Thermal Multi-Sensor Assessment of the Spatial Sampling Behavior of Urban Landscapes Using 2D Turbulence Indicators

Abstract

Understanding spatial variations in land surface temperature (LST) is critical for analyzing urban climate dynamics, especially within the framework of two-dimensional (2D) turbulence theory. This study assesses the spatial sampling behavior of urban thermal fields across eight metropolitan areas, encompassing diverse morphologies, surface materials, and Köppen–Geiger climate zones. We analyzed thermal infrared (TIR) imagery from two remote sensing platforms—MODIS (1 km) and Landsat (30 m)—to evaluate resolution-dependent turbulence indicators such as spectral slopes and breakpoints. Power spectral analysis revealed systematic divergences across spatial scales. Landsat exhibited more negative breakpoint values, indicating a greater ability to capture fine-scale thermal heterogeneity tied to vegetation, buildings, and surface cover. MODIS, in contrast, emphasized broader thermal gradients, suitable for regional-scale assessments. Seasonal differences reinforced the turbulence framework: summer spectra displayed steeper, more variable slopes, reflecting increased thermal activity and surface–atmosphere decoupling. Despite occasional agreement between sensors, spectral metrics remain inherently resolution-dependent. MODIS is better suited for macro-scale thermal structures, while Landsat provides detailed insights into intra-urban processes. Our findings confirm that 2D turbulence indicators are not fully scale-invariant and vary with sensor resolution, season, and urban form. This multi-sensor comparison offers a framework for interpreting LST data in support of climate adaptation, urban design, and remote sensing integration.

1. Introduction

Urbanization has significantly altered the Earth’s surface energy balance, generating the urban heat island (UHI) effect and reshaping land surface temperature (LST) patterns across various climates [1,2,3]. Urban morphology, land cover heterogeneity, and seasonal dynamics strongly influence LST variability, often creating sharp thermal contrasts between urban and rural areas [4,5,6]. This complexity calls for advanced methods to understand urban thermal environments, especially for UHI, energy balance, and climate adaptation planning [7]. As cities grow, thermal patterns become more complex due to interactions among land cover, anthropogenic heat, and built morphology [8,9], resulting in turbulent LST variability [10,11].

The theory of 2D turbulence, from Batchelor [12] and Kraichnan [13], identifies two cascade regimes—an inverse energy cascade at larger scales and a direct enstrophy cascade at smaller scales. Spatial power spectra help detect these regimes and interpret the surface UHI. Spectral analyses have been used to study turbulence anisotropy in the urban boundary layer [14,15], aiding the mapping of energy and heat transfer shaped by urban form and surface properties [16,17]. In a recent research work, we applied 2D turbulence theory to urban thermal environments to explain how energy is transferred across spatial scales [18]. The key metrics, spectral slope and spectral breakpoint, indicated the rate of energy decay and the transition between large and small-scale features. These rely on scale invariance within the inertial subrange, though sensor resolution limits the detectable range. Coarse sensors miss fine variability, while high-resolution ones risk noise interference, raising questions about how resolution affects spectral indicators’ reliability.

Integrating remote sensing of multiple spatial resolutions is central to environmental analysis. Landsat TM/ETM+/TIRS/TIRS2 (Thematic Mapper, Enhanced Thematic Mapper Plus, Thermal Infrared Sensor, and its second generation) offers high spatial resolution (~30 m) but with a 16-day revisit, while Aqua/Terra MODIS (Moderate Resolution Imaging Spectroradiometer) provides daily coverage at coarser resolution (~500–1000 m). For clarity, we hereafter refer to these data sources as MODIS and Landsat, respectively. Understanding how spatial resolution affects interpretation is essential for many applications. It was found that in heterogeneous terrain, Landsat 7 ETM+ detected LST variations at sub-kilometer scales (10–20 K), which MODIS failed to capture these due to its coarse resolution [19]. The same study found that Landsat-derived LSTs showed much better agreement with ground-based measurements (RMSE = 1.7 K, bias = −0.5 K) than MODIS (RMSE = 4.2 K, bias = 3.2 K), confirming Landsat’s greater accuracy for fine-scale thermal variability. LST simulations from the Simple Biosphere Model (SiB2) proved to be significantly closer to Landsat-derived LSTs (RMSD: 2.92–4.46 K) than to MODIS-derived LSTs (RMSD: 4.36–11.53 K), demonstrating Landsat’s superior capability in detecting localized thermal variation in urban environments [20]. It was observed that while MODIS effectively captures large-scale and continuous LST trends, Landsat’s finer spatial resolution enables more precise mapping of urban density and land cover. Their integration enhances the analysis of urban thermal patterns by combining spatial detail with temporal frequency [21]. It was also found that MODIS underrepresents spatial variability in evapotranspiration (ET) due to its coarser resolution, while Landsat’s finer resolution captured heterogeneity more effectively, resulting in more accurate ET estimates in complex landscapes [22]. It has been demonstrated that Landsat better detects local-scale phenological changes, whereas MODIS is more suited to regional-scale monitoring, highlighting their complementary use [23]. Research has shown that Landsat-8’s higher spatial resolution provided more precise chlorophyll-a and water temperature mapping in small or nearshore water bodies, where MODIS lacked details [24]. Similarly, it was reported that Landsat, when aggregated, aligned better with theoretical models of ocean submesoscale dynamics, while MODIS showed greater variance due to higher pixel-level noise [25]. Research found that Landsat and SPOT yielded strong leaf area index (LAI) estimates (R² > 85%) in mixed grasslands, while MODIS performed poorly (R² ≈ 28–37%), failing to capture fine-scale variability [26]. An investigation has emphasized that Landsat ETM+ resolves surface heterogeneity and overcomes classification ambiguities caused by mixed pixels, whereas MODIS requires supplementary high-resolution input for accuracy [27]. It was found that Landsat ETM+ achieved near-perfect accuracy (k = 1.00, a = 99.5%) for mapping burned areas, while MODIS’s performance declined at coarser resolution, reinforcing Landsat’s reference value [28]. A study reported that Landsat-TM detected invasive species patterns in fragmented landscapes more effectively than MODIS, whose coarse resolution missed key spatial details [29]. The comparative analysis of MODIS and Landsat data across various environmental applications underscored the significance of spatial resolution in remote sensing. While MODIS offers the advantage of frequent observations, its coarse resolution may not capture fine-scale spatial variability, which is critical in heterogeneous landscapes. Landsat’s finer resolution provides more detailed spatial information but with less frequent temporal coverage. The fusion of both data sets to obtain a high-spatial-frequency and high-temporal-frequency-resolution product was also tested in different downscaling studies related to LST [30,31], ET [32] and UHI [33].

This study evaluates the scale sensitivity of turbulence indicators derived from thermal infrared (TIR) imagery by comparing land surface temperature (LST) fields from MODIS (1-km) and Landsat (30-m) sensors. Eight metropolitan regions were selected to represent five major Köppen–Geiger climate zones—desert, humid continental, humid subtropical, mediterranean, and oceanic—capturing a range of urban forms and climate conditions. The analysis includes both summer and winter imagery to assess seasonal variability. Urban cores and their rural surroundings are jointly analyzed to examine thermal gradients and interface dynamics. The main objective is threefold: (1) to compare the spectral slopes and breakpoints of LST fields obtained from MODIS and Landsat; (2) to interpret these metrics within the framework of 2D turbulence theory, focusing on forward and inverse cascade regimes; and (3) to assess the degree to which turbulence indicators are resolution-dependent or sensor-consistent. Power spectral analysis is applied to each image to quantify scale-dependent thermal behavior and examine the validity of using different sensors in urban turbulence studies.

2. Materials and Methods

2.1. Study Area Selection

This work focuses on a subset of our previous research [18] which encompasses 8 large metropolitan regions (Figure 1): Paris (France), Madrid (Spain), Nashville (Tennessee, USA), Las Vegas (Nevada, USA), Wuhan (China), Santiago (Chile), Amman (Jordan), and Ankara (Turkey). These cities were chosen based on their substantial urban footprint, which ensures sufficient pixel coverage for meaningful and valid spatial frequency analysis using both Landsat (high-resolution) and MODIS (moderate-resolution) thermal imagery. The subset of cities chosen accounts for the coarser spatial resolution of MODIS, which requires larger urban extents to capture robust spatial patterns in LST fields. The sample maintains a wide climatic and geographic diversity characterizing different urban morphologies, structures and materials. A summary of the subset of selected cities together with their Köppen–Geiger climate classifications [34] is provided in

Table 1. Climatic zones of the locations under study according to Köppen–Geiger classification.

City	Country	Köppen–Geiger Classification	Climate Zone
Santiago	Chile	Csb	Temperate
Madrid	Spain	Csa	Mediterranean
Paris	France	Cfb	Oceanic
Ankara	Turkey	Csa	Mediterranean
Amman	Jordan	BSh	Arid
Las Vegas	USA	BWh	Desert
Nashville	USA	Cfa	Humid Subtropical
Wuhan	China	Cfa	Humid Subtropical

.

2.2. Preprocessing and Image Standardization

Thermal infrared data were obtained from Landsat 5 Thematic Mapper (TM), Landsat 8 Thermal Infrared Sensor (TIRS), and Landsat 9 TIRS2 imagery. Although the original thermal bands for Landsat 5 TM and Landsat 8/9 TIRS were acquired at a native spatial resolution of 120 m (Landsat 5 TM) and 100 m (Landsat 8/9 TIRS), respectively, these bands were resampled to 30 m to match the spatial resolution of the multispectral bands in the standard U.S. Geological Survey (USGS) LST products. The LST imagery from Landsat 5, 8, and 9 as well as the MODIS was used to analyze the spatial thermal structure of each city. For both sensor types, imagery selection followed similar criteria to the previous study [18]: scenes were selected from summer and winter periods across three decades (1990s, 2000s and 2010s) with minimal cloud cover and complete urban coverage, including sufficient non-urban buffer zones. For both sensor types, imagery selection followed similar criteria to the previous study [18]: scenes were selected from summer and winter periods across three decades (1990s, 2000s and 2010s) with minimal cloud cover and complete urban coverage by means of the construction of square bounding boxes centered on the urban extent. These boxes were extended outward to the nearest integer dimension to ensure symmetry in the Fourier domain, while also capturing the urban core, its periphery, and a buffer zone of non-urban pixels. This standardized approach preserved the radial structure of urban-to-rural transitions and allowed consistent spectral and spatial analysis across cities (Figure 2 and Figure 3). Six images per city were selected where possible to ensure interdecadal comparability. Each LST scene was processed through Google Earth Engine (GEE) for cloud and snow masking. To minimize the introduction of artificial gradients that could bias spectral slope estimation, missing pixels due to cloud masking or invalid values were filled using nearest-neighbor interpolation. This conservative approach avoids artificial smoothing that could distort the spatial frequency content in the Fourier domain. It was applied systematically to all images containing relatively minor gaps, with masking thresholds kept low to ensure minimal impact on the thermal field’s structure. As in the prior work [18], square bounding boxes were constructed for each city, centered on the urban extent with extended margins to preserve symmetry in the Fourier domain and to allow radial averaging. This ensured standardized image dimensions suitable for 2D Fourier analysis across both sensor types.

2.3. Two-Dimensional Spectral Analysis of LST Fields

To analyze the spatial characteristics of LST variability and detect turbulence-like structures, we applied the same methodology as in our previous work [18] where we computed the two-dimensional (2D) power spectrum of each LST image via discrete Fourier transform (DFT). The codes were written in MATLAB Ver. 2024a [35], although the approach employed was partly inspired by the implementation of the Python (v.3.11.5) statistical analysis package TURBUSTAT [36,37]. This approach captures the distribution of spatial frequencies and allows the identification of scaling regimes via spectral slope and spectral breakpoint metrics. Prior to Fourier analysis, missing pixels (resulting from cloud or snow masking) were filled using nearest-neighbor interpolation to minimize spectral distortion. The 2D DFT was computed using the following equation:

$$F\left(u,v\right)=\sum _{x=0}^{N_x-1}\sum _{y=0}^{N_y-1}f\left(x,y\right)e^{-2\mathsf{\pi }i\left({\displaystyle \frac{ux}{N_x}}+{\displaystyle \frac{vy}{N_y}}\right)}$$

where f(x,y) is the pixel value at spatial location (x,y), and F(u,v) represents its frequency-domain transform. N_x and N_y are the total number of pixels along the x- and y-axes, respectively. The spatial frequency components in the horizontal (x) and vertical (y) directions are represented by u and v. The complex exponential function representing the Fourier basis is represented by $e^{-2\pi i\left({\displaystyle \frac{ux}{N_x}}+{\displaystyle \frac{vy}{N_y}}\right)}$. The power spectrum, which measures the strength (magnitude) of the frequency components in the transformed domain, is visualized to study the distribution of frequencies in an image. The power spectrum is derived as follows:

$$P\left(u,v\right)={\left|F\left(u,v\right)\right|}^2={\left[\mathrm{R}\mathrm{e}\left(F\right(u,v\left)\right)\right]}^2+{\left[\mathrm{I}\mathrm{m}\left(F\right(u,v\left)\right)\right]}^2$$

where P(u,v) is the squared magnitude, representing the power or energy contained at each frequency (u,v), with u as the frequency in the horizontal direction and v as the frequency in the vertical direction. The term ∣F(u,v)∣ corresponds to the magnitude of the Fourier transform at frequency (u,v). The real part of F(u,v), denoted Re(F(u,v)), corresponds to the contribution of cosine waves to the frequency component, and it can be interpreted as the x-coordinate of the complex number in the complex plane. Conversely, the imaginary part, Im(F(u,v)), represents the contribution of sine waves and is the y-coordinate of the same complex number. Together, these components provide a complete description of the frequency’s behavior within the system. We then computed the radial spatial frequency:

$$\mathrm{r}=\sqrt{{\left({\displaystyle \frac{u}{N_x}}\right)}^2+{\left({\displaystyle \frac{v}{N_y}}\right)}^2}.$$

Subsequently, we performed radial averaging to obtain the 1D isotropic power spectrum, P(r), where all frequency components at the same radius are averaged. To ensure appropriate frequency resolution, an optimal bin width for radial averaging was determined using Scott’s rule:

$$h=3.5{·\sigma ·n}^{-{\displaystyle \frac{1}{3}}}$$

where σ is the standard deviation of the frequency values, and n is the number of points. A logarithmic binning scheme was used to better represent the scaling behavior at low spatial frequencies.

2.4. Two-Segment Linear Fitting and Breakpoint Detection

To identify scale transitions in the spectral structure, the log-transformed radial power spectrum log10 P(r) vs. log10 r was modeled using a two-segment linear fit:

$$\mathrm{l}\mathrm{o}\mathrm{g}P(r)=\left\{\begin{array}{c}{m}_1·\log r+b_1, r<r_b\\ m_2· \log r+b_2, r\ge r_b\end{array}\right.$$

where $m_1$ and $m_2$ are the spectral slopes before and after the breakpoint $r_b$, and $b_1$, $b_2$ are intercepts. The optimal breakpoint $r_b$ was identified by iteratively testing candidate points (avoiding the edges) and minimizing the total residual error E:

$$E=\sum _{i=1}^{r_b}{\left(P_i-y_1\left(i\right)\right)}^2+\sum _{\mathrm{i}=r_b+1}^n{\left(P_i-y_2\left(i\right)\right)}^2$$

where $y_1\left(i\right)$ and $y_2\left(i\right)$ are predicted values from the two linear fits. The breakpoint that produced the lowest error was selected as the optimal transition point in scaling behavior. To assess the quality of each segment fit, we computed the coefficient of determination R² for both segments:

$$R^2=1-{\displaystyle \frac{{\mathit{SS}}_{\mathit{res}}}{{\mathit{SS}}_{\mathit{tot}}}} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} {\mathit{SS}}_{\mathit{res}}=\sum {\left(P_i-y_i\right)}^2,\hspace{1em}{\mathit{SS}}_{\mathit{tot}}=\sum {\left(P_{\mathrm{i}}-\overline{P}\right)}^2$$

where $\overline{P}$ is the mean power value for the respective segment. The spectral slopes m₁ and m₂, along with the breakpoint $r_b$, provide insight into changes in spatial organization and variability scales in the LST field.

For MODIS imagery, which has a coarser spatial resolution, we used an enhanced fitting algorithm with constraints to ensure monotonicity and continuity at the breakpoint, preventing artificial discontinuities and overfitting. This adjustment maintains comparability across resolutions while improving segmentation stability for lower-resolution data.

3. Results

3.1. Spectra Profiles’ Comparison

We comparatively analyzed the spectral profiles of Landsat and MODIS data. A comparative analysis revealed significant differences in both the shape of the spectra and the behavior of spectral breakpoints, offering insights into how sensor resolution and data structure influence the thermal characterization of urban environments. In the Landsat LST spectra, spectral breakpoints—marked by dashed black lines (Figure 4)—show considerable temporal variation, reflecting seasonal and intra-annual shifts. For instance, in Amman, the Landsat-derived spectral breakpoints range from −1.02832 to −1.27492, and in Ankara, from −1.27862 to −1.32574. These shifts illustrate Landsat’s sensitivity to dynamic surface processes such as vegetation change, moisture availability, and solar angle variations throughout the year. Similarly, cities like Paris and Wuhan show wide breakpoint ranges (−1.21623 to −1.4427 and −1.11127 to −1.607, respectively), reinforcing Landsat’s capability for capturing fine-grained seasonal thermal transitions across heterogeneous urban surfaces. By contrast, MODIS LST spectra (Figure 5) exhibit a remarkably consistent breakpoint for each city, regardless of the acquisition date or season. For instance, Amman and Ankara both share an identical MODIS breakpoint of −1.20598, while Las Vegas, Nashville, and Santiago (Chile) all converge at −1.14685. Similarly, Paris maintains a constant MODIS breakpoint at −1.41923, and Wuhan at −1.00823. This constancy indicates that MODIS-derived breakpoints are largely invariant to short-term environmental changes, instead reflecting stable, structural attributes of the urban climate envelope, such as dominant land cover types and urban form at a coarser spatial scale. These differences are accentuated when examining the shape of the spectral profiles. Landsat spectra tend to be more irregular and textured, showing steeper gradients and clearer thresholds between thermal zones. This is a direct consequence of Landsat’s higher spatial resolution (~30 m), which enables it to detect localized thermal heterogeneity, from small vegetated patches to asphalted surfaces or differences between building materials and street canyons. In contrast, MODIS, with its coarser resolution (~1 km), smooths spatial variability, producing more continuous and aggregated spectra that emphasize a broad-scale thermal structure rather than localized anomalies. From a methodological standpoint, this distinction is critical. Landsat LST data are ideal for analyzing microclimatic variation, intra-urban heterogeneity, and seasonal thermal dynamics, particularly when investigating land use, morphology, and heat mitigation strategies at neighborhood or street level. On the other hand, MODIS LST data offer robust, temporally stable baselines, which are highly effective for comparing urban heat characteristics across cities or identifying persistent morphological or climatic patterns at the meso- or macro-urban scale. In summary, the data reveal that while Landsat provides temporally dynamic and spatially granular insight into urban thermal behavior, MODIS emphasizes structural consistency and enables inter-city comparisons. Their complementary nature reinforces the importance of multi-scale LST analysis: Landsat is suitable for local precision and temporal variability, and MODIS for spatial generalization and climatic benchmarking.

3.2. Spatial Resolution Dependency

The spectral breakpoints, obtained from the two-segment linear fit of the spatial power spectrum, serve as indicators of the transition scale between large-scale and small-scale variability in land surface temperature (LST) fields, which are interpretable within the framework of 2D turbulence theory. The spectral breakpoint demarcates the transition from large-scale (synoptic) structures to smaller, turbulence-dominated eddies. In the context of 2D turbulence, this breakpoint delineates the energy injection and energy dissipation regimes. Results reveal a systematic divergence in breakpoint behavior between spatial scales. Landsat exhibits slightly more negative breakpoint values overall, with city-averaged medians of approximately −1.32 in summer and −1.30 in winter, indicating its enhanced ability to resolve fine-scale thermal heterogeneity. MODIS breakpoints, by contrast, are more stable and less negative, with seasonal medians of −1.21 in summer and −1.18 in winter, reflecting its emphasis on broader-scale thermal structure. Landsat consistently exhibits lower (more negative) spectral breakpoints than MODIS (e.g., in Amman in winter 2022, they were −1.275 for Landsat vs. −1.206 for MODIS), reflecting its higher spatial resolution (~30 m vs. 1 km), which allows it to resolve finer-scale thermal variability linked to vegetation, urban built-up structures, and land cover (Figure 6). This suggests that Landsat captures smaller turbulent eddies more effectively, pushing the spectral breakpoint toward higher spatial frequencies (lower wavelengths). MODIS, in contrast, due to coarser spatial resolution, aggregates smaller-scale features, causing earlier breakpoints indicative of a lower turbulence threshold in the observed spectrum. This is in line with the 2D turbulence theory, which posits that higher-resolution measurements push the observed spectral energy cascade deeper into the inertial subrange, thus enabling more accurate retrieval of urban-induced temperature fluctuations at smaller scales. Cross-sensor and seasonal observation analysis of spectral breakpoints derived four main properties associated with the 2D turbulence theoretical approach.

3.3. Cross-Sensor and Seasonal Patterns

A clear and systematic difference is evident between the Landsat and MODIS results across all cities and both seasons. Landsat-derived breakpoints tend to exhibit lower values (i.e., more negative) than their MODIS counterparts. This is consistent with the expectation that higher spatial resolution imagery (Landsat) captures finer-scale temperature variability, thereby pushing the breakpoint toward smaller spatial scales (lower wavenumber or more negative log values). This observed disparity highlights the sensor-resolution dependency of turbulence metrics and supports the theoretical concern that the estimation of turbulence indicators like spectral breakpoints may not be scale-invariant unless the sensor adequately resolves the inertial subrange of the 2D turbulent cascade.

For most cities, summer breakpoints (red and yellow bars Figure 6) are lower than those observed in winter (blue and purple bars), regardless of the sensor. This seasonal effect suggests enhanced fine-scale thermal heterogeneity during the summer months, likely driven by stronger surface–atmosphere coupling, reduced moisture content, and increased urban–rural thermal contrasts. Notably, Madrid, Paris, and Wuhan display a particularly marked seasonal shift, especially in the Landsat observations, with summer breakpoint values being significantly more negative. This aligns with previous studies showing that urban heat island dynamics intensify in summer, increasing the energy in higher spatial frequencies and hence shifting the breakpoint.

Cities exhibit heterogeneous breakpoint values, even when observed with the same sensor and season, reinforcing the notion that urban morphology, climate zone, and vegetation structure modulate the spatial structure of surface temperatures. For instance: Santiago and Wuhan (Landsat, in winter) show some of the lowest breakpoint values, indicating strong fine-scale variability. Ankara and Las Vegas, by contrast, show higher breakpoints (less negative) for all combinations, possibly reflecting more homogeneous land cover or smoother urban thermal fields. This variability underlines the importance of analyzing turbulence indicators not only in terms of sensor and season, but also within the context of local urban and environmental characteristics.

In some cases, notably Amman and Las Vegas, the differences between MODIS and Landsat breakpoints are relatively small, suggesting partial agreement despite spatial resolution differences. However, for cities like Wuhan, Santiago, and Paris, the gap widens considerably. The error bars—standard error of the mean (SEM) or confidence intervals—are larger in cities with more heterogeneous landscapes or lower image quality (e.g., Wuhan and Santiago). Nevertheless, non-overlapping error bars between sensors in multiple cases suggest that the differences are not merely stochastic but systematic and significant, reinforcing the notion that turbulence-derived metrics are resolution-dependent.

The findings of this study underscore the critical influence of spatial resolution on the detection and interpretation of scale-dependent urban thermal patterns. Building on the hypothesis that sensor resolution directly governs the visibility of seasonal and structural thermal dynamics, we observed a marked contrast between the behavior of Landsat and MODIS-derived spectral breakpoints. Landsat-based analyses revealed clear seasonal sensitivity, with breakpoint frequencies shifting in response to intra-annual variations in urban surface temperatures. These shifts likely correspond to dynamic factors such as vegetation phenology, moisture availability, shading effects, and anthropogenic heat, which are more spatially heterogeneous and thus more visible at Landsat’s higher resolution (~30 m). In contrast, MODIS-derived spectral breakpoints remained invariant across seasons for each city, suggesting that coarser-resolution data (~1 km) primarily captures persistent, structural components of the urban thermal landscape. This points to MODIS being less sensitive to short-term or localized changes, instead reflecting more stable features such as overall land use configuration, dominant block sizes, street grid patterns, and large-scale land cover heterogeneity. A particularly compelling observation emerged when examining cross-city MODIS breakpoint values. Despite differing climates, urban forms, and topographies: Amman (Jordan) and Ankara (Turkey) consistently exhibited identical spectral breakpoints. Likewise, Las Vegas (Nevada), Nashville (Tennessee), and Santiago (Chile) shared the same MODIS-derived breakpoint value. These shared breakpoint values across diverse cities suggest that, at MODIS’s resolution, certain macrostructural patterns are recurrent across urban systems. These may include commonalities in sprawl geometry, landscape fragmentation, or coarse-scale zoning distributions that manifest similarly in spatial frequency spectra—despite differing climatic contexts. This convergence across cities supports the notion that thermal structure is not solely climate-dependent but is also strongly mediated by the geometric and topological properties of urban form, especially when observed at coarser scales. The stability of MODIS spectral breakpoints thus provides a kind of urban “signature” at a structural level, insensitive to seasonal dynamics but reflective of the broader urban configuration. Conversely, the seasonal variability observed in Landsat data offers a complementary perspective, revealing how finer-scale thermal behaviors fluctuate across time, driven by environmental and socio-ecological processes. Together, these findings reinforce a multi-scalar theoretical framework for urban climate analysis, wherein

a.

Coarse-resolution thermal imagery (MODIS) captures persistent structural thermal patterns linked to form and morphology.

b.

High-resolution imagery (Landsat) captures seasonal and functional dynamics, shaped by phenology, albedo changes, and energy balance variations.

This discussion highlights the importance of matching sensor resolution to research objectives in urban thermal studies and supports the integration of multi-resolution datasets to bridge structural and seasonal thermal phenomena across scales.

Implications for 2D Turbulence Characterization

The observed trends lend support to the hypothesis that 2D turbulence in urban surface temperatures may be only partially scale-invariant, with sensor resolution exerting a non-negligible influence on breakpoint estimation. While MODIS can offer valuable insights at larger scales, the resolution cutoff imposed by its coarser footprint may prevent full resolution of the inertial subrange, limiting its standalone use for turbulence characterization. The consistent directionality of differences between Landsat and MODIS, however, opens the door for developing correction models or cross-scale transfer functions to bridge results between sensors. This would be particularly valuable for long-term or large-scale studies that rely on MODIS continuity but seek Landsat-like detail in thermal structure.

In conclusion, this analysis confirms the theoretical expectation that turbulence indicators such as spectral breakpoints are sensitive to spatial resolution and season. The results support the inclusion of both high-resolution (Landsat) and coarse-resolution (MODIS) imagery in turbulence studies, not as interchangeable sources, but as complementary tools whose joint interpretation can reveal the multi-scale structure of urban thermal fields. Furthermore, the findings strengthen the case for developing methods that account for resolution effects when interpreting or integrating results across satellite platforms.

3.4. Seasonal Spectral Slope Distribution

Slope values represent the temporal rate of change in LST at specific spatial scales, derived from spectral analysis of long-term thermal signals. A more negative slope indicates a steeper declining trend, implying a more pronounced cooling process. Figure 7 presents boxplots illustrating the seasonal distribution of LST trend slopes derived from the Landsat (lower panels) and MODIS (upper panels) datasets, distinguishing between summer and winter conditions. Each subplot contrasts the slopes before and after the identified spectral breakpoint, corresponding, respectively, to the inverse energy cascade and the forward enstrophy cascade regimes, in accordance with the theoretical framework of two-dimensional (2D) turbulence.

Across both datasets and seasons, a clear asymmetry between the inverse and forward cascade regimes is evident. In the MODIS summer data, the median slope before the breakpoint is K^−4.60, compared to K^−2.79 after, indicating a steeper cooling trend associated with the inverse cascade. The range of values (min: K^−7.99, max: K^−1.59) and interquartile ranges (K^−6.65 to K^−2.42) further highlight the broader variability and stronger negative gradients in the pre-breakpoint regime. This suggests that, at coarser spectral scales, energy transfer mechanisms linked to broader urban thermal processes are associated with stronger cooling trends. Conversely, Landsat summer data displays an opposite trend, with a median slope after the breakpoint of K^−4.35, which is significantly more negative than the median before the breakpoint (K^−1.96). This suggests that the forward cascade processes captured by Landsat are more dominant at finer spatial resolutions, likely reflecting small-scale urban heterogeneities such as surface materials, vegetation patches, and built-up density. The summer season, conversely, exhibits steeper and more variable slopes, indicative of intensified thermal activity and more robust cascade processes, especially in urban areas characterized by heterogeneous land cover and surface materials. In winter, both datasets again show systematic differences. For MODIS, the median slope before the breakpoint is K^−3.79, compared to K^−2.84 after, reinforcing the summer pattern where the inverse cascade contributes to stronger cooling dynamics. Meanwhile, Landsat winter data follows the same behavior as in summer; the median slope after the breakpoint (K^−3.92) is again more negative than before (K^−1.89). Importantly, the inter-seasonal comparison indicates that winter slope distributions are generally less negative (or closer to zero), suggesting a seasonal suppression of thermal gradients and weaker turbulent transfer processes during colder periods. These findings suggest that forward cascade processes captured at higher resolution remain relevant in colder seasons, albeit with reduced variability. The consistent pattern across both seasons—with MODIS showing steeper slopes before the breakpoint and Landsat showing steeper slopes after—emphasizes the scale-dependent nature of thermal energy redistribution in urban landscapes. MODIS appears to better capture the broader-scale dynamics of the inverse cascade, while Landsat detects finer-scale forward cascade behavior related to enstrophy transfer within heterogeneous urban structures. These results not only validate the application of 2D turbulence spectral theory to urban thermal environments but also confirm the complementarity of MODIS and Landsat in revealing multi-scale LST trends. They highlight how spectral slopes can be linked to physical processes of energy accumulation, dispersion, and surface heterogeneity, offering a new lens through which to interpret the thermal behavior of urban systems.

3.5. Spectral Steepness as a Resolution-Driven Metric

The analysis of average spectral slopes values of the decadal samples encompassing eight cities representing diverse climatic and geographic contexts is presented in Figure 8. The average spectral slopes values before the breakpoint for Landsat during winter range from a minimum of K^−2.20 to a maximum K^−1.69 and for summer from a minimum of K^−2.04 to a maximum of K^−1.59. Meanwhile, for MODIS they range for winter from a minimum of K^−7.96 to a maximum of K^−1.79 and for summer from a minimum of K^−6.68 to a maximum of K^−1.30. Spectral slope values after the breakpoint for Landsat during winter range from a minimum of K^−4.60 to a maximum of K^−3.72, while during summer they range from a minimum of K^−4.31 to a maximum of K^−3.33. MODIS present values for winter from a minimum of K^−3.17 to a maximum of K^−2.58 and for summer from a minimum of K^−3.27 to a maximum of K^−2.45. Under 2D turbulence theory, the slope before the breakpoint typically represents the energy-containing range, while the slope after the breakpoint reflects the inertial subrange. Landsat-derived spectra exhibit consistently steeper slopes after the breakpoint than MODIS (e.g., Paris 2019: Landsat K^−3.80 vs. MODIS K^−2.49), suggesting that Landsat resolves a greater diversity and intensity of sub-grid turbulent structures. This steepness indicates a more rapid loss of energy at small scales, consistent with theoretical expectations where finer spatial resolution detects stronger temperature gradients across building edges, street canyons, and vegetated zones. Conversely, MODIS tends to underestimate post-breakpoint steepness due to pixel averaging, which smears out small-scale heterogeneity. Before the breakpoint, slope differences are less pronounced but still visible. MODIS often exhibits exaggerated negative slopes (e.g., Madrid 2013: MODIS K^−6.84 vs. Landsat K^−1.92), possibly due to aliasing effects and oversampling larger structures, inflating the apparent contrast between energy-dominated and transition regimes.

Seasonal Variability

Seasonal contrasts reinforce the resolution-driven disparities. During summer, when thermal gradients are sharper (due to enhanced surface–atmosphere decoupling), and Landsat captures steeper slopes post-breakpoint (e.g., Las Vegas 2020: K^−4.48), while MODIS maintains a relatively dampened spectral response (K^−2.74). The cooling effect of vegetation and shadows from built structures is more heterogeneous and localized in summer, which Landsat can resolve but MODIS cannot. In winter, temperature gradients are more subdued, leading to smaller differences in slope between sensors. However, Landsat still often detects more pronounced post-breakpoint spectral changes (e.g., Ankara 2024: Landsat K^−3.99 vs. MODIS K^−2.63), confirming the persistent advantage of high spatial resolution even in less turbulent thermal contexts.

Across both panels of Figure 8, Landsat-derived slopes (blue for winter and red for summer) appear generally more moderate in magnitude compared to MODIS-derived slopes (purple for winter and yellow for summer), particularly in the pre-breakpoint period. This difference likely reflects the finer spatial resolution and shorter time coverage of Landsat, contrasted with the coarser but temporally denser MODIS product. In both datasets, winter slopes are typically steeper (more negative), indicating a more pronounced cooling trend in winter months compared to summer, especially before the breakpoint. A significant observation from the pre-breakpoint slopes is the dominance of strongly negative trends in the MODIS winter series. Cities like Ankara, Las Vegas, Madrid, and Paris show extremely steep MODIS winter slopes, exceeding ~K⁻⁶, with some reaching nearly ~K⁻⁸ (e.g., Ankara and Madrid). This suggests a robust cooling pattern in winter LSTs captured by MODIS prior to the breakpoint. Conversely, Landsat slopes during the same period are shallower (generally between K^−1.5 and K^−3.5), indicating a more conservative detection of temperature trends. In summer, the MODIS-derived pre-breakpoint slopes also show a more pronounced downward trend than Landsat, though the contrast is less extreme than in winter. Cities like Ankara, Madrid, and Paris again exhibit notable MODIS summer slope magnitudes (~K⁻⁵ or more), possibly reflecting broader seasonal sensitivity or urban heat island dynamics captured at the MODIS scale. After the breakpoint, slope magnitudes generally decrease across all series, with fewer extremely steep declines. Landsat winter slopes still show the strongest negative trends (often between ~K⁻⁴ and ~K⁻⁶), notably in Amman, Madrid, and Paris, suggesting continued or even accelerating cooling during the post-breakpoint period. Interestingly, MODIS-derived slopes, particularly in summer, become more stable and less extreme, often ranging between ~K⁻² and ~K⁻³ across most cities. This change may indicate a shift toward temperature stabilization or a response to climate adaptation strategies, such as greening and albedo enhancement in urban environments.

We found a series of city-specific observations and divergences:

• Amman exhibits one of the steepest Landsat winter slopes post-breakpoint, suggesting continued wintertime cooling. MODIS values are more conservative here.
• Ankara and Madrid show strong agreement in MODIS slopes before the breakpoint but diverge notably in Landsat data, hinting at possible spatial-scale discrepancies in temperature trend detection.
• Las Vegas, despite its arid setting, reveals relatively moderate post-breakpoint slopes, potentially reflecting effective urban mitigation efforts or saturation of warming trends.
• Paris maintains steep slopes across both sensors and periods, particularly in winter, aligning with documented heat stress mitigation challenges in the urban core.
• Wuhan stands out for its relatively small variation between sensors and between pre- and post-breakpoint slopes, indicating a potentially smoother climatic transition or differing land surface dynamics.

Figure 8 underscores a clear sensor-based divergence; MODIS generally records steeper pre-breakpoint declines, especially in winter, possibly due to higher temporal frequency capturing short-term climate variability. Landsat, with its finer spatial granularity, may better represent urban-specific processes, but with less temporal nuance. The post-breakpoint convergence of slopes between sensors in many cities suggests improved agreement over time or a true flattening of LST trends under climate adaptation.

In sum, the comparative slope analysis reveals important temporal and sensor-specific insights into urban LST evolution across diverse climates. The clear breakpoints and differing slope magnitudes suggest that MODIS and Landsat, while both valuable, capture complementary aspects of urban thermal dynamics. The general trend toward reduced slope steepness post-breakpoint hints at either climate stabilization or effective local urban heat mitigation. Further studies should integrate land use, vegetation, and socio-economic indicators to disentangle the drivers behind these spectral slope trends.

3.6. Climatic Zone Sensitivity

A comprehensive visualization of how temporal trends in spectral responses—indicative of land surface dynamics such as vegetation phenology or land cover transformations—vary not only by satellite platform but also by climatic regime and seasonal context is offered in the dual-panel plot below (Figure 9). Cities in different climatic zones show varied sensor sensitivities. In desert cities like Las Vegas, both sensors exhibit sharper transitions, but Landsat consistently maintains steeper slopes and more negative breakpoints. This may reflect urban–desert edge effects and fine-scale surface temperature mosaics that are spatially compressed and lost in MODIS products. In Mediterranean and oceanic cities (e.g., Madrid and Paris), urban morphology (compactness, green spaces) generates a complex thermal pattern that is more detectable by Landsat. MODIS often fails to fully capture these microclimatic heterogeneities, resulting in shallower post-breakpoint slopes (e.g., Paris 2020: MODIS K^−2.73 vs. Landsat K^−4.35). In humid continental climates (e.g., Nashville), the difference between sensors is less pronounced in winter, likely due to homogenized snow/soil surface cover, though summer patterns again favor Landsat’s finer spatial delineation.

Prior to the spectral breakpoint, slope values generally exhibit steeper (i.e., more negative) trends, particularly for Landsat data in summer and MODIS data in both seasons across certain climatic zones. This trend suggests more pronounced declines in spectral reflectance or vegetation indices over time in the earlier part of the study period.

• Desert and humid subtropical zones: In these zones, Landsat summer data show the most dramatic pre-breakpoint decline, with slope values nearing or exceeding ~K⁻⁶, especially in the desert zone. MODIS winter slopes in the desert zone also register steep declines (below ~K⁻⁵), reinforcing the idea that desert environments experienced significant spectral changes likely due to vegetation stress or land degradation processes that predate the breakpoint.
• Mediterranean and oceanic zones: These zones show a more moderate slope range before the breakpoint (between ~K⁻² and ~K⁻⁵), with MODIS summer and Landsat winter being less steep, potentially reflecting greater seasonal stability or less anthropogenic disturbance in these regions during the initial time segment.

Interestingly, the relatively smaller error bars for Landsat winter and MODIS winter in most zones suggest a higher degree of temporal consistency and potentially stronger reliability in slope estimates during the dormant season. After the breakpoint, slope values are generally less steep across all climatic zones, indicating a significant shift in the trajectory of spectral dynamics. This flattening trend could be interpreted as a stabilization in land surface reflectance patterns, potentially attributable to climate adaptation measures, reforestation efforts, or reaching saturation in land cover changes.

• Cross-sensor comparisons: MODIS data—especially in winter—exhibit a pronounced reduction in slope steepness after the breakpoint, though still remaining negative in most climatic zones. In contrast, Landsat winter shows more moderate and consistent slope values across zones, hovering around ~K⁻³ to ~K⁻⁴, which may point to its higher spatial resolution capturing subtler variations in land cover.
• Climatic zone differences: The humid continental and mediterranean zones show a marked reduction in slope magnitude for both sensors and seasons, suggesting a regionally consistent break in land cover change trajectories, possibly due to large-scale policy or ecological shifts. The oceanic zone, while also exhibiting a reduction in slope steepness, retains relatively strong spectral change rates in Landsat winter, indicating possible persistent disturbances or natural seasonality effects in this maritime climate.

The reduction in standard errors for MODIS summer across zones post-breakpoint may indicate increased data stability or improved signal consistency, aligning with MODIS’s finer temporal resolution benefits in capturing phenological trends after stabilization.

3.7. Interpretative Implications

The breakpoint likely represents a temporal threshold coinciding with a major environmental, climatic, or socio-economic shift, such as widespread adoption of land management practices, climate anomalies (e.g., extended droughts), or policy interventions. The differences in slope behavior between MODIS and Landsat also highlight the influence of spatial vs. temporal resolution; while MODIS offers higher temporal frequency (capturing transient phenological signals), Landsat’s finer spatial detail may better detect localized land cover changes. Furthermore, the seasonal split reveals how vegetation and land surfaces respond differently to climatic pressures in growth (summer) versus dormant (winter) seasons. The overall pattern of steeper pre-breakpoint slopes transitioning to flatter post-breakpoint values across climatic zones suggests an overarching trend of decelerating land surface changes, which may reflect both climate resilience strategies and natural ecological thresholds.

4. Discussion

This study aimed to evaluate the spatial scaling behavior of land surface temperature (LST) in urban environments by analyzing spectral breakpoints and slopes derived from 2D turbulence theory using both Landsat and MODIS satellite data. It provides a comprehensive multi-scale analysis of LST dynamics across urban environments using high-resolution Landsat (~30 m) and coarse-resolution MODIS (1 km) data. By leveraging the theoretical framework of 2D turbulence, we systematically evaluated spectral breakpoints and slope behaviors to discern how thermal energy is distributed and dissipated across spatial scales, seasons, cities, and climatic zones.

The analysis revealed that sensor resolution plays a defining role in determining the detectability of turbulent regimes and thermal variability. Landsat consistently exhibited more negative spectral breakpoints and steeper post-breakpoint slopes, especially during summer, confirming its sensitivity to fine-scale urban thermal structures. These include localized variations in surface materials, vegetation, urban morphology, and human activity. Conversely, MODIS provided a more temporally stable and spatially generalized perspective, with breakpoint values that remained largely invariant across seasons and cities, reflecting persistent structural attributes of the urban thermal envelope.

Spectral breakpoint asymmetry, a hallmark of cascade behavior in turbulent systems, was evident across all observations, validating the application of 2D turbulence theory to urban thermal environments. The slope analysis, particularly before and after spectral breakpoints, further demonstrated asymmetric cascade regimes, with Landsat favoring forward (enstrophy) cascade dynamics and MODIS highlighting inverse (energy) cascade signatures. These contrasting behaviors not only reflect sensor-specific sensitivities but also illustrate the multi-scalar complexity of urban climate processes.

Seasonal differences observed across both datasets underscore the dynamic nature of urban thermal environments. Summer spectra displayed steeper and more variable slopes, indicative of enhanced thermal activity and intensified cascade processes due to reduced moisture, higher surface–atmosphere decoupling, and greater anthropogenic influence. In contrast, winter patterns were more subdued, with shallower slopes and less variability, suggesting seasonal suppression of turbulent energy transfer. These findings were consistent across cities and climatic zones, further supporting the robustness of the results.

A key outcome of this work is the clear demonstration of Landsat and MODIS as complementary tools. Rather than treating them as interchangeable sources, we advocate for their integrated use. Landsat offers critical insight into intra-urban heterogeneity and temporal shifts, making it ideal for neighborhood-scale planning, green infrastructure assessment, and climate adaptation studies. MODIS, with its broader spatial footprint and dense temporal coverage, provides valuable continuity for inter-city comparisons and long-term climate monitoring. The combination of both sensors could be very useful in downscaling algorithms, since the results of the generated slopes could help to improve the spatial resolution of low-resolution sensors, thus complementing existing methods used in urban regions [38].

Additionally, this study highlights that urban thermal behavior is not solely climate-dependent but also profoundly influenced by urban form, morphology, and land management practices. The similarity of MODIS-derived breakpoints across geographically distinct cities suggests that at coarse scales, shared structural features—such as zoning patterns, street grids, or development typologies—may produce similar thermal signatures.

Other works have addressed the impacts of thermal infrared (TIR) imagery’s spatial resolution on the SUHI effect through the analysis of LST spatial patterns. Based on degradation of airborne imagery’s spatial resolution, previous research [39] established a threshold of approximately 50 m for capturing urban landscape heterogeneity. This aligns with our findings using Landsat (~30 m) data. Therefore, 2D turbulence indicators demonstrate strong potential for spatial scaling analysis in the context of SUHI studies and complement traditional methods (e.g., spatial degradation techniques).

The practical implications of these findings are significant for urban planning, climate adaptation, and public health strategies. By distinguishing the complementary strengths of MODIS and Landsat, urban policymakers can tailor thermal mitigation interventions to different spatial and temporal scales. For instance, Landsat-based slope and breakpoint analyses can be integrated into zoning regulations to identify high-risk thermal hotspots at the neighborhood level and prioritize cooling interventions such as tree planting, green roofs, or reflective surfaces. At the same time, MODIS-derived indicators can inform broader-scale strategies, such as regional urban heat management, land use planning, or climate-sensitive infrastructure design.

The 2D turbulence framework also provides a physics-informed basis for monitoring the effectiveness of these interventions over time. Municipalities and environmental agencies could operationalize these metrics in remote sensing dashboards for real-time surveillance of urban heat dynamics, thereby enhancing resilience planning in the face of the increasing frequency and intensity of urban heatwaves.

5. Conclusions

This study contributes to advancing our understanding of how spectral metrics derived from remote sensing can serve as proxies for urban energy processes. The confirmed spectral asymmetries, sensor-specific breakpoint behaviors, and seasonally modulated slope dynamics provide a robust foundation for future research in urban climatology, remote sensing, and turbulence characterization.

Future research should focus on integrating socioeconomic, land use, and vegetation data to better explain spatial variability in thermal patterns. Additionally, efforts should be directed toward developing resolution-aware correction functions to bridge MODIS and Landsat results across scales. The use of newer sensors, such as ECOSTRESS and Sentinel, should also be explored to enable higher-frequency or higher-resolution tracking of land surface temperature. Finally, applying machine learning and physical modeling approaches could enhance the simulation and prediction of urban thermal dynamics by leveraging multi-sensor inputs.

In a warming world where cities face increasing heat-related risks, understanding the multi-scalar structure of urban thermal fields is crucial. This study contributes to that understanding by demonstrating how satellite-derived spectral indicators can reveal the spatial and temporal complexity of urban heat, informing both theory and practice in urban climate resilience.