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## Appendix A. Inclination and Azimuth

We first transform our gravimetric and magnetic data, which are in left-handed Cartesian coordinates, to more familiar right-handed Cartesian coordinates. The transformation is shown in Fig. 9 and produces the right-handed gravity vector **A** and magnetic vector **M**.

Fig. 9. Transformation of the left-handed board coordinate system to the right-handed coordinate system. The orientation of the LSM303C breakout board (red) as the tilt sensor is installed upright in the borehole is shown. To mimic the conventional positive vertical axis, we transform the downward pointing board *y*-axis to an upward pointing *z*-axis. The board *z* and *x*-axes are oriented similarly to the right-handed *x* and *y*-axes respectively, so we can transform them without sign changes.

Inclination *θ* is the angle from the vertical gravity axis **A** from which the sensor's vertical axis **Z**^{′} has tilted, as illustrated in Fig. 10. We can find *θ* by calculating the angle between **A** and **Z**^{′} using the dot product

Azimuth *ϕ* is the direction of the sensor's vertical axis **Z**^{′} relative to magnetic north (0° azimuth). To measure the azimuth on a plane normal to the vertical gravity axis **A**, we project **Z**^{′} and the magnetic vector **M** onto the plane normal to **A** as illustrated in Fig. 11. The **Z**^{′} projection is **Z**^{′}⊥ **A**

and the **M** projection is **M**^{′}, representing the component of the magnetic north vector on the plane normal to **A**

**Z**^{′}⊥ **A** and **M**^{′} lie in the same plane and are normal to **A**. We can find *ϕ* by calculating the angle of rotation from **Z**^{′}⊥ **A** to **M**^{′} about **A** using the dot product

To determine the direction of rotation of *ϕ*, we calculate the dot product between (**Z**^{′}⊥ **A**) × **M**^{′} and **A**. The cross product (**Z**^{′}⊥ **A**) × **M**^{′} yields a vector that will be parallel, anti-parallel or perpendicular to **A**. A positive dot product indicates (**Z**^{′}⊥ **A**) × **M**^{′} and **A** are parallel and *ϕ* is being calculated in our intended counterclockwise rotation direction. A negative dot product indicates (**Z**^{′}⊥ **A**) × **M**^{′} and **A** are anti-parallel and *ϕ* is being calculated in the clockwise rotation direction, so we subtract 360° from *ϕ* to flip the rotation direction to counterclockwise. A dot product of zero indicates (**Z**^{′}⊥ **A**) × **M**^{′} and **A** are perpendicular, meaning **Z**^{′}⊥ **A** and **M**^{′} lie on top of each other on the plane normal to **A** which makes *ϕ* = 0°.

Fig. 10. The inclination of the sensor's tilt *θ* (yellow) is the angle from the vertical gravity axis **A** (cyan) to the sensor's vertical axis **Z**^{′} (purple).

Fig. 11. The azimuth of the sensor's tilt *ϕ* is the counterclockwise angle of rotation from **Z**^{′}⊥ **A** (purple) to **M**^{′} (green) about **A** (blue) on the plane normal to **A** (pink). **Z**^{′}⊥ **A** and **M**^{′} are projections of **Z**^{′} (yellow) and **M** (red) respectively onto the plane normal to **A**.

## Appendix B. Shear Strain Rate

Glen's Flow Law (Glen, 1955) defines the relation of strain rate $\dot {\varepsilon }$ to deviatoric stress *σ*. In our two-dimensional analysis, we restrict measurements of the sensor's tilt to the down-flow *xz*-plane and are only interested in the tensor component $\dot {\varepsilon }_{zx}$

where *E* is the enhancement factor, *A* is the rate factor in an isothermal case determined by the average borehole temperature over the data collection period and *n* is the stress exponent. The down-flow shear stress component is σ_{zx} = ρgzsinα, where *z* is borehole depth, *ρ* is density of glacier ice (917 kg/m^{3}), *g* is gravitational constant (9.81 m/s^{2}) and *α* is surface slope derived from GPS data. We assume a constant strain rate $\dot {\varepsilon }_{zx}$ and relate $\dot {\varepsilon }_{zx}$ to the net angle of rotation *ψ* of the tilt sensor through the horizontal down-flow displacement ℓ

From Fig. 12, we trigonometrically calculate *ψ* using ℓ and sensor length *L* (0.205 m)

We then calculate $\dot {\varepsilon }$ from *ψ*

Fig. 12. The tilt sensor rotates in the down-flow direction at $\dot {\varepsilon }_{zx}\, {\rm s}^{-1}$, measurable by the horizontal down-flow displacement ℓ of one end of the sensor relative to the other over time *t* from an initial upright position.