practice to repeat measurements several times. Evaluating them at
once leads to a statistically firmer result and facilitates the
identification of outliers. Investigating the calculated error bar allows
measuring conditions to be assessed. As shown in Fig.

The MATLAB source code implementing the method presented above is available
from GFZ Data Services

The authors declare that they have no conflict of interest.

This article is part of the special issue “The Earth's magnetic field: measurements, data, and applications from ground observations (ANGEO/GI inter-journal SI)”. It is a result of the XVIIth IAGA Workshop on Geomagnetic Observatory Instruments, Data Acquisition and Processing, Dourbes, Belgium, 4–10 September 2016.

We thank Anna Willer, Truls Lynne Hansen and Peter Crostwaite. They provided their experiences in numerically modelling a DI-flux theodolite and solving the needed non-linear system of equations.

We thank Christopher Turbitt for the editing. We thank two anonymous reviewers for an intense and fruitful reviewing effort. Kirsten Elger helped to publish the software as a digital object. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Christopher Turbitt Reviewed by: two anonymous referees