Example - From absorption

Sample example calculation: How Much CO₂ Do Satellites Measure?

What Do Satellites Actually Measure?

Satellites like: OCO-2, Sentinel-5P, do not directly measure ppm.

They measure:

$$\text{Spectral radiance} ~~I$$

at specific CO₂ absorption wavelengths (~1.6 µm, 2.0 µm).

Basic Physical Principle (Beer–Lambert Law)

For a single absorption band:

$$I= I_0 e^{-\tau}$$ Where:
  • \(I\) = measured radiance
  • \(I_0\) = incoming radiance
  • \(\tau = n\sigma L\) = optical depth
  • \(\sigma\) = absorption cross section
  • \(n\) = CO₂ number density
  • \(L\) = path length

Now assume, incoming radiance: \(I_0\) = 100 units.

and let's assume Satellite measures: \(I\) = 85 units.

Now using Beer-Lambert law: \(I= I_0 e^{-\tau}\)

$$85 = 100 e^{-\tau} \Rightarrow \tau = 0.163$$

Relating Optical Depth to CO₂ Concentration:

Now using formula: \(\tau = n\sigma L\), and considring \(\sigma =2\times 10^{-23} ~\text{m}^2\) and \(L= 8000 ~m\), then

$$n =\frac{\tau}{\sigma L} = 1.02\times 10^{18}~\text{molecules}/\text{m}^3$$

This is number density of CO₂ molecules per cubic meter.

What is XCO₂?

Now we need to calculate how many CO₂ molecules exist per million dry air molecules in the entire atmospheric column and is represented by XCO₂. It is measured in terms of ppm (parts per million). So, XCO₂ is not number density.

Mathematical Definition:

$$XCO_2 = \frac{\text{Column CO}_2}{\text{Total air column}}$$ In simple terms: $$XCO_2 = \frac{\text{Column CO}_2 ~\text{Molecules}}{\text{Total air molecules}}$$ expressed in ppm. So we must compare CO₂ number density with total air number density.

What is total air number density?

At surface (approximate): \(n_{\rm air} \approx 2.5\times 10^{25} ~\text{molecules}/\text{m}^3\). This comes from ideal gas law at 1 atm, 288 K.

We start from: \(PV = N kT\). Rearranging for number density \( n = N/V\). Therefore, \(n = P / kT\).

Now for presure \(P = 1 ~\text{atm} = 1.013\times 10^5~{\rm Pa}\) and temperature: \(T=288 K\) and Boltzman constand \(k = 1.38\times 10^{-23} ~J/K\). Therefore from these values we can calculate \(n\): $$n \approx 2.55 \times 10^{25} ~\text{molecules}/\text{m}^3$$

Therefore the mixing ratio,

$$\text{Mixing ratio} = \frac{n_{{\rm CO}_2}}{n_{{\rm air}}} = 4.0 \times 19^{-8}.$$

Now convert to ppm:

$$1~{\rm ppm} = 10^{-6}$$ so: $$XCO_2 = 0.0408 ~\text{ppm}.$$

This is far too small compared to real atmospheric CO₂ (~420 ppm).

Why?

Because: We used a very simplified path length. We did not integrate over full atmospheric column. Real retrieval uses column integration. So, whatever, we just calculated, it is just a conceptual, not physically consistent.

What is XCO₂?

XCO₂ tells us how many CO₂ molecules exist per million dry air molecules in the entire atmospheric column, and it is expressed in terms of ppm (parts per million).

Real \(\text{XCO}_2\) computed from:

$$ XCO_2 = \frac{\int_0^\infty n_{{\rm CO}_2}(z) dz}{\int_0^\infty n_{{\rm air}}(z) dz} $$ Where:
  • \(n_{{\rm CO}_2}(z)\) = CO₂ number density (molecules per m\(^3\)) at height \(z\). At each height \(z\), this tells us, how many CO₂ molecules exist in one cubic meter of air at that height.
  • \(n_{{\rm air}}\) = dry air number density
  • Integral over height gives column amount
So XCO₂ is a ratio of column amounts. In above equation, integrals are done from surface \(z=0\), to top of atmosphere.

Why Do We Convert \(n\) to XCO\(_2\)?

Because number density alone is not meaningful globally. Now from ideal gas law:

$$n = \frac{P}{k~T}$$ So:
  • At sea level → n is high
  • At high altitude → n is low
  • In warm regions → n changes
  • In cold regions → n changes

Therefore, \(n\) varies strongly with altitude, pressure, temperature n varies strongly with altitude, pressure, temperature. So it is NOT a good quantity to compare globally.

Now, when we consider the mixing ratio i.e. XCO₂, it removes pressure dependence. When we divide by total air molecules, we can calculate local mixing ratio:

$$ xCO_2 = \frac{n_{{\rm CO}_2}}{n_{\rm air}} $$

So, pressure cancels out and we get pure mixing ratio. This tells us, What fraction of the atmosphere is CO₂ independent of altitude and pressure.

Deriving Column Air Using Hydrostatic Balance:

We start from hydrostatic equilibrium: $$\frac{dP}{dz}= -\rho ~g$$ where:
  • \(P\)=pressure
  • \(\rho\) = air density (kg/m³)
  • \(g\) = gravity

Now, converting mass density to number density:

$$\rho = m_{\rm air} n_{\rm air}$$ where \(m_{\rm air}\) = mean molecular mass of air and \(n_{\rm air}\) = number density (molecules/m³)

Now substituting in the equation:

$$\frac{dP}{dz}= -m_{\rm air} n_{\rm air}~g \Rightarrow n_{\rm air} dz= -\frac{1}{m_{\rm air}g} dP$$ Total column air can be calculated by integrating this equation: $$ \int_0^\infty n_{\rm air} dz= -\frac{1}{m_{\rm air}g} \int_0^{P_s} dP$$

Because pressure decreases upward, we can simplify above equation as:

$$N_{\rm air} = \frac{P_s}{m_{\rm air}g}$$

Why Satellites Retrieve XCO₂ (Not n)?

Define local mixing ratio:

$$ x_{{\rm CO}_2} = \frac{n_{{\rm CO}_2}}{n_{\rm air}} \Rightarrow n_{{\rm CO}_2} = x_{{\rm CO}_2}(z) ~ n_{\rm air}(z) $$

Now global mixing ratio is:

$$ XCO_2 = \frac{\int_0^\infty n_{{\rm CO}_2}(z) dz}{\int_0^\infty n_{{\rm air}}(z) dz} = \frac{\int_0^\infty x_{{\rm CO}_2}(z) ~ n_{\rm air}(z) dz}{\int_0^\infty n_{{\rm air}}(z) dz} dP = \frac{\int_0^{P_s} x_{{\rm CO}_2}(P) dP}{- \int_0^{P_s} dP} $$ Which can be simplified to: $$XCO_2 =\frac{1}{P_s}\int_0^{P_s} x_{{\rm CO}_2}(P)dP$$ where \(P_s = m_{\rm air}g / \sigma\).

Satellites like: OCO-2, OCO-3, GOSAT measure total column absorption. Absorption depends on:

$$\tau = \int \sigma n_{{\rm CO}_2} (z) dz$$

Now substituting \(n_{{\rm CO}_2} = x_{CO_2}~n_{\rm air}\),

$$\tau = \int \sigma x_{CO_2}(z)~n_{\rm air}(z) dz$$

Convert to pressure coordinate:

$$\tau = \frac{\sigma}{m_{\rm air}g}\int_0^{P_s} x_{CO_2}(P)~dP$$ Therefore, $\tau \propto \int_0^{P_s} x_{CO_2}(P)~dP$ But earlier we derived \(XCO_2\): $$\tau \propto P_s XCO_2$$
Since from satellite, we measure \(I\) and hence \(\tau \sim \text{column}~\text{CO}_2 \). Using hydrostatic balance: $$\text{column}~\text{CO}_2 \sim P_s XCO_2$$

So retrieval algorithm effectively solves:

$$\text{Radiance} \rightarrow \tau \rightarrow XCO_2.$$

XCO₂ is the pressure-weighted vertical average of CO₂ mixing ratio, derived directly from hydrostatic balance and the radiative transfer equation.

So what they retrieve is: $$N_{{\rm CO}_2} = \int n_{{\rm CO}_2}(z) dz $$

Then they normalize by total air column and gives \(XCO_2\).

Reference

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