# A Designed Calibration Approach for the Measurement -While-Drilling Instrument
## 2 Materials and Methods  
### Sensor Model and Error Model  
The sensor model of the magnetometer-based MWD instrument is mathematically considered as three sets of a coordinate frame: the coordinate frame of carrier platform (\(x_{\text{b}}y_{\text{b}}z_{\text{b}}\), _abbr. car_ frame, and the index is \(b\)), the coordinate frame of the tri-axial accelerometer (\(x_{\text{a}}y_{\text{a}}z_{\text{a}}\), _abbr. acc_ frame, and the index is \(a\)) and the coordinate frame of the tri-axial magnetometer (\(x_{\text{m}}y_{\text{m}}z_{\text{m}}\), _abbr. mag_ frame, and the index is \(m\)). By default, the _car_ frame \(x_{\text{b}}y_{\text{b}}z_{\text{b}}\) is thought of as a standard Cartesian coordinate frame. However, the other two sensor coordinate frames are all non-standard (the axes are non-orthogonal and unequal to each other). The situation is rendered in Figure 1. Moreover, the index \(so\) denotes a mirror coordinate frame of a sensor triplet which is a standard Cartesian coordinate frame in any direction, such as \(x_{\text{a}}y_{\text{a}}z_{\text{a}}\) or \(x_{\text{m}}y_{\text{m}}z_{\text{m}}\) for tri-axial accelerometer or tri-axial magnetometer, respectively. After the matrix rotation transformation, the mirror coordinate frames (\(x_{\text{a}}y_{\text{a}}z_{\text{a}}\) or \(x_{\text{m}}y_{\text{m}}z_{\text{m}}\)) would be aligned with the _car_ frame (\(x_{\text{b}}y_{\text{b}}z_{\text{b}}\)). The attitude information (i.e., azimuthal angle \(\psi\), pitch angle \(\theta\) and tool-face angle \(\varphi\)) for this sensor model is the three Euler angles which refer to the transformation from the geographic coordinate frame (i.e., North-East-Down coordinate frame; _abbr._ NED frame) to the _car_ frame. The detailed calculation of these angles has been given by [PERSON] in [33]. Based on the sensor model, the error model of these sensors can be formulated as the equation of Equation (1). As shown in this equation, the vector \(\mathbf{v}_{k}^{s}\) represents one element in the data set of raw measurements \(\mathbf{D}_{k}[\mathbf{v}_{\mathbf{k}}^{s}=(v_{kxn}^{s},\,v_{kyn}^{s},\,v_{kxn}^{s})^{\text {T}}]\)\(n=1,\,\ldots,\,N\)). The vector \(\mathbf{u}_{\mathbf{k}}^{b}\) represents the local field vector in the _car_ frame, \(\mathbf{u}_{k}^{b}=(u_{kxn}^{b},u_{kyn}^{b}+u_{kyn}^{b})^{\text{T}}\). In the equation, the superscripts \(s\) and \(b\) represent the categories of the coordinate frame, in which \(s\) denotes the coordinate frame of the sensor and \(b\) denotes the _car_ frame. Similarly, the subscript \(k\) denotes the type of vector (i.e., \(k=a\) or \(k=m\) denotes the acceleration vector of gravity or the vector geomagnetic field), and the resulting physical quantities \(\mathbf{v}_{\mathbf{k}}^{s}\) and \(\mathbf{u}_{\mathbf{k}}^{b}\) in metric units denote the \(n\)th measured value of the vector \(\mathbf{v}_{\mathbf{k}}^{s}\) and \(\mathbf{u}_{\mathbf{k}}^{b}\) in their respective coordinate frame. \[\mathbf{v}_{\mathbf{k}}^{s}=K_{kb}^{s}\mathbf{u}_{\mathbf{k}}^{b}+b_{k}+\varepsilon_{k} \tag{1}\]  
where \(K_{kb}^{s}\) is a 3 \(\times\) 3 matrix which aggregates all errors with a linear transformation, \(b_{k}\) is a 3 \(\times\) 1 bias vector which includes all the additive errors and \(\varepsilon_{k}\) is Gaussian white noise. Although it is impractical to separately consider the error sources [31], the error matrix \(K_{kb}^{s}\) could still be divided into the internal component and the external component, according to the error sources, which can be formulated as the following equation. \[K_{kb}^{s}=C_{kso}^{s}R_{kb}^{so} \tag{2}\]  
Figure 1: The schematic diagram of coordinate frame for MWD instrument. where \(R^{so}_{kb}\) represents the error parts outside a sensor triplet, and \(C^{s}_{kso}\) represents that inside. Then, the error model for a MWD instrument can be separately rewritten as Equation (3).