Quantifying Uncertainty in Spatial Energy Models

In the first part of this series, we mapped the spatial variation of renewable energy potential across the UK — revealing a tenfold range in wind energy and a surprisingly narrow band for solar. But producing an estimate is only half the job. The harder question is: how much should you trust it?

Any spatial model that interpolates values from a network of observation points is, at some level, guessing. The quality of that guess depends on the density and distribution of the observation network, and on the statistical behaviour of the thing being measured. This post describes a practical framework for quantifying that uncertainty and communicating it to end users.

Why a single number is dangerous

The whole motivation for this work was to move beyond misleading manufacturer benchmarks — single numbers presented without context, qualification, or regional adjustment. Replacing those with our own single numbers, however precisely modelled, would have been no improvement. A homeowner in coastal Wales and a homeowner in central Lincolnshire might receive very different energy estimates, but without knowing the confidence behind each, they'd have no way to judge how seriously to take them.

We needed a system that could answer not just "how much energy?" but "how sure are we?" — and that could do so differently for every location and for each energy type.

Two dimensions of uncertainty

Uncertainty in a spatially interpolated estimate comes from two fundamentally different sources.

Network uncertainty: the observation infrastructure

The first source is the observation network itself. Every interpolated value is derived from nearby stations, weighted by distance. The quality of that derivation depends on:

How many stations contributed — a value informed by twelve stations is more trustworthy than one informed by two
How much those stations agree — if neighbouring stations report similar values, the interpolation is reliable; if they diverge sharply, the estimate is a compromise between contradictory evidence

For wind, we used a search radius of 78 km — the distance that guaranteed every point in the UK would be influenced by at least one station. For solar, that radius was 106 km, reflecting the sparser solar station network. Within those radii, some locations benefited from up to seventeen contributing stations, while others relied on as few as one.

The standard deviation of energy values among the contributing stations gave us a direct measure of local agreement. For solar, this was almost always below 135 kWh — stations in a region tended to tell the same story. For wind, standard deviations regularly exceeded 2,000 kWh, especially along the coast where a sheltered valley station and an exposed headland station might sit within the same search radius.

Data uncertainty: the statistical fingerprint

The second source of uncertainty lives within each station's own data. Even if you have a dense, agreeing network, the underlying measurements might be inherently volatile — and that volatility affects how confidently you can project from a year of observations to a long-term expectation.

We analysed the probability distributions at each station by binning hourly readings into histograms and fitting distribution curves. The shape of these distributions tells you a great deal about predictability:

Tight, well-peaked distributions — where most readings cluster around a central value — indicate stable, predictable conditions. Solar stations almost universally showed this pattern.
Wide, heavy-tailed distributions — where readings are spread across a broad range with significant probability mass in the extremes — indicate volatile, hard-to-predict conditions. Wind stations in exposed locations consistently showed fat tails, with the shape parameter of their fitted curves reflecting high variability.

The energy pattern factor we calculated for wind — the ratio of the mean of the cubes to the cube of the mean — captures this directly. A factor of 1.6 indicates a relatively well-behaved distribution. A factor approaching 3.0 indicates extreme variability, where gusts dominate the energy picture and the mean wind speed tells you almost nothing useful on its own.

Critically, these two dimensions of uncertainty are independent. You can have a dense, agreeing network of stations that all show volatile distributions (high network confidence, low data confidence). Or a sparse network where the few available stations show very stable, predictable patterns (low network confidence, high data confidence). A useful uncertainty framework needs to account for both.

Building the confidence rating

We combined these dimensions into a single 1-to-10 confidence score, split evenly between network quality and data characteristics.

The network component scored from 1 to 5 based on station count within the search radius — from two or fewer stations (score of 1) up to twelve or more (score of 5). This was deliberately kept the same scale for both wind and solar, so that the sparser solar network would naturally show lower network scores than wind, accurately reflecting the fewer observation points available.

The data component also scored from 1 to 5, based on the standard deviation of energy values among the contributing stations. Tight agreement (under 300 kWh deviation) scored 5. Wide disagreement (over 2,200 kWh) scored 1. Again, the same scale was applied to both energy types, letting the inherent characteristics of wind and solar speak for themselves rather than being normalised away.

The result was striking in its consistency. Solar estimates routinely scored 8 to 10 — high network agreement combined with stable data distributions. Wind estimates typically scored 4 to 7, pulled down by the high variance between stations and the volatile nature of wind data itself. Coastal wind locations often scored lowest of all, despite having the highest energy potential — a useful reminder that high opportunity and high uncertainty often go hand in hand.

What uncertainty tells users that estimates alone cannot

Presenting confidence ratings alongside energy estimates transformed how the results could be interpreted:

A wind estimate of 4,000 kWh with a confidence of 4/10 tells the user: this location probably has good wind resource, but conditions are highly variable and the actual yield could differ significantly. Proceed with caution; consider a site-specific assessment before investing.
A solar estimate of 1,800 kWh with a confidence of 9/10 tells the user: this is a reliable estimate. The station network agrees, the data is stable, and you can plan around this number with reasonable assurance.
A wind estimate of 1,200 kWh with a confidence of 8/10 tells the user: we're quite sure about this — and unfortunately, what we're sure about is that this isn't a good location for a wind turbine.

The third case is arguably the most valuable. High confidence in a low estimate is more useful than low confidence in a high one. It prevents wasted investment and redirects attention to where it'll actually pay off.

Applicability beyond energy

The framework described here — combining network-based and distribution-based uncertainty into a composite confidence score — is applicable wherever spatial interpolation is used to estimate continuous variables from point observations:

Air quality modelling — sensor networks with uneven coverage and pollutants that vary sharply over short distances (much like wind)
Noise mapping — where measurement points are sparse and local topography creates dramatic variation
Property valuation — where comparable sales are the "stations" and market volatility is the distribution characteristic
Agricultural yield estimation — where soil sample points inform predictions across entire fields, but soil composition can change within metres

In each case, the principle holds: the most responsible thing a model can do is quantify its own ignorance. An estimate without a confidence rating is a claim without evidence. Adding uncertainty doesn't weaken the output — it makes it honest, and honest estimates are the ones people can actually make decisions from.