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Detecting Atmospheric CO2 Trends as Population-Level Signatures for Long-Term Stable Water Oceans and Biotic Activity on Temperate Terrestrial Exoplanets

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that a future space-based mid-infrared interferometer could detect the carbonate-silicate CO2 trend in a population of as few as 30 temperate rocky exoplanets, even at S/N = 10 and R = 50.

desk verdict Solid mission-concept simulation for LIFE CO2 trend detection, but the headline detectability claim needs a flat-null control before it is clean. read the letter →

arxiv 2505.23230 v1 pith:4ZDJ5K3O submitted 2025-05-29 astro-ph.EP astro-ph.IM

Janina Hansen , Daniel Angerhausen , Sascha P. Quanz , Derek Vance , Björn S. Konrad , Emily O. Garvin , Eleonora Alei , Jens Kammerer
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Felix A. Dannert
This is my paper
classification astro-ph.EPastro-ph.IM
keywords exoplanetatmospherescarbonate-silicatecyclehabitablezonebiosignaturesthermalemissionspectroscopyatmosphericretrievalpopulation-leveltrendsnullinginterferometry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tests whether a future mid-infrared interferometer such as LIFE could measure a population-level trend in atmospheric CO2 partial pressure versus incident stellar flux among temperate terrestrial exoplanets. Such a trend is the observational fingerprint of the carbonate-silicate weathering cycle, the climate thermostat that requires long-lived surface liquid water and is strengthened by land life. The authors build synthetic planet populations from geochemistry-climate model predictions, simulate noisy thermal emission spectra, retrieve CO2 abundances, and apply a hierarchical Bayesian trend inference. They find decisive evidence for a decreasing CO2 trend over a flat trend with as few as 30 planets at the lowest spectrum quality considered, in both biologically enhanced and mostly abiotic weathering scenarios. They also find that retrieval biases flatten the recovered trend and must be corrected before biotic and abiotic populations can be told apart, which then requires at least 100 planets.

What carries the argument

The argument runs through a chain of coupled models and a hierarchical importance-sampling retrieval. Synthetic survey populations are drawn from the Lehmer et al. (2020) coupled geochemistry-climate predictions of the 2D $p\mathrm{CO}_2$-$S$ distribution, with the biotic scenario reproduced from that work and an abiotic scenario obtained by restricting the biological weathering fraction to $[0.1, 0.25]$. A grid of 12 self-consistent climates spanning $p\mathrm{CO}_2 = 10^{-4}$ to $10$ bar and $S/S_\oplus = 0.4$ to $1.0$ is used to generate thermal emission spectra with the ATMOS and petitRADTRANS models, noise-realistic LIFE observations with the LIFEsim simulator, and CO2 posteriors with a Bayesian retrieval. The retrieved posteriors are compressed into maps of Gaussian width $\sigma$ and offset $\Delta$ in the $p\mathrm{CO}_2$-$S$ plane, which assign an observational uncertainty to every synthetic planet. The population trend is then inferred with the hierarchical Bayesian atmospheric retrieval (HBAR) formalism, which reweights each planet's CO2 posterior samples under a semi-logarithmic trend model $\log_{10}(p\mathrm{CO}_2) = \alpha + \beta\,S/S_\oplus$ with scatter $\sigma_N$; comparing the evidence against a flat model yields the Bayes factor used to declare detection.

What would settle it

Run the same pipeline on synthetic populations drawn from a null model with no CO2-flux correlation (uniform pCO2 across the habitable zone, as the paper describes from Lehmer et al. 2020); if the hierarchical retrieval still returns decisive Bayes factors for a linear trend, the detection claim would be an artifact of the retrieval or prior rather than of a real population trend.

Watch

Extended reading notes

Core claim

The central claim is that CO2 trends predicted by the carbonate-silicate cycle are observable as population-level signatures with a LIFE-like mid-infrared nulling interferometer. For populations of $N_P \ge 30$ Exo-Earth Candidates and spectrum qualities as low as $S/N = 10$ and $R = 50$, a linear semi-logarithmic trend in $\log_{10}(p\mathrm{CO}_2)$ versus $S/S_\oplus$ is decisively preferred over a flat trend by Bayes factors $\log_{10}(K) > 2$ in both the biotic scenario (biological weathering fraction $f_\mathrm{bio} = 0.1$ to $1.0$) and the abiotic scenario ($f_\mathrm{bio} = 0.1$ to $0.25$). Detecting the trend does not require high spectral resolution or signal-to-noise; population size is the dominant driver. Distinguishing the biotically enhanced from the abiotic trend is harder: systematic offsets in retrieved CO2 partial pressures, traced mainly to the assumption of vertically constant H2O profiles, bias slope estimates toward flatter values, and accurate differentiation would require $N_P \ge 100$ once those biases are corrected.

Load-bearing premise

The whole detectability test assumes the Lehmer et al. (2020) carbonate-silicate model's predicted range of CO2 partial pressures across the habitable zone, and the chosen biotic and abiotic parameter ranges, actually bracket real temperate exoplanet populations; if the real spread is wider, flatter, or shaped differently, the injection-recovery test overstates how easily the trend would be seen.

Editorial extensions

If this is right

  • With as few as 30 moderately characterized Exo-Earth Candidates, a LIFE-class mission could decide whether the carbonate-silicate thermostat is operating across a planetary population.
  • Detection of the CO2 trend does not hinge on the highest spectral resolutions or signal-to-noise; population size dominates confidence, relaxing mission requirements for this observable.
  • If retrieval biases are corrected, populations of about 100 temperate rocky planets could separate trends shaped by global-scale biotic weathering enhancement from abiotic weathering.
  • The identified bias from vertically constant H2O profiles affects abundance and pressure constraints across HZ atmospheres, so retrieval frameworks used for population studies must be tested against diverse, non-constant atmospheric profiles.
  • Characterizing planets at the hot edge of the habitable zone is intrinsically limited by optically thick, water-rich lower atmospheres, narrowing the accessible population for surface-linked trend studies.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extension suggested by the paper's own logic is to apply the same injection-recovery test to a uniform pCO2-S population (the Lehmer null case), to verify that the Bayes factor does not prefer a linear trend when no trend is injected.
  • The trend-slope difference between biotic and abiotic scenarios implies that the slope itself, not just its presence, could serve as a population-level biosignature; correcting the H2O-profile treatment is a concrete path toward using the slope quantitatively.
  • The same population-level HBAR machinery could be applied to other spatially or chemically correlated atmospheric signatures, such as CH4-O2 disequilibrium or O3, provided their geochemical null distributions are known.
  • Because the paper draws one population per scenario, the reported detection thresholds in the low-$N_P$ regime carry sampling variance; repeating with many population draws would quantify how often a 30-planet survey succeeds.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper presents an injection-recovery study of population-level atmospheric CO2 trends as a function of incident stellar flux, using simulated LIFE thermal-emission observations of temperate terrestrial exoplanets. The authors generate synthetic survey populations from LIFE yield simulations, assign pCO2 values from Lehmer et al. (2020) carbonate-silicate cycle predictions for biotic and abiotic scenarios, and compute pCO2 posteriors from 12 atmospheric model scenarios via retrievals. They interpolate these into sigma/delta posterior maps, then use hierarchical Bayesian retrieval following Lustig-Yaeger et al. (2022) to infer a semi-logarithmic trend and compare linear versus flat trend models via Bayes factors. The headline results are that the linear model is decisively preferred for all population sizes and spectral qualities in the biotic case, and for N_P>=30 in the abiotic case, with the claim of 'robust detection'; differentiation of biotic from abiotic trends requires bias correction and N_P>=100. The paper identifies retrieval biases, especially the constant-H2O-profile assumption, as the main source of trend inaccuracy.

Significance. If the detection claim survives the missing null control, the result is significant for the design and science case of LIFE and comparable mid-infrared interferometers: it would show that Cb-Si driven CO2 trends are accessible with about 30 planets even at S/N=10 and R=50, making population-level habitability characterization a realistic goal. The paper is methodologically transparent and reproducible: it uses established public tools (LIFEsim, petitRADTRANS, pyMultiNest, HBAR), provides source data on Zenodo, and clearly separates detection from differentiation while explicitly listing limitations in Section 4.3. Credit is also due for honest discussion of the single-draw caveat, the 12-point grid, the idealized bias correction, and the model-dependence of the biotic/abiotic distinction. However, because the linear-versus-flat comparison has not been tested against a retrieval-bias-only null population, the central quantitative claim is not yet established; the paper's own discussion of S-dependent pCO2 offsets in Section 4.2.1 and Figures 6 and 15 makes such a control necessary.

major comments (3)
  1. [4.1.1, Eq. (9), Figures 6 and 15] The headline detection claim is not controlled against a retrieval-bias null. The sigma and delta maps in Figures 6 and 15 show that pCO2 posterior offsets are strongly structured in S/S_sun: for pCO2 < 1 bar the offsets grow with incident flux (up to roughly 1 dex at S/S_sun = 1.0), while for pCO2 >= 1 bar they are negative. A true population that is flat in log pCO2 would, after passing through these delta maps and the HBAR likelihood in Eq. (4), produce retrieved posterior medians that trend with S. The linear-over-flat Bayes factor from Eq. (9) could therefore become decisive even when no Cb-Si trend is present. The manuscript only runs the linear-versus-flat comparison on trend-injected populations; a flat-null population has not been pushed through the same sigma/delta maps and HBAR pipeline. Please add this control and report the resulting Bayes factors; if a bias-only population also yields decisive evidence, the N_P>=30 detection claim must be substantially revised or restricted to bias-corrected retrievals.
  2. [4.1.1, Figure 9] The robust-detection claim for N_P>=30 is supported by a single population draw per (N_P, S/N, R) cell. The text acknowledges that a different draw in the low-N_P regime could change trend detectability, but the abstract and Section 5 state the N_P>=30 result without this caveat. Please report the distribution of log10(K) over many population realizations, at least for the lowest spectrum quality (S/N=10, R=50) and N_P=30, and give the fraction of draws that reach decisive evidence. This is necessary to support the word 'robust' in the central claim.
  3. [3.2, 4.3] The trend inference relies on sigma and delta maps interpolated from only 12 atmospheric retrievals, with linear interpolation between grid points and nearest-neighbor extrapolation outside them. The injected Lehmer populations place most planets between the three S/S_sun columns and the six pCO2 rows used in the grid, so the interpolated maps are doing substantial work in assigning every planet's posterior. The paper acknowledges in Section 4.3 that the accuracy of this interpolation remains uncertain, but the detection claim depends on it. Please validate the interpolation by running direct retrievals on a sample of off-grid (pCO2, S) pairs, or by comparing interpolated posteriors with a denser grid, and quantify how any interpolation error propagates into the Bayes factors.
minor comments (3)
  1. [Eq. (9)] The printed expression log10(K) = ln(ZA)-ln(ZB)/ln(10) is missing parentheses; the intended formula is log10(K) = (ln ZA - ln ZB)/ln(10). Please correct the typography.
  2. [2.4, Eq. (7)] The notation N1/2(0,1) for the half-normal prior on sigma_N is not defined in the text; please add a one-sentence definition or use a more standard notation such as HalfNormal(0,1).
  3. [A.1] The hypothesis labels H0,bio and H0,abio are confusing because H0,bio tests the biotic sample against the abiotic slope, not against the biotic slope. Please rename the hypotheses or add a sentence clarifying the naming convention to avoid ambiguity in the Appendix.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the detection claim is a forward-model injection-recovery consistency test; the missing flat-null control is a validity gap, not a definitional reduction.

full rationale

This paper is an injection–recovery study, not a derivation that reduces to its inputs. The chain is: (i) Lehmer et al. (2020) carbonate–silicate model generates pCO2–S distributions for biotic (f_bio = 0.1–1.0) and abiotic (f_bio = 0.1–0.25) populations; (ii) 12 representative atmospheres are forward-modeled with ATMO/petitRADTRANS and observed with LIFEsim; (iii) retrievals produce pCO2 posteriors summarized as sigma and Delta maps; (iv) hierarchical Bayesian trend inference (Lustig-Yaeger et al. 2022) compares linear vs. flat trend models on population draws. The headline claim—linear trend decisively preferred for N_P >= 30—is a consistency check that injected trends survive the retrieval-plus-HBAR pipeline. No fitted parameter is renamed as a prediction: the beta slopes are free parameters in the HBAR fit, and the injected Lehmer trends are explicitly labeled ground truth, not independent predictions. The 'abiotic' population is a restricted f_bio subset of the same model; the paper states 'the abiotic distribution corresponding to a subset of this' and acknowledges the distributions 'likely do not provide an accurate representation of reality,' so the modeling choice is disclosed rather than smuggled in. The most substantive concern, raised by the skeptic, is that the S-dependent retrieval bias (Delta maps, Section 4.2.1) could produce a spurious linear-over-flat preference for a truly flat population because no flat-null injection through the bias maps is reported. That is an experimental-control gap and a threat to the interpretive strength of the detection claim, but it is not circularity: no equation or fitted parameter reduces by definition to another. Self-citations to LIFEsim, P-Pop, the Konrad et al. (2022) retrieval framework, and Kammerer et al. (2022) population data are tool and method citations; their outputs are independently simulated and do not import the target conclusion. Verdict: no circularity (score 0).

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the Lehmer et al. (2020) geochemical model and on a specific operational definition of biotic versus abiotic scenarios. No new physical entities are introduced. The main free choices are the abiotic f_bio range, the coarse 12-point grid, the discrete population sizes, and the HBAR hyperprior, each of which could shift the quantitative conclusions.

free parameters (4)
  • Abiotic biological weathering fraction range = fbio = [0.1, 0.25]
    The abiotic scenario is defined by restricting the biological weathering fraction parameter in the Lehmer et al. (2020) model to values below 0.25, shifting the pCO2-S distribution to higher partial pressures. This choice is based on catchment studies but is not uniquely determined, and the resulting distribution is a subset of the biotic one.
  • Atmospheric grid sampling = pCO2 in {1e-4, 1e-3, 1e-2, 1e-1, 1, 10} bar; S in {0.4, 0.7, 1.0} S_sun
    The 12-scenario grid defines the retrieval posterior maps; results are interpolated across the continuous pCO2-S plane, and the paper notes that extrapolation beyond the grid is uncertain.
  • Survey population size = Np = [10, 30, 50, 100]
    Population sizes are chosen to bracket theoretical requirements from earlier work and yield estimates; the detection threshold is derived from these discrete values.
  • Half-normal hyperprior width = sigma_N ~ N1/2(0,1)
    The HBAR trend inference uses a half-normal prior on the scatter of CO2 about the trend, adopted from Lustig-Yaeger et al. (2022); this affects posterior width and could influence detectability.
assumptions (6)
  • domain assumption Lehmer et al. (2020) carbonate-silicate cycle model produces realistic pCO2-S distributions for temperate terrestrial planets.
    The entire synthetic population is generated from this model via its public software; if the model's parameter ranges or weathering formulations are unrepresentative, the injected trends and the detection test are not meaningful.
  • ad hoc to paper The biotic and abiotic scenarios are adequately represented by fbio ranges [0.1,1.0] and [0.1,0.25], respectively.
    This operational definition of 'abiotic' is a choice made for this study, and the resulting abiotic distribution is a subset of the biotic one, making differentiation intrinsically hard.
  • domain assumption The CO2 trend is semi-log linear in incident flux (Eq. 5).
    The trend inference uses log10(pCO2) = alpha + beta*(S/S_sun), following Lehmer et al. (2020) and earlier work; if the true trend is nonlinear, the test may be mis-specified.
  • domain assumption Retrieval forward models with vertically constant H2O profiles are adequate except for the biases identified in Section 4.2.
    This simplification is common but is shown here to cause systematic offsets in pCO2; the paper argues these biases could be reduced with improved parametrization, but the current results depend on this assumption.
  • domain assumption LIFE mission parameters (2 m aperture, 5% throughput, 4-18.5 um) are representative of a future mid-infrared interferometer.
    The detectability results are tied to the specific performance assumptions of LIFE; other mission concepts with different parameters might yield different thresholds.
  • domain assumption Linear interpolation between 12 retrieval posteriors approximates the true behavior across the pCO2-S plane.
    The posterior maps are built by interpolating between grid points, and the paper states that accuracy outside the grid boundaries is uncertain (Section 4.3).

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Cite this review

Pith. "Pith review of Detecting Atmospheric CO2 Trends as Population-Level Signatures for Long-Term Stable Water Oceans and Biotic Activity on Temperate Terrestrial Exoplanets." pith.science (2026). https://pith.science/paper/4ZDJ5K3O

@misc{pith2026250523230,
  author       = {Pith},
  title        = {Pith review of: Detecting Atmospheric CO2 Trends as Population-Level Signatures for Long-Term Stable Water Oceans and Biotic Activity on Temperate Terrestrial Exoplanets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ZDJ5K3O}},
  note         = {Machine review of arXiv:2505.23230}
}
abstract

Identifying key observables is essential for enhancing our knowledge of exoplanet habitability and biospheres, as well as improving future mission capabilities. While currently challenging, future observatories such as the Large Interferometer for Exoplanets (LIFE) will enable atmospheric observations of a diverse sample of temperate terrestrial worlds. Using thermal emission spectra that represent conventional predictions of atmospheric CO2 variability across the Habitable Zone (HZ), we assess the ability of the LIFE mission - as a specific concept for a future space-based interferometer - to detect CO2 trends indicative of the carbonate-silicate (Cb-Si) weathering feedback, a well-known habitability marker and potential biological tracer. Therefore, we explore the feasibility of differentiating between CO2 trends in biotic and abiotic planet populations. We create synthetic exoplanet populations based on geochemistry-climate predictions and perform retrievals on simulated thermal emission observations. The results demonstrate the robust detection of population-level CO2 trends in both biotic and abiotic scenarios for population sizes as small as 30 Exo-Earth Candidates (EECs) and the lowest assessed spectrum quality in terms of signal-to-noise ratio, S/N = 10, and spectral resolution, R = 50. However, biased CO2 partial pressure constraints hinder accurate differentiation between biotic and abiotic trends. If these biases were corrected, accurate differentiation could be achieved for populations with $\geq$ 100 EECs. We conclude that LIFE can effectively enable population-level characterization of temperate terrestrial atmospheres and detect Cb-Si cycle driven CO2 trends as habitability indicators. Nevertheless, the identified biases underscore the importance of testing atmospheric characterization performance against the broad diversity expected for planetary populations.

Figures

Figures reproduced from arXiv: 2505.23230 by the authors.

Figure 1
Figure 1. Schematic of our trend survey and inference methodology. We generate synthetic atmospheric populations by associating each planet with an incident flux S, injecting the predicted Cb-Si cycle-driven pCO2 variability across the HZ, and assigning observational uncertainties δpCO2. These populations then serve as input for the trend retrieval routine. To produce the HZ detection yield population, LIFEsim estimates the i… view at source ↗
Figure 2
Figure 2. Distribution of simulated EEC yield (solid con￾tours) and underlying planet population (dotted contours) around FGK-type stars within 20 pc. We show the plane￾tary radius over incident flux distribution, where the color shading represents the relative density of planet occurrences in this parameter space. The colored contours enclose 10% to 100% of the total planet counts, with the innermost con￾tour indicating the … view at source ↗
Figure 3
Figure 3. 2D pCO2-S distributions using coupled climate and Cb-Si cycle models (Krissansen-Totton & Catling 2017; Krissansen-Totton et al. 2018; Lehmer et al. 2020). Left panel: Biotic distribution reproducing Lehmer et al. (2020). Right panel: Abiotic distribution showing a limited biosphere contribution to the Cb-Si weathering feedback. Black lines represent weighted least squares (WLS) trend fits, yielding βbio = 3.96±0.18… view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: Atmospheric profiles and emission spectra for the 12 atmospheric scenarios covered by our grid. (a) Atmospheric grid structure visualization. (b) Simulated emission spectra with LIFEsim noise estimates for S/N = 20 and R = 100. (c)-(e) VMR profiles for CO2, H2O, and N2…
Figure 5
Figure 5. Figure 5: Posterior distributions of CO2 abundance derived from atmospheric spectrum retrievals. The panels are ar￾ranged to reflect the atmospheric grid within the pCO2-S parameter space. Columns represent different stellar insolation values, S/S⊕ = 1.0, 0.7, 0.4, while rows co…
Figure 6
Figure 6. Figure 6: Maps of posterior standard deviations (σ) and offsets (∆) for observational sensitivity case S/N = 20 and R = 100. Color shading indicates the magnitude of σ (top) and ∆ (bottom), with blue and purple representing positive and negative offsets, respectively. Black hori…
Figure 7
Figure 7. Figure 7: Retrieved population-level trends for biotic (top) and abiotic (bottom) scenarios for spectrum quality case S/N = 20 and R = 100. We differentiate between population sizes NP = [10, 30, 50, 100]. Dotted blue lines indicate median retrieved trends while blue shading ind…
Figure 8
Figure 8. Figure 8: Retrieved posterior distributions of the slope parameter β for biotic (top) and abiotic (bottom) scenarios for spectrum quality case S/N = 20 and R = 100. We differentiate between population sizes NP = [10, 30, 50, 100]. Dashed vertical lines indicate theoretical bioti…
Figure 9
Figure 9. Figure 9: Bayes’ factor comparison of linear and flat trend models for biotic (top) and abiotic (bottom) scenarios and for various combinations of population size NP = [10,30,50,100] and spectrum quality , S/N = [10,20] and R = [50,100]. Posi￾tive log10(K) values (blue) indicate…
Figure 10
Figure 10. Figure 10: Comparison of trend distinction performance across four model assumptions in biotic (top) and abiotic (bottom) scenarios: 1) Base scenario, 2) No offset scenario, 3) Fixed standard deviation scenario, and 4) Reduced scatter scenario. We limit the model comparison to p…
Figure 11
Figure 11. Figure 11: Comparison of atmospheric retrieval results between variable (top) and constant (bottom) H2O profiles in the input spectrum of the pCO2 = 10−4 bar and S/S⊕ = 1.0 scenario, for an observational sensitivity of S/N = 20 and R = 100. We show, from left to right, retrieved…
Figure 12
Figure 12. Figure 12: Atmospheric retrieval results for three CO2-dominated scenarios from our atmospheric population. Case 1 (left): pCO2 = 1 bar, S/S⊕ = 1.0; Case 2 (center): pCO2 = 10 bar, S/S⊕ = 0.4; and Case 3 (right): pCO2 = 10 bar, S/S⊕ = 0.4. The top row illustrates the posterior d…
Figure 13
Figure 13. Figure 13: Minimum population sizes required to detect and differentiate biotic and abiotic CO2 trends. Left: Visualization of the minimum population size inference approach. Right: Map of minimum required population sizes NP to achieve statistical power between 95% and 99.9% ac…
Figure 14
Figure 14. Figure 14: As top panel in [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: As bottom panel in [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: As [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: Posterior distributions of H2O abundance derived from atmospheric spectrum retrievals. The panels are ar￾ranged to reflect the atmospheric grid within the pCO2-S parameter space. Columns represent different stellar insolation values, S/S⊕ = 1.0, 0.7, 0.4, while rows c…
Figure 18
Figure 18. Figure 18: P − T profiles derived from atmospheric spectrum retrievals for S/N = 10 and R = 50. The panels are arranged to reflect the atmospheric grid within the pCO2-S parameter space. Columns represent different stellar insolation values, S/S⊕ = 1.0, 0.7, 0.4, while rows corr…
Figure 19
Figure 19. Figure 19: Same as [PITH_FULL_IMAGE:figures/full_fig_p029_19.png]
Figure 20
Figure 20. Figure 20: Same as [PITH_FULL_IMAGE:figures/full_fig_p030_20.png]
Figure 21
Figure 21. Figure 21: Same as [PITH_FULL_IMAGE:figures/full_fig_p031_21.png]
Figure 22
Figure 22. Figure 22: As [PITH_FULL_IMAGE:figures/full_fig_p032_22.png]
Figure 23
Figure 23. Figure 23: As [PITH_FULL_IMAGE:figures/full_fig_p033_23.png]
Figure 24
Figure 24. Figure 24: As [PITH_FULL_IMAGE:figures/full_fig_p034_24.png]

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