How the published carbon figures are produced

01 / DEFINITIONS

Quantities and units

An emission is a mass released over a stated period. A concentration describes how much CO₂ is present in air. Converting a concentration into an emission estimate requires a model of transport, background concentration and instrument response.

Notation used on this page
Symbol	Quantity	Unit or support
E	Emitted mass within a reporting boundary	kg CO₂ or kg CO₂e per reporting period
A, a	Activity amount, activity rate	For example kWh, kWh per second
ρ	Emission factor per unit of activity	For example kg CO₂e per kWh
q, Q	Spatial emission flux, individual source rate	kg CO₂ m⁻² s⁻¹, kg CO₂ s⁻¹
c, χ	CO₂ mass concentration, dry-air mole fraction	kg CO₂ m⁻³, mol mol⁻¹ or ppm
y, H	Observed signal, observation operator	Instrument-specific units
Ω, T	Geographic region, time interval	Declared polygon and start/end times

CO₂e combines greenhouse gases under a specified global warming potential assessment and time horizon. It is not a concentration of CO₂. Surface mole fraction and satellite column-averaged mole fraction have different vertical support.

02 / DATA TYPES

Data types and how to use them

Every record the platform publishes is one of four data types. The type states what the number is, which question it can answer and which it cannot. The type is separate from the source's status, freshness and licence.

The four data types
Type	What it is	Quantity and unit	Answers	Must not be used to
Observed	A signal measured by an instrument or provider, carried with a timestamp and a stated spatial support.	Intensity (gCO₂e/kWh), power (MW), concentration (ppm), column XCO₂ (ppm)	What is happening now, or at a known time	State an annual emissions mass, or be summed with an inventory
Inventory	An annual, compiled estimate of emitted mass over a declared boundary and period.	kg CO₂ or kg CO₂e per reporting period	Where emissions are concentrated and how they change between years	Be read as a live measurement, or be summed across sources with different GWP bases
Modelled	Output of a numerical model, reanalysis or inversion. An estimate, not a measurement.	Flux, meteorology and derived fields, declared per model	Fill gaps, supply transport context and test attribution	Be presented as observed, or establish a real-world measurement
Calculated	An activity amount multiplied by a compatible, versioned emission factor.	kg CO₂e, with the factor's unit and version	Attribute emissions to the user's own activity	Be mixed with a published inventory without stating the scope and boundary

A source also carries a status that is independent of its type: measured, provider estimated, forecast, modelled, synthetic, stale, unavailable and not in plan. A measured status does not establish freshness. The published period and cadence do. A modelled status is never shown as observed.

Which type to use for an analysis

For regional or structural analysis, use Inventory. It is the only type with a consistent period, boundary, gas and GWP basis, so it is the correct basis for comparing places and years and for setting a target.
For time-sensitive or operational analysis, use Observed per signal. Treat intensity and concentration as their own quantities. They are not emissions.
For gap filling, transport context or source attribution, use Modelled and label the result modelled.
For the user's own footprint, use Calculated and keep the activity, factor and version with the result.

03 / INVENTORIES & ACTIVITIES

Carbon accounting

Multiply each activity amount by a compatible emission factor. Select the factor by activity type, geography, reporting year and accounting scope. Retain its version with the result.

E=\sum_{i=1}^{n} A_i\rho_i,\qquad E_T=\int_T a(t)\rho(t)\,\mathrm{d}t

(1) Activity-based emissions

The sum uses activity amounts. The integral uses an activity rate. For example, 100 kWh multiplied by an illustrative factor of 0.20 kg CO₂e/kWh gives 20 kg CO₂e. This arithmetic example is not a current UK factor.

I_T=\frac{\sum_t E_t}{\sum_t K_t}=\frac{\sum_t I_tK_t}{\sum_t K_t},\qquad \sum_t K_t>0

(2) Electricity intensity weighted by energy

Intensity is grams of CO2e per kilowatt hour when emissions E are in grams and energy K is in kilowatt hours. An unweighted average of interval intensities gives the wrong total when energy use varies. The public explorer currently has no connected numeric electricity-intensity feed.

E_{\mathrm{CO_2e}}=\sum_g m_g\,\mathrm{GWP}_{g,h},\qquad m_{\mathrm{CO_2}}=\frac{44}{12}m_{\mathrm{C}}

(3) Gas aggregation and carbon conversion

Here g indexes gases and h is the time horizon. The 44/12 factor converts carbon mass into CO₂ mass. It does not convert other greenhouse gases into CO₂e. Inventories using different GWP assessments or overlapping boundaries are displayed separately.

04 / GEOGRAPHIC SUPPORT

Spatial grids and conservation

The London map uses the NAEI 2023 inventory on its native 1 km grid. A cell represents a published estimate for its area and year. Zooming changes the display scale, not the resolution of the estimate.

E_{i,T}=\int_T\int_{\Omega_i}q(x,t)\,\mathrm{d}A\,\mathrm{d}t,\qquad \bar q_{i,T}=\frac{E_{i,T}}{|\Omega_i||T|}

(4) From flux to a cell total

Dividing an annual total by a year gives a period average. It does not reconstruct hourly variation. Each map layer therefore needs its own quantity, unit, period and colour scale.

E'_j=\sum_i w_{ji}E_i,\quad w_{ji}=\frac{|\Omega_i\cap\Omega'_j|}{|\Omega_i|},\quad \sum_j E'_j=\sum_i E_i

(5) Conservative transfer of cell totals

This overlap rule assumes uniform density within each source cell and target cells that partition the same domain. Areas require a suitable projected or geodesic system. Cropping changes the retained total and must be recorded. Concentrations are intensive quantities and require a separate interpolation rule.

05 / SIMULATION

Simulation methods

The workspace scenario builder turns component parameters into a daily emission mass for each accounting ledger. The calculation is deterministic arithmetic on the assumptions declared with each component. It is not a calibrated plant model and it does not describe a real facility. The model is versioned as component-scenario-1.

Symbols used in the simulation
Symbol	Quantity	Unit
P, Pₑ	Rated output of a combustion component, electrical demand	kW
h, λ, η	Running time, average load fraction, useful efficiency	h/day, dimensionless
ρ_fuel	Fuel emission factor on a net calorific basis	kg CO₂/TJ
d, e	Daily distance and fuel energy per kilometre	km/day, kWh/km
Wₑ, ρ_grid	Purchased electricity and its location-based generation factor	kWh/day, kg CO₂e/kWh
m	Mass of carbonate feed, material or dry biomass	kg/day
p, c, γ	Carbonate purity, decomposed fraction, carbon mass fraction	dimensionless
ρ_material	Product-stage fossil global warming potential factor	kg CO₂e/kg
U, U′	Wall thermal transmittance before and after added insulation	W m⁻² K⁻¹
A, t, k	Wall area, added insulation thickness, insulation conductivity	m², mm, W m⁻¹ K⁻¹
Λ, Λ₀	Heat-loss coefficient of the house and its non-wall part	W/K
θ_in, θ_out	Indoor and outdoor temperature	°C
Q_h, Q_i	Daily useful heat demand and the heat delivered by component i	kWh/day
s	Start hour on the 24-hour strip	h
E_L	Ledger total for one accounting boundary	kg CO₂ or kg CO₂e per day

The generic emission factor ρ from Section 1 appears in this section as ρ_fuel, ρ_grid or ρ_material with the matching unit. All other subscripts are local to the simulation.

5.1 Component emission models

Combustion components convert useful output into fuel energy with the useful efficiency, then into CO₂ with the fuel factor. The factor is stated on a net calorific basis in kg CO₂ per terajoule, and one kilowatt hour equals 3.6×10⁻⁶ TJ.

F=\frac{Ph\lambda}{\eta},\qquad E=3.6\times10^{-6}F\rho_{\mathrm{fuel}}

(6) Combustion fuel energy and emission

Travel components replace the output and the efficiency with a daily distance and a fuel energy per kilometre.

F=de,\qquad E=3.6\times10^{-6}F\rho_{\mathrm{fuel}}

(7) Transport fuel energy and emission

Purchased electricity uses the grid generation factor directly. The factor keeps its own source, region and year, and the component refuses to calculate until both the factor and its reference are supplied.

W_e=P_eh,\qquad E=W_e\rho_{\mathrm{grid}}

(8) Purchased electricity emission

Process emissions use reaction stoichiometry. Limestone decomposition releases one mole of CO₂ per mole of calcium carbonate, so the mass ratio of the products is 44.009 divided by 100.087.

E=mpc\,\frac{M_{\mathrm{CO_2}}}{M_{\mathrm{CaCO_3}}},\qquad \frac{M_{\mathrm{CO_2}}}{M_{\mathrm{CaCO_3}}}=0.43971

(9) Carbonate process emission

A material batch uses a product-stage fossil global warming potential from a declared declaration and edition.

E=m\rho_{\mathrm{material}}

(10) Material batch emission

Gross biogenic CO₂ uses the carbon mass balance of the oxidised dry mass. The result is a gross release and carries no carbon-neutrality claim.

E=m\gamma\,\frac{44}{12}

(11) Gross biogenic CO₂ from carbon content

Each component is evaluated on its own unless an example house is present. A value outside the stated parameter range is refused and shown as incomplete rather than clipped to the limit.

5.2 Building heat demand

An example house sets a heat demand for the heating components that serve it. Added wall insulation lowers the wall transmittance and therefore the demand. The example day is a fixed winter temperature profile and the indoor temperature is held at 20 °C.

U'=\frac{1}{\dfrac{1}{U}+\dfrac{t}{1000\,k}}

(12) Wall transmittance after added insulation

The wall and the added layer act as resistances in series, which is the standard component method for a plane building element. The heat-loss coefficient adds the remaining losses, which cover the roof, windows, floor and ventilation and stay fixed in this model.

\Lambda=AU'+\Lambda_0,\qquad Q_h=10^{-3}\sum_{t=1}^{24}\Lambda\max\!\left(0,\ \theta_{\mathrm{in}}-\theta_{\mathrm{out}}(t)\right)\Delta t

(13) Heat-loss coefficient and daily heat demand

Temperatures are in degrees Celsius and the sum runs over one-hour steps, so Δt is 1 h and the 10⁻³ converts watts to kilowatts. A house emits nothing itself. It changes the fuel that the serving components use, and the change is the difference between two runs of the same model.

5.3 Heat allocation across serving components

A heating component marked as serving the house receives a share of the demand in proportion to its rated output. Its delivered heat cannot exceed its own available output, which is the rated output times the running time times the load. The fuel follows from the useful efficiency.

Q_i=\min\!\left(Q_h\frac{P_i}{\sum_j P_j},\ P_ih_i\lambda_i\right),\qquad F_i=\frac{Q_i}{\eta_i},\qquad Q_{\mathrm{unmet}}=\max\!\left(0,\ Q_h-\sum_j P_jh_j\lambda_j\right)

(14) Served heat, fuel and unmet demand

When more than one component serves the house, the components compete for the same demand and the ledger adds the resulting fuel masses. This is a shared-demand case, so the result differs from the sum of the same components run independently. The unmet term is stated rather than hidden when the serving capacity falls short of the demand.

5.4 Time and operating windows

A component with a running time carries a start hour and occupies a window from the start hour to the start hour plus the running time. The workspace draws the windows on one 24-hour strip and marks a window that passes midnight. The card order can set the start hours in sequence, and the strip then shows which component runs before another and where the windows overlap.

The daily mass does not depend on the start hour or on the card order. The same fuel mass released in the morning and in the evening has the same daily total. Time changes a system answer only when a factor or an interaction changes with time, such as the marginal grid intensity. The workspace holds no hourly intensity series, so it states that time-of-day grid intensity is not modelled instead of applying an average factor to a marginal question.

5.5 Ledger addition and comparison

Components are added within a ledger and never across ledgers. Direct, grid, material and biogenic masses use different boundaries and gases, so a combined total would mix quantities that are not comparable.

E_L=\sum_i E_{i,L},\qquad \Delta E_L=E_L^{\mathrm{scenario}}-E_L^{\mathrm{baseline}}

(15) Ledger addition and baseline change

The workspace attaches no calibrated interval to a scenario total. The uncertainty methods in subsection 5.10 are not applied to the simulation output.

5.6 Transport and observation operators

A forward model predicts what an instrument would measure for a specified emission field. The advection-diffusion equation below is a standard physical reference. It is not the atmospheric solver currently running behind the map.

\frac{\partial c}{\partial t}+\nabla\cdot(\mathbf{u}c)=\nabla\cdot(\mathbf{K}\nabla c)+s-\ell

(16) Mass-concentration transport

Velocity u is in m/s, diffusivity K is in m²/s, and the volumetric source s and sink ℓ are in kg CO₂ m⁻³ s⁻¹. A surface flux q enters as a boundary flux rather than being substituted for s. Initial conditions, boundary conditions and atmospheric density must be specified.

y=H(q,\mathbf{u},b)+\varepsilon,\quad H=\mathcal{S}\circ\mathcal{T},\quad y=GQ+b+\varepsilon\ \text{(linear case)}

(17) Transport followed by instrument sampling

T denotes transport and S denotes instrument sampling. The background b captures incoming CO₂. G maps source rates Q to observation units for a fixed transport state. Noise ε includes measurement error. Instrument averaging and missing observations belong in this mapping.

\chi_{\mathrm{CO_2}}=\frac{c_{\mathrm{CO_2}}/M_{\mathrm{CO_2}}}{\rho_d/M_d},\qquad \chi_{\mathrm{ppm}}=10^6\chi_{\mathrm{CO_2}}

(18) Mass concentration to dry-air mole fraction

M denotes molar mass and ρd dry-air density. This conversion requires compatible dry-air quantities. Satellite XCO₂ further averages through the column and requires the retrieval's pressure weights, averaging kernel and prior profile.

5.7 Estimating source rates

Source estimation reverses the observation process. Many emission fields can fit the same readings. A standard constrained inverse problem illustrates the required components.

\hat Q=\underset{Q\in\mathcal{C}}{\operatorname{arg\,min}}\ \frac12\|GQ+b-y\|_{\Sigma^{-1}}^2+\lambda R(Q)

(19) Illustrative constrained least-squares baseline

Σ is a positive-definite observation-error covariance and the weighted squared norm is rᵀΣ⁻¹r. R is a declared regularizer and λ sets its weight. The feasible set C specifies admissible rates, known totals and source support. Nonnegative rates are appropriate for an emissions-only model. Net ecosystem exchange needs signed fluxes.

This is a conventional baseline, not a validated CO2Locality source-completion algorithm. The handoff asks whether unexplained residuals justify an additional source after accounting for transport error, background error and uncertainty. Reallocating a fixed total does not establish additional emissions.

Research outputs require total, attributable and unresolved emissions, uncertainty and observability status. They must not write directly into an accounting ledger.

5.8 Identifiability and unresolved sources

E=A\rho+U,\qquad \rho'=\rho+\delta,\qquad U'=U-A\delta

(20) Activity-response decomposition

The same total E results from both pairs whenever they satisfy the constraints. For A = 2 and E = 10, the pairs (ρ, U) = (3, 4) and (4, 2) are both nonnegative solutions. Total emissions alone cannot choose between them.

A smooth residual field U is not automatically singular with respect to an activity measure. On a finite grid with strictly positive activity in every cell, any cell-supported finite residual is absolutely continuous with respect to that measure. This activity-response split should not be called a proven singular Lebesgue decomposition.

G_1=G_2\quad\Longrightarrow\quad G_1Q_1+G_2Q_2=G_1(Q_1+Q_2)

(21) Indistinguishable source signatures

Identical columns of G constrain a sum, not individual rates. Report an unresolved source group. For nonlinear models, inspect the local sensitivity J = ∂H/∂Q. Small singular values or strongly correlated columns indicate unstable attribution even when numerical rank is full.

Extra observations help only when they add independent information. A stronger regularizer can select a solution without proving that the selected attribution is true.

5.9 Neural operators

A neural operator learns a mapping between input and output fields. The Fourier neural operator (FNO) combines a local linear transformation with a learned transformation of Fourier coefficients. Li et al., 2021.

v_{\ell+1}(x)=\sigma\!\left(W_\ell v_\ell(x)+\mathcal{F}^{-1}[R_\ell(k)\,\mathcal{F}[v_\ell](k)](x)+b_\ell(x)\right)

(22) Fourier neural operator layer

v is a hidden field, W is a local weight matrix, F is the Fourier transform and R contains learned weights for retained modes. The activation σ supplies nonlinearity. An input projection creates v₀ and an output projection returns the target field.

The handoff requires a direct FNO, an activity-response model, sparse source inversion, a conditional neural operator and a physical baseline to use comparable inputs and training budgets. Fourier axes, real-transform symmetry and boundary treatment must be checked. Finer output pixels alone do not demonstrate finer physical accuracy.

Activity, weather and source metadata must have been available at the forecast issue time. Future observations and proxies derived from the target would invalidate the comparison.

5.10 Uncertainty and calibrated intervals

\operatorname{Var}(A\rho)\approx\rho^2\operatorname{Var}(A)+A^2\operatorname{Var}(\rho)+2A\rho\operatorname{Cov}(A,\rho)

(23) First-order uncertainty for one activity-factor product

This approximation uses local linearization about the mean values. Shared factors and correlated activities also create covariance between inventory terms. Treating all terms as independent can understate uncertainty.

\widehat{\mathrm{coverage}}_{1-\alpha}=\frac1N\sum_{i=1}^{N}\mathbf{1}\{y_i\in[L_i,U_i]\}

(24) Empirical interval coverage

An interval labelled 90% should contain about 90% of held-out targets under the evaluation conditions. Report interval width as well as coverage, split by region, season and missing-data regime. Wide intervals can achieve coverage while providing little practical information.

5.11 Validation requirements

Use chronological and geographic holdouts. Freeze source vintages and transformations before evaluating the holdout. Record dataset hashes, seeds, model versions and the exact target definition with each result.

\mathrm{MAE}=\frac1N\sum_i|\hat y_i-y_i|,\qquad \mathrm{RMSE}=\sqrt{\frac1N\sum_i(\hat y_i-y_i)^2}

(25) Prediction error in the target's unit

Tests required before a source-estimation claim
Test	Required evidence
No added source	False source activation rate under background and measurement noise
Identical signatures	Unresolved group rather than falsely certain separate estimates
Unseen locations and activity	Held-out source supports without target-derived proxies
Transport and missingness	Sensitivity to wind error, instrument gaps and background uncertainty
Grid changes	Conserved integrated mass within a declared numerical tolerance
Baseline comparison	Matched information, parameter budgets, optimization steps and seeds

Report failed cases and baseline wins. No universal model-accuracy percentage is justified by these requirements alone.

06 / APPLICATION STATUS

What the application currently uses

Published inventory map: NAEI 2023 grid with source-specific units and sector layers. The map does not run an atmospheric inversion.
Activity accounting: The backend has factor-based calculation and ledger routes. Access, factor availability and source licences are checked separately from signing in.
Workspace simulation: The scenario builder runs the component emission models, the example house heat balance and the serving allocation from declared inputs. Every result is a calculated value with its assumptions shown, and no simulation result enters the published inventory.
Atmospheric source estimation: Simplified observation fixtures exist. They are not a validated live service connected to the public map.

Global map navigation does not imply global emissions coverage. Current integration is limited to UK data.

07 / SOURCES

References and method provenance

CO2Locality implementation handoff v2. Stages 14.04 to 14.09 define decomposition limits, observation physics and conservative transfer. Stages 15 and 24 define fair comparisons and identifiability tests. Stage 22.08 separates observations from emissions.
CO2Locality workspace scenario model, version component-scenario-1. Equations (6) to (15) implement the component, house heat and allocation rules described in Section 5.
UK National Atmospheric Emissions Inventory. The map's dataset record identifies the extract, year, units and source licence.
IPCC 2006 Guidelines for National Greenhouse Gas Inventories. Inventory methods, activity data, emission factors and uncertainty. The default fuel factors and the 44/12 carbon conversion.
UK Government greenhouse gas conversion factors for company reporting (DESNZ). The UK-published factor set a user can supply in place of the IPCC default fuel factors.
EN ISO 6946:2017, Building components and building elements. Thermal resistance and thermal transmittance. The series-resistance rule in equation (12).
ISO 13790:2008, Energy performance of buildings. Calculation of energy use for space heating and cooling. The steady heat balance and degree-hour form in equation (13).
GHG Protocol Scope 2 Guidance. The location-based grid factor and the separation of purchased electricity from direct combustion.
EN 15804:2012+A2:2019, Sustainability of construction works. Environmental product declarations. The product-stage fossil global warming potential factor in equation (10).
IUPAC Commission on Isotopic Abundances and Atomic Weights. The molar masses behind the 0.43971 ratio in equation (9).
Li et al. Fourier Neural Operator for Parametric Partial Differential Equations, ICLR 2021. The standard FNO baseline in subsection 5.9.

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