In this project, the linear version of spectral mixture analysis (SMA) was used to calculate satellite spectral response to burned areas. SMA is a widely used technique for modeling the composite nature of pixels [
22]. For studies on wildfires, it has been used to estimate fire severity (e.g., [
23,
24]), intensity (e.g., [
25]), and vegetation recovery (e.g., [
26]). The technique is relatively simple, has been extensively tested, and has proven to be reliable [
22]. The approach in SMA is to model the spectral signatures of a pixel as a mixture of pure land covers, called endmembers. A burned pixel may, for example, be a combination of the endmembers grass, bare soil, and charcoal. Typically, SMA is used to estimate abundances of such endmembers in pixels imaged by satellites. Here, we did not use satellite imagery; instead, we created artificial pixel samples from combining these endmembers and used these to determine satellite response. To our knowledge, no prior study has modeled satellite detectability of burned area in such a way.
2.1. Study Area
The study area for which endmembers were selected is Yellowstone National Park (shortened here to Yellowstone), a nature preserve in Wyoming, USA [
27]. The park covers 8903 km
2, and is a landscape characterized by high volcanic plateaus eroded by glaciers and rivers and flanked by mountains. Lakes cover around 5% of the park’s area. Elevation ranges between 1637 m and 3512 m, averaging at 2479 m [
28]. The area is volcanically active and contains three large calderas [
29]. The park was selected as a study area because the available vegetation spectral data are complete and easily accessible. Additionally, wildfires occur often in the park during the summer season. The burned environments for this region were modeled using a linear combination of three endmembers: vegetation, substrate, and charcoal.
The dominant vegetation community in Yellowstone is coniferous forest, which represents around 85% of the surface area. The United States National Park Service classifies them under four communities. Of them, forests dominated by lodgepole pine (
Pinus contorta subsp.
latifolia) are by far the most common. In addition, there are forests dominated by spruce fir (
Picea engelmannii and
Abies lasiocarpa), whitebark pine (
Pinus albicaulis), and douglas fir (
Pseudotsuga menziesii). This last vegetation community is interesting as it has the highest frequency of fire [
27]. The vegetation patterns in the region vary greatly with elevation and topography, with douglas fir being located at lower elevations, lodgepole pine between 2000 and 2400 m, and the other communities reaching to the upper tree line [
30]. Other vegetation communities in the park are grasslands, meadows, sagebrushes, and hydrothermal areas. These are classified under non-forested vegetation. Finally, scattered throughout the park are smaller communities of aspen forests, wetlands, and streamside vegetation [
27].
2.2. Endmember Selection
The reflectance spectra of the environments in Yellowstone were modeled using three categories of endmembers: vegetation, substrate, and charcoal. Vegetation spectral data were obtained from the United States Geological Survey’s (USGS) spectral library, which includes data collected from laboratory, field, and airborne spectrometers. Covering wavelengths between 0.2 and 200
m, the library contains measurements of vegetation, minerals, chemical compounds, and manmade materials. Among this library, averaged spectral reflectance data of the dominant vegetation communities in Yellowstone National Park are available. These are averaged top-of-canopy measurements of vegetation communities, previously collected by the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) [
31]. The used spectra span wavelengths between 370 and 2500 nm at an interval of 10 nm. The vegetation samples that were used in this study are summarized in
Table 1. It is common in spectral mixture analysis to subdivide the vegetation endmember into two: green and non-photosynthetic parts of the vegetation cover. However, as the vegetation samples used are averages of plant communities, they represent spectra that contain both green and non-photosynthetic parts of the plants. Thus, the vegetation endmember samples from AVIRIS represent the vegetation spectra as they would be measured by a satellite instrument.
The endmembers that were used to simulate the substrates in Yellowstone were obtained from the NASA Jet Propulsion Laboratory’s ECOSTRESS spectral library. This library is a collection of laboratory measurements covering a wavelength range between 350 and 15,400 nm of (green and non-photosynthetic) vegetation, rocks, soils, minerals, and some manmade materials [
32]. From this library, both rock and soil samples were obtained to simulate the substrate conditions of the park. Rock spectral samples were selected according to the geology of Yellowstone, as described by Keefer [
33]. At the wavelengths used in this work (up to 2350 nm), the spectral samples from this library had a measurement interval of 1 nm.
The inclusion of soil samples is important, as some soil-forming processes influence the spectral signature. For example, increases in organic matter result in lower reflection in the NIR domain [
34]. Furthermore, in the presence of charcoal, the Normalized Burn Ratio is highly sensitive to soil type [
35]. However, selecting soils is less straightforward compared to rocks, as the characteristics of soils are highly variable. Apart from the lithology in which the soil is formed, it is highly dependent on the local climate characteristics [
36]. Soils in Yellowstone are typically in the frigid or cryic temperature regime, although warmer regimes form around hydrothermal areas. The climate of the park is characterized as moderately dry [
28], and mostly falls within the udic moisture regime [
36]. The most common soil types in the park are mollisols (thick, organic soils) and inceptisols (weakly developed mineral soils); however, entisols, andisols, and alfisols are present as well. For a full overview of the soils in Yellowstone, see Rodman et al. [
28].
We only selected soils for the more common lithologies, as these have a higher impact on the park-wide results. Initially, the selected soils matched parent materials with the rocks in the park, could form in the temperature and moisture regimes of Yellowstone [
36], and are similar to the soils described by Rodman et al. [
28]. The rock and soil samples were grouped in accordance with the geological units, as shown in
Table 2.
However, some very common parent materials were greatly underrepresented in the soil samples due to a lack of soils that fit all the criteria. In these cases, some soils were chosen that are technically outside of this climatic regime, but fit well with the soil descriptions and parent material. Most notably, there was no soil sample found for volcanic ash that also fit the climatic regimes. Thus, the otherwise well-fitting “inceptisol xerumbrept” (which forms in deserts) was added.
The charcoal endmembers were collected by Veraverbeke et al. [
37] from the 2011 Canyon fire scar in Kern County, California. Their spectra were measured at NASA’s Jet Propulsion Laboratory, in the same fashion as the used rock and soil samples from their ECOSTRESS library. Three charcoal endmember samples were available and used.
2.3. Sample Preparation
The first step of the analysis was to calculate the near-infrared (NIR) and shortwave infrared (SWIR) response of the endmember samples (
Figure 1). To perform this, we followed the procedure outlined by Barsi et al. [
38]. For every wavelength of an endmember sample, the response of a satellite band was calculated:
where
is the spectral reflectance of an endmember sample
as observed by a satellite instrument band
b at wavelength
.
is the spectral reflectance of the endmember sample from library (again at wavelength
), and
is the spectral response of a satellite at that wavelength. Both vary between 0 and 1. The satellites for which detectability was assessed are Landsat 8, Sentinel-2, and the Moderate Resolution Imaging Spectroradiometer (MODIS) (
Table 3). Their response functions are shown in
Figure 2. The spectral response functions were linearly interpolated to match the spectral resolution of the endmember sample.
The responses to these reflectances were then averaged over the band’s wavelength range:
where
is the spectral response for the endmember sample
ems of a given instrument band
b,
and
are the wavelength boundaries of the instrument band, and
is the wavelength distance between the measurements of the endmember sample. In this case, multiplying the endmember reflectance with the satellite response functions automatically sets the wavelength range (as the responses are set to 0 outside of the band of the instrument). If
is constant over the wavelength domain, Equation (3) can be simplified:
For every endmember sample, Equation (4) was used to calculate the NIR and SWIR band values for the different satellite sensors.
2.4. Modeling Burned Area Response
In linear spectral mixture analysis, spectral endmembers are combined linearly to simulate the spectrum of a pixel [
42]:
Here, is the reflectance of a pixel, which is a weighted average of the reflectances of the endmembers used to simulate the pixel (), with weights given by their spectral contribution fractions (). In linear SMA, these fractions can also be interpreted to represent the surface area covers of the endmembers, and therefore are non-negative and have to add up to unity.
In this work, pixel spectra were modeled using three endmembers: vegetation, substrate, and charcoal. This yields the following calculation for the pixel reflectances:
In this equation, , , , and are the reflectance values of the pixel and its vegetation (v), substrate (g), and charcoal (c) endmembers, respectively. , , and are the respective cover contribution fractions. The letter g (from ground) is used for substrate endmembers to disambiguate from the s used for starting values in later equations.
The endmember reflectances in this equation (and further equations) are those obtained by Equation (4). Thus, the obtained pixel reflectance is for a given instrument band and is already scaled with the band’s response function. This means that the NIR and SWIR values obtained from this equation are the reflectances of the pixel as would be obtained from the satellites.
Before a fire, a pixel only contains vegetation and substrate endmembers (
); thus, Equation (6) yields
is the pixel’s (
p) pre-fire (starting,
s) reflectance.
is the vegetation cover of the pixel sample before the fire. This value was varied between 0.05 and 1 in steps of 0.05, yielding 20 pixel samples per ground-vegetation endmember combination. The substrate endmember takes up the rest of the area of the pixel; therefore, its contribution fraction is equal to
. From this pre-fire pixel reflectance, the Normalized Burn Ratio before fire (
) could be calculated:
The cover contributions for the pixels were then altered by replacing parts of the ground and vegetation with charcoal, simulating a burn in the pixel. The model calculates a decrease in the contribution of vegetation (
) using a variable fraction of burned vegetation (
). The fraction of burned vegetation (
) gives the fraction of the original vegetation (
) that is currently burned. The fraction of vegetation (
) could then be calculated using
Along with changes in
, the contribution of ground (
) and charcoal (
) also need to change, as the total fraction of the pixel needs to be 1. The parameter
(or
char) gives the change in charcoal cover fraction per unit of vegetation lost (burned). The remaining contributions could thus be calculated using
represents vegetation loss without charcoal input. represents a fire in which all vegetation is replaced by charcoal; the contribution of ground stays constant. represents an increase of both charcoal and bare ground. indicates that part of the ground will be covered by charcoal in addition to vegetation loss. The parameter can not be negative, as this would result in an illogical mechanism where forest cover increases as it is burned. In the model, results were calculated for ’s between 0 and 1 in steps of 0.25.
Vegetation was removed until the vegetation contribution became 0, or the burned area became detectable. The detectability was assessed using the differenced Normalized Burn Ratio (dNBR), which is calculated by taking the difference between the pre- and post-fire NBR. After an increase in the burned vegetation fraction, the pixel’s NBR (
) and, subsequently, dNBR were calculated:
If the dNBR was greater than a certain threshold, the fire was set to be detectable and the cover fractions of the different endmembers , , and were saved. A burned pixel was set to be undetectable if the burned fraction reached 1 before the dNBR threshold was exceeded. In the model code, multiple threshold values were used to be able to assess the influence of this parameter. The dNBR thresholds varied between 0.05 and 0.25 in steps of 0.05.
The cover fractions were calculated for every substrate sample combined with every vegetation and charcoal sample. However, we were only interested in the range of these values for a certain combination, not necessarily the outputs for the individual library samples. For example, the combination of gneiss (19 samples) with lodgepole pine forest (11 samples) and charcoal (3 samples) has a total of model output values. However, the interest is only in the detectability of the gneiss–lodgepole pine combination. Therefore, of these 627 results, only the minimal, mean, and maximal values were saved. This reduced the number of results for this combination to three per variable.
From the burned fraction
, the endmember contributions can be calculated. The goal of the model was then to find the value of
for which the burn could be detected. This can be achieved using a root-finding algorithm; for example, iteratively, by increasing stepwise
, testing the resulting dNBR and checking if it is above the threshold. Then
could be tweaked to attain closer approximations of the threshold. However, it is possible to directly calculate the burned fraction needed for detection. The methodology is explained in
Appendix A. The direct calculation is faster and more precise, allowing more samples to be processed within a reasonable amount of time.
2.5. Data Aggregation
With a high number of endmember sample combinations, aggregation of the results was required to draw conclusions. The results were aggregated at different levels (
Table 4), allowing conclusions to be drawn at different scales.
The direct result outputs are defined at level 0. At this level, the results are given per endmember sample combination; each one is calculated using a combination of one substrate, vegetation, and charcoal spectral sample.
As we are not interested in the variation between the samples of a particular endmember (e.g., between basalt samples), we can reduce the amount of data. We perform this by only saving some statistical properties of a collection of these sample combinations. Here, we save only the mean, minimal, and maximal values of the model results in a collection or grouping, essentially giving one value with uncertainty. This yields results at aggregation level 1. In this case, spectral samples are grouped according to their soil name, rock name (see
Table 2), or vegetation community (see
Table 1). Every grouping is then a combination of a named substrate with a vegetation community. For example, one of the grouping combinations is basalt with lodgepole pine, containing all model results that are a combination of basalt and lodgepole pine.
Higher aggregation levels than the groupings level require some information on the area covers of the substrate and vegetation combinations. However, the surface area covers of rocks or soils within a geological unit are largely not documented. For aggregation level 2, this is accounted for as follows. We assign every rock and soil to a geological unit (per
Table 2). For the mean value of a unit, we assume the rock and soil types to be equally abundant on the surface. The high uncertainty in this assumption is taken into account by obtaining the minimal and maximal value of combinations assigned to a geological unit. These extreme values thus assume that the entire unit is dominated by the most extreme spectral samples within them, providing a constraint on the variability.
The final aggregation level, aggregation level 3, assesses burned area detectability at a park-wide scale. To achieve this, the results for the geological units were weighed with their surface area. To find these weights, a geological map [
43] and vegetation habitat map [
44] of Yellowstone National Park were reclassified and combined to show the geological unit–vegetation community combinations. Area sizes of these combinations were subsequently set as the weights in the result analysis. The classes and corresponding weights are shown in
Table 5.
At aggregation level 3, the model contains three results (min, mean, max) for the entire park. However, parameterization adds additional dimensions; thus, the park-wide results are dependent on a given dNBR threshold (5 options), char (5), instrument (4), and starting vegetation fractions (10), totaling 6000 result values.