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<h1>Example - From absorption</h1>
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alt="Active Passive"
style="width: 100vw; height: auto; display: block;">
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<section>
<h3>Sample example calculation: How Much CO₂ Do Satellites Measure?</h3>
<p><b>What Do Satellites Actually Measure?</b></p>
<p>Satellites like: OCO-2, Sentinel-5P, do not directly measure ppm.</p>
<p>They measure:</p>
$$\text{Spectral radiance} ~~I$$
<p>at specific CO₂ absorption wavelengths (~1.6 µm, 2.0 µm).</p>
<h4>Basic Physical Principle (Beer–Lambert Law)</h4>
<p>For a single absorption band:</p>
$$I= I_0 e^{-\tau}$$
Where:
<ul>
<li>\(I\) = measured radiance</li>
<li>\(I_0\) = incoming radiance</li>
<li>\(\tau = n\sigma L\) = optical depth</li>
<li>\(\sigma\) = absorption cross section</li>
<li>\(n\) = CO₂ number density</li>
<li>\(L\) = path length</li>
</ul>
<p>Now assume, incoming radiance: \(I_0\) = 100 units.</p>
<p>and let's assume Satellite measures: \(I\) = 85 units.</p>
<p>Now using Beer-Lambert law: \(I= I_0 e^{-\tau}\)</p>
$$85 = 100 e^{-\tau} \Rightarrow \tau = 0.163$$
<p><b>Relating Optical Depth to CO₂ Concentration:</b></p>
<p>Now using formula: \(\tau = n\sigma L\), and considring \(\sigma =2\times 10^{-23} ~\text{m}^2\) and \(L= 8000 ~m\), then</p>
$$n =\frac{\tau}{\sigma L} = 1.02\times 10^{18}~\text{molecules}/\text{m}^3$$
<p>This is number density of CO₂ molecules per cubic meter.</p>
<p><b>What is XCO₂?</b></p>
<p>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.
</p>
<p><b>Mathematical Definition:</b></p>
$$XCO_2 = \frac{\text{Column CO}_2}{\text{Total air column}}$$
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.
<p><b>What is total air number density?</b> </p>
<p> 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.</p>
<p></p>
<div class="grey-box">
We start from: \(PV = N kT\). Rearranging for number density \( n = N/V\). Therefore, \(n = P / kT\).
<p>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 \(n\):
$$n \approx 2.55 \times 10^{25} ~\text{molecules}/\text{m}^3$$
</p>
</div>
<p>Therefore the mixing ratio, </p>
$$\text{Mixing ratio} = \frac{n_{{\rm CO}_2}}{n_{{\rm air}}} = 4.0 \times 19^{-8}.$$
<p>Now convert to ppm: </p>
$$1~{\rm ppm} = 10^{-6}$$
so:
$$XCO_2 = 0.0408 ~\text{ppm}.$$
<p>This is far too small compared to real atmospheric CO₂ (~420 ppm). </p>
<p>Why?</p>
<p><b>Because:</b> 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.
</p>
<!-- new heading -->
<h4>What is XCO₂?</h4>
<p>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 <b>ppm (parts per million)</b>.</p>
<p>Real \(\text{XCO}_2\) computed from:</p>
$$
XCO_2 = \frac{\int_0^\infty n_{{\rm CO}_2}(z) dz}{\int_0^\infty n_{{\rm air}}(z) dz}
$$
Where:
<ul>
<li>\(n_{{\rm CO}_2}(z)\) = 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.
</li>
<li>\(n_{{\rm air}}\) = dry air number density</li>
<li>Integral over height gives column amount</li>
</ul>
So XCO₂ is a ratio of column amounts. In above equation, integrals are done from surface \(z=0\), to top of atmosphere.
<p><b>Why Do We Convert \(n\) to XCO\(_2\)?</b></p>
<p>Because number density alone is not meaningful globally. Now from ideal gas law:</p>
$$n = \frac{P}{k~T}$$
So:
<ul>
<li>At sea level → n is high</li>
<li>At high altitude → n is low</li>
<li>In warm regions → n changes</li>
<li>In cold regions → n changes</li>
</ul>
<p>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.
</p>
<p>Now, when we consider the mixing ratio i.e. <b>XCO₂</b>, it removes pressure dependence. When we divide by total air molecules, we can calculate local mixing ratio:</p>
$$
xCO_2 = \frac{n_{{\rm CO}_2}}{n_{\rm air}}
$$
<p>So, pressure cancels out and we get <b>pure mixing ratio</b>. This tells us, What fraction of the atmosphere is CO₂ independent of altitude and pressure.</p>
<p></p>
<div class="grey-box">
<p><b>Deriving Column Air Using Hydrostatic Balance:</b></p>
We start from hydrostatic equilibrium:
$$\frac{dP}{dz}= -\rho ~g$$
where:
<ul>
<li>\(P\)=pressure</li>
<li>\(\rho\) = air density (kg/m³)</li>
<li>\(g\) = gravity</li>
</ul>
<p>Now, converting mass density to number density:</p>
$$\rho = m_{\rm air} n_{\rm air}$$
where \(m_{\rm air}\) = mean molecular mass of air and \(n_{\rm air}\) = number density (molecules/m³)
<p>Now substituting in the equation:</p>
$$\frac{dP}{dz}= -m_{\rm air} n_{\rm air}~g \Rightarrow n_{\rm air} dz= -\frac{1}{m_{\rm air}g} dP$$
Total column air can be calculated by integrating this equation:
$$ \int_0^\infty n_{\rm air} dz= -\frac{1}{m_{\rm air}g} \int_0^{P_s} dP$$
<p>Because pressure decreases upward, we can simplify above equation as:</p>
$$N_{\rm air} = \frac{P_s}{m_{\rm air}g}$$
</div>
<p></p>
<h4>Why Satellites Retrieve XCO₂ (Not n)?</h4>
<p>Define local mixing ratio:</p>
$$
x_{{\rm CO}_2} = \frac{n_{{\rm CO}_2}}{n_{\rm air}} \Rightarrow n_{{\rm CO}_2} = x_{{\rm CO}_2}(z) ~ n_{\rm air}(z)
$$
<p>Now global mixing ratio is:</p>
$$
XCO_2 = \frac{\int_0^\infty n_{{\rm CO}_2}(z) dz}{\int_0^\infty n_{{\rm air}}(z) dz}
= \frac{\int_0^\infty x_{{\rm CO}_2}(z) ~ n_{\rm air}(z) dz}{\int_0^\infty n_{{\rm air}}(z) dz} dP
= \frac{\int_0^{P_s} x_{{\rm CO}_2}(P) dP}{- \int_0^{P_s} dP}
$$
Which can be simplified to:
$$XCO_2 =\frac{1}{P_s}\int_0^{P_s} x_{{\rm CO}_2}(P)dP$$
where \(P_s = m_{\rm air}g / \sigma\).
<p>Satellites like: OCO-2, OCO-3, GOSAT measure total column absorption. Absorption depends on:</p>
$$\tau = \int \sigma n_{{\rm CO}_2} (z) dz$$
<p>Now substituting \(n_{{\rm CO}_2} = x_{CO_2}~n_{\rm air}\), </p>
$$\tau = \int \sigma x_{CO_2}(z)~n_{\rm air}(z) dz$$
<p>Convert to pressure coordinate:</p>
$$\tau = \frac{\sigma}{m_{\rm air}g}\int_0^{P_s} x_{CO_2}(P)~dP$$
Therefore,
$\tau \propto \int_0^{P_s} x_{CO_2}(P)~dP$
But earlier we derived \(XCO_2\):
$$\tau \propto P_s XCO_2$$
<div class="important-box">
Since from satellite, we measure \(I\) and hence \(\tau \sim \text{column}~\text{CO}_2 \). Using hydrostatic balance:
$$\text{column}~\text{CO}_2 \sim P_s XCO_2$$
<p>So retrieval algorithm effectively solves:</p>
$$\text{Radiance} \rightarrow \tau \rightarrow XCO_2.$$
<p>XCO₂ is the pressure-weighted vertical average of CO₂ mixing ratio, derived directly from hydrostatic balance and the radiative transfer equation.</p>
</div>
So what they retrieve is:
$$N_{{\rm CO}_2} = \int n_{{\rm CO}_2}(z) dz $$
<p>Then they normalize by total air column and gives \(XCO_2\).</p>
<p></p>
<p></p>
<hr>
<h3>Reference</h3>
<ul>
<li><a href="https://arunp77.github.io/Remote-sensing.html" target="_blank">Fundamentals of Remote sensing</a></li>
<li><a href="https://arunp77.github.io/electromagnetic-waves.html" target="_blank">Relevance of Electromagnetic waves in the context of earth observation</a></li>
<li><a href="https://arunp77.github.io/Remote-sensing-content.html" target="_blank">Concept of the orbits for a satellite (non scientific discussion)</a></li>
<li><a href="https://arunp77.github.io/Ground-segment-systems.html" target="_blank">How various teams works in close collaboration for the ground data processing?</a></li>
<li><a href="https://arunp77.github.io/Satellite-data.html" target="_blank">How raw satellite data is processed to do a level where you do your scientifc research?</a></li>
<li><a href="https://arunp77.github.io/toa-reflectance.html" target="_blank">In depth understandingof the satellite data (op-of-atmosphere reflectance)</a></li>
<li><a href="https://arunp77.github.io/Satellites-sensors.html" target="_blank">Resolution and calibration</a></li>
<li><a href="https://arunp77.github.io/OLCI.html" target="_blank">Understanding how OLCI data is processed</a></li>
<li><a href="https://www.nesdis.noaa.gov/news/transforming-energy-imagery-how-satellite-data-becomes-stunning-views-of-earth" target="_blank">Transforming Energy into Imagery: How Satellite Data Becomes Stunning Views of Earth</a></li>
</ul>
</section>
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