EC&V Pty Ltd: TROPOMI CSF: Introduction: Equations

Equations

A box model represents conservation of mass for a chemical species, augmented by terms describing emissions, chemical reactions, and removal processes. For ${CH}_{4}$ emissions from a coal mine over timescales of hours, chemical loss processes can be neglected. The fastest relevant reaction is oxidation by the hydroxyl radical (OH), which gives ${CH}_{4}$ an atmospheric lifetime of approximately nine years Seinfeld, et al., 2016 (Section 6.4), far longer than the timescale of interest here. Similarly, removal processes such as soil uptake by methanotrophic bacteria and archaea occur on much longer timescales Jackson, et al., 2021 and can be ignored in this context.

An Eulerian box model applies the principle of mass conservation to a control volume of dimensions Δx × Δy × Δh that encloses the emission source, with the coordinate system defined such that the x-axis is aligned with the mean wind direction. In finite difference form, this formulation is expressed by Equation 1 ( see Equation 25.1 in Seinfeld, et al., 2016 ). The equation states that the emission rate equals the time rate of change of mass within the box plus the net flux of mass through its boundaries.

Q = \frac{d}{dt} (C_{{CH}_{4}} Δ x Δ y Δ h) + u Δ y Δ h (C_{{CH}_{4}} - {C_{{CH}_{4}}}^{0})

(1)

where:
$C_{{CH}_{4}}$ - is concentration ( $\frac{kg}{m^{3}}$ )
${C_{{CH}_{4}}}^{0}$ - is background concentration ( $\frac{kg}{m^{3}}$ )
$Δ x$ - is the length of the box parallel to the direction of wind ( $m$ )
$Δ y$ - is the length of the box perpendicular to the direction of wind ( $m$ )
$Δ h$ - is the height of the box ( $m$ )
$Q$ - is the mass emission rate entering the box ( $\frac{kg}{s}$ )
$u$ - is wind speed perpendicular to the side of the box ( $\frac{m}{s}$ )

This formulation assumes that both the plume concentration field and the background concentration field are known. In practice, the equation is commonly applied in two simplified forms, depending on which term on the right hand side is assumed to be negligible. The first approximation, expressed by Equation 2 , assumes that the total mass within the box remains constant over time, so that the storage term can be set to zero.

Q = u Δ y Δ h (C_{{CH}_{4}} - {C_{{CH}_{4}}}^{0})

(2)

To obtain the sense of magnitude for variables in this equation:
If:

the enhancement is

(C_{{CH}_{4}} - {C_{{CH}_{4}}}^{0}) = 0.001

(

\frac{g}{m^{3}}

),
side of the box perpendicular to the flow direction is

Δ y Δ h = 1000 * 1000

(

m^{2}

)

wind speed is

u = 5

(

\frac{m}{s}

Then:

the emission rate is 5 (

\frac{kg}{s}

The second approximation, expressed by Equation 3 , assumes that all emissions are fully contained within the box, with no mass flux across the lateral boundaries. Under this assumption, the accumulation term is retained, and the total mass within the box increases over time in response to the emission rate.

Q = \frac{d}{dt} (C_{{CH}_{4}} Δ x Δ y Δ h)

(3)

To obtain the sense of magnitude for variables in this equation:
If:

the change in enhancement is

\frac{{dC}_{{CH}_{4}}}{dt} = 0.001

(

\frac{g}{m^{3} s}

),
volume of the box is

Δ y Δ h = 1000 * 1000 * 1000

(

m^{3}

)

Then:

the emission rate is 1000 (

\frac{kg}{s}

The equations above are expressed in terms of local concentration fields. In contrast, satellite derived products provide column averaged quantities, representing the mean ${CH}_{4}$ concentration integrated over the atmospheric column from the surface to the satellite sensor for each pixel. By vertically integrating the concentration from the surface to the satellite altitude H, we can define the ${CH}_{4}$ mass per unit area as:

Ω = \int_{Surface}^{H} C_{{CH}_{4}} dh

(4)

This formulation allows the approximations expressed in Equations ( 2 ) and ( 3 ) to be rewritten in terms of colum integrated mass, rather than local concentration, by substituting the vertically integrated mass per unit area for the concentration field, yielding Equations ( 5 ) and ( 6 ) respectively.

Q = u Δ y (Ω_{{CH}_{4}} - {Ω_{{CH}_{4}}}^{0})

(5)

Q = \frac{d}{dt} (Ω_{{CH}_{4}} Δ x Δ y)

(6)

These box model approximations have been applied in a wide range of contexts and are referred to by different names depending on the specific assumptions and implementation.

Source pixel method The “source pixel method” as termed by Varon, et al., 2018 and originally attributed to Jacob, et al., 2016 , applies a single box model over a large spatial domain to characterise detectability rather than to estimate emissions for individual plumes. In this approach, detection thresholds are derived as a function of the minimum source emission rate under assumed meteorological conditions. For TROPOMI, Jacob, et al., 2016 estimated a minimum detectable emission rate of approximately 4.2 ( $\frac{t}{hour}$ ) at a wind speed of 5( $\frac{km}{hour}$ ).

Cross Sectional Flux method (CSF) The "Cross Sectional Flux method", applied to both aircraft and satellite observations, estimates emissions by computing the flux of mass through one or more plume cross sections oriented perpendicular to the plume axis ( Varon, et al., 2018 , Section 2.3). Each cross section is referred to as a transect. Conceptually, this approach extends the discrete box model formulation to continuous spatial coordinates and is based on the steady state assumption expressed in Equations ( 2 ) and ( 5 ). The emission rate is obtained by integrating the mass flux across each transect, where the coordinate y parameterises the cross sectional surface.

Q = \int u (x, y) Δ Ω_{{CH}_{4}} (x, y) dy

(7)

Varon, et al., 2018 propose that the effective wind speed used in the Cross Sectional Flux method, denoted $u_{eff}$ , should be derived from the wind speed measured at 10 m above ground level. This height corresponds to the standard level for surface wind observations and is also a commonly available input in numerical weather prediction models. The relationship used to compute $u_{eff}$ is described in Equation (6) of Varon, et al., 2018 :

Q = u_{eff} \int Δ Ω_{{CH}_{4}} (x, y) dy

(8)

This formulation of the inverse problem provides the basis for the method applied by Sadavarte, et al., 2021 . Their implementation determines transect locations and background concentration by computing an upwind box average and identifying a box enclosing the plume. Equation ( 8 ) is then applied with two key modifications. First, emission rates are averaged across up to 12 transects, with a requirement that at least three transects meet the data quality criteria. Second, the effective wind speed $u_{eff}$ is replaced by the pressure averaged boundary layer wind derived from ERA5 reanalysis data.

This emission estimation approach has been applied in several studies, including Lauvaux, et al., 2022 and Sadavarte, et al., 2021 . A closely related variant of the method, employing a different approach to plume identification, has also been used by Schneising, et al., 2023 .

Total Mass method (TM) The "Total Mass" method is based on the second box model approximation expressed by Equations ( 3 ) and ( 6 ). It assumes that there is no net mass flux across the boundaries of the plume and that all emitted mass remains contained within the analysis domain. Under these conditions, the mean emission rate is given by the rate of increase of total mass within the plume, equal to the change in mass over a time interval T divided by T:

Q_{ave} = \frac{IME (T) - IME (0)}{T}

(9)

where Integrated Mass Enhancement function $IME$ is defined as:

IME (t) = \int_{plume(t)}^{} ΔΩ (x,y,t) dx dy

(10)

When applying this method, the time T must be determined independently, using a CTM, local observations, or another appropriate estimation approach.

The TM approach introduces an additional term corresponding to the value of the IME function at the initial time $t = 0$ . In practice, this initial mass term is often assumed to be negligible and omitted from the calculation. However, neglecting this contribution introduces a positive bias and can lead to an overestimation of the emission rate.

Integrated Mass Enhancement method (IME) The "Integrated Mass Enhancement" method is a special case of the TM approach in which the time is expressed as the plume residence time, τ. This residence time is estimated as a function of the effective wind speed and the characteristic length scale of the plume, while the initial mass term is again assumed to be zero. By adopting the 10m effective wind speed $u_{eff}$ and converting the area integral of column mass to a discrete summation over N satellite pixels, this formulation leads to Equation (8) of Varon, et al., 2018 .

Q_{ave} = \frac{1}{τ} IME (τ) = \frac{u_{eff}}{L} IME (τ) = \frac{u_{eff}}{L} \sum_{i = 1}^{N} Δ Ω_{i} A_{i}

(11)

This approach is used in Maasakkers, et al., 2022 .

All of the equations above are formulated in terms of column mass per unit area. In contrast, TROPOMI measurements are based on observations of solar radiation scattered by the Earth' atmosphere and report ${CH}_{4}$ abundance in terms of the column averaged ${CH}_{4}$ mixing ratio. This quantity, commonly denoted $X_{{CH}_{4}}$ , represents the ratio of the number of ${CH}_{4}$ molecules to the total number of air molecules in the atmospheric column and is typically expressed in parts per billion (ppb).

Conversion from a mixing ratio representation to a mass based concentration requires knowledge of the atmospheric column mass and molecular weights and can be performed using relationships derived from Avogadro's law.

Ω_{{CH}_{4}} = M_{{CH}_{4}} X_{{CH}_{4}} \frac{Ω}{M}

(12)

where:
$X_{{CH}_{4}}$ - is ${CH}_{4}$ mixing ratio
$M_{{CH}_{4}} = 16$ - is ${CH}_{4}$ molecular mass ( $\frac{g}{mol}$ )
$M$ - is air molecular mass ( $\frac{g}{mol}$ )
$Ω_{{CH}_{4}}$ - is the mass of the column of ${CH}_{4}$ ( $\frac{kg}{m^{2}}$ )
$Ω$ - is the mass of the column of air ( $\frac{kg}{m^{2}}$ )

The total mass of the atmospheric air column, Ω, can be computed as the surface pressure divided by the acceleration due to gravity.

Ω_{{CH}_{4}} = \frac{M_{{CH}_{4}}}{M} \frac{p}{G} X_{{CH}_{4}}

(13)

where:
$p$ - is surface pressure ( $Pa$ )
$G$ - is graviational acceleration ( $\frac{m}{s^{2}}$ )

The TROPOMI retrieval processor explicitly accounts for atmospheric water vapour (see Section 5.1.1. Eqn. 11 Hasekamp, et. all ATBD ). As a result, the atmospheric column is represented as a mixture of dry air and water vapour rather than dry air alone. For the purposes of this algorithm, the effective molecular mass of air can therefore be computed as a weighted average of the molecular masses of dry air and water vapour, proportional to their respective contributions to the column.

M = \frac{N_{α} M_{α} + N_{H_{2} O} M_{H_{2} O}}{N_{α} + N_{H_{2} O}}

(14)

where:
$N_{α}$ - is the mass of dry air column ( $\frac{mol}{m^{2}}$ )
$N_{H_{2} O}$ - is the mass of dry water column ( $\frac{mol}{m^{2}}$ )
$M_{α} = 28.97$ - is the molecular mass of dry air column ( $\frac{g}{mol}$ )
$M_{H_{2} O} = 18$ - is the molecular mass of water column ( $\frac{g}{mol}$ )

By combining Equations ( 13 ) and ( 14 ), we obtain Equation ( 15 ), which provides a general expression that can be applied under various simplifying assumptions to estimate emission rates.

Ω_{{CH}_{4}} = M_{{CH}_{4}} \frac{N_{α} + N_{H_{2} O}}{N_{α} M_{α} + N_{H_{2} O} M_{H_{2} O}} \frac{p}{G} X_{{CH}_{4}}

(15)

All quantities required by Equation (15) are available from the ERA5 and TROPOMI datasets:

$G$ - gravitational acceleration, available from ERA5 constants.
$p$ - surface pressure, provided by the TROPOMI variable surface_pressure ( $Pa$ ).
$N_{α}$ — dry air column density, provided by the TROPOMI variable dry_air_subcolumns ( $\frac{mol}{m^{2}}$ ).
$N_{H_{2} O}$ — total water vapour column density, provided by the TROPOMI variable water_total_column ( $\frac{mol}{m^{2}}$ ).
$X_{{CH}_{4}}$ - ${CH}_{4}$ column averaged mixing ratio, provided by the TROPOMI variable app-methane_mixing_ratio_bias_corrected ( $ppb$ ).

To obtain the sense of magnitude for variables in this equation.
If:

atmosphere is dry

N_{H_{2} O} = 0

(

ppb

),
mixing ratio

X_{{CH}_{4}} = 1

(

ppb

)

= 10^{-9}

the surface pressure is equal to the standard atmospheric pressure

p = 1013.25

(

hPa

)

Then:

the mass of

{CH}_{4}

per unit area is

5.076 * 10^{-6}

(

\frac{kg}{m^{2}}

Addition of water which has lower molecular mass then the reference molecular mass of dry atmosphere will increase ${CH}_{4}$ mass. Addition of $N_{H_{2} O} = 700$ ( $\frac{mol}{m^{2}}$ ) will change the mass of ${CH}_{4}$ per unit area to $5.711 * 10^{-6}$ ( $\frac{kg}{m^{2}}$ ).

<<>>