From the first part of our solar storms series, you now know what’s happening above and around the UK. But what happens next? How do we go from observing magnetic chaos to actually protecting our power network?
In this next part, we’ll show how machine learning, data analysis, and detailed Earth models can predict magnetic fields across the country, calculate the resulting electric fields, and identify where geomagnetically induced currents could be strongest.
Predicting Earth’s Magnetic Field
So, how do we move from observation to prediction? Since the solar wind directly influences Earth’s magnetic field, predicting magnetic changes could allow us to prepare before the data even reaches ground stations. That’s the first step.
To do this, we’re using a Random Forest model to predict the magnetic field at the three UK magnetometer locations: Eskdalemuir, Hartland, and Lerwick. But three stations alone aren’t enough to infer how the magnetic field fluctuates across different locations, especially since we now know that latitude affects field strength. We’ll have to deal with this data sparsity.
Here’s where SuperMAG proves invaluable again. By selecting magnetometers from around the world at similar latitudes to those across the UK, we can expand our dataset and improve model accuracy. Using historical data from 30 SuperMAG stations, we trained our Random Forest model to predict magnetic field strength based on seven solar wind parameters.
The results are promising: our model’s predictions (blue) align closely with observed data (red) at all three UK stations during the most recent solar storm which took place between 10th-12th May 2024. This provides a strong foundation for scaling up to a full UK-wide model.
Figure 1: Random Forest model prediction of the horizontal magnetic field for the three UK stations over the period of the 2024 May storm.
Top to bottom: Model’s prediction in blue compared to the ground-truth horizontal magnetic field (in nT) in red for the Eskdalemuir, Hartland, and Lerwick station, respectively. The model is capable of predicting drastic fluctuations in the magnetic field for each of the three stations.
However, the Random Forest model is not an interpolation method. It does exactly what you ask of it, to predict the magnetic field for the three UK stations.
So, we have to use geospatial interpolation methods to model the magnetic field in between the three magnetic stations to cover the entire UK. We’re using an inverse distance weighting algorithm to do exactly this. It operates on the principle that a point’s value can be estimated by a weighted average of the known values around it, where the influence of each known point diminishes with distance, and the weights being inversely proportional to the distance from the known points. In other words, the prediction for a location in Scotland will be influenced far more by the Lerwick and Eskdalemuir data than by the Hartland data. With this interpolation, we have appropriate measurements of the magnetic field for all locations in the UK, not just the three magnetometer stations.
The British Geological Survey has installed three additional stations in the UK to monitor magnetic field fluctuations, located at Florence Court (FLO), market Harborough (LEI), and Herstmonceux Castle (HTX). These stations, known as variometers, which measure changes in the magnetic field over time, are located in “magnetically quiet” locations in the UK, meaning that they are far from industry or infrastructure that could cause magnetic disturbances. More information regarding these stations can be found on the BGS SAGE website.
We can all agree that the more data, the merrier! The data that the variometers measure are not publicly available, so we cannot use that data for our model. However, we can use the additional BGS variometers as a method of validating our magnetic field interpolation, and compare them to their measurements. So fear not – with our interpolation algorithm, Figure 2 shows that we are able to predict the magnetic field for the three public stations, and infer the magnetic field at the three additional variometers.
Figure 2A: Magnetic field time series for the three UK magnetic stations (ESK, HAD, LER), and the three additional variometers (FLO, LEI, HTX) over the period of the May 2024 storm. BGS results published in BGS response to the May 2024 storm.
Figure 2B: Our results using the Random Forest model to predict the magnetic field at ESK, LER, and HAD, and using our interpolation method for FLO, LEI, and HTX.
We are not able to recover the measurements from the private variometers supported by the BGS, but with a comparison to the BGS response to the May 2024 storm, we can confidently state that the power of advanced data techniques, and machine learning, should not be overlooked. By confirming the capability of our interpolation algorithm, we are now ready to investigate GICs.
From Magnetic Field to Electric Field
Now that we can estimate the magnetic field across the UK, the next question is: what about the induced currents, the GICs? Calculating the GIC that flows in a specific transformer in the UK power network is a two-part process. First, we must determine the geoelectric field, E, produced at the Earth’s surface by the storm. This field is the “driver” of the GIC, and it’s calculated using two key inputs:
- Geomagnetic Data: the time-series measurements of the magnetic field variations during the storm.
- Earth Conductivity Models: a model of how conductive the ground is beneath the power grid. This is crucial because the Earth’s structure significantly changes the resulting electric field.
Once we have the geoelectric field, we can apply it to a model of the UK power network, and by using system information such as transmission line connections and transformer resistances, we can calculate GICs in specific transformer connections. We’ll come back to the discussion of the GIC calculation and focus, here, on calculating the geoelectric field.
We’ve discovered that a Random Forest model is a powerful tool at predicting the time-series magnetic field values at the three UK magnetic stations, and that using a geospatial interpolation method allows us to infer the magnetic field values throughout all locations in the UK. So, we have our first ingredient for calculating the electric field. Now, we need to focus on how Earth’s conductivity (or resistivity), specifically for the UK, in conjunction with our magnetic field affects the electric field.
It’s a good time to get a coffee because we’ll have to get down and dirty with some equations. The following sections might give you flashbacks to your maths lessons… but we promise that everything will come together!
Earth’s Response to the Magnetic Field: A Story of Impedance
To get the geoelectric field, we need to understand how the Earth responds to the magnetic disturbances. The storm’s source currents flow in the ionosphere, over 100 km up. These magnetic variations penetrate the Earth, inducing electric currents deep within the crust and mantle. Since transmission lines are minuscule by comparison, we can assume that the electric field at their height is effectively the same as that at Earth’s surface. The electric field we see at the surface is the combined result of the storm’s “external” field and the Earth’s “internal” induced field. The depth these fields penetrate is governed by the skin depth:
Equation 1: The skin depth, describing how deeply an electromagnetic wave can penetrate a conductor. A higher frequency of the wave, and a greater Earth conductivity, result in a smaller skin depth.
Where 𝜔 is the angular frequency, 𝜇 is the magnetic permeability, and 𝜎 is the Earth’s conductivity. At the very low frequencies of GIC (e.g., 0.001 Hz), the skin depth can be hundreds of kilometres. This means our model can’t just look at the surface soil; it must account for the conductivity of the deep crust and mantle.
In the frequency domain, the relationship between the surface geoelectric field, E, and the geomagnetic field, H, is defined by a variable called the Earth’s surface impedance, Z(𝜔) and is written as follows:
Equation 2: The fundamental relationship between the geoelectric field and the geomagnetic field at the Earth’s surface.
The Earth’s surface impedance represents the ground’s electrical resistance to the induced currents. This value depends on the underlying geology (the conductivity model of the Earth).
Here, x and y represent the Northward and Eastward components of both the geoelectric field and the geomagnetic field intensity, respectively. The magnetic field density B (what is often measured) is related to the intensity H by:
Equation 3: The conversion factor used to turn the measured magnetic field data (B) into the magnetic field intensity (H), needed for geoelectric field calculations. Where 𝜇0 is the permeability of free space.
The negative sign arises as a result of Maxwell’s equations of electromagnetism (for a mathematical deep-dive, take a look at NASA’s cautionary message!)
The key, then, is to find the surface impedance, Z(𝜔). We do this by modeling the Earth as a stack of flat layers (also known as a 1-D model), each with its own thickness and conductivity. Calculating the surface impedance uses a recursive method that starts at the deepest layer and works its way up. For any given layer n, the propagation constant for working our way to the next layer is given by:
Equation 4: The propagation constant describes how the electromagnetic wave from the geomagnetic disturbance travels through and is weakened by that specific layer of rock.
This determines how an electromagnetic wave travels and attenuates in the layer. This is an important step in calculating the surface impedance, and the reflection at the boundary with the layer below it. Now let’s get started with our impedance calculations. For the bottom-most layer (which we assume has infinite thickness), the impedance is:
Equation 5: Impedance of the bottom deepest layer, which is assumed to be infinitely deep. Because it’s infinitely deep, there are no reflections coming back up from below it.
Here comes the recursive step: for all layers above, we calculate the impedance at the top of the layer below and use it to find the reflection coefficient of that layer, which we’ll call r, at the boundary between the two layers. Take a deep breath because this is a long one. The reflection coefficient, r, is given as follows:
Equation 6: The reflection coefficient, r, is used to calculate the total impedance at the top of the current layer. It describes how much of the electromagnetic wave is reflected back up when it hits the boundary between two layers.
This reflection coefficient, along with the depth of each layer, is then used to find the impedance at the top of the current layer. We repeat this process for every layer until we reach the surface, giving us the final surface impedance we need, Z(𝜔).
The surface impedance depends on the conductivity profile beneath the area of interest, which means that we somehow need to have a model of the Earth’s layers for the UK. For Europe, we rely on the European RHO Model (EURHOM) developed by Adam et al. (2012), based on magnetotelluric measurements. The model divides the continent into sub-regions, each with its own 1D resistivity profile that describes how electrical resistance changes with depth. Since resistivity is the inverse of conductivity, we can use this.
Several of these sub-models cover the UK, which is shown in the image below, so we have to select the appropriate sub-model for each location. Beneath the surface, the crust is generally more resistive, while deeper mantle layers become more conductive due to increasing pressure and temperature. At infinite depths of the Earth, resistance is set to a theoretical constant.
Figure 3: European Resistivity models calculated from magnetotelluric measurements. Each model contains a list of depths into the Earth with corresponding resistivity values.
From Equations to the Electric Field
Well done for getting this far, but we’re not quite finished with our calculation just yet! We need to put everything together to calculate the electric field. With the surface impedance Z(𝜔) calculated, we can use the frequency domain technique to find the geoelectric field using the following process:
- We start with the time-series magnetic field data from the storm, B(t). This is the output from our Random Forest model and interpolation.
- We use a powerful mathematical tool called a Fast Fourier Transform (FFT) to deconstruct this time-domain signal into its fundamental frequency components, B(𝜔). Think of this as turning a piece of music into a graphic equaliser display, showing you the strength of the bass, mid-tones, and treble. This is the “frequency domain”.
- We convert B(𝜔) to H(𝜔) (using the magnetic permeability 𝜇0) and apply the impedance relationship we described above in Equation 2. This gives us the electric field spectrum. This is like adjusting the sliders on the equaliser.
- We finally use what’s called an Inverse Fast Fourier Transform (IFFT) to reassemble these modified frequency components back into a new time-domain signal. This new signal is the resulting electric field time series, E(t).
The result is our electric field, E(t), a time-series of the geoelectric field during the storm. This is the critical input we need for modelling the GIC in the UK power network itself.
Mapping, Modelling, and Visualising the UK’s Electric Field
To calculate the electric field at each point, we need both the magnetic field, the Earth’s layers, and resistivity in the UK. To do this, we created a UK map at 10 km × 10 km resolution, which serves as input to our Random Forest model, along with the seven solar wind parameters. This model is part of a pipeline for calculating and visualising the electric field across the UK. The pipeline takes the magnetic field predictions and feeds them into our equations above, together with the UK Earth models, to compute the electric field. By running this process over time-series solar wind data, we can track how the electric field evolves – critical for identifying the maximum values during solar storms.
During quiet periods – typical day-to-day conditions without a solar storm – the electric field values remain small and stable. For the example of the 2024 May storm, the electric field at the Eskdalemuir station was relatively stable the days post-storm, between 13th-15th May.
Figure 4: Electric field time-series for the Eskdalemuir station the days following the May 2024 storm, calculated using our pipeline.
Top: Northward electric field (in mV/km)
Bottom: Eastward electric field (in mV/km)
Going back in time, during the May 2024 storm between 10-12th, however, values spike in both magnitude and range.
Figure 5: Electric field time-series for the Eskdalemuir station over the period of the May 2024 storm, calculated using our pipeline.
Top: Northward electric field (in mV/km)
Bottom: Eastward electric field (in mV/km)
Our goal is to be able to mitigate the risk to critical infrastructure, which is the UK power grid, so appropriately visualising the electric field on a map of the UK is crucial to pinpoint locations that have the potential to experience high GIC values. Since the highest electric fields correspond directly to the largest currents in power systems, we would not have to flag risks with stable low values of the electric field. Here, we see a relatively typical electric field in both northward and eastward directions.
However, an intense spike in the electric field is a cause for concern for power systems, and areas experiencing high electric field values become a priority. The resulting visualisation provides a clear picture of how geomagnetic disturbances propagate and intensify across the UK, causing spikes in the geoelectric field. In this example, areas in Scottish islands are most severely affected, and where we would have to make a priority for possible GIC disturbances.
From Modelling to Safeguarding…
The past three months have been a deep dive into solar storm physics, magnetic field prediction, and mapping UK resistivity. During this time, we have developed the tools to predict the magnetic field across the UK and to calculate the resulting electric field based on Earth resistivity and solar wind data.
The next step is to take these electric field calculations and combine them with datasets of the UK’s power systems to compute GICs across the national network. This will allow us to identify where GICs are likely to be strongest, and which parts of the network are most at risk. From there, we can plan strategies to mitigate the most severe GIC cases, reducing the potential impact on the UK’s electricity infrastructure.
This stage will bridge our modelling work with practical applications, showing how our electric field predictions translate into potential effects on real-world systems. It will also provide the foundation for the next phase: testing and implementing measures to protect the power network.
Stay tuned to find out if we can help the UK’s power systems withstand a major solar storm… or if we will leave the UK to a power outage!
