Level 2 data from the CrIS instrument onboard NOAA's JPSS satellites are used to construct realistic profiles of trace gases (N₂O, CH₄, H₂O) and temperature as well as surface temperature and emissivity useful for the TIR radiative transfer modeling. This study leverages 100 CrIS soundings over the US Midwest acquired on 23 August 2023. The selected date provides a large number of clear-sky soundings spanning a wide range of thermal contrast (2–12 K), providing a representative sample of US Midwest summertime conditions for evaluating retrieval sensitivity. Figure  shows the spatial distribution of the selected CrIS pixels, color-coded by thermal contrast, which is defined as the difference between retrieved surface temperature and that of the lowest atmospheric layer. The selected region in Fig. is dominated by agricultural land cover, primarily corn and soybean croplands. Thermal contrast is most important for the TIR band, as it governs the strength of the upwelling signal and thus influences vertical sensitivity. The prior error matrices for H₂O and temperature profiles available in the CrIS Level 2 data are used to set a priori constraints in the linear sensitivity study (see Sect. ).

Airborne in-situ measurements of N₂O were conducted over the state of Iowa, located within the corn belt of United States (41 to 43.5° N and 92 to 95° W), in 2022 as part of the MAIZE campaign. A total of eight research flights took place between 18 and 30 May 2022 using a Mooney aircraft operated by Scientific Aviation, Inc. Flights were conducted during growing season under fair weather conditions, avoiding active precipitation, low visibility events, and on days with steady winds. This period, characterized by recent fertilizer application and warm and moist conditions was conducive to elevated N₂O emissions from cropland soils. Each flight lasted around 5–6 h and was conducted between 11:00 and 18:00 local time to ensure sampling within a well-developed PBL. The aircraft flew at an average altitude of ∼ 475 m above ground level (a.g.l.), with transects oriented perpendicular to the prevailing wind direction to enhance sensitivity to surface fluxes. Two vertical profiles were captured during each flight to characterize the PBL structure. The primary target species, N₂O, was measured using a Los Gatos Research (LGR) N₂O/CO Analyzer (model 916‐0015), which also recorded H₂O and CO. In addition, CH₄ and CO₂ were measured using a Picarro G2401–m analyzer . The N₂O in situ measurements are used to approximate column-integrated mixing ratio of N₂O and analyze its spatial variability. Figure a shows the area coverage for each flight and Fig. b indicates the corresponding in situ N₂O mixing ratios measured during each flight. Figure b has two vertical axes, with the left and right axes showing in situ N₂O mixing ratio in ppb for alternating flights to improve readability. In both panels, faint lines indicate vertical profile segments and bold lines mark the horizontal PBL transects used to quantify XN2O variability for the semivariogram-based detectability analysis.

Hourly N₂O fluxes were measured from May 2022 to April 2023 in a conventionally-tilled maize field near Villa Grove, Illinois, using 16 automated chambers. To capture spatial and temporal variability of emissions, the chambers were distributed across four sampling nodes within ∼ 5 ha area, with nodes spaced 50–100 m apart (Fig. a inset). Each node contained four chambers radially positioned 12 m from a central N₂O gas analyzer (LI-7820, LI-COR Biosciences), which sequentially sampled the fluxes using an automated multiplexer (LI-8250). Chamber collars (20 cm diameter) were permanently installed adjacent to crop rows, ensuring that no vegetation was present within the chamber footprint and that chambers remained fully open between measurements in order to avoid shading or disturbance. Figure a shows the spatiotemporal variation of measured N₂O flux across four nodes, where chambers in the same node are averaged and then aggregated to 6 h intervals to enhance visualization. Data are missing for approximately three weeks in October–November 2022 due to instrument maintenance and crop harvest . Figure b zooms in over a high emission event of 21–25 May 2022 to highlight the hourly variability of N₂O emissions. The node-wise average time series are shown as solid lines, while the maximum and minimum chamber values within each node are shown as dashed lines. The thick grey line represents the overall hourly flux averaged over all nodes which remain mostly above 5 nmol m⁻² s⁻¹ and occasionally exceed 10 nmol m⁻² s⁻¹ in this period.

When spatially aggregating XN2O observations, the measurement error component propagated linearly from detector noise decreases with the target length scale d. This relationship assumes that the detector noise between individual soundings is independent and uncorrelated. 14σXN2O(d)=σXN2O(d0)d0d, where σXN2O(d0) is the native measurement error at sounding footprint size d0, obtained from the linear sensitivity analysis (Sect. ). At a length scale d≥d0, it is meaningful to compare the aggregated measurement error σXN2O(d) with the real-world XN2O variability δXN2O(d). With a known XN2O spatial distribution, its spatial variability can be quantified through a semivariogram : 15δXN2O(d)=12(ΔdXN2O)2, where δXN2O(d) is the standard deviation that represents variability at length scale d. Δd is an operator that differentiates all XN2O values separated by distances in a range of d±ε, where ε controls the granularity of the distance binning. The angular brackets 〈〉 average all point pairs within the specified bin.

In reality, spatially distributed XN2O measurements that can support semivariogram calculation are very rare. Nonetheless, we can approximate the semivariogram of XN2O using the semivariogram of in situ measured N₂O mixing ratio in the PBL: 16ΔdXN2O=1-exp⁡(-hb/H)1-exp⁡(-ht/H)ΔdμN2O, where hb is PBL height, ht is observation altitude determined by the instrument design parameters listed in Table  and H=7.5 km is the assumed atmospheric scale height. ΔdμN2O is the spatial differentiation of PBL N₂O mixing ratio between observation pairs separated by distance d, where μN2O represents the in situ measurements collected along horizontal flight legs within the well-mixed PBL. Because ht is significantly larger than hb, the variability of XN2O is smaller than that of μN2O. Equation () holds when the variability of XN2O is dominantly driven by the variability in the PBL, and the PBL N₂O mixing ratio is approximately uniform in the vertical direction. Combining Eqs. () and (), we obtain: 17δXN2O(d)=1-exp⁡(-hb/H)1-exp⁡(-ht/H)12(ΔdμN2O)2. Here, the semivariogram is directly calculated from extensively measured N₂O mixing ratios (μN2O) along horizontal flight legs within the well-mixed PBL during the MAIZE campaign, and the PBL height is inferred from collocated spiral profiles by identifying the sharp vertical transition of trace gases, temperature, and relative humidity between the well-mixed boundary layer and the free troposphere . The XN2O variabilities calculated using Eq. (17) differ between the airborne and spaceborne instruments because they sample different portions of the atmospheric column due to their different observational height (ht). The semivariograms are calculated for each flight from 0 to 50 km with 0.25 km bin width.

Since XN2O measurement error decreases with d (Eq. ), and XN2O variability typically increases with d (Eq. ), a critical spatial scale for variability can be numerically solved from the following equation: 18q×σXN2O(d∗)=δXN2O(d∗), where q is a positive scalar. The solution d∗(q) is the aggregated spatial scale beyond which XN2O variability exceeds q times the measurement error. q=1 means the variability equals the measurement error, and q=2 means the variability is twice the measurement error, which is typically assumed as the threshold of detectability .

A complementary way of understanding N₂O detectability is to characterize the critical spatial scales of detectable N₂O emission sources. Here, we adopt the approach from for CH₄ detectability. Similar to Sect. , we aim to determine a critical spatial scale for emission, d+(q,E), beyond which the XN2O enhancement due to emissions at value E is q times the measurement error.

Assuming a uniform emission flux E and wind speed U, the enhancement of N₂O column amount due to emission at aggregated spatial scale d is 19ΔΩ=EdU. The N₂O enhancement increases with emission intensity and accumulating distance and decreases with wind speed. We assume U=5 km h⁻¹ following . At the same length scale d, the measurement error of the observable column amount, σΩ, can be derived from Eq. () by converting XN2O to a column amount: 20σΩ=σXN2Ops1-exp⁡(-ht/H)Mgd0d, where ps is surface pressure assumed to be 1000 hPa, M is air molar weight, and g is gravity and H is the assumed atmospheric scale height. The single-sounding measurement error σXN2O (at d0, notation dropped for simplicity), sounding footprint size d0, and observation altitude ht are inherent properties of the instrument. ht is the only vertical parameter in this formulation and is used to determine the vertical extent of the atmospheric column observed by the instrument. Other parameters in Eqs. () and () can be held constant except emission E and aggregation length scale d. Then, the detectability metric q can be calculated by dividing Eqs. () and (): 21q(E,d)=ΔΩσΩ=Ed2MgUσXN2Od0ps1-exp⁡(-ht/H). Equation () indicates that q increases linearly with emission, E, and quadratically with length scale, d, consistent with the intuition that more intensive and expansive emission sources are easier to detect. Alternatively, we can fix q to a certain value, e.g., 1 or 2, and solve for the critical length scale for emission level E: 22d+(q,E)=qUσXN2Od0ps1-exp⁡(-ht/H)EMg.

We evaluate the effectiveness of airborne and spaceborne instruments in retrieving XN2O by analyzing their measurement error and vertical sensitivity. The measurement error is calculated using Eq. () and vertical sensitivity is calculated using Eq. (). Both metrics reflect the combined effect of observational constraint imposed by the instrument design and a priori constraint, as detailed in Sect. . Figure  illustrates the dependence of XN2O measurement error and near-surface sensitivity on the strength of a priori constraint for airborne and spaceborne instruments specified in Table  and compare results of three spectral settings; SWIR-only, TIR-only and joint SWIR–TIR setting. The near-surface sensitivity here is defined as the mean averaging kernel value of the lowest two atmospheric layers, representing the lowest ∼ 1 km of the atmosphere (corresponding pressure values are 962 and 901 hPa). In Fig.  the a priori constraint is adjusted by scaling the GOSAT/MethaneSAT-based N₂O prior standard deviation profile (see Fig. a) by a factor γ.

The sensitivity analysis is performed using logarithmically spaced γ values between 0.03 and 10, spanning retrieval regimes from strong domination by the prior regularization (small γ) to those controlled primarily by the observation constraint (large γ). The applied γ only modulates N₂O a priori covariance matrix and does not affect the a priori covariance matrices of all other state vector elements. The results presented in Fig.  are the mean values calculated separately using the environmental conditions at 100 CrIS sounding locations shown in Fig. . These results correspond to single-shot observations at native instrument footprint size (d0=20 m for airborne and d0=0.7 km for spaceborne). Across the 100 selected CrIS soundings, the surface temperature ranges from 302.8 to 318.8 K (mean: 308.8 K), while the thermal contrast ranges from 2.8 to 12.3 K (mean: 6.6 K). Consistent across instruments and spectral settings, XN2O precision generally improves with tighter a priori constraint at small γ at the expense of degrading near-surface sensitivity.

For the airborne instrument (Fig. a–b), the XN2O measurement error shown in panel (a) reveals three distinct regimes. In the rightmost side with large γ, the retrieval is largely dependent on the amount of information provided by observations, with random detector noise dominating the error. In this regime, SWIR–TIR joint setting delivers the lowest measurement error as it combines information from two bands which increase the effective observational constraint. In contrast, the SWIR and TIR bands alone have fewer spectrum data points and thus provide weaker observation constraints than the joint band case, leading to larger errors. In a transition regime where γ is roughly 2–5, dual-band case briefly exhibit higher measurement errors than those from the SWIR or TIR bands alone, because the retrieval is less stabilized by the a priori constraint as compared to the single-band cases. In other words, SWIR-only and TIR-only cases begin to “feel” the a priori constraint earlier due to weaker observational constraints. As γ continues decreasing, all three cases become increasingly controlled by the a priori constraint, so the errors converge and the dual-band case falls between the SWIR-only and TIR-only cases. For the SWIR-only case, this transition to a prior-dominated regime is particularly abrupt, resulting in a rapid reduction in the XN2O measurement error that is synchronized with the simultaneous decrease in near-surface sensitivity. This behavior is also consistent with prior covariance tuning behavior reported for SWIR CH₄ retrievals in and likely reflects the inherent differences between shortwave and longwave radiative transfer, including both the strength and vertical distribution of the N₂O Jacobians.

Figure b highlights the trade-off value of γ between measurement error and near-surface sensitivity. At a very large γ (rightmost side) the retrieval is primarily observation dominated, resulting in relatively strong TIR near-surface sensitivity due to the stronger TIR observational constraint. Under this regime, the SWIR-only retrieval exhibits the weakest near-surface sensitivity, while the joint SWIR–TIR retrieval benefits from complementary information from both bands and therefore shows the highest sensitivity. A very strong prior (leftmost side, small γ) suppresses the SWIR near-surface information which increases rapidly as the prior is relaxed and then reaches a plateau. A value of γ=0.5 (vertical black line) provides a balanced operating point because it sits on this plateau and SWIR near-surface sensitivity at this point is nearly saturated without entering the weak-prior regime where its value degrades quickly. This choice also keeps the dual-band case sensitive to observations in both bands, resulting in low measurement error while retaining sensitivity to the near-surface layers.

For the spaceborne instrument (Fig. c–d), the same qualitative order holds but the curves are less structured. XN2O measurement error (Fig. c) is overall lower than that from the airborne instrument largely due to the significantly wider TIR spectral range enabled by the spaceborne detector. The SWIR produces the largest measurement errors across all prior strengths. Both TIR-only and joint SWIR–TIR settings achieve comparable errors, but the integrated approach improves near-surface sensitivity relative to the TIR band alone (Fig. d). This added sensitivity is particularly beneficial given the inherently reduced surface sensitivity in satellite observations due to greater path lengths. We choose the same value of γ=0.5 for the spaceborne instrument because it provides a balanced operating point between observational and a priori constraint with retaining sensitivity to near-surface layers along with achieving low XN2O measurement error. At this selected operating point γ, single sounding XN2O measurement error for the joint SWIR–TIR retrieval spans the interval 2.8–3.8 ppb (mean: 3.2 ppb) for the airborne instrument at a footprint size of d0=20 m, whereas that for the spaceborne instrument ranges 0.6–1.8 ppb (mean: 1.1 ppb) at d0 of 0.7 km.

Figure  shows the vertical structure of the N₂O column averaging kernel for three spectral settings under different a priori strengths, illustrating how measurement information is spread throughout the observed atmospheric column. For the airborne instrument (Fig. a–c), the SWIR-only exhibits stronger sensitivity to the column with a discontinuity at the observation altitude of 9.25 km . This behavior follows the solar-backscattered nature of the viewing geometry where the incident sunlight samples the atmosphere only once above the aircraft, whereas the photons traverse the air mass twice (downward to the surface and then upward to the sensor) below the aircraft. In contrast, the TIR-only relies on thermal emissions and therefore show no sensitivity above the aircraft altitude, with its response confined to layers below 9.25 km. Within this region, TIR band shows an enhanced sensitivity in the layers just below the aircraft, which is plausibly related to the vertical weighting function in the retrieval. The joint SWIR–TIR setting alleviates this localized amplification and distributes the information more evenly by integrating complementary strengths of both bands.

For the spaceborne instrument (Fig. d–f), the overall vertical patterns are similar but smoother, because satellite geometry at orbital altitude avoids the discontinuity that appears in the case of airborne instrument. SWIR-only setting maintains strong column-wide sensitivity approaching near-unity kernel values for sufficiently relaxed a priori constraints. The TIR-only and the joint SWIR–TIR settings exhibit similar vertical response structures broadly, owing to the dominant contribution from the TIR band. Nevertheless, the inclusion of the SWIR band provides a visible improvement over the TIR-only case, particularly in enhancing sensitivity in the PBL, thereby highlighting the advantage of SWIR and TIR synergy for XN2O retrievals. For both instruments and all spectral settings, as the a priori constraint is relaxed (increasing γ), the averaging kernels start incorporating information from the observational constraint and shift from prior-dominated to measurement-informed.

We determine the critical spatial scales for the airborne and spaceborne instruments at which XN2O variability becomes distinguishable from measurement error. Here, the observed spatial structure of XN2O variability is inferred from in situ PBL N₂O observations, and the instrument-specific XN2O measurement errors are estimated from the linear sensitivity analysis (see Sect. ).

Figure  presents the semivariograms of PBL N₂O mixing ratios derived from eight MAIZE flights (i.e., 12〈(ΔdμN2O)2〉 in Eq. ) along with their ensemble mean. Although the semivariance for all days generally increases with the separation distance, substantial variability is evident across individual flights. Several flights exhibit abrupt changes, which align with spatial intermittency in the PBL N₂O due to patchy sources and evolving boundary-layer conditions. The semivariogram sill, i.e., the maximum variance, also shows significant variation across flights, reflecting substantial differences in the overall intensity of heterogeneity. The extent of variability can also be attributed to meteorological conditions such as wind speed. Flights with stronger winds (e.g. 20220520, 20220521, 20220529, 20220530) show reduced semivariance as enhanced horizontal mixing diminishes spatial gradients, whereas flights with weaker winds (e.g. 20220518, 20220527) allow localized emission contrasts to persist, leading to higher semivariance. The ensemble-mean semivariogram, shown as the black line in Fig. , exhibits a smooth, steady rise and is utilized as a representative model of typical N₂O PBL variability for the subsequent detectability analysis.

Built on the MAIZE semivariograms, the observed PBL N₂O variability translates to expected XN2O variability through Eq. () and is then compared with XN2O measurement error for airborne and spaceborne instruments. Measurement error here refers to the single-sounding uncertainty at the native footprint size d0, whereas the uncertainty after averaging independent soundings to the spatial aggregation scale d is referred to as XN2O precision. Figure  shows the trends of XN2O variability and XN2O precision as functions of the spatial aggregation scale (d). The two solid lines with circular markers in Fig.  represent the inferred XN2O variability for airborne (blue) and spaceborne (black) instruments, respectively. The circular markers represent the distance bins used in the semivariogram calculation. Although the same semivariogram is used in calculating the XN2O variability for both instruments, the results differ, as this variability depends on the observational altitude (ht) of the instrument (see Eq. ) and thus the relative weight of PBL variability in the observed atmospheric column. The higher ht for the spaceborne instrument dilutes the boundary-layer heterogeneity more, yielding smaller XN2O variability as compared to that for airborne instrument at a given horizontal distance. The blue solid line in Fig.  corresponds to the aggregated precision σXN2O(d) of airborne instrument, and black solid line corresponds to that of spaceborne instrument, computed from single-sounding XN2O measurement error (σXN2O(d0)=3.2 ppb for the airborne instrument and σXN2O(d0)=1.1 ppb for the spaceborne instrument) obtained in Sect. , using Eq. (). Airborne and spaceborne precision lines do not overlap because σXN2O(d0) for airborne is larger (3.2 ppb) as compared to that for the spaceborne (1.1 ppb), and this quantity decreases quickly for the airborne instrument with the spatial averaging due to much smaller footprint size (d0=20 m for airborne and d0=0.7 km for spaceborne, see Table ). As formulated in Eq. (), the XN2O precision decreases log-linearly with d due to averaging of independent soundings, while the XN2O variability increases with d following the behavior shown in Fig. . The intersection points of these two quantities (marked with red stars in Fig. ) define the critical spatial scale d∗(q), at which atmospheric variability is q times the measurement error, as defined in Eq. (). The dashed lines represent a more conservative detectability requirement by plotting q×σXN2O(d) with a commonly used criterion of q=2. The corresponding intersection points locate the spatial scales where XN2O variability is twice the XN2O precision. For q=2, the critical spatial scale (d∗(q=2)) is approximately 2.5 km for the airborne and 22 km for the spaceborne instrument. This reflects how XN2O variability can be resolved from measurement noise under typical conditions observed during MAIZE.

In Sect. , the critical spatial scales for variability are inferred by comparing XN2O precision against XN2O variability that implicitly reflects the influence of surface emissions, but this variability does not provide a quantitative mapping between N₂O emission strength and XN2O enhancements. Here, we make that connection explicit by modeling expected XN2O enhancement resulting from a uniform emission strength E and comparing it to the corresponding XN2O precision at matching spatial scales. We then quantify the emissions detectability using the q(E,d) metric, introduced in Sect. . Here q(E,d) is a continuous variable, defined as the ratio of emission-induced enhancement in XN2O to the aggregated measurement error at spatial aggregation scale d (Eq. ). Figure  shows q(E,d) as a function of emission strength (E) and spatial aggregation scale (d) as in Eq. () for the airborne (panel a) and spaceborne (panel b) instruments. Consistent across the instruments, detectability increases linearly with E and quadratically with d, implying that stronger sources can be detected at much finer scales while weaker emissions require substantial spatial aggregation to achieve the same level of confidence q.

White regions in Fig.  correspond to the detectability values below the minimum plotted contour level (q<2-8). The black contour lines in Fig.  indicate the solutions of Eq. () for q=1 and q=2; these provide a convenient way to read off the level of spatial aggregation needed to achieve a chosen detectability for a given emission strength. Here, q=1 means that an emission-induced enhancement is equal to the aggregated precision and q=2 indicates XN2O enhancement becomes twice the aggregated precision at a given spatial aggregation scale d. Using a representative uniform emissions of 5 nmol m⁻² s⁻¹, consistent with the mean value of all nodes from 21–24 May in Fig. b, Fig.  implies critical spatial scale d+ is ∼2.1 km for the airborne instrument at q=1 and ∼2.9 km at q=2. For the spaceborne instrument, the corresponding values are substantially larger with d+ ∼8.4 km at q=1 and ∼12 km at q=2. For a higher uniform emissions of 10 nmol m⁻² s⁻¹, the d+ reduces to 1.5 km at q=1 and 2.1 km at q=2 for the airborne instrument, and the corresponding values decreases to 5.9 km (q=1) and 8.5 km (q=2) for the spaceborne instrument. d+ values are different for airborne and spaceborne instruments as it is dependent on instrument properties such as footprint size (d0), observational altitude (ht) and single-sounding measurement error (σXN2O(d0)). Overall, weaker emissions require larger-scale spatial aggregation to achieve a certain detectability level for both instruments.

This section provides a combined view of critical spatial scales reported as d∗ and d+ in Sect.  and , respectively. Table  summarizes these scales for the airborne and spaceborne instruments at two detectability levels, q=1 and q=2, illustrating how the required aggregation scale depends on detectability threshold and the approaches to estimate XN2O variability. In all cases, the airborne instrument consistently achieves smaller critical spatial scales as compared to spaceborne instrument. Although the airborne single-sounding error is larger, its much finer footprint size d0 of 20 m allows more rapid reduction of random error with aggregation length scale d and detectability is achieved quickly under favorable conditions. In contrast, the spaceborne instrument with footprint size d0 of 0.7 km, generally requires aggregation over several kilometers to reach the same detectability threshold, emphasizing that its strength lies in detecting broader regional patterns rather than localized enhancements.

For the variability-driven estimates, the choice of semivariogram primarily controls the inferred critical spatial scales. Using the mean semivariogram of all 8 flights conducted in May 2022 yields more conservative critical spatial scales than using the semivariogram of highest-variability flight (27 May 2022), which represents a more favorable scenario associated with relatively weak wind and reduced atmospheric mixing conditions. When the XN2O variability is twice the XN2O precision, d∗ reduces from 2.5 to 1.7 km for the airborne instrument and from 22 to 16 km for the spaceborne instrument when switching from the mean semivariogram to the high-variability semivariogram. For the emission-driven cases, the critical spatial scales d+ are reported for uniform emissions of E=1, 5, 10 and 20 nmol m⁻² s⁻¹ in Table . As anticipated, d+ decreases systematically with increasing uniform emission strength. For instance, increasing E from 1 to 20 nmol m⁻² s⁻¹, reduces the d+(2,E) from 6.6 to about 1.5 km for the airborne, and from 26 to around 5.9 km for the spaceborne instrument. Together, the variability- and emission-driven results bracket the spatial scales over which XN2O signals are expected to become detectable under typical MAIZE-like boundary-layer variability and plausible agricultural emissions.