A Reduced One-Dimensional Source–Transport–Observation Analysis for Soil-Gas Interpretation at the Soil–Atmosphere Interface

Abstract

Soil-gas observations can retain a source-related response while remaining ambiguous with respect to source amplitude, source depth, transport state, water state, and measurement support. This study develops a reduced one-dimensional source–transport–observation analysis for gas-continuous unsaturated soils. Integral-normalized source kernels, a finite-volume solver, a raw scaled-coordinate Jacobian, common-threshold nuisance projection, and an auxiliary noise-scaled check are combined in a reproducible workflow. At a display tolerance of 0.05, the primary structural convention normalizes parameter columns over the complete 11-row observation universe before extracting observation subsets. Under this convention, a carbon dioxide (CO₂) surface-flux observation retains one projected amplitude direction; within-set normalization reduces that restricted result to zero. The CO₂ profile, ideal-state-constraint set, and full diagnostic set retain ranks of 1/2, 2/3, and 3/4, respectively, under both conventions. The noise-scaled calculation places one projected source response at or above the illustrative one-standard-deviation threshold within a numerical tolerance of 1 × 10⁻¹⁰. A published-data worked example shows why water and carbonate context are required before a gas-phase CO₂ deficit is interpreted as a source decrease. The result is a pre-field screening method, not a site-calibrated inversion.

1. Introduction

Soil-gas measurements are used to infer biological activity, geogenic input, radiogenic production, leakage, and exchange at the soil–atmosphere interface. The inference is indirect because concentration, flux, isotope, noble-gas, radon, and thoron signals are transformed by biological production, water state, pore structure, gas-specific transport, atmospheric forcing, and measurement support before they are observed.

The diagnostic question addressed here is narrower than general source inversion: under which conditions does an observation set retain source-related directions after plausible transport, forcing, water-state, and support sensitivities are represented? The individual process modules are established. The methodological gap lies in combining them within a reproducible local screening analysis that separates source amplitude, source geometry, and nuisance controls without presenting structural rank as field-scale source identifiability [1,2,3].

The contribution is a one-dimensional, quasi-steady source–transport–observation analysis built around four elements. First, spatial source kernels are integral-normalized so that a geometry perturbation does not change the integrated source amplitude. Second, local sensitivities are retained in raw scaled-coordinate form before either structural normalization or noise scaling. Third, the source blocks are projected onto the complement of a nuisance subspace truncated with the same absolute threshold used for the augmented matrix. Fourth, structural separability is reported separately from an illustrative noise-scaled detectability check. This combination, rather than any individual transport law or singular-value method, is the methodological novelty. The reduced transport assumptions follow established treatments of pressure-driven advection, Fickian trace-gas transport, and soil–atmosphere exchange [4,5,6,7].

The domain is a gas-continuous unsaturated soil column. Water is prescribed as a state variable rather than solved with a Richards equation. It affects gas connectivity and effective diffusivity in every snapshot; it affects steady radionuclide profiles through the decay-storage term; and it would produce dynamic retardation for soluble gases only in a transient calculation. Saturated or gas-disconnected states lie outside the numerical domain. The air-filled-porosity closure and the treatment of interfacial and hysteretic effects are reduced representations of established porous-media relations [8,9,10,11].

The same ambiguity is relevant to vegetated slopes, bioengineered and evapotranspirative covers, landfill covers, and other eco-geotechnical systems, where roots, evapotranspiration, suction, and layered hydraulic structure alter air-filled porosity and gas transport. These settings are discussed as applications of the diagnostic logic, not as processes explicitly simulated by the present quasi-steady column.

Three hypotheses organize the analysis. H1: a restricted carbon dioxide (CO₂) observation set can retain a source-related direction yet remain attribution-ambiguous because source amplitude, source geometry, transport, and support can produce similar responses. H2: decay-separated tracers and independently supplied nuisance-state constraints can add projected source directions in the synthetic design, although the gain is not a universal guarantee for any specific sensor package. H3: the practical value of the same observation set depends on the soil end-member. H3 is examined through a literature-anchored heuristic soil translation rather than through a separate Jacobian sweep for every soil class.

The study reports synthetic experiments, common-threshold projected-rank diagnostics, grid-refinement and closure-sensitivity checks, a quantitative worked example based on published regression coefficients and open dataset metadata, and a targeted pattern-level literature comparison. No field calibration or full raw time-series inversion is attempted.

Figure 1 follows the study sequence from the source question through the soil filter and observation support, the reduced model, synthetic outputs, local-sensitivity/Jacobian analysis, published evidence and literature patterns, and the final interpretation. Nuisance-projected structural rank and the separate noise-scaled check are derived from the Jacobian in Section 3.8.

The article is organized as follows. Section 2 defines the source–transport–observation problem. Section 3 presents assumptions, equations, numerical implementation, and diagnostics. Section 4 translates soil classes into heuristic end-members. Section 5 reports the synthetic and robustness results. Section 6 gives the published-data worked example. Section 7 reports the targeted literature consistency check. Section 8, Section 9 and Section 10 discuss implications, field translation, limitations, and conclusions.

2. Source–Transport–Observation System

The reduced observation model is

$$\mathbf{y}=\mathbf{h}({\mathbf{S}}_A,{\mathbf{G}}_S,\mathit{\eta }),$$

(1)

where $\mathbf{y}$ is the observation vector, ${\mathbf{S}}_A$ contains source-amplitude parameters, ${\mathbf{G}}_S$ contains source-geometry parameters, and $\mathit{\eta }$ contains transport, gas-continuity, forcing, and observation-support nuisance parameters.

Local linearization gives sensitivity blocks ${\mathbf{J}}_A$, ${\mathbf{J}}_G$, and ${\mathbf{J}}_{\eta }$. Source informativeness is evaluated after truncating the nuisance subspace with a declared absolute singular-value threshold $ϵ$. If ${\mathbf{U}}_{\eta ,ϵ}$ contains the retained nuisance left singular vectors, then

$${\mathbf{P}}_{\eta ,ϵ}={\mathbf{U}}_{\eta ,ϵ}{\mathbf{U}}_{\eta ,ϵ}^{\mathsf{T}},{\mathbf{J}}_{A,\perp }=(\mathbf{I}-{\mathbf{P}}_{\eta ,ϵ}){\mathbf{J}}_A.$$

(2)

A nonzero numerical rank of ${\mathbf{J}}_{A,\perp }$ indicates a locally retained source-amplitude direction beyond the nuisance subspace at the selected threshold. It does not establish a unique field source, practical identifiability, or detectability under measurement noise. Source geometry is treated analogously and is reported separately because effective depth is part of the source hypothesis rather than a transport nuisance parameter.

This definition resolves two distinct questions. Structural analysis asks whether source sensitivities are non-collinear with retained nuisance sensitivities. A separate noise-scaled analysis asks whether the magnitude of the raw projected response lies at or above an illustrative one-standard-deviation observation scale within a declared numerical tolerance. The two results must not be conflated.

The formal derivation is reported in Appendix A.

3. Methods

The methods define the reduced soil column, the gas-specific steady forward problem, water-state closure, source normalization, observation operator, raw sensitivity calculation, common-threshold projected diagnostics, numerical robustness checks, and literature-data treatment.

Table 1. Principal assumptions of the reduced theoretical–numerical analysis.

Assumption	Implementation	Implication
Geometry	one-dimensional column, $0\le z\le 2$ m	vertical screening only; no lateral heterogeneity
Time treatment	steady diagnostic snapshots derived from a transient process template	no infiltration, evapotranspiration, or source-history reconstruction
Gas domain	gas-continuous unsaturated soil	gas-disconnected and saturated states require a multiphase model
Water state	prescribed ${\theta }_a$, with ${\theta }_w=\varphi -{\theta }_a$ and $\varphi =0.45$	hydrological state is not a Richards-equation solution
Spatial coefficients	${\theta }_a$, ${\theta }_w$, $D_i^{\mathrm{eff}}$, $R_i$, and $v_g$ are uniform within each snapshot	depth-variable coefficients and layered flow require an extended model
Diffusivity closure	normalized power law $D_i^{\mathrm{eff}}=D_i^0{D}_{\mathrm{scale}}{({\theta }_a/{\theta }_{a,0})}^{{\beta }_g}$	controlled closure, not a universal pedotransfer relation
Source representation	each Gaussian or uniform source kernel has unit depth integral before amplitude scaling	geometry changes support, not integrated source strength
Lower boundary	baseline $J_{z,i}\left(L\right)=0$ with internal sources	deep scenarios are effective internal source supports
State constraints	pressure- and water-state rows are idealized independent nuisance constraints	they are not universal direct field measurements
Structural sensitivity	raw finite differences, diagnostic row weighting, then column normalization	evaluates local sensitivity geometry, not detectability
Noise-scaled check	raw Jacobian divided by declared observation standard deviations before any column normalization	illustrative detectability check, not a calibrated field error model

consolidates the assumptions that delimit the analysis.

3.1. Synthetic Soil Column, Boundary Conditions, and Observation Design

The synthetic domain is a one-dimensional column $z\in [0,L]$, with the surface at $z=0$, lower boundary at $L=2$ m, and positive z downward. The baseline state is gas-continuous and unsaturated. Total porosity is fixed at $\varphi =0.45$, and the prescribed water-filled porosity is ${\theta }_w=\varphi -{\theta }_a$, so the phase fractions remain internally consistent.

The coordinate flux is   

$$J_{z,i}=-D_i^{\mathrm{eff}}{\displaystyle \frac{\partial c_i}{\partial z}}+v_g{c}_i,$$

(3)

where positive values are downward. The upward surface flux is

$$F_{\mathrm{up},i}=-J_{z,i}\left(0\right)=k_{\mathrm{ex},i}[c_i\left(0\right)-c_{i,\mathrm{atm}}].$$

(4)

The scaled anomaly calculations use $c_{i,\mathrm{atm}}=0$. The baseline lower boundary is no-through-flow,   

$$J_{z,i}\left(L\right)=0.$$

(5)

An optional imposed lower-boundary input may be written as $J_{z,i}\left(L\right)=J_{i,L}$, with upward input represented by negative $J_{i,L}$; this option is part of the sign convention and is not used in the baseline numerical experiments.

Three symbolic profile supports, a surface-flux support, and idealized contextual constraints define the observation design. Observation depths satisfy $A<B<C$. Exact support centers and widths are reported in the Supplementary Materials.

Figure 2 summarizes the domain geometry, source supports, observation depths, upper-boundary forcing, surface exchange, and baseline no-through-flow lower boundary.

3.2. Gas-Specific Reduced Transport Equation

The process template is

$$R_i({\theta }_a,{\theta }_w){\displaystyle \frac{\partial c_i}{\partial t}}={\displaystyle \frac{\partial }{\partial z}}\left(D_i^{\mathrm{eff}}{\displaystyle \frac{\partial c_i}{\partial z}}\right)-v_g{\displaystyle \frac{\partial c_i}{\partial z}}+S_i(z,t)-{\lambda }_i{R}_i{c}_i.$$

(6)

Here $c_i$ is gas-phase concentration or activity, $D_i^{\mathrm{eff}}$ is effective diffusivity, $v_g$ is a reduced spatially uniform gas velocity, $S_i$ is the source term, and ${\lambda }_i$ is the decay constant. The numerical experiments solve the steady form. The non-conservative advection term is therefore a lumped approximation for uniform $v_g$; a depth-variable velocity would require $-\partial \left(v_g{c}_i\right)/\partial z$.

The storage descriptor is   

$$R_i={\theta }_a+H_i^{(aq/g)}{\theta }_w,$$

(7)

with no solid-associated term in the numerical experiments. The rounded diagnostic descriptors for CO₂, radon/thoron (Rn/Tn), and helium (He) are $H_{{\mathrm{CO}}_2}^{(aq/g)}=0.75$, $H_{\mathrm{Rn}/\mathrm{Tn}}^{(aq/g)}=0.25$, and $H_{\mathrm{He}}^{(aq/g)}=0.03$; they are documented model descriptors rather than universal Henry constants. For steady CO₂ and He, ${\lambda }_i=0$ and $R_i$ cancels from the governing balance, so the reported steady profiles are controlled by source geometry, effective diffusivity, advection, boundary exchange, and observation support. For Rn/Tn, $R_i$ remains active through the decay term. Moisture-dependent emanation, pH-dependent carbonate speciation, isotope exchange, and transient dissolved-gas storage are not resolved.

The implemented source term is

$$S_i\left(z\right)=A_i{g}_i\left(z\right),{\int }_0^L{g}_i\left(z\right)\mathrm{d}z=1,$$

(8)

so changes in source depth or support do not alter the integrated source amplitude $A_i$.

CO₂ can combine biological and geogenic sources and can partition into water; He is highly diffusive and field provenance normally requires companion noble-gas ratios; ²²²$\mathrm{Rn}$ integrates a larger near-surface volume than ²²⁰$\mathrm{Rn}$, whose short half-life makes its profile strongly support- and resolution-dependent [12,13,14,15,16]. Methane (CH₄) is retained only as an extension case.

Table 2. Dimensional and numerical conventions used in the reduced transport equation.

Quantity	Convention	Comment
$c_i$	concentration/activity per gas volume	positive model observable
$S_i$	amplitude times unit-integral depth kernel	separates source amplitude from geometry
$D_i^{\mathrm{eff}}$	spatially uniform within a snapshot	reduced air-filled-porosity closure
$v_g$	uniform reduced velocity, positive downward	negative values indicate upward motion
$k_{\mathrm{ex},i}$	surface exchange velocity	defines $F_{\mathrm{up},i}$
$J_{i,L}$	optional downward-positive lower-boundary flux	not used in the baseline experiments

summarizes the dimensional, sign, and flux conventions used in the reduced transport formulation.

Table 3. Gas-specific interpretation used in the synthetic experiments. Abbreviations: CO₂, carbon dioxide; CH₄, methane; He, helium; Rn, radon; Tn, thoron. CH₄ is not simulated.

Gas/Tracer	Retained Property	Interpretive Consequence
CO2	mixed biological/geogenic sources; water/mineral partitioning possible	gas-phase observations may not equal total production
CH4	buoyancy and oxidation sensitivity	extension relevant to landfill-cover systems
He	high diffusivity; atmospheric contamination	single concentration is illustrative, not a provenance proof
222$\mathrm{Rn}$	decay-limited transport over a larger support than thoron	transport-sensitive near-surface tracer
220$\mathrm{Rn}$	very short half-life	near-surface support and grid resolution are critical

summarizes the gas-specific properties retained in the synthetic experiments.

3.3. Literature Anchoring of Synthetic Parameters

The numerical values used in the synthetic experiments are literature-anchored end-member values rather than site calibrations. They are chosen to span common soil physical conditions. Soil class terminology is aligned with international soil classification and survey references [17,18]. Porosity, water retention and hydraulic end-member ranges are guided by general soil-physics and pedotransfer literature [19,20,21,22,23,24,25]. Effective gas diffusivity is constrained by classical and modern gas-transport relations in unsaturated media [8,26,27,28,29,30]. Gas–water partitioning is interpreted with reference to Henry-law compilations [31]. Biological source masking is constrained by the soil-respiration literature [32,33,34,35]. Radon/thoron behavior is anchored to radiometric transport and decay literature [14,15,16,26,36,37]. Soil-gas geological end-members involving CO₂, Rn, He and faults are constrained by published soil-gas studies [38,39,40,41].

The model therefore uses synthetic values in a controlled way: they are not fitted to a single site, but they are not arbitrary. The Supplementary Materials include a parameter–literature anchoring table that identifies the literature basis, intended use and limitation of each numerical or ordinal parameter group.

Table 4. Main parameter groups and literature anchoring used in the synthetic experiments. Values are end-member or diagnostic choices rather than site-calibrated measurements; the full parameter table is provided in the Supplementary Materials.

Parameter Group	Symbol/Example	Use	Literature Basis	Limitation
Soil end-member labels	soil class	applicability matrix	[17,18]	not a formal site classification
Water and air-filled porosity	${\theta }_a,{\theta }_w,\varphi$	gas-continuity and storage states	[19,20,21,22,23,24,25]	end-member ranges, not measurements
Gas diffusivity	$D_i^{\mathrm{eff}}$	transport attenuation	[8,26,27,28,29]	reduced closure; regime-dependent
Gas–water partitioning	$H_i^{(aq/g)}\left(T\right)$	storage/retardation	[31]	convention-dependent
Biological CO2 masking	$S_{{\mathrm{CO}}_2}$, ordinal score	source ambiguity in topsoil	[32,33,34,35]	does not separate root and microbial dynamics
Radon/thoron behavior	${\lambda }_i$, radiometric range	decay-limited sensitivity	[14,15,16,26,36,37]	not a site-calibrated radon model
Geogenic soil-gas ambiguity	CO2, Rn, He, source depth	deep/shallow ambiguity	[38,39,40,41,42,43,44,45,46]	pattern-level only
Rank diagnostics	$\tau ,\Delta r,I_{S,k}^{\perp }$	local separability analysis	[1,2,3]	local and tolerance-dependent

links the synthetic parameter groups to their literature basis and modeling role.

3.4. Parameterization of Water-State Filtering

Effective diffusivity is prescribed as

$$D_i^{\mathrm{eff}}=D_i^0{D}_{\mathrm{scale}}{\left({\displaystyle \frac{{\theta }_a}{{\theta }_{a,0}}}\right)}^{{\beta }_g},{\theta }_{a,0}=0.18,{\beta }_g=2.1.$$

(9)

The equation is a normalized power-law closure, not the Millington–Quirk equation. Classical and modern gas-diffusivity models motivate the dependence on air-filled porosity, while the exponent and normalization are tested rather than treated as universal constants.

Total porosity is $\varphi =0.45$, and all water-state cases satisfy ${\theta }_w=\varphi -{\theta }_a$. The dry case retains water but has high gas connectivity; the baseline case represents the principal gas-continuous domain; the near-limit case is a warning state approaching gas disconnection. These are diagnostic end-members, not calibrated hydraulic states.

Table 5. Prescribed water-state cases. The phase fractions satisfy ${\theta }_a+{\theta }_w=\varphi =0.45$.

Case	${\mathit{\theta }}_{\mathit{a}}$	${\mathit{\theta }}_{\mathit{w}}$	Role
Dry gas-continuous	0.32	0.13	high gas connectivity with water retained
Unsaturated gas-continuous	0.18	0.27	baseline numerical state
Near gas-continuity limit	0.10	0.35	boundary-of-validity attenuation state

summarizes the three prescribed water-state cases.

3.5. Synthetic Scenarios

The scenarios connect H1–H3 to explicit outputs (

Table 6. Synthetic scenarios, hypothesis links, and evidential role.

Scenario	Changed Quantity	Hypothesis	Evidence Supplied
CO2 ambiguity paths	$S_{{\mathrm{CO}}_2}$, $D_{\mathrm{scale}}$, support width	H1	similar restricted-support responses can arise from different mechanisms
Water-state filtering	${\theta }_a$, with ${\theta }_w=\varphi -{\theta }_a$	H1/H3	gas connectivity changes transmitted steady CO2 response
Radiometric profiles	decay and effective diffusivity	H2	gas-specific depth response and short thoron length scale
Observation-set diagnostics	gas, profile, tracer, ideal state-constraint rows	H2	projected source directions beyond a common-threshold nuisance subspace
Soil end-member translation	ordinal biological, water, attenuation, pathway, deep-source scores	H3	heuristic, literature-anchored design implications

). H1 is examined through overlapping CO₂ source, transport, and support paths rather than through a claim that CO₂ has zero source rank. H2 is examined with tracer and ideal state-constraint rows. H3 is examined through the heuristic soil end-member translation and literature evidence, not through a separate Jacobian for every soil class.

Table 7. Numerical implementation of the synthetic scenarios. Outputs are steady diagnostic snapshots rather than transient site simulations.

Scenario	Source/Boundary Treatment	Changed Quantity	Reported Output
CO2 ambiguity	unit-integral internal source; fixed lower boundary	source amplitude, diffusivity, support width	normalized support response paths
Water filtering	moving unit-integral CO2 kernel	${\theta }_a$, ${\theta }_w$, source depth	normalized support response
Tracer profiles	fixed internal source kernels; decay retained	gas property and decay	normalized profiles
Structural diagnostics	linearization around baseline state	observation rows and parameter blocks	projected ranks, residuals, conditioning
Noise-scaled check	raw Jacobian scaled by declared observation uncertainty	same parameter blocks	singular response in one-sigma units

links each synthetic experiment to its numerical implementation and reported output.

3.6. Numerical Implementation and Reproducibility

The steady finite-volume solver uses 241 baseline nodes on $0\le z\le 2$ m. A 961-node solution is used as the mesh-convergence reference. All source kernels are integral-normalized before multiplication by their amplitude. Within a snapshot, $D_i^{\mathrm{eff}}$, $R_i$, ${\theta }_a$, ${\theta }_w$, and $v_g$ are spatially uniform.

The baseline parameter vector is

$${\mathit{\theta }}_0={[S_{{\mathrm{CO}}_2},S_{\mathrm{Rn}},S_{\mathrm{He}},z_s,D_{\mathrm{scale}},v_g,{\theta }_a,w_{\mathrm{supp}}]}^{\mathsf{T}}.$$

(10)

Central differences are calculated in dimensionless scaled coordinates $u_k$ with step $h_u=0.05$. Log-coordinate parameters satisfy $p_k=p_{k,0}exp\left(s_k{u}_k\right)$; additive parameters satisfy $p_k=p_{k,0}+s_k{u}_k$. Gas observations and ratios are log-transformed, whereas ideal state constraints remain on their native dimensionless scales.

The raw scaled-coordinate Jacobian is computed first:

$${\left[{\mathbf{J}}_{\mathrm{raw}}\right]}_{qk}\approx {\displaystyle \frac{{\tilde{y}}_q(u_k+h_u)-{\tilde{y}}_q(u_k-h_u)}{2h_u}}.$$

(11)

Two derived matrices are then kept separate. For structural geometry, rows are multiplied by diagnostic scale factors and each parameter column is normalized to unit norm over the complete 11-row observation universe before any observation-set subset is extracted. This global-before-subset convention is the primary scaling used in the reported structural ranks. A companion within-set normalization check is reported in results/normalization_sensitivity.csv. For the auxiliary detectability check, the unnormalized raw rows are divided by declared independent observation standard deviations:

$${\mathbf{J}}_{\mathrm{noise}}={\mathbf{\Sigma }}_y^{-1/2}{\mathbf{J}}_{\mathrm{raw}}.$$

(12)

Here ${\mathbf{\Sigma }}_y=diag({\sigma }_1^2,\dots ,{\sigma }_Q^2)$ is an illustrative diagonal observation covariance. The noise-scaled matrix preserves response magnitude in one-standard-deviation units but is not a site-specific covariance or instrument-error model.

Table 8. Numerical settings used by the corrected solver.

Component	Choice	Purpose
Domain/discretization	2 m; 241 baseline nodes; 961-node reference	steady column and convergence test
Surface boundary	$F_{\mathrm{up}}=-J_z\left(0\right)=k_{\mathrm{ex}}[c\left(0\right)-c_{\mathrm{atm}}]$	upward flux observable
Lower boundary	$J_z\left(L\right)=0$	internal-source diagnostic baseline
Source kernels	unit-integral Gaussian/uniform mixtures	isolate amplitude from geometry
Structural singular-value-decomposition (SVD) floor	${ϵ}_{\mathrm{abs}}={10}^{-10}$	suppress numerical-null nuisance directions
Display tolerance	$\tau =0.05$, with six-value scan	transparent numerical-rank display
Finite differences	scaled-coordinate step $h_u=0.05$	raw local Jacobian

lists the numerical settings used by the corrected solver. The structural rank calculation uses a singular value decomposition (SVD) with the stated absolute and display thresholds.

Table 9. Local sensitivity coordinates. The finite-difference step is $h_u=0.05$ in each dimensionless coordinate.

Block	Parameters	Coordinate Scale	Mode
Source amplitude	$S_{{\mathrm{CO}}_2},S_{\mathrm{Rn}},S_{\mathrm{He}}$	10% log scale	multiplicative
Source geometry	$z_s$	0.25 m	additive
Transport	$D_{\mathrm{scale}},v_g$	10% log scale	multiplicative
Gas continuity	${\theta }_a$	0.05 volumetric fraction	additive
Observation support	$w_{\mathrm{supp}}$	0.10 m	additive

defines the scaled local sensitivity coordinates and perturbation rules.

Table 10. Observation rows and diagnostic structural weights. The values are scale factors, not calibrated measurement-error standard deviations.

Observation Row	Weight
CO2 flux	1.00
CO2 A, B, C	0.85, 0.82, 0.75
222$\mathrm{Rn}$ A; 220$\mathrm{Rn}$ A; He C	0.80, 0.70, 0.70
pressure-state constraint; water-state constraint	0.55, 0.55
profile ratio; support ratio	0.65, 0.55

lists the observation rows and diagnostic structural weights.

The pressure- and water-state rows are idealized independent constraints on nuisance state. They test the consequence of having contextual information; they are not direct observations of $v_g$, $D_{\mathrm{scale}}$, or ${\theta }_a$, and they do not imply that any generic pressure or moisture sensor produces the same gain. The released run_all.py regenerates the raw, structural, and noise-scaled matrices; common-threshold diagnostics; the global-versus-within-set normalization check; mesh and closure tests; the supplementary symbol glossary; the published-case table; figure inputs; and the exact numerical manuscript figures identified in data/figure_index.csv.

3.7. Observation Operator

The observed signal is represented as a support-weighted quantity,

$$y_q\left(t\right)={\int }_0^L{w}_q\left(z\right)c_q(z,t)dz+{\epsilon }_q\left(t\right),$$

(13)

where $w_q\left(z\right)$ is the observation support and ${\epsilon }_q$ is measurement error. The support function is normalized as $w_q\left(z\right)\ge 0$ and ${\int }_0^L{w}_q\left(z\right)dz=1$, so that changing the support changes the sampled depth weighting rather than the dimensional meaning of the observable. For a surface-flux observable, the reduced form is

$$F_{\mathrm{up},i}\left(t\right)=D_i^{\mathrm{eff}}{\left.{\displaystyle \frac{\partial c_i}{\partial z}}\right|}_{z=0}-v_g{c}_i(0,t)=-J_{z,i}(0,t).$$

(14)

Here $F_{\mathrm{up},i}$ is the upward emission flux; it is not the downward-positive coordinate flux $J_{z,i}$. This representation follows the principle that measurement architecture is part of the forward problem. Published probe and chamber studies show that sampling support, chamber pressure equilibration, line transfer, and detector response can influence the signal that is interpreted [47,48,49,50,51].

3.8. Sensitivity, Common-Threshold Projection, and Diagnostic Scores

The structural matrix is partitioned into nuisance ${\mathbf{J}}_{\eta }$, source-amplitude ${\mathbf{J}}_A$, and source-geometry ${\mathbf{J}}_G$ blocks. For a target block ${\mathbf{J}}_T$, a common absolute threshold is defined from the augmented matrix:   

$${ϵ}_T\left(\tau \right)=max\{\tau {\sigma }_1\left(\left[{\mathbf{J}}_{\eta }{\mathbf{J}}_T\right]\right),{ϵ}_{\mathrm{abs}}\}.$$

(15)

The nuisance basis retains only singular vectors of ${\mathbf{J}}_{\eta }$ whose singular values exceed ${ϵ}_T$. The target is then projected onto the orthogonal complement of that same truncated nuisance basis. The projected source-amplitude and combined source ranks are

$$\begin{array}{cc}\hfill \Delta r_A^{\perp }\left(\tau \right)& ={rank}_{{ϵ}_A}\left[(\mathbf{I}-{\mathbf{P}}_{\eta ,{ϵ}_A}){\mathbf{J}}_A\right],\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \Delta r_{A+G}^{\perp }\left(\tau \right)& ={rank}_{{ϵ}_{A+G}}\left[(\mathbf{I}-{\mathbf{P}}_{\eta ,{ϵ}_{A+G}})\left[{\mathbf{J}}_G{\mathbf{J}}_A\right]\right].\hfill \end{array}$$

(17)

Using one threshold for the nuisance and augmented blocks prevents a numerical-null nuisance direction from acquiring rank merely because it is normalized relative to itself.

For a source parameter k, the projection residual uses the nuisance basis truncated with the common threshold of the corresponding source block (amplitude or geometry):

$$I_{S,k}^{\perp }={\displaystyle \frac{\parallel (\mathbf{I}-{\mathbf{P}}_{\eta ,ϵ}){\mathbf{j}}_{S,k}\parallel }{\parallel {\mathbf{j}}_{S,k}\parallel }}.$$

(18)

The same truncated nuisance construction is used for the rank and residual calculations. These metrics report retained local sensitivity geometry. A value of one projected direction means that the observation set responds to at least one source-related combination; it does not identify a source count, distinguish amplitude from geometry, or establish field detectability.

The auxiliary noise-scaled analysis projects ${\mathbf{J}}_{\mathrm{noise}}$ using a one-standard-deviation absolute threshold. A projected source singular response is counted when it is at or above $1-{10}^{-10}$ in standard-deviation units; the tolerance prevents a floating-point value numerically equal to one from being described as meaningfully greater than the threshold. This check is kept separate from $\Delta r_A^{\perp }$.

The soil score

$$D_{\mathrm{design}}=0.25(6-B_{\mathrm{bio}})+0.20(6-W_{\mathrm{stor}})+0.20(6-T_{\mathrm{att}})+0.18P_{\mathrm{path}}+0.17G_{\mathrm{deep}}$$

(19)

is a separate ordinal, expert-defined screening index. It is neither a rank metric nor a physical soil property.

Table 11. Robustness of the heuristic $D_{\mathrm{design}}$ ranking to independent $\pm 20\%$ weight perturbations and renormalization over 10,000 draws.

Soil End-Member	Median Rank	Rank Range	Top-3 Prob.	Bottom-3 Prob.
Fractured regolith	1	1–1	1.0000	0.0000
Sandy soil	2	2–3	1.0000	0.0000
Shrink–swell cracked	3	2–4	0.9999	0.0000
Carbonate-rich	4	3–4	0.0001	0.0000
Volcanic ash	5	5–5	0.0000	0.0000
Clayey compacted	6	6–6	0.0000	0.0000
Organic topsoil	7	7–8	0.0000	1.0000
Seasonally wet	8	7–8	0.0000	1.0000
Peat/wet organic	9	9–9	0.0000	1.0000

reports the sensitivity of the heuristic ranking to weight perturbations.

3.9. Forcing-Aware Interpretation Changes the Diagnostic Class

Forcing variables are not secondary covariates. They can change an observation even when the source is fixed. Pressure fluctuations alter gas-transport phase and amplitude. Moisture changes reduce air-filled porosity and can alter partitioning or decay-weighted storage, depending on the gas and time treatment. Observation support may then integrate, amplify, or suppress the affected part of the profile. For this reason, each observation set is classified not only by the gases measured, but also by whether the relevant forcing variables are observed.

In the numerical experiments, the addition of pressure-state and water-state constraints does not itself identify the source. Instead, it constrains nuisance directions and reduces their ability to mimic a source change. This is why the richest observation set has a higher source-amplitude rank gain: part of the transport and forcing variability is no longer unconstrained. The practical implication is that a forcing-aware design can be more valuable than adding another gas that responds in the same direction as the target gas.

3.10. Observation Support Controls What Part of the Soil Filter Is Sampled

The observation operator can change the diagnostic class of a signal. A shallow thoron-sensitive support emphasizes near-field pathway opening. A deeper support suppresses thoron and favors radon or CO₂ information. A chamber support integrates surface flux, but it may also alter the pressure condition at the boundary. A line or detector with a long residence time can attenuate high-frequency or short-lived signals. Model outputs must therefore be read together with the support functions that generated them. The same soil column can lead to different ambiguity classes if it is sampled with different support volumes.

3.11. Targeted Pattern-Level Literature Consistency Protocol

The literature component is a targeted pattern-level consistency set, not a systematic review and not a calibration dataset. Each included study contributes an observable/process pattern that can be mapped to a model component. The extracted fields are: reference, digital object identifier (DOI) when available, system type, observable, controlling process, model analogue, evidence level, comparison basis, agreement outcome, and limitation. Evidence level E1 denotes a qualitative mechanism or behavior class, E2 denotes a semi-quantitative constraint such as range, depth dependence, frequency response, or measurement-support effect, and E3 would denote a directly comparable quantitative case. Most cases used here are E1 or E2 because the published studies were not designed as a common calibration dataset.

3.12. Treatment of Published Literature Cases

The treatment of published cases follows the protocol above. The objective is to test whether the reduced source–transport–observation system reproduces the same classes of ambiguity and the same directions of sensitivity reported in independent soil-gas, vadose-zone, radiometric and chamber-based studies.

A study was included when it met at least one of the following criteria: it reported soil-gas or vadose-zone gas behavior; it documented a source, transport, forcing or measurement-support control; it allowed an interpretive ambiguity to be identified; or it provided a process class that could be mapped to one of the model components. Studies were not used when the reported information did not allow the signal, the controlling process and the measurement support to be separated even qualitatively.

For each selected study, the following information was extracted: system type, soil or medium type, gas or tracer, observable, reported controlling process, dominant ambiguity, corresponding model component, evidence level, comparison basis, agreement outcome, DOI and limitation. The comparison was then assigned an evidence level. Level 1 indicates a qualitative behavioral pattern. Level 2 indicates a semi-quantitative constraint, such as a reported range, dominant frequency, depth dependence or tracer response. Level 3 indicates a quantitative case directly comparable with a model output. Most cases used here are Level 1 or Level 2 because the published studies are heterogeneous and were not designed for common calibration.

The procedure provides a pattern-level external consistency check. The reduced model should reproduce the same dominant ambiguity classes that have been observed in independent studies. The full extraction matrix is provided in the Supplementary Materials.

Detailed derivations of the common-threshold criterion, the soil-column model, the synthetic experiments, the site-scale data requirements, and the symbol glossary are provided in Appendices A–F.

4. Soil Classes as Information Filters

The numerical model is more useful for Soil Systems when its reduced parameters are tied to recognizable soil settings. The categories used here are not a formal soil classification, but they are consistent with major soil-classification and survey references [17,18] and with soil-physical ranges reported in the pedotransfer and hydraulic-parameter literature [19,20,21,22,23,24,25]. The same equation has different interpretive value in an organic topsoil, a sandy soil, a compacted clay, a shrink–swell soil, a peat, or a shallow fractured regolith; gas-flow observations in cracked clays provide an empirical example of how structural state affects air permeability [52]. Pressure-pumping studies document the related role of atmospheric forcing in unsaturated and fractured media [53,54,55]. Recent reduced studies of deep gas sources, hydrothermal memory, and diffusive isotope interpretation further motivate the separation of source amplitude, source history, transport, and observation effects [56,57,58]. This section does not introduce a full pedotransfer model. It maps common soil end-members onto the filtering mechanisms that control soil-gas interpretation.

The classification is operational. Each soil setting is described by five controls: biological source masking, water-state filtering, transport attenuation, pathway activation, and deep-source ambiguity. Scores from 1 to 5 indicate low to high expected influence. They are not site-calibrated coefficients. They define end-member ambiguity classes for the synthetic experiments and for the pattern-level consistency checking.

Table 12. Soil-type applicability matrix for the source–transport–observation system. Scores indicate expected influence from 1, low, to 5, high. Model validity refers to the reduced gas-continuous formulation used in this study.

Soil Setting	Bio.	Water	Atten.	Path.	Deep	Model Validity	Best Diagnostic Observations
Organic-rich topsoil/A horizon	5	3	3	2	2	Conditional	CO2 flux, ${\delta }^{13}$C-CO2, moisture, temperature
Sandy well-aerated soil	2	1	1	1	2	Strong	CO2 profile, 222$\mathrm{Rn}$, pressure forcing
Clayey or compacted soil	3	4	5	3	3	Conditional if gas-continuous	Moisture, suction, pressure, Rn/Tn
Shrink–swell cracked soil	3	4	3	5	3	Strong with pathway term	Crack state, moisture, pressure, 220$\mathrm{Rn}$/222$\mathrm{Rn}$
Volcanic ash/Andisol-like soil	3	5	3	2	4	Conditional; water-retention context important	CO2, Rn, water content, temperature
Peat/organic wet soil	5	5	5	2	2	Limited without hydrology	CO2/CH4, water table, redox, temperature
Calcareous or carbonate-rich soil	3	3	3	2	4	Conditional; carbonate context required	CO2, ${\delta }^{13}$C-CO2, pH/carbonate context
Shallow fractured regolith/bedrock soil	2	2	2	5	5	Strong for mixed source–transport cases	CO2, He, Rn/Tn, pressure forcing
Seasonally wet or near-saturated soil	4	5	5	3	3	Outside reduced model when gas disconnected	Water table, redox, flux interpreted cautiously

and Figure 3 summarize the matrix.

Table 13. Synthetic soil end-member parameters used to translate soil class into model behavior. Values are illustrative, literature-anchored, and ordinally constrained rather than site-calibrated.

Soil End-Member	${\mathit{\theta }}_{\mathit{a}}$	${\mathit{\theta }}_{\mathit{w}}$	Bio. Factor	Diff. Factor	Path. Factor	Primary Diagnostic Consequence
Organic-rich topsoil	0.18	0.22	high	moderate	low	CO2 masking; check temperature, moisture and isotopes
Sandy well-aerated soil	0.32	0.08	low	high	low	efficient gas transfer; screen with profiles and pressure
Clayey or compacted soil	0.11	0.30	moderate	low	moderate	verify gas continuity and hysteresis before attribution
Shrink–swell cracked soil	0.16	0.26	moderate	conditional	high	pathway activation may dominate anomaly
Volcanic ash/ Andisol-like soil	0.20	0.28	moderate	conditional	low–moderate	water retention and gas connectivity control interpretation
Peat/organic wet soil	0.08	0.55	very high	very low	low	limited unless water-table control is measured
Carbonate-rich soil	0.20	0.18	moderate	moderate	low	use carbonate context; carbon isotopes
Fractured regolith	0.24	0.12	low–moderate	high	high	Rn/Tn/He and pressure are diagnostic
Seasonally wet soil	0.06	0.38	high	very low	conditional	wet gas disconnection can invalidate interpretation

shows why soil type cannot be treated as background metadata. It enters the numerical problem through storage, effective diffusivity, biological source masking, and the probability that preferential pathways dominate the observation. The table is not a soil classification system. It translates soil description into observation design.

The matrix also shows why one diagnostic rule cannot fit all soils. In organic-rich topsoil and peat, biological production may mask smaller geogenic or transport signals. In sandy well-aerated soils, the system is closer to a gas-diffusion limit, so pressure and radon observations can be highly informative. In compacted clay and seasonally wet soils, the reduced model is valid only while the gas phase remains connected. If gas connectivity collapses, the problem becomes a different multiphase problem. In shrink–swell or fractured regolith, pathway activation may dominate the signal even if the source is unchanged.

Soil type must therefore be considered before the monitoring design is chosen. A minimum observation set can describe an anomaly. An enhanced or diagnostic set is needed when the dominant ambiguity is biological masking, water-state filtering, pathway activation, forcing, or support distortion.

4.1. How Individual Soil End-Members Filter Gas Information

The ordinal matrix in Table 12 is deliberately compact. For field interpretation, however, each end-member has a different reason for weakening, strengthening, or redirecting source information before it reaches the observation support. The following short interpretations make the matrix operational.

Organic-rich topsoil and biologically active A horizons. In these soils the biological term may dominate the CO₂ budget, particularly when root respiration and microbial decomposition respond to temperature and moisture. A small geogenic or transport-related change can therefore be hidden beneath a larger biological signal. The model is applicable only if temperature, moisture and biological activity are retained as possible non-source controls; otherwise the source term is over-interpreted.

Sandy, well-aerated soils. These soils represent the closest field analogue to the high-connectivity gas-continuous limit. Storage and water filtering are relatively weak, and the effective diffusivity remains high. This does not mean that the source is directly observable. It means that the transport operator is simpler, so profiles, pressure forcing and radiometric companions can more efficiently test whether a surface anomaly is source-informative.

Clayey and compacted soils. Here the central filter is reduced air-filled porosity and high tortuosity. A source may exist but be strongly attenuated; conversely, a small change in connectivity can produce a large change in the measured signal. The reduced equation remains useful only when the gas phase is still connected. If gas pathways are disconnected, the problem is no longer a limiting case of the present gas-continuous model.

Shrink–swell and cracked soils. These soils are controlled by pathway memory. Drying may open cracks, wetting may close them, and repeated cycles need not return the medium to the same permeability branch. The main danger is to interpret pathway activation as a source pulse. Short-range tracers, pressure response and structural observations are therefore diagnostic rather than optional.

Volcanic ash and Andisol-like soils. These settings combine high porosity with high water retention and reactive surfaces. A gas signal may be delayed or attenuated without implying a weaker source. Water retention and aqueous or mineral partitioning are therefore important for field interpretation. In the present steady CO₂ snapshots, however, the implemented water-state effect enters through air-filled porosity, effective diffusivity, and observation support rather than dynamic dissolved-gas storage.

Peat and organic wet soils. These soils sit near the limit of the present model. Biological production may be strong, water-table control may be dominant, and redox state may alter the gas family itself. The reduced gas-continuous equation is useful only during intervals when connected gas pathways persist. During wet or near-saturated conditions, water-table and redox observations become mandatory.

Carbonate-rich and calcareous soils. These soils require additional geochemical context because CO₂ may be affected by carbonate buffering, dissolution, precipitation and isotopic exchange. In such cases a concentration or flux anomaly should not be interpreted without carbon isotopes, carbonate context, and water-state information.

Shallow fractured regolith and bedrock soils. These settings are pathway-dominated. Deep or lower-boundary inputs may coexist with shallow biological and radiogenic signals. The strongest observation sets are therefore multi-gas and forcing-aware: CO₂, He, radon/thoron and pressure response supply complementary constraints.

Seasonally wet and near-saturated soils. These are not simply “wet cases” of the same gas model. When air-filled porosity drops below a gas-continuity threshold, the model changes class. Gas observations made during those intervals should be treated as hydrologically conditioned anomalies rather than direct source indicators.

Table 14. Pedological interpretation of the synthetic end-members. The table translates soil description into the main filter acting on source information and the practical observations needed before source attribution is attempted.

Soil Setting	Dominant Information Filter	Main Risk If Ignored	Minimum Practical Control
Organic-rich topsoil	biological CO2 masking	weak non-biological signal misread	temperature, moisture, isotopes
Sandy well-aerated soil	fast gas connectivity	profile read as source without forcing	profile + pressure forcing
Clayey or compacted soil	attenuation and tortuosity	missed source or false transport change	air-filled porosity; suction/moisture
Shrink–swell cracked soil	pathway activation and memory	crack opening read as source pulse	pathway proxy; wet/dry state
Volcanic ash/ Andisol-like soil	water retention and gas-connectivity filtering	attenuation read as weak source	water content; partitioning- or decay-sensitive tracers
Peat/organic wet soil	water table and redox control	model used outside validity range	water table, redox, temperature
Carbonate-rich soil	carbonate buffering and isotope exchange	CO2 assigned to wrong source	carbonate context; carbon isotopes
Fractured regolith	preferential pathways and lower input	pathway read as deep-source change	pressure; Rn/Tn; He or multi-gas
Seasonally wet soil	intermittent gas disconnection	gas anomaly read in invalid state	gas continuity and hydrological state

translates the soil end-members into practical observation priorities.

4.2. From Soil Description to Numerical End-Members

The solver baseline uses $\varphi =0.45$ only for the reduced numerical column. The soil end-member table below is not constrained to that single baseline porosity. For high-porosity settings such as volcanic ash and peat, ${\theta }_a$ and ${\theta }_w$ should be read as end-member state descriptors associated with soil-specific total porosity, not as components required to sum below the solver baseline $\varphi$.

The soil classes above are translated into model behavior through the end-member parameters in Table 13. The key variables are not intended as pedotransfer estimates; they are reduced end-members constrained by published soil physical, soil respiration, gas-diffusion and soil-gas evidence [8,17,18,19,20,21,22,23,24,25,26,27,28,29,31,32,33,34,35,37,40,41]. They are reduced controls that connect soil description to the numerical experiment. Air-filled porosity sets the gas-continuity and diffusivity scale. Water content controls gas connectivity and effective diffusivity and, in the transient template or for decaying species, also affects storage and gas–water partitioning. The biological factor controls the likelihood that biological CO₂ covers smaller signals. The diffusivity factor controls attenuation. The pathway factor controls whether a structural change can be confused with source variability.

This translation step is essential because a single mathematical equation has different evidential meaning in different soils. In a sandy soil the same equation may act as a nearly gas-diffusive diagnostic model. In peat it may act only as a warning that gas observations are hydrologically conditioned. In cracked clay it becomes a pathway-state test. In carbonate-rich soils it becomes a source-class test that requires isotopic support. The numerical model is therefore applied through soil end-members rather than as one generic soil column.

5. Results

The results are organized from soil end-members to solver outputs and observation-set diagnostics. The first subsections show how soil type, water state, and support affect the signal; the later subsections evaluate whether multiparameter observation sets add local source-informative directions.

The results also track the three hypotheses introduced in the Introduction: H1 concerns CO₂-only ambiguity, H2 concerns the added value of decay-separated tracers and ideal state constraints, and H3 concerns soil-end-member dependence.

5.1. Soil Type Determines the Dominant Ambiguity Before the Gas Is Measured

The soil-type matrix provides the heuristic test of H3 and changes the interpretation of the synthetic experiments. In a sandy gas-continuous soil, the dominant ambiguity is often source strength versus transport efficiency. In an organic-rich topsoil or peat, the first ambiguity is biological masking: a strong biological CO₂ component can cover smaller geogenic or radiometric signals. In compacted clay and seasonally wet soil, the dominant ambiguity is whether the gas phase remains connected enough for the reduced model to be meaningful. In shrink–swell and fractured regolith settings, the dominant ambiguity is pathway activation, because a change in connected aperture or crack state can resemble a source pulse. The numerical experiments below therefore should not be read as one generic soil case. They are end-member tests of how different soils filter the same kind of source information.

The end-member ranking in Figure 4 provides a second result: the settings with the highest diagnostic-design opportunity are not necessarily those with the largest gas anomalies. Fractured regolith and sandy gas-continuous soils score highly because the observation design can add non-redundant constraints on pathway state and pressure forcing. Peat and seasonally wet soils score lower because hydrological and biological filters dominate unless water-table and redox information are added. Organic topsoil has a strong signal but weak source specificity. This result is important for practice: signal magnitude and source informativeness are not the same quantity.

Figure 5 also shows why the anhydrous interpretation is avoided. The model is not built around a dry column. It is built around a gas-continuous unsaturated domain in which water can be present. The validity issue is therefore not whether water exists, but whether gas connectivity persists. This is the point at which a soil-science description becomes a model condition.

5.2. Water State Changes the Transmitted Steady Response

Figure 6 compares the same unit-integral moving CO₂ source under three prescribed phase states. Because the CO₂ snapshots are steady and ${\lambda }_{{\mathrm{CO}}_2}=0$, the separation of the curves is produced by air-filled-porosity control of effective diffusivity and by support averaging; it is not a dynamic storage-retardation result. Reduced gas connectivity attenuates the shallow observable from a deeper source even though integrated source amplitude is unchanged.

The result supports the water-state part of H1: a change in the soil filter can alter a shallow observation without a source-amplitude change. It also defines the domain boundary for H3: near gas disconnection, gas measurements require explicit hydrological qualification.

5.3. Restricted CO₂ Observations Retain a Source Direction but Remain Attribution-Ambiguous

Figure 7 compares source-amplitude, transport, and support perturbations using the same shallow CO₂ support. The response paths overlap or have the same sign over part of the perturbation range. This supports H1: a restricted CO₂ design can respond to a source-related change while remaining unable to assign the response uniquely to source amplitude, effective source depth, transport, or support.

Under the primary global-before-subset normalization, the CO₂ surface-flux row retains one source-amplitude direction at $\tau =0.05$, but a single row cannot distinguish amplitude from source geometry or source class. Under within-set normalization, this one-row result becomes zero. The attribution conclusion is unchanged: a profile adds spatial information, not automatic uniqueness.

5.4. ²²²Rn and ²²⁰Rn Provide Gas-Specific Depth Response

Figure 8 compares normalized steady profiles under the same reduced column formulation. CO₂ reflects source geometry, effective diffusivity, advection, and support. ²²²$\mathrm{Rn}$ is additionally shaped by decay over a larger near-surface support, whereas the much larger decay constant of ²²⁰$\mathrm{Rn}$ confines its profile close to the surface. The figure therefore illustrates gas-specific depth sensitivity, not a direct source-depth inversion.

Tracer contrast alone does not guarantee an additional projected amplitude rank in every restricted observation set. Its value emerges when the tracer response is combined with profile, state, or support information that reduces nuisance collinearity, as tested under H2.

5.5. Multiparametric Observations Increase Projected Structural Richness

The common-threshold projection changes the interpretation of the observation sets. At $\tau =0.05$, under the primary global-before-subset normalization, the CO₂ surface-flux set retains one projected source-amplitude direction. The within-set normalization check reduces this restricted result to zero. The observation is therefore responsive under the adopted global scaling, but neither convention permits source strength, effective depth, or source class to be distinguished. The CO₂ profile retains one amplitude direction and two combined geometry–amplitude directions. Adding Rn/Tn without independent state information does not increase the amplitude rank beyond one in this synthetic design. The gain becomes larger when ideal nuisance-state constraints are supplied, and the full set retains three amplitude directions and four combined source directions (Figure 9;

Table 15. Projected structural diagnostics at $\tau =0.05$ under the primary global-before-subset normalization. $\Delta r_A^{\perp }$ refers to source-amplitude directions; $\Delta r_{A+G}^{\perp }$ includes source geometry.

Observation Set	Interpretation	$\mathbf{\Delta }{\mathit{r}}_{\mathit{A}}^{\perp }$	$\mathbf{\Delta }{\mathit{r}}_{\mathit{A}+\mathit{G}}^{\perp }$	$\mathit{\kappa }$
CO2 surface flux	one source-related response; no attribution separation	1	1	–
CO2 profile	profile support adds a combined geometry direction	1	2	6.78
CO2 + 222$\mathrm{Rn}$	one source-related direction with decay contrast	1	1	3.03
CO2 + Rn/Tn	radiometric contrast, but no extra amplitude rank alone	1	1	4.15
CO2 + Rn/Tn + ideal state constraints	two amplitude directions after nuisance-state constraints	2	3	3.65
full diagnostic set	three amplitude and four combined source directions	3	4	4.77

).

The full tolerance scan and projected singular spectra have been moved to the Supplementary Materials (results/rank_tolerance_scan.csv and results/projected_singul ar_values.csv), reducing repetition among closely related tables. The structural pattern at $\tau =0.05$ is stable across equal weights and downweighted ideal state constraints, although conditioning changes.

A separate normalization-sensitivity calculation compares the primary global-before-subset convention with normalization repeated within each observation set. At $\tau =0.05$, the CO₂ surface-flux set changes from $1/1$ amplitude/combined rank under global normalization to $0/0$ under within-set normalization. The CO₂ profile remains $1/2$; the two intermediate radiometric sets remain $1/1$; the ideal-state-constraint set remains $2/3$; and the full diagnostic set remains $3/4$. Thus, the restricted CO₂-only result is scaling-dependent, whereas the principal multiparametric conclusions are unchanged. The complete six-tolerance comparison is provided in results/normalization_sensitivity.csv.

The auxiliary noise-scaled calculation gives a more cautious result (

Table 16. Illustrative noise-scaled projected source response. The calculation uses the raw Jacobian divided by independent declared observation standard deviations before projection. Dimensions are counted at or above $1\sigma$ using a numerical tolerance of ${10}^{-10}$. It is not a site-specific instrument model.

Observation Set	Dimensions $\ge 1\mathit{\sigma }$	Largest $\mathit{\sigma }$-Scaled SV	Smallest $\mathit{\sigma }$-Scaled SV
CO2 surface flux	1	1.000	1.000
CO2 profile	1	1.169	0.000
CO2 + 222$\mathrm{Rn}$	1	1.059	0.000
CO2 + Rn/Tn	1	1.077	0.000
CO2 + Rn/Tn + ideal state constraints	1	1.292	0.000
full diagnostic set	1	1.472	0.476

). Under the declared illustrative standard deviations, one projected source dimension lies at or above the one-standard-deviation threshold in each tested set when a numerical tolerance of ${10}^{-10}$ is used. The full set has the largest projected singular response (1.472 standard-deviation units), but its smallest projected source singular response is 0.476. Structural rank therefore describes sensitivity geometry, whereas practical detection remains limited by response magnitude and uncertainty.

Figure 10 shows the structural matrix used for the richest set. Rows are weighted observations or ideal state constraints; columns are source amplitude, source geometry, and nuisance parameters. Column normalization makes directions comparable but removes absolute detectability information.

The result supports H2 only in its stated form: independently supplied nuisance-state information and a richer tracer/support design can add projected directions in this synthetic analysis. It does not establish that any generic pressure or moisture sensor will deliver the same gain.

5.6. Pattern-Level Consistency Outcomes in Brief

The targeted pattern-level consistency set supports the model behavior classes without being used for site calibration. Pressure-forced gas-transport studies support the forcing–transport ambiguity class. Probe and chamber studies support the observation-operator class. Radon/thoron and multi-gas studies support the use of non-redundant tracer families. The detailed DOI-traceable consistency matrix is reported in Section 7 and in the Supplementary Materials.

5.7. End-Member Soils Imply Different Monitoring Designs

The end-member matrix produces a practical result that is not visible from the one-dimensional equation alone: the same observation set has different interpretive value in different soils. A CO₂ concentration profile is more source-informative in a sandy gas-continuous soil than in an organic topsoil because biological masking is smaller and transport attenuation is lower. Conversely, a CO₂ flux observation in peat or seasonally wet soil may describe a surface exchange rate but may not constrain the source unless water-table state, redox state and temperature are also measured. In a shrink–swell or fractured regolith setting, the limiting ambiguity is different again: a pathway opening event can increase surface response without any source increase. In that case ²²⁰$\mathrm{Rn}$/²²²$\mathrm{Rn}$ and pressure forcing provide more diagnostic value than an additional CO₂ replicate alone.

The distinction helps avoid both unnecessary instrumentation and unsupported source attribution. The model does not imply that every site requires CO₂, He, Rn, Tn, isotopes, moisture, temperature and pressure. Instead, the soil end-member first identifies the expected ambiguity. The observation set is then selected to break that ambiguity. For biological masking, the companion variables are temperature, moisture and isotopic composition. For water-state filtering, the companion variables are air-filled porosity, water content and possibly suction or water-table state. For pathway activation, the companion variables are short-range radiometric signals, pressure, structural state and depth support. For deep/shallow ambiguity, helium, radiocarbon or carbon isotopes may add non-redundant source sensitivity.

The practical consequence is that a source–transport–observation system should be designed as a soil-specific test rather than as a universal sensor list. Minimum monitoring describes the anomaly; enhanced monitoring separates the common alternatives; diagnostic monitoring is justified only when source attribution is the objective.

5.8. Numerical Robustness Checks

The corrected mesh study uses 61, 121, 241, 481, and 961 nodes, with 961 nodes as the reference. The projected rank pattern is unchanged at $\tau =0.05$ for all observation sets. At the 241-node baseline, the maximum transformed-observation difference is 0.149%, and the maximum normalized-profile differences are 0.018% for CO₂, 0.073% for ²²²$\mathrm{Rn}$, and 1.285% for ²²⁰$\mathrm{Rn}$. The 481-node thoron profile error falls to 0.462%. Because the thoron diffusion–decay length is approximately 0.0123 m and the 241-node spacing is 0.00833 m, support-averaged thoron observables are considered converged for the present diagnostic purpose; pointwise near-surface profiles require a finer or locally refined mesh.

The closure sweep spans 27 combinations of $D_{\mathrm{scale}}$, ${\theta }_{a,0}$, and ${\beta }_g$, all six observation sets, and all six rank tolerances. At $\tau =0.05$, every closure combination preserves the baseline projected ranks: 1/1 for CO₂ flux, 1/2 for the CO₂ profile, 1/1 for CO₂ + ²²²$\mathrm{Rn}$, 1/1 for CO₂ + Rn/Tn, 2/3 with ideal state constraints, and 3/4 for the full set (amplitude/combined). At the highest tolerance, some combined geometry ranks change, so the conclusion is stated as tolerance- and closure-dependent rather than universal.

Table 17. Numerical robustness summary. Full outputs are provided in results/mesh_convergence.csv, results/thoron_resolution_check.csv, and results/closure_sensitivity.csv.

Check	Range	Outcome
Grid refinement	61–961 nodes	all $\tau =0.05$ projected ranks unchanged
Baseline profile convergence	241 versus 961 nodes	CO2 0.019%, 222$\mathrm{Rn}$ 0.073%, 220$\mathrm{Rn}$ 1.285% maximum normalized-profile difference
Thoron resolution	0.0123 m characteristic length	support averages stable; pointwise near-surface profile needs finer resolution
Closure sweep	27 closures, 6 sets, 6 tolerances	$\tau =0.05$ rank pattern unchanged; some high-tolerance combined ranks vary

summarizes the mesh, thoron-resolution, and closure-sensitivity checks.

6. Published-Data Worked Example

A quantitative worked example uses published soil-gas carbon dioxide partial-pressure (pCO₂)–oxygen partial-pressure (pO₂) regression coefficients from the Cole Farm watershed and the metadata of the associated open PANGAEA dataset [59,60]. Hodges et al. reported slopes of $-0.65\pm 0.08$ at the non-carbonate ridgetop (CFRT), $-0.37\pm 0.05$ at the carbonate-bearing east midslope (CFEMS), and $-0.48\pm 0.04$ at the west midslope (CFWMS), based on 0.20 and 0.40 m soil-gas observations. Soil moisture and mineralogical context were reported with the gas measurements.

Using the ridgetop slope magnitude as the local gas-phase aerobic reference, the fraction missing from the gas-phase CO₂ response is calculated as

$$f_{\mathrm{miss}}=1-{\displaystyle \frac{|m_{\mathrm{site}}|}{|m_{\mathrm{CFRT}}|}}.$$

(20)

This gives $0.431\pm 0.104$ at CFEMS and $0.262\pm 0.110$ at CFWMS, consistent with the published interpretation that aqueous/mineral partitioning removes part of respired CO₂ from the gas phase. The calculation is reproduced in data/published_case_hodges2021.csv.

Table 18. Published -data worked example based on Hodges et al. [59] and the associated open dataset [60]. Uncertainties are propagated from the reported regression standard errors.

Site	Context	Published Slope	${\mathit{f}}_{\mathbf{miss}}$	Diagnostic Reading
CFRT	non-carbonate ridgetop reference	$-0.65\pm 0.08$	0	aerobic/diffusive reference
CFEMS	carbonate-bearing east midslope	$-0.37\pm 0.05$	$0.431\pm 0.104$	gas deficit requires water/mineral context
CFWMS	carbonate-bearing west midslope	$-0.48\pm 0.04$	$0.262\pm 0.110$	partial gas deficit; source-only reading unsupported

summarizes the published-data worked example and propagated uncertainties.

The example applies the diagnostic logic to published quantitative observations: a lower gas-phase CO₂ response is not interpreted as lower biological production without considering water state, carbonate mineralogy, and aqueous export. It does not calibrate the reduced partial differential equation (PDE), estimate its transport coefficients, or re-fit the full raw time series. The associated open dataset contains 462 tabular data points with pCO₂ and pO₂ observations across the reported sites and depths, but the present worked example intentionally uses the peer-reviewed regression statistics and metadata to avoid implying a site-specific inversion that the reduced model was not designed to perform.

7. Targeted Pattern-Level Literature Consistency Check

The numerical experiments are not fitted to individual field sites. The comparison with the literature is used in a more conservative way. Published studies are treated as independent evidence of behavior classes. Each case contributes an extracted pattern, such as biological masking, pressure-driven transport, observation-support distortion, decay-limited radiometric response or multi-gas source ambiguity. The model is then checked against the corresponding behavior class.

Table 19. Condensed targeted consistency matrix. The comparison tests whether the reduced model reproduces published behavior classes, not whether it reproduces individual sites. The complete DOI-traceable matrix is provided in the Supplementary Materials.

Literature Case	Extracted Pattern	Model Component	Comparison Type	Evidence
Hanson et al. [32]	root and microbial respiration can dominate soil CO2	biological source term and topsoil masking	pattern-level	Level 1
Deepagoda et al. [26]	diffusivity and air permeability depend on air-filled porosity and structure	$D_i^{\mathrm{eff}}\left({\theta }_a\right)$ and attenuation	semi-quantitative	Level 2
Rouf et al. [11]	soil-gas transport parameters can be hysteretic	history-dependent transport state	mechanistic	Level 2
Kuang, Takle and Harp [53,54,55]	pressure forcing can drive vadose-zone gas motion	reduced forcing velocity $v_g\left(t\right)$	pattern-level	Level 1–2
Gil-Loaiza et al. [47]	probe design and flow affect representativeness	observation support $w_q\left(z\right)$	methodological	Level 2
Nazaroff, the International Atomic Energy Agency (IAEA), and Spoto [14,15,36]	radon/thoron signals depend on production, emanation, transport, decay and closure	decay-limited radiometric module	process-level	Level 1–2
Chiodini, Bini, Lombardi and Voltattorni [39,42,43]	CO2, Rn, He and isotopes reduce deep/shallow ambiguity	non-redundant tracer family	interpretive	Level 1–2
Pumpanen, Kutzbach and Xu [48,49,50]	chamber and pressure conditions affect flux estimates	surface observation operator [51]	metrological	Level 2

gives the condensed extraction matrix used in the main text. The complete matrix, including limitations and assigned evidence level, is provided in the Supplementary Materials. Figure 11 shows the comparison workflow.

The comparison is consistent with four interpretation principles. First, biological source variability can dominate the topsoil signal and can mask transport information. Second, air-filled porosity and structure can change gas diffusivity enough to mimic a source change. Third, forcing and measurement support can alter the measured signal even when source strength is unchanged. Fourth, radon/thoron and multi-gas observations can add complementary process sensitivity when combined with profile, state, or support information; no single tracer is treated as a direct source proxy.

The literature comparison remains deliberately conservative. It shows that the reduced analysis is consistent with the main ambiguity classes that motivate the study; it is not a validation of the complete workflow.

8. Discussion

8.1. What the Corrected Diagnostics Add

The corrected analysis distinguishes a source-responsive observation from a source-attributing design. A single CO₂ flux retains one projected source direction, but the same row cannot separate amplitude, effective depth, or source class. A profile can add a geometry-sensitive direction. Tracer and support diversity become most useful when independent state information constrains nuisance behavior. This is the central methodological result: source informativeness is evaluated after common-threshold nuisance projection, whereas detectability is assessed separately from raw response magnitude.

The noise-scaled check tempers the structural result. Although the full set has three projected amplitude directions structurally, only one lies at or above the illustrative one-standard-deviation threshold within the declared numerical tolerance. A rich observation design can therefore improve geometry without guaranteeing practical field resolution. Measurement covariance, natural parameter variability, model discrepancy, and instrument detection limits remain necessary for any site-specific inference.

8.2. Interpretation of H1–H3

H1 is supported as an attribution statement, not as a claim of zero CO₂ source sensitivity. Restricted CO₂ responses can be reproduced by different source, transport, and support paths. H2 is supported within the synthetic design: decay-separated tracers and ideal independent state constraints can increase projected rank, but the state constraints are deliberately favorable and should not be equated with generic sensors. H3 is supported only at the level claimed: the end-member matrix and $D_{\mathrm{design}}$ translate literature-informed soil properties into different monitoring priorities. A future study would need a separate parameterized Jacobian for each soil class to test H3 numerically.

8.3. Published Evidence and Practical Relevance

The Cole Farm example demonstrates why source interpretation cannot rely on gas-phase CO₂ alone. Published regression slopes imply gas-phase deficits of approximately 43% and 26% at carbonate-bearing midslopes, while water and mineralogical observations support aqueous/mineral partitioning rather than a simple decrease in biological production [59,60]. The example is quantitative, but it is not a calibration of the present transport equation.

The broader literature comparison remains a pattern-level check. It supports pressure-forcing, support, biological, radiometric, and multi-gas ambiguity classes without establishing field validation of the complete workflow.

8.4. Eco-Geotechnical and Engineered-Cover Contexts

Vegetation can alter suction, infiltration, and hydraulic conductivity in slopes, thereby changing the gas-continuity state that accompanies a soil-gas observation [61]. Evapotranspirative landfill-cover experiments likewise show that root-water uptake and drying can increase gas permeability and gas emission [62]. These studies support the relevance of water-state and support constraints in vegetated or engineered covers. They do not mean that root architecture, methane oxidation, or transient cover hydraulics are represented by the present solver.

8.5. Applicability and Limitations

The method applies to gas-continuous unsaturated soils when a one-dimensional support approximation is defensible. Preferential pathways, cracked soils, and fractured regolith may be screened through end-member and pathway-state logic, but strongly localized 2D/3D flow requires an extended model. Near gas disconnection, source attribution from gas observations should be suspended or explicitly qualified.

Other limitations are: prescribed rather than dynamically solved water state; spatially uniform coefficients within each snapshot; Fickian diffusion and uniform reduced advection; no moisture-dependent radon emanation; no carbonate reaction network; no transient source or forcing history; local finite-difference sensitivities; idealized state constraints; and illustrative rather than site-specific observation covariance. The thoron support averages are numerically stable at the baseline mesh, but pointwise near-surface thoron profiles require finer resolution.

8.6. Practical Use

A field design should begin with the dominant ambiguity. Gas-only screening may be sufficient for anomaly detection. Source attribution requires companion observations that constrain the relevant alternative: water state for gas-connectivity filtering, pressure for forcing, depth support for geometry, isotope or noble-gas ratios for source class, and transfer information for chamber or line artifacts. The rank calculation is then used as a design screen; it is not reported as the number of field sources.

9. Practical Field Translation

Table 20. Practical translation of the diagnostic analysis. The evidence basis is stated explicitly.

Ambiguity	Typical Setting	Minimum Context	Diagnostic Upgrade	Basis
Biological masking	organic topsoil, peat	CO2, temperature, moisture	carbon isotopes/radiocarbon when source age matters	literature and expert
Water-state filtering	clayey, ash, seasonally wet, compacted soils	gas profile, ${\theta }_a$ or moisture/suction	repeated wetting–drying and water-table context	numerical and literature
Transport/forcing	sandy or structured gas-continuous soil	profile, pressure, surface flux	sensitivity screen for $D_i^{\mathrm{eff}}$ and $v_g$	numerical
Pathway activation	cracked clay, fractured regolith	pressure, moisture, depth-resolved gas	Rn/Tn and repeated event monitoring	literature and expert
Deep/shallow source class	volcanic, tectonic, carbonate settings	CO2, Rn, pressure/geology	He/Ne, He isotopes, carbon isotopes	literature and expert
Observation support	probe/chamber/line systems	support geometry, flow, pressure, timing	transfer and support-sensitivity correction	numerical and literature

separates recommendations that follow directly from the numerical experiments from those supported primarily by published evidence or expert field interpretation.

The table is not a universal sensor prescription. A minimum set describes the anomaly; an enhanced set constrains the dominant nuisance process; a diagnostic set is justified only when source attribution is the objective. Soil type, water state, forcing, support, and uncertainty should be reported with the gas observation rather than as secondary metadata.

10. Conclusions

The corrected analysis treats soil-gas interpretation as a local source–transport–observation screening problem. Unit-integral source kernels separate amplitude from geometry, and a common absolute threshold is used to truncate and project away nuisance sensitivity before source rank is calculated.

A restricted CO₂ observation can retain a source-related direction while remaining attribution-ambiguous. At $\tau =0.05$, the CO₂ surface-flux set retains one projected amplitude direction; the CO₂ profile retains one amplitude and two combined geometry–amplitude directions; ideal state constraints increase the projected ranks to two and three; and the full set reaches three amplitude and four combined directions. These are structural directions, not identifiable field sources. Under illustrative noise scaling, one projected source dimension lies at or above the one-standard-deviation threshold in each set within a numerical tolerance of ${10}^{-10}$.

Grid refinement and a 27-combination closure sweep support the display-tolerance rank pattern. A published-data worked example shows that gas-phase CO₂ deficits of approximately 43% and 26% at carbonate-bearing sites require water and mineralogical context and cannot be read as simple source decreases.

The practical outcome is a pre-field design screen. Its use is justified only within gas-continuous unsaturated conditions and with explicit reporting of water state, forcing, observation support, and uncertainty. Site-specific source attribution still requires calibrated covariance, richer process representation, and field evidence.