Broadband longwave irradiance exchange at the land-atmosphere interface is governed principally by surface temperature, surface emissivity, and atmospheric downwelling emission (Rodgers, 2000). For an opaque surface and for measurements sufficiently close to the surface that additional atmospheric path terms between the surface and radiometer can be neglected, the measured upwelling irradiance L can be written as 1L=ϵσTs4+(1-ϵ)l. Here, l is the measured downwelling longwave irradiance, ϵ is broadband surface emissivity, Ts is physical surface temperature, and σ=5.670374419×10-8 Wm-2K-4 is the Stefan–Boltzmann constant. The first term in Eq. (1) represents thermal emission by the surface, whereas the second represents the reflected fraction of atmospheric downwelling irradiance. If ϵ is externally specified, Eq. (1) can be solved directly for Ts. If both Ts and ϵ are unknown, however, a single observation provides one radiometric equation for two coupled surface variables.

Taking the surface as a unit-emissivity emitter provides an observable radiometric temperature directly from the measured upwelling irradiance: 2Ts′=Lσ1/4. Defining the equivalent radiative temperature of the downwelling irradiance as Tl=(l/σ)1/4, Eqs. (1) and (2) give 3Ts′4=ϵTs4+(1-ϵ)Tl4. Thus, Ts′ represents an emissivity-weighted radiometric state formed from the physical surface and atmospheric longwave contributions. It should not in general be interpreted as the physical Ts, nor as a universal upper or lower bound on Ts. Its principal value here is diagnostic: because Ts′ is directly observable, changes in Ts′ between two nearby observations indicate whether the combined surface radiative state has changed appreciably. The retrieval therefore uses the difference Ts′(t2)-Ts′(t1) as a quasi-steady screening variable rather than assuming that temporal proximity alone guarantees thermal stability.

For two observations, the same emissivity can be written in the two radiative balances L1=ϵσTs,14+(1-ϵ)l1 and L2=ϵσTs,24+(1-ϵ)l2. Over a sufficiently short interval, broadband emissivity can be treated as invariant. If the physical surface-temperature change is also sufficiently limited, the two observations provide additional information through their differing radiative states.

The existence of two observations does not by itself make emissivity identifiable. Define ΔL=L2-L1 and Δl=l2-l1. If the change in L closely follows the change in l, the pair may contain little additional surface-sensitive information even though both irradiances are measured accurately. Conversely, a pair becomes more informative when the upwelling change contains a resolvable component beyond the contemporaneous downwelling change. This motivates the distinction between temporal admissibility and irradiance identifiability.

Observability refers to whether the measurements contain information about the unknown surface state; irradiance identifiability refers more specifically to whether a candidate pair contains a resolvable surface-sensitive contrast relative to changes in atmospheric downwelling irradiance; and numerical conditioning describes the sensitivity of the nonlinear solution to perturbations in the measurements or state variables. These concepts are treated separately in the retrieval. Physical and radiometric information are assessed before the final nonlinear inversion, and Newton residuals and local sensitivities are interpreted as diagnostics of an already selected production state rather than as substitutes for observability.

When emissivity is known, Eq. (1) can be rearranged to give 4Ts=L-(1-ϵ)lϵσ1/4. When ϵ is unknown, Eq. (4) emphasizes the nonlinear coupling between emissivity and temperature. Apparent temperature from Eq. (2) is therefore used only as a short-term state diagnostic, not as the retrieved physical temperature.

The measured L(t) and l(t) series are processed with a third-order Savitzky–Golay filter using an 11-sample frame. Missing-value flags are converted to missing values before processing. Where smoothing requires a complete local sequence, missing samples are temporarily filled by linear interpolation with nearest-value end treatment and the original missing locations are restored immediately after filtering; interpolation therefore enables smoothing but does not create valid observations at originally missing times.

For each field dataset, an initial network-specific time pair defines a validation midpoint tm. The midpoint is used only as the temporal reference for the subsequent population-based search. All distinct candidate pairs (ti, tj), with j > i, are constructed subject to both endpoints lying within ±10 min of tm and to a maximum pair separation of 10 min, 5|ti-tm|≤10min⁡,|tj-tm|≤10min⁡,60<|tj-ti|≤10min⁡. The 10 min interval defines the maximum search geometry rather than a prescribed production separation. The actual search proceeds progressively through radii of 1, 2, …, 10 min.

All four irradiances L1, l1, L2, l2 must be finite. A candidate must also satisfy the quasi-steady apparent-temperature condition 7|Ts′(t2)-Ts′(t1)|≤1.0K. Equation (7) limits thermodynamic-state change; satisfying this criterion alone, however, does not establish that the inverse problem is sufficiently observable.

For each surviving pair, introduce the intermediate quantities 8N=L1-l1,9C1=14l1-3/4,C2=14l2-3/4,10M=12((L1-l1)C1+(L2-l2)C2),11J=L11/4-12(L11/4+L21/4-l11/4-l21/4), and D=J4-l1, the candidate linear emissivity is 12ϵlin=N-4MJ3D. Numerical screening then uses the relative denominator-conditioning measure 13κD=max⁡(|J4|,|l1|)|D|, and candidates with κD>106 are excluded. The resulting linear estimate must additionally satisfy 0<ϵlin<1; no clipping and no empirical lower emissivity bound are imposed.

Irradiance identifiability is evaluated independently of the emissivity value. With ΔL=L2-L1 and Δl=l2-l1, the production condition is 14|ΔL|>|Δl|. Under Eq. (14), the temporal change in measured upwelling irradiance must exceed the contemporaneous change in atmospheric downwelling irradiance. This criterion therefore tests for a resolvable surface-sensitive radiative contrast rather than merely reflecting atmospheric forcing.

Candidate pairs satisfying Sect. 3.3 are evaluated progressively over temporal radii r=1,2,…,10 min. At radius r, only identifiable candidates whose midpoint lies within r min of tm and whose temporal separation does not exceed r are included.

For candidate j, the uncertainty of ϵlin is propagated from the four irradiances using 15σϵ,j2=gjTCygj, where gj=∇yϵlin for yj=(L1,l1,L2,l2)T. The covariance matrix is 16Cy=σL21ρcrossρLρcrossρcross1ρcrossρlρLρcross1ρcrossρcrossρlρcross1, Here, σL denotes the standard uncertainty assigned to each longwave irradiance measurement; ρL and ρl denote the temporal correlations in the upwelling and downwelling measurements, respectively; and ρcross denotes the assumed cross-channel correlation. Candidate uncertainty contributes the baseline weight 17wD,j=1σϵ,j2+σfloor2,σfloor=10-4. To account for the information that remains available for emissivity after allowing Ts to vary, a local two-parameter Fisher matrix is calculated. With the candidate sensitivity matrix Kj and the temporal upwelling covariance CF, the Fisher matrix is 18Fj=KjTCF-1Kj. Marginalizing over temperature leaves the emissivity information 19Iϵ|T,j=Fϵϵ,j-FϵT,j2FTT,j. The scene-wise median then normalizes the valid information values: 20RF,j=Iϵ|T,jmedian(Iϵ|T),wj=wD,jRF,j. If marginalized Fisher information is unavailable or non-positive, RF,j=1 and the uncertainty-based weight is retained. No candidate is rejected solely because of its Fisher information and no fixed Fisher-information cutoff is introduced.

At each radius, the representative local emissivity state is the weighted median of the candidate ϵlin population. Because candidate pairs overlap in time, information sufficiency is evaluated from both the number of distinct source observations and an effective pair count. A stage is information sufficient when 21Ndistinct≥4,Neff≥2.5. Stability additionally requires the robust emissivity centre to satisfy 22|ϵ̃r-ϵ̃r-1|≤0.005, for two consecutive stage transitions. Once an information-sufficient stable stage is reached, the production pair is selected independently of its emissivity value, ranked first by midpoint proximity to the validation midpoint, then by preference for a separation of at least 2 min when otherwise tied, then by longer informative span, and finally by candidate uncertainty. If stability is not demonstrated, a prescribed fallback uses the best information-sufficient stage or, where necessary, the complete identifiable population within the 10 min search geometry.

Once the production pair is fixed, the robust population estimate ϵ̃ is retained as the production anchor. Two state-ordered linear emissivities are evaluated from the same selected pair, and the selected-pair thermal coefficient is 23KT=ΔLΔL-Δl. Applying this coefficient gives the thermal correction and corrected anchor: 24ΔϵT=KT(ϵL2-ϵL1),ϵT=ϵ̃+ΔϵT. The correction is applied only when the ordered emissivities, denominator, and corrected result are finite and physically admissible; otherwise ϵT=ϵ̃.

The local radiative environment of the selected pair is then characterized using the pair-mean irradiances L‾ and l‾, the surface-sensitive contrast A‾=L‾-l‾, and 25S‾T=l‾+A‾ϵT,RM=(1-ϵT)A‾ϵT. For valid surrounding observations in the same production neighbourhood, define the normalized state by 26x=S-S‾TRM,S=l+L-lϵT. A second characterization follows from Stefan–Boltzmann scaling. With TT=(S‾T/σ)1/4, Tsky=(l‾/σ)1/4, t=Tsky/TT, and G=l‾/A‾, define 27ΨSB=2G-(1+t+t2+t3),Z=ϵT(1+t+t2+t3). The framework then considers two exact reciprocal transformations: 28R1(ϵT)=2ϵT1+ϵT,R2(ϵT)=ϵT2-ϵT. A conservative backbone selector determines the default mapping: R1 is selected for ϵT≤0.963 and R2 for ϵT≥0.976. Within 0.963<ϵT<0.976, R1 is selected when the outward normalized upwelling contribution is negative or the preceding normalized state satisfies xbefore<-2/5; otherwise R2 is retained.

The backbone decision is overridden only for two specific physical-state configurations. In the Stefan–Boltzmann-negative configuration, the mapping is governed by rT=(1-ϵT)/ϵT relative to rSB=(1-0.96246)/0.96246. In the reciprocal-conductance configuration, the physical substitution is applicable only when 29xbefore<-16, The corresponding branch criterion is 30Z<2+2. Outside these physically applicable configurations, the backbone mapping is retained. The selected reciprocal result becomes the initial emissivity ϵ0 for nonlinear refinement; if a transformed result is non-finite or falls outside 0<ϵ0<1, the conservative fallback ϵ0=ϵT is used.

Starting from ϵ(0)=ϵ0, define P1=L11/4 and P2=L21/4. At Newton iteration k, 31M(ϵ(k))=12(P1+P2-(L1-l1ϵ(k)+l1)1/4-(L2-l2ϵ(k)+l2)1/4). The nonlinear residual takes the form 32f(ϵ)=ϵ((P1-M(ϵ))4-l1)-(L1-l1). Equation (32) retains the established production sign convention; the correction enters through P1-M(ϵ) and is not replaced by the alternative positive-sign form. The derivative is 33f′(ϵ)=(P1-M)4-l1-4ϵ(P1-M)3M′(ϵ), where 34M′(ϵ)=-18((L1-l1ϵ+l1)-3/4L1-l1ϵ2+(L2-l2ϵ+l2)-3/4L2-l2ϵ2). Newton–Raphson then advances the emissivity according to 35ϵ(k+1)=ϵ(k)-f(ϵ(k))f′(ϵ(k)). A fixed sequence of up to eight updates is applied. The sequence is interrupted if the derivative becomes non-finite or zero, or if the next emissivity is non-finite or lies outside 0<ϵ(k+1)<1. The eight-update structure provides a common numerical trajectory and is not described as an empirical stopping tolerance. The Newton–Raphson formulation follows standard nonlinear root-finding practice (Ortega and Rheinboldt, 2000; Nocedal and Wright, 2006).

After the final valid update, compute the pair-mean apparent temperature as 36Ts′=P1+P22σ1/4, The associated emissivity-induced temperature correction becomes 37ΔT=P1+P2-(L1-l1ϵ+l1)1/4-(L2-l2ϵ+l2)1/42σ1/4. Finally, the retrieved surface temperature follows the established convention 38Ts=Ts′-ΔT.

The final nonlinear residual is reported as Rf=|f(ϵ)|, and the minimum local sensitivity encountered over the Newton trajectory is 39Jmin⁡=min⁡k|f′(ϵ(k))|. Using the full covariance structure gives the production emissivity uncertainty 40σϵ,corr=aTCyaJmin⁡,a=(1,0,1,0)T. An independent-error limiting form is retained only as a diagnostic. Differentiating Eq. (38) with respect to emissivity gives the local temperature sensitivity ∂Ts/∂ϵ, from which the emissivity-driven contribution is 41σTs,ϵ=∂Ts∂ϵσϵ,corr. The correlated irradiance-driven contribution follows from 42σTs,L=L1-3/4+L2-3/48σ1/4σL111ρLρL111, Combining the irradiance- and emissivity-driven terms gives the total propagated temperature uncertainty 43σTs,corr=σTs,L2+σTs,ϵ2. These quantities are propagated retrieval uncertainties and are not assumed to be calibrated confidence intervals; empirical coverage is evaluated separately using the known-truth experiment. The interpretation of local sensitivity and uncertainty follows standard inverse-problem reasoning (Tarantola, 2005; Menke, 2012; Aster et al., 2019).

The field population combines observations from seven SURFRAD stations and two BSRN stations: Bondville (BON), Desert Rock (DRA), Fort Peck (FPK), Goodwin Creek (GWN), Penn State (PSU), Sioux Falls (SXF), Table Mountain (TBL), Cabauw (CAB), and Gobabeb (GOB). Their composition is summarized in Table 1.

A total of 637 source datasets entered the workflow audit. Five datasets (one BON case, one CAB case, and three GOB cases) did not contain the required finite initial irradiances and were excluded, leaving 632 usable field datasets. All 632 usable cases produced final finite production retrievals.

The field comparison uses a ground-based validation-reference temperature Ts,ref. It is derived from station-calibrated ground-based longwave thermal-radiometer observations combined with externally specified broadband emissivity, 44Ts,ref=Lref-(1-ϵref)lrefϵrefσ1/4. The field reference and retrieval are therefore not statistically independent in their irradiance measurements: both ultimately involve ground-observed longwave irradiance. The distinction is that ϵref is externally specified and is not supplied to the proposed emissivity retrieval. The field arm consequently validates combined emissivity-temperature retrieval behaviour under real observations rather than providing independent broadband-emissivity truth.

For the SURFRAD reference records, externally supplied broadband emissivity information was used to derive the validation-reference temperatures. For Cabauw and Gobabeb, the validation-reference temperature was constructed from the nearest 1 min upwelling and downwelling longwave observations at the Landsat acquisition time together with the corresponding Landsat-8 3×3 mean emissivity.

The externally specified emissivities used only to construct the field temperature reference are themselves subject to spectral, spatial, temporal, and representativeness uncertainty. The field comparison is therefore interpreted as a realistic benchmark of the combined emissivity–temperature retrieval rather than as independent emissivity truth; direct broadband-emissivity accuracy is evaluated only in the prescribed-truth experiment.

For each field retrieval, the signed temperature error is eT,i=Ts,iret-Ts,iref. Performance is summarized by bias, MAE, RMSE, standard deviation of signed error, and squared Pearson correlation coefficient R2. A separate read-only audit tracks candidate geometry, finite irradiance, quasi-steady screening, numerical/physical admissibility, irradiance identifiability, information sufficiency, production-pair selection, Newton progression, and final retrieval. The audit does not impose additional rejection criteria and does not modify the production retrieval. Detailed scene- and pair-level accounting is given in Appendix A.

Across the 632 usable field scenes, the audit contains 88 154 geometry-valid candidate pairs, of which 88 135 have finite irradiances, 81 682 satisfy the quasi-steady and denominator-validity stages, 68 525 retain a physically admissible linear-emissivity state, and 63 190 satisfy the irradiance-identifiability condition. At scene level, all 632 usable datasets contain at least one admissible and identifiable pair, all 632 enter Newton–Raphson refinement, and all 632 complete eight valid Newton updates. Appendix A provides the corresponding station-wise and stage-by-stage accounting.

The synthetic validation contains Nsyn=6000 independent truth scenes. A fixed random seed, rng(858), is used for reproducibility. Exactly 2000 scenes are generated in each of three emissivity regimes: 0.85≤ϵtrue<0.92, 0.92≤ϵtrue<0.97, and 0.97≤ϵtrue≤0.99. The initial true surface temperature is prescribed independently over 270–330 K. Synthetic temporal forcing is empirically grounded by bootstrap sampling observable temporal-driver states derived from the 632 field cases.

For each time sample, the true surface emission is S(t)=σTstrue(t)4, and exact upwelling irradiance is generated as 45L(t)=ϵtrueS(t)+(1-ϵtrue)l(t). Thus, ϵtrue enters only through the forward model and is never derived from, fitted to, or conditioned on the retrieval result.

Each truth scene is used for two paired experiments. G1 contains the exact physically generated longwave observations with no additional measurement perturbation and therefore isolates retrieval/inversion behaviour. G2 uses the same 6000 truth scenes but adds a correlated production-noise realization drawn from the covariance structure defined in Eq. (16) before unchanged Savitzky–Golay processing. Apart from this perturbation, G1 and G2 use identical truth states and the same retrieval implementation.

All 6000 generated scenes remain in the denominator used to compute retrieval success: 46fsuccess=Nsuccessful6000. Bias, MAE, and RMSE require a finite retrieved value and are therefore calculated over successful retrievals only. For Ts, error is evaluated against the true temperature corresponding to the selected production pair.

Figure 3 characterizes the progressive selector over the complete 632-case field population. Of the 632 retrievals, 628 were selected through the stable-centre pathway and only four required fallback selection. The median selected production-pair separation was 3.0 min. These results show that, for the overwhelming majority of cases, the progressive search identified a locally stable emissivity population without exhausting the prescribed fallback pathway.

The relationship in Fig. 3d is used as an empirical check for systematic displacement of the emissivity solution with increasing pair separation. Within the retained population, no strong monotonic shift is evident. Quasi-steady screening is therefore interpreted as one component of the selection chain rather than a complete observability test; stable production selection also requires irradiance identifiability and sufficient independent information.

Following selected-pair thermal correction, the retrieval evaluates whether the surrounding radiative state supports one of the exact reciprocal emissivity transformations. Figure 4 presents the two physical configurations for which the applicability boundaries have a direct radiative interpretation.

The Stefan–Boltzmann-controlled configuration occurs in 102 of the 632 field cases, while 58 cases occupy the reciprocal-conductance configuration. The remaining 472 cases exhibit other production-state signatures and are not represented in these two diagnostic panels. The reciprocal transformation is therefore state dependent rather than universally applied, and the branch choice is made before comparison with the field validation reference.

Figure 5 evaluates Newton–Raphson behaviour after candidate screening, production-pair selection, and physical-state conditioning. Figure 5a relates the final absolute residual to the minimum local emissivity sensitivity, while Fig. 5b shows the distribution of that sensitivity across the retained field population.

The figure characterizes the numerical state of accepted production retrievals and should not be interpreted as evidence that arbitrary temporal pairs are well conditioned. Weakly identifiable and numerically inadmissible candidates are addressed earlier in the workflow. Small residuals establish consistency with the nonlinear equation at the retrieved state, while the derivative magnitude quantifies local sensitivity; neither quantity alone is treated as a complete observability criterion.

Figure 6 links the selected-pair upwelling contrast |ΔL| to local nonlinear sensitivity and propagated uncertainty. Across the 632 retrievals, the Spearman correlation between |ΔL| and correlated emissivity uncertainty is -0.4026, while the correlation between |ΔL| and minimum sensitivity is +0.4026. Minimum sensitivity and correlated emissivity uncertainty have a rank correlation of -1.0000.

The first two relationships show that larger selected-pair upwelling contrast is generally associated with greater emissivity sensitivity and lower propagated emissivity uncertainty. The perfect rank anticorrelation between sensitivity and σϵ follows directly from the mathematical structure of the local uncertainty formulation and is not treated as an independent validation statistic. Figure 6d extends the conditioning pathway to Ts through its irradiance- and emissivity-driven uncertainty components. These are propagated uncertainties; their empirical calibration is evaluated separately in Appendix C.

Figure 7 shows retrieved broadband-emissivity distributions for the nine stations. All 632 production retrievals are included, with station medians and interquartile ranges. The distributions are interpreted descriptively rather than as field emissivity-validation errors because a common independent broadband-emissivity truth is not available for the complete field population.

Figure 8 compares retrieved Ts with the ground-based validation-reference Ts for all 632 cases. Across the complete field population, the production retrieval has a bias of -0.162 K, MAE of 0.432 K, RMSE of 0.558 K, and R2=0.9984.

The low overall bias indicates little systematic displacement relative to the field reference, while the MAE and RMSE show that the typical retrieval-reference difference remains well below 1 K. Station-wise values demonstrate that the aggregate result is assembled from distinct site populations rather than repeated observations from one homogeneous environment. Station-wise validation statistics are summarized in Table . The reference temperature nevertheless shares ground-based longwave measurements with the retrieval and is therefore not treated as fully independent radiometric truth.

The principal known-truth test is the G1 noiseless experiment shown in Fig. 9 and summarized in Table 3. Of the 6000 independently prescribed truth scenes, 5924 yield successful finite retrievals, corresponding to 98.73 %. All 6000 generated scenes remain in the success denominator.

Across successful retrievals, overall emissivity bias, MAE, and RMSE are 0.03129, 0.04908, and 0.06578, respectively. The corresponding Ts bias, MAE, and RMSE are -0.335, 1.307, and 3.213 K. Performance varies across emissivity regimes: the largest positive emissivity bias occurs in the lowest-emissivity interval, while the highest-emissivity interval has a smaller absolute bias and lower emissivity RMSE. The method therefore maintains high retrieval availability across 0.85≤ϵtrue≤0.99, but low-emissivity conditions remain the most demanding part of the tested retrieval space.

The paired G2 experiment applies the production-noise model to the same truth scenes. Detailed G1–G2 comparisons are retained in Appendix B because they quantify robustness to realistic measurement perturbations rather than constituting an additional independent validation population.

The results support a conditioning pathway in which retrieval reliability arises from complementary controls rather than Newton behaviour alone. Temporal screening limits strong thermodynamic-state changes; irradiance identifiability requires surface-sensitive contrast; progressive population analysis requires sufficient independent information and centre stability; physical-state diagnostics determine reciprocal applicability; and Newton refinement solves the resulting nonlinear emissivity equation.

The field and synthetic experiments provide complementary evidence. The field observations retain real instrument behaviour, atmospheric variability, temporal variability, and surface heterogeneity, but the validation-reference Ts shares longwave radiometric information with the retrieval. The synthetic experiment provides independently prescribed emissivity and temperature truth but necessarily represents a controlled physical model of the observed environment. Confidence in the retrieval therefore derives from the consistency of the two lines of evidence rather than from either experiment alone.

Two further limitations are important. First, the conditioning and Newton diagnostics describe the screened production population and should not be extrapolated to arbitrary longwave temporal pairs. Second, propagated uncertainty quantifies local measurement and emissivity sensitivity but is not automatically a calibrated confidence interval. Known-truth uncertainty coverage is therefore evaluated separately in Appendix C. The synthetic validation was evaluated without subsequent retuning of production thresholds, pair-selection rules, reciprocal-state logic, or uncertainty parameters using the synthetic outcomes.