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Illustration of the compound event indicesBuilding on earlier studies24,25, we develop two univariate indices to model concurrent climate conditions, i.e., a CWD index that varies from compound cold-wet conditions to CWD conditions, and a CCD index that varies from compound warm-wet conditions to CCD conditions (see “Methods”). The two indices incorporate the dependence between temperature and precipitation and are a measure of how warm/cold and dry a point is relative to the distribution of climate conditions at a given location. We illustrate the two indices on two grid points that have strong but opposite temperature-precipitation correlation. In the case where temperature and precipitation are strongly negatively correlated, the CWD index is well aligned with the primary axis of the bivariate distribution (Fig. 1a). In the case where temperature and precipitation are strongly positively correlated, the same holds for the CCD index (Fig. 1d). As illustrated for several concurrent hot-dry and cold-dry events that occurred around the globe, the two indices well capture these events (Supplementary Figs. 1 and 2).Fig. 1: The relationship between precipitation and temperature and compound indices.a Scatter plot of summer precipitation and temperature anomalies (z-score) with corresponding CWD index in color (see “Methods”). The location is at 97.25°W and 33.75°N from 1901 to 2018. b The same as a but for spring at 84.75°E and 66.75°N. c Same distribution as in a but colored based on the CCD index. d Same distribution as in b but colored based on the CCD index.Full size imageNotably, in the case where precipitation and temperature are strongly positively correlated, the CWD index indicates the relative anomalies of bivariate joint distribution, and some counterintuitive situations might occur relative to the univariate marginals (Fig. 1b). For instance, points might be labeled as strong CWD events (CWD index > 1.5) even though temperature is anomalously cold (temperature anomalies < 0, red dots in lower left quadrant of Fig. 1b). The CCD index exhibits similar behavior (Fig. 1c). This indicates an interesting property of the compound indices to identify strong compound conditions relative to bivariate distribution that are not necessarily extreme from a univariate perspective3,24,26,27.Widespread direct and lagged impacts of springtime compound climate conditionsTo evaluate the lagged summer vegetation responses to spring compound climate conditions, we compute partial correlation between CWD (CCD) spring and subsequent summer vegetation variation by controlling for the influence of summer compound climate conditions on these correlations (see “Methods”). Results show widespread negative associations between CWD spring and subsequent summer vegetation in the mid-latitudes (50°N).a–c The average standardized anomalies (z-score) of GPP during CWD spring but subsequent non-CWD summer (a), non-CWD spring but subsequent CWD summer (b), and consecutive CWD spring and summer (c) for areas in Fig. 2a where summer vegetation responds positively (r ≥ 0.22) to spring CWD climate conditions. d–f The same as a–c, but for soil moisture. g–i The same as a–c, but for runoff. The bar plots with dash lines (without dash line) indicate the average anomalies of multiple observation-based (model-based) products, and the circles indicate the average anomalies of each product. GLASS, LUE, NIRv, Flux-CRU, and Flux-ERA5 are observation-based GPP products, while model simulations are taken from TRENDYv6. GLEAM is observation-based soil moisture. GRUN represents observation-based runoff. GLDAS-VIC, GLDAS-Noah, GLDAS-Catchment, and FLDAS indicate assimilatory soil moisture and runoff that incorporate satellite- and ground-based observational products.Full size imageFig. 4: The responses of vegetation productivity and hydrological variables to CWD events in mid-latitudes (23.5–50°N/S).a–c The average standardized anomalies (z-score) of GPP during CWD spring but subsequent non-CWD summer (a), non-CWD spring but subsequent CWD summer (b), and consecutive CWD spring and summer (c) for areas in Fig. 2a where summer vegetation responds negatively (r ≤ −0.22) to spring CWD climate conditions. d–f The same as a–c, but for soil moisture. g–i The same as a–c, but for runoff. The bar plots with dash lines (without dash line) indicate the average anomalies of multiple observation-based (model-based) products, and the circles indicate the average anomalies of each product. For details on data see Fig. 3.Full size imageFig. 5: The effects of CCD events on vegetation productivity and hydrological variables in mid-to-high latitudes.a–c The average standardized anomalies (z-score) of GPP during CCD spring but subsequent non-CCD summer (a), non-CCD spring but subsequent CCD summer (b), and consecutive CCD spring and summer (c) for areas in Fig. 2b where summer vegetation responds negatively (r ≤ −0.22) to spring CCD climate conditions. d–f The same as a–c, but for soil moisture. g–i The same as a–c, but for runoff. The bar plots with dash lines (without dash line) indicate the average anomalies of multiple observation-based (model-based) products, and the circles indicate the average anomalies of each product. For details on data see Fig. 3.Full size imageCWD events increase vegetation productivity in high latitudesWe first analyze the direct responses of productivity to springtime and summertime CWD events across high latitudes ( >50°N, Fig. 3). Productivity increases during CWD spring and summer (Fig. 3a–c), which is consistent with vegetation responses (Supplementary Fig. 8a–c). Despite elevated spring greenness, spring water overall shows positive anomalies during CWD spring (Fig. 3d, f, g, i). This result indicates that spring greenness during CWD conditions is not associated with dry spring across high latitudes, which is further confirmed by similar anomalies in springtime TWS (Supplementary Fig. 8d, f). In contrast, severe water reduction is found in CWD summer (Fig. 3e, f, h, i). This suggests that despite the beneficial effects of CWD events on productivity in summer, they are associated with summer water deficit.Next, to analyze the lagged effects of springtime CWD events, we investigate the productivity anomalies in summer under three cases, namely CWD spring but non-CWD summer, non-CWD spring but CWD summer, and consecutive CWD spring and summer. Our results indicate that springtime CWD events have positive lagged effects on summer productivity across high latitudes (Fig. 3). Specifically, we find that during non-CWD summer (that is not favorable for summer vegetation growth) preceded by CWD spring, positive anomalies are still found in summer productivity (Fig. 3a). In contrast, during CWD summer (preceded by non-CWD spring), some models and observation-based products exhibit a reduction in summer productivity (Fig. 3b). We further find that summer productivity highly increases during consecutive events (Fig. 3c). Vegetation anomalies show similar behaviors (Supplementary Fig. 8a–c). Regarding the lagged responses of hydrological variables, CWD springs followed by non-CWD summers do not lead to water dryness, despite increased vegetation greenness (Fig. 3d, g). The magnitude of summer water deficit is similar for both cases that include CWD summer (Fig. 3e, f, h, i) and is consistent with summer TWS anomalies (Supplementary Fig. 8e, f). These results imply that in high latitudes, summer water reductions characterized by TWS, soil moisture, and runoff are not associated with increased spring greenness but are primarily caused by summer precipitation deficit.The productivity responses to compound climate conditions may be stronger than that to individual events through the synergistic effects of temperature and precipitation28. To investigate this, we compute the average anomalies in GPP and soil moisture associated with univariate events across the focus areas, which are then compared with the effects of CWD and CCD events in high latitudes (see “Methods”). Warm events can not only directly increase productivity but also show positive lagged effects (Supplementary Fig. 9a, b). In contrast, dry events reduce productivity (Supplementary Fig. 9e, f). This indicates that the direct and lagged positive effects of CWD events across high latitudes are mainly dominated by the warm component, while dry conditions have negative effects. Therefore, the warm-induced increase in productivity slightly exceeds that associated with CWD events (Supplementary Fig. 9b). Soil moisture under warm springs shows positive anomalies (Supplementary Fig. 9c, d), while they slightly decline during dry spring (Supplementary Fig. 9g, h). This suggests that the positive anomalies in soil water during CWD spring are driven by the warm component.CWD events reduce vegetation productivity in mid-latitudesHere, we first investigate the direct effects of springtime and summertime CWD events across mid-latitudes (23.5–50°N/S). Springtime productivity exhibits little changes during CWD spring (Fig. 4a, c), despite dry spring (Fig. 4d, f, g, i). When considering the direct effects of CWD events in summer, the results are similar, whereas the negative magnitude of productivity in summer is larger than that in spring (Fig. 4b, c). This difference suggests CWD conditions in summer show more adverse effects on productivity than that in spring in mid-latitudes. The anomalies in vegetation and TWS are consistent (Supplementary Fig. 10).Next, the lagged effects of springtime CWD events in mid-latitudes are assessed. In cases with CWD spring but non-CWD summer, summer productivity exhibits slight anomalies (Fig. 4a), with slightly decreased summer water (Fig. 4d, g). Summer productivity and water show much higher reductions in case with consecutive events (Fig. 4c, f, i) than for the case with only CWD summer (Fig. 4b, e, h). These results are supported by the responses of vegetation indices and TWS (Supplementary Fig. 10), revealing that springtime CWD events in mid-latitudes have negative lagged effects on summer productivity and water availability.The direct and lagged effects of individual events are finally compared with that of CWD events in mid-latitudes. Dry conditions in spring and summer directly decrease productivity and cause soil water dryness (Supplementary Fig. 11a–d). Moreover, dry spring depletes soil moisture earlier, which, in turn, causes dry summer and reduction in productivity during non-dry summer (Supplementary Fig. 11a, c). This indicates that dry springs have negative lagged effects on summer productivity. In contrast, productivity and soil water show positive anomalies during warm springs, while they show negative anomalies in summer (Supplementary Fig. 11e–h). These results suggest that the direct and lagged negative effects of CWD springs are dominated by the dry component in mid-latitudes, while the warm component mitigates the negative effects of the dry component in spring. Accordingly, the decline in productivity due to dry conditions thus exceeds that triggered by CWD events (Supplementary Fig. 11b).Decreased vegetation productivity due to the negative synergistic effects of CCD eventsHere, we first investigate the direct effects of CCD events across mid-to-high latitudes. Productivity reductions are found during springtime and summertime CCD events (Fig. 5a–c) concurrent with water reductions (Fig. 5). Vegetation and TWS show similar behaviors during CCD spring and summer (Supplementary Fig. 12). These results reveal that CCD events in spring and summer can impose direct adverse impacts on productivity and soil water across mid-to-high latitudes. The productivity reductions in spring and summer are similar in magnitude (Fig. 5a, b), indicating that CCD events between spring and summer can cause similar damage to productivity.We then analyze the lagged effects of springtime CCD events. Our results indicate that springtime CCD events show negative lagged effects on summer productivity and cause summer water reductions in mid-to-high latitudes (Fig. 5). Specifically, we find that in cases with CCD spring but non-CCD summer, summer productivity and water exhibit strongly negative anomalies (Fig. 5a, d, g). Moreover, summer anomalies are higher during consecutive events (Fig. 5c, f, i) than the cases including only CCD summer (Fig. 5b, e, h). Vegetation indices and TWS show similar responses (Supplementary Fig. 12). Our results further indicate that CCD spring has more severe negative lagged effects on productivity than CWD spring. That is, we find that in comparison to cases with preceding CWD spring and consecutive CWD events, summer productivity shows higher reduction in cases with preceding CCD spring and consecutive CCD events (Fig. 4a, c versus Fig. 5a, c). Moreover, in cases with CCD spring but non-CCD summer (Fig. 5a, d, g), summer anomalies are close to those in scenarios with non-CCD spring but CCD summer (Fig. 5b, e, h). The vegetation and TWS anomalies further confirm this situation (Supplementary Fig. 12a, b, d, e). These results suggest that the lagged effects of CCD spring can be of similar magnitude as their direct adverse effects.We finally compare the direct and lagged effects of individual events with that of CCD events in mid-to-high latitudes. Cold conditions in spring and summer directly reduce productivity but show weak effects on soil moisture (Supplementary Fig. 13a–d), and cold spring shows negative lagged effects on summer productivity (Supplementary Fig. 13a). Dry events show direct and lagged negative effects on productivity and soil moisture (Supplementary Fig. 13e–h). These results imply that the negative lagged effects of CCD springs are dominated by both cold and dry components. The effects of CCD events on productivity mostly exceeds the individual dry or cold impacts (Supplementary Fig. 13a, b, e, f). More

OverviewIn this study, we used VIIRS active fire detections to track the dynamic evolution of all fires in California from 2012 to 2020 (Fig. 1). We developed an approach that has the following steps. First, after reading the satellite fire pixel data at each 12-hour time step, the new fire pixels are aggregated into multiple clusters using the fire pixel locations and an automatic clustering algorithm. These clusters are then spatially compared to existing fire objects. If a cluster is not close to any existing active fire object, we use all fire pixels within the cluster to form a new fire object. If a cluster is located near an existing fire object which is still active, we view the cluster as an extension of the existing fire. In this case, we append all pixels within the cluster to the corresponding existing fire object, allowing the existing object to grow. When a fire expands and gets close enough (within a pre-defined distance threshold) to an existing active fire object, we merge the two objects. For each time step (12 hours in this case for the two overpasses), we derive or update a suite of attributes and status indicators associated with each fire event, including pixel-level attributes of fire and surface properties, vector geometries related to the fire shape, and meta-attributes characterizing the entire fire object.Data inputSatellite remote sensing instruments provide active fire detections with accurate geographical location and broad spatial coverage. The primary data for this fire tracking system are active fire locations and the fire radiative power (FRP) recorded by the VIIRS instrument aboard the Suomi National Polar-orbiting Partnership (Suomi-NPP) satellite24. VIIRS observes Earth’s surface twice each day in low and mid latitude regions, with local overpass times of approximately 1:30 am and 1:30 pm. Compared to its predecessor, the MODIS sensors on the Terra and Aqua satellites, VIIRS has a higher spatial resolution and can detect smaller and cooler fires24. Also, the VIIRS instrument provides a more consistent pixel area across the image swath25, resulting in more accurate estimates of active fire location. Therefore, compared with MODIS, the VIIRS active fire products can be used to map fire event progression with higher accuracy21. Two streams of VIIRS active fire data are operationally produced using a contextual fire detection algorithm24, drawing upon VIIRS moderate resolution band (M-band) and imaging band (I-band) reflectance and radiance data layers. In this fire tracking system, we used the Suomi-NPP VIIRS I-band fire location data product (VNP14IMGML, Collection 1 Version 4) that contains the centre location, FRP, scan angle, and other attribute fields associated with each pixel. The I-band fire detection product has a 375-m spatial resolution at nadir (the sub-satellite point) and an average resolution across the full swath of about 470 m. Theoretical estimates of fire detection efficiency for the VIIRS sensor indicate that during the day, VIIRS can detect 700 K fires with 50% probability that have a size of about 200 m2 (a 15 m × 15 m fire area)24. During night, the detection efficiency increases, and VIIRS can detect 700 K fires as small as 40 m2. From a fire spread tracking perspective, these detection efficiencies imply that in many instances, the area of a fire pixel that is covered with flaming fire combustion is several orders of magnitude smaller than the overall pixel size. The VNP14IMGML data, available from 2012 onwards, were downloaded from the University of Maryland VIIRS Active Fire website (https://viirsfire.geog.umd.edu/).Land cover data are an additional input in the system required to classify different fire types and determine the spatial connectivity threshold. Here we use the U.S. National Land Cover Database (NLCD 2016)26 that is available from the Multi-Resolution Land Characteristics (MRLC) Consortium website (https://www.mrlc.gov/national-land-cover-database-nlcd-2016). We aggregated the original 30-m data to match the spatial resolution of VIIRS active fire data, and merged the original 16 classes into several groups: ‘Water’, ‘Urban’, ‘Barren’, ‘Forest’, ‘Shrub’, ‘Grassland’, and ‘Agriculture’. We used the 1000-hour dead fuel moisture from the high-resolution (4 km) gridMET product27 for the purpose of separating wildfires and management fires. This gridMET dataset was computed from 7–day average conditions composed of day length, hours of rain, and daily temperature and humidity ranges. Regularly updated gridMET data are available from the Climatology Lab website (http://www.climatologylab.org/gridmet.html).Other ancillary and validation datasets used in this study included a shapefile of California borders and fire perimeters from the California Forestry and Fire Protection’s Fire and Resource Assessment Program (FRAP) dataset (https://frap.fire.ca.gov/mapping/maps/).Fire object hierarchyFire detections from VIIRS are dynamically tracked within the framework of a three-level object hierarchy (Fig. 1). The lowest level is the fire pixel object, which includes the geographical location (latitude and longitude), the FRP value, and the origin (first assigned fire object id). The second level is the fire object, which includes all attributes associated with each individual fire event at a particular time step (Table 2). Each fire object includes one or more fire pixel objects, a unique identification number (id), and a set of attributes associated with the whole fire. Two types of fire attributes are derived and recorded for each fire object. The first type encompasses temporal (e.g., ignition time, duration) and spatial (e.g., centroid, ignition location) characteristics of the object as well as general properties (e.g., size, type, active status). The second type is the geometric information related to the fire object, including the fire perimeter, the active fire front line, and the newly detected fire pixel locations (stored as vectors). All fire objects in the State of California are combined to form an allfires object, to characterize the whole-region fire situation at a specific time step. The allfires object comprises a list of fire objects, and also contains meta information representing the statistics of all fires and the records describing fire evolution. A full list of the attributes associated with the pixel object, the fire object, and the allfires object is presented in Table 2.Table 2 List of main attributes associated with pixel, fire and allfires objects.Full size tableFire event trackingThe fire records (locations and FRPs) from the monthly VIIRS active fire location products (VNP14IMGML) are read into the system at each half-daily time step (roughly 1:30 am and 1:30 pm local time). We apply spatial and temporal filters to the data to extract active fire pixels recorded in California during each 12-hour time interval. We also apply quality flag filters (thermal anomaly type of ‘0: presumed vegetation fire’ in VNP14IMGML)) to ensure the use of only pixels likely associated with vegetation fires. The fire location and FRP values are used to create fire pixel objects. To speed up the calculation, the newly detected active fire pixels after filtering are first aggregated to specific clusters using the distances between them and an automatic clustering algorithm. In this initial aggregation algorithm, a ball tree28 is created to partition all newly detected active fire pixels into a nested set of hyperspheres in a 2-D space (latitude and longitude). This space partitioning data structure can be used to expedite nearest neighbours search29 and allow for quick cluster grouping. Here we refer to a cluster as a collection of pixel objects that are recorded at the same time step and are also spatially nearby. In the following steps, all pixels within a cluster are considered as a whole for fire merging and creation.We define an extended area for every existing fire object as the fire vector perimeter (see the section of Calculating and recording fire attributes for detail) plus a radial buffer that depends on the fire type property of the object. The buffer is set to 5 km for forest fires and 1 km for other fire types (shrub, crop, urban), considering that the fire spread rate can differ across biomes13. We then evaluate the spatial distance between the perimeters of a newly classified cluster and all existing active fire objects (a fire object keeps an active status if one or more active fire pixels associated with it are detected during the past 5 days), and calculate the shortest distance. If the shortest distance is smaller than the buffer of the associated existing active fire (i.e., new cluster overlaps with the extended area of an existing fire object), we assume all fire pixels in the new cluster are associated with the growth of the existing fire object at the current time step (Fig. 2). The existing fire object is updated by appending all fire pixel objects within the new cluster. If a newly classified cluster does not overlap with the extended area of any existing active fire object, we assume this is a new fire. A new fire object (by assigning a new fire id) is created using all fire pixel objects in the cluster.With the addition of new fire pixels, an existing fire object may expand and touch the extended area of another existing active fire object. If this happens, we assume that these two existing fire objects merge into a single object at this time step. All fire pixels in the fire object with a higher id number (a later start date, termed as the ‘source fire’) are appended to the fire object with lower id number (earlier start date, termed as the ‘target fire’) in this case. We record the id of the target fire in a list of fire mergers, and update all attributes associated with this fire (Fig. 3). In order to avoid double counting, the source fire object (with all pixels being transferred to the target fire object) is flagged as invalid, and is excluded from statistical analysis of fire events.Fig. 3The time series of growth for the SCU Lightning Complex fire (2020). Panel (b) shows the fire size of the SCU fire (total area within the fire object perimeter) at half-daily time steps. A fraction of the fire growth (shown in orange) was due to the addition of newly detected fire pixels. Panel (a) shows the number of new fire pixels (associated with the SCU fire object) detected at each time step. The other part of the fire growth (shown in red) was due to the merging with existing fire objects. Panel (c) shows the number of fire pixels in the existing objects that were merged to the SCU fire object.Full size imageCalculating and recording fire attributesOther than individual fire pixels contained in a fire object, several core attributes (properties and geometries) are also dynamically updated at each time step and are used for fire tracking and characterization.Important time-related attributes include the fire ignition time (the time step at which the first fire pixel within the fire object was detected), the fire end time (the latest time step with an active fire observation), and the fire duration (the time difference between the ignition time and end time). If a fire object does not have new active fire pixels appended during 5 consecutive days (i.e., the fire end time is more than 5 days before the present time step), its status is set to inactive. Once inactive, a fire object is no longer evaluated for use in future clustering (i.e., new active fire detections later will form new fire objects, even if they are spatially close to the inactive fire object).Each fire object is assigned to a specific fire type. The fire type is identified using the major land cover type within the fire perimeter (Table 3). In an initial analysis, we found that prescribed fires, on average, have higher coarse fuel moisture levels than wildfires. Therefore, we also record the 1000-hour fuel moisture (fm1000) from the gridMET dataset27 for each fire object (corresponding to the ignition time step) and use this value to divide forest and shrub fires further to wildfire and prescribed types.Table 3 Classification of fire types based on dominant land cover type (from the US National Land Cover Database) within each fire perimeter and the 1000-hr fuel moisture (FM-1000, from gridMET dataset) at the time of ignition.Full size tableAn essential step in this object-based fire tracking system is to determine the vector shape of the fire perimeter. In this system, we use an alpha shape30 algorithm to derive bounding polygons containing fire pixels in a fire object. For an alpha shape, the radius of the disks forming the curves in the polygon is determined by the alpha parameter α. Compared with the commonly used convex hull, the alpha shape hull is able to capture the irregular shapes around the fire perimeter more accurately22.To identify the optimal values for the α parameter, we performed the following analysis. First, we derived the final fire perimeters for all large fires that occurred in California during the 2018 wildfire season using a set of α values ranging from 500 m to 10 km and compared the results with more refined fire perimeters from the Fire and Resource Assessment Program (FRAP) dataset (Fig. 4). Large magnitude α values tended to overestimate the total burned area, while small α values often fragmented a large fire event. We found that a value of α = 1 km was optimal in terms of balancing the ability of the hull to catch the boundary shape and to keep the integrity of a fire object. For each time step, we applied the alpha shape algorithm to all fire pixel locations associated with a fire object since the time of ignition. This processing step resulted in a concave hull with the shape of polygon or multipolygon. To account for the pixel size, we expanded the concave hull to the fire perimeter using a buffer size equal to half of the VIIRS nadir cross-track pixel width (187.5 m). The alpha shape algorithm does not work when the total number of fire pixels (npix) is less than 4. If npix equals 3, we used a convex hull algorithm and the same 187.5 m buffer to determine a polygon perimeter. If npix is 1 or 2, circles centered on the fire pixel location with radius of 187.5 m were used.Fig. 4Optimization of the alpha shape parameter (α). For all large fires (final size  > 4 km2) in California during 2018, fire perimeters were estimated using VIIRS active fires and different alpha parameters. By comparing (a) the burned area (BA) and (b) the number of fire objects with the FRAP data, an optimal alpha parameter of 1 km was identified for use in this study (shown in red). The vertical bars and lines show the mean and 1-std variability from all fires. The dashed blue lines indicate the ideal values when compared to FRAP. Panels (c)–(h) show the fire perimeters derived using different alpha shape parameters for two sample fires in 2018. The shapes with pink color are final FEDS fire perimeters derived from VIIRS active fires using the alpha shape algorithm. The blue shapes represent the corresponding fire perimeters from the FRAP dataset. Overlap between FRAP and FEDS is shown in purple.Full size imageWe also calculate the active front line for each fire object at each time step. The active fire front consists of the segments of the fire perimeter that are actively burning and releasing energy and emissions. The position of the active fire line is critical in evaluating the fire risk, estimating the fire emissions, and predicting fire spread. We derive the active portion of the fire perimeter as segments that are within a 500 m radius of newly detected fire pixel locations. We found that this threshold allowed for a continuous projection of the active fire front in rapidly expanding areas of large wildfires during the 2018 fire season; this threshold may be optimized in future work to maximize performance metrics for fire model forecasts. The resulting active line for each fire at each time step has the shape of a linestring (object representing a sequence of points and the line segments connecting them), a multi-linestring (a collection of multiple linestrings), or a linear ring (closed linestring). Figure 5 shows an example map of the fire perimeters and active fire front lines on September 8 during the 2020 wildfire season.Fig. 5An example map of fire perimeters and fire active fronts in California. The map was created using the fire event data suite (FEDS) as of the Suomi-NPP afternoon overpass (~1:30 pm local time) of Sep 8, 2020. The background is the Aqua MODIS Corrected Reflectance Imagery (true color) recorded at the same day (provided by the NASA Global Imagery Browse Services). The active front line of a fire is shown in yellow, active fire areas are shown in red, and the area of inactive (extinguished) fires are shown in dark red.Full size imageAdditional fire properties, such as the fire area and active fire line length, are also derived using these geometries of the fire object (see Table 2). Note this list can be easily expanded to include more user-defined properties with the help of the fire object core vector data.The allfires object contains a list of all existing fire objects at a time step. This object also records the ids of fire objects that have been modified (including fires newly formed, fires that expanded with new pixel additions, fires with pixels addition due to merging, and fires that just became invalid) at the current time step.Creating the fire event data suite (FEDS)By tracking the spatiotemporal evolution of all fire objects in California, we derived a complete dataset of fire events for each calendar year (Jan 1 am – Dec 31 pm) during the Suomi-NPP VIIRS era (2012–2020). The dataset contains four products that represent the fire information in California at multiple spatial scales and from different perspectives (Fig. 1 and Table 4), ranging from the most detailed and memory-intensive data format (Pickle) to the most high-level format (CSV).Table 4 Data structure of the FEDS.Full size tableThe first product is the direct serialization result of the allfires object at each time step (twice per day). The product is stored as a Pickle file31 which allows for analysis of the complex allfires object structure (including all attributes associated with all fire objects it contains). This file also serves as the restart file for continued fire tracking at any time step, which is essential for the operational mode using the near-real-time fire data. By restoring an exact copy of the previously pickled allfires object, any attribute in the allfires object can be deserialized from the saved files. The Pickle file is the most basic data product in the dataset, and is created at each half-day time step.The second product (Snapshot) represents a more accessible and self-explanatory variant of the Pickle serialization product. In this product, we tabulated important diagnostic attributes for each fire and saved them in GeoPackage32 data files. Each GeoPackage file includes three data layers: one contains the properties and the fire perimeter geometry, another contains the active fire line geometry, and a third contains the new fire pixel location geometry. This product, created at a half-daily time step, allows for a more straightforward interpretation of regional fire status at a particular time step. We also created a GeoPackage file that summarizes the final fire perimeters and attributes for all fires during the whole study period (2012–2020).The third product (Largefire) focuses on the temporal evolution of individual large fires with an area greater than 4 km2. At each time step, the time series of properties and geometries (fire perimeter, active fire line, and new fire pixel locations) for each of the large fires are extracted and saved to GeoPackage files. This product facilitates the visualization and analysis for an individual targeted fire (Fig. 6) and is particularly useful in the near-real-time evaluation, forecasting, and policy making.Fig. 6The spatiotemporal evolution of the Creek fire (2020). Contours and dots reflect the fire perimeters and newly detected fire pixels at each 12-hour time step. Data for the period of Sep 5 am–Nov 6 am, 2020 are shown.Full size imageThe fourth product (Summary), which is stored as NetCDF and CSV files and created at the end of a fire season, records the all-year time series of fire statistics (including major fire attributes such as number, size, duration, fire line length, etc.) over the whole State of California. This product provides a feasible regional summary of the temporal evolution of fires.Potential for near-real-time (NRT) fire event trackingWhile the main objective of this paper is to apply the object-based fire tracking system to historical VIIRS fire detections and create a retrospective multi-year FEDS, we note that this system has the potential to be used for tracking fire events in near-real-time, providing rich and valuable information for fire management and short-term risk assessment. We have experimented with the use of this system for NRT fire event tracking in California using the daily NRT Suomi-NPP VIIRS active fire detection product (VNP14IMGTDL, collection 6) as the main data source. The VNP14IMGTDL product is routinely produced and is publicly available at the NASA Fire Information for Resource Management System (FIRMS). Since the NRT product undergoes less rigorous quality assurance, we use only fires with ‘nominal’ or ‘high’ confidence levels from the NRT product for fire tracking. Some active fire detections from the NRT data are potentially associated with static non-vegetation fires (e.g., fires from gas flaring in oil and gas or landfill industries or false detections due to reflection from solar panels) and are not the main interest for vegetation fire studies. To avoid the unnecessary computation associated with these static fires, we record and evaluate the fire pixel density for each fire object at each time step. When a small fire ( 20 per km2), it is considered to be a static fire and subsequently labelled as invalid.Similar to the retrospective FEDS, we use the active fire detections to create an object serialization product, a regional snapshot GIS product, and a time series product of large fire evolution twice daily. This experimental NRT data will be available upon publication through a university hosted server. More

Study areaGrassland herbage samples are from Shaerqin base, institute of grassland research of CAAS (Chinese Academy of Agricultural Sciences). We obtained the permission of the institution to take HSI of the grassland sample. Our work did not cause damage to grassland. Researcher Weihong Yan of the institute provided us with relevant information about grassland. The land use type in the study area is mainly grassland, which is composed of forage species, most of which are representative species of typical grassland. We take this area as an example to conduct research on grass classification. By enriching the relevant recognition technology, it can also be used as a reference for the pastures of other grasslands. The grass species Grass1 for the experiment is shown in Table 1. The official introduction of plant materials is detailed in the flora of China15.Table 1 Samples information for Grass1 dataset.Full size tableThe field hyperspectral platformWe assemble a system for collecting HSI in the field: HyperSpec©PTU-D48E HSI instrument, high-precision scanning PTZ, tripod, data analysis software Hyperspec, etc. The light source is natural light. The imaging instrument is in line scanning mode. Table 2 shows the technical parameters.Table 2 Technical parameters of hyperspectral instrument.Full size tableData collectionIn July 2021, the data was collected during the lush grass growth period. Collect data from 11:00 a.m. to 2:00 p.m. every day. At this time, it is sunny, cloudless and the wind force does not exceed level 2. So as to ensure the consistency of the acquisition time line and avoid the influence of different degrees of light on the reflectivity as far as possible. The measuring points are arranged facing the sun and the opposite direction of the shadow. We collect data from different angles of the grassland, which is based on the growth of various types of forages, and selects relatively concentrated places within the study area. Each shot is a single category of grass. The image resolution is 1166 × 1004 pixels (Fig. 1). The imaging spectrometer is fixed with scanning head when shooting. Data acquisition and transmission are executed on Hyperspec software. Then save it as a BIL file. The ENVI5.3 software was used to extract the forage spectrum to establish the dataset Grass1. Well balanced regions with a clear image, uniform spectral distribution are selected for further segmentation. The average value of spectral reflectance of grass pixels was taken as the reflectance spectrum of a single type of grass.Figure 1True color map of grass samples.Full size imageMethodologyIn Fig. 2, we present the framework of visible-NIR hyperspectral classification of grass based on multivariate smooth mapping and extreme active learning (MSM–EAL). Specifically, we first introduce the proposed MSM algorithm for global enhanced spectral reconstruction, which utilizes smooth manifold projection technology to alleviate the problems of difficult feature selection and redundant data. Then, the EAL framework is proposed to address the matter of hyperspectral labeled samples and spectral classification. In the following, each step of this method will be presented in detail.Figure 2Proposed MSM–EAL framework for grass HSI classification.Full size imageThe proposed MSM algorithmIn the process of field HSI acquisition, on the one hand, the surface distribution of grass is uneven and the plant height is different, causing certain scattering effect and coverage spectrum change. On the other hand, HSI is easy to be disturbed by external natural factors such as light, wind and shadow, resulting in a certain degree of distortion. Multiplicative scatter correction (MSC) is a scattering correction effect, which helps to eliminate the scattering effect caused by the above reasons and enhance the spectral variability. The moving window smooth spectral matrix (Nirmaf) belongs to the smooth effect, which improve the signal-to-noise ratio of the spectrum and reduce the influence of random noise16,17. Preprocessing methods are different and related to each other. We design an enhanced preprocessing multivariate smooth (MS) method that fusing MSC and smooth Nirmaf to target grass spectral signal features. In the follow-up, a model will be established to verify the validity of MS.Most of the high-dimensional spatial data have the characteristics of being embedded in a manifold body, so the manifold learning isometric feature mapping (Isomap) based on spectral theory is adopted. Isomap preserves the global geometric features of the initial data and extracts features by reconstructing the underlying smooth manifold of HSI. It is nonlinear dimensionality reduction based on linear and multidimensional scaling transformation18. Isomap has been applied in image and HSI classification19,20, but there is no report on visible-NIR hyperspectral classification of grass.In view of the above, we proposed the multivariate smooth mapping (MSM) spectral reconstruction algorithm, which can be represented as follows:$$ MSM_{z} { } = { }frac{{left( {P_{j} – b_{j} } right)left( {2n + 1} right) + n_{j} cdot mathop sum nolimits_{j = – n}^{n} C_{j} P_{k + j} }}{{n_{j} left( {2n + 1} right)}} + V_{Z} F_{Z}^{frac{1}{2}} { } $$
(1)
where Pj, bj, and Cj represent the raw reflectance value of spectrum j, baseline shift amount, and weight factor, respectively, k and nj represent the polynomial degree and offset, respectively. MSMz is the feature cube reconstructed to Z dimension from the spectrum calculated by 2n + 1 moving window width, V eigenvector matrix and F eigenvalue matrix.In Isomap equidistant mapping, the shortest path of edge Pi Pj needs to be solved, and the representation matrix is:$$ D_{G} = [d_{G}^{2} (P_{i} ,P_{j} )]_{i,j = 1}^{n} $$
(2)
where d (Pi, Pj) is the weight of the edge Pi Pj calculated from the neighborhood graph G and its side Pi Pj.The proposed EAL frameworkLabeling hyperspectral samples is expensive in terms of time and cost, at the same time, the lower spatial resolution and more bands increase the difficulty of labeling. Active learning (AL) provides an efficient labeling strategy, which only needs to label a relatively small number of samples to learn a more accurate model21. The pool-based AL selects the most informative samples according to the query strategy for limited labeling through iteration, so as to facilitate model improvement. Commonly used query strategies are uncertainty criteria, such as least confidence22, the bayesian active learning disagreement (BALD), the entropy sampling23, etc.Due to there is still an over-fitting problem, different strategies such as hybrid prediction and regularization need to be used for non-recursive datasets24. The research25 proposed that extreme gradient boosting algorithm (XGBoost) based on gradient boosting. As a classification method, XGBoost has been successfully applied in Kaggle competition and other fields. Its most important feature for visible-NIR hyperspectral classification is that can easily and directly classify according to features, and the physical interpretation of features can help understand the electronic nature behind spectral classification. XGBoost is a machine learning algorithm based tree structure that integrates multiple weak classifiers to achieve flexible and high-precision classification. It is an upgraded version of gradient boosting decision tree. The optimization process of XGBoost entailed: (1) Expanding the objective function to the second order, and finds a new objective function for the new base model to improve the calculation accuracy. (2) L2 regularization term is added to the loss function to prevent over-fitting. (3) Using blocks storage structure realize automatic parallel computing26,27. The algorithm steps are as follows:The objective function:$$ Lleft( Phi right) = mathop sum limits_{i} lleft( {y^{i} ,widehat{{y^{i} }}} right) + mathop sum limits_{k} Omega left( {f_{k} } right) $$
(3)
In formula (3), the first and second terms are the loss function term and the regularization term, respectively. Where,$$ Omega left( {f_{k} } right) =upgamma {text{T}} + frac{1}{2}lambda left| w right|^{2} $$
(4)
γ and λ are regularization parameters which are used to adjust complexity of the tree.Next, second derivative Taylor expansion of the objective function. Where (g_{i}) and (h_{i}) are the first derivative and second derivative, respectively.$$ L^{left( t right)} = mathop sum limits_{i = 1}^{n} lleft( {y_{i} ,widehat{{y_{i}^{t – 1} }} + f_{t} left( {x_{i} } right)} right) + Omega left( {f_{t} } right) $$
(5)
$$ g_{i} = partial_{{hat{y}_{i} (t – 1)}} lleft( {y_{i} ,widehat{{y_{i}^{t – 1} }}} right) $$
(6)
$$ h_{i} = partial_{{widehat{{y_{i} }}(t – 1)}}^{2} lleft( {y_{i} ,widehat{{y_{i}^{t – 1} }}} right) $$
(7)
$$ {text{L}}^{left( t right)} approx mathop sum limits_{i = 1}^{n} left[ {lleft( {y_{i} ,widehat{{y_{i}^{t – 1} }}} right) + g_{i} f_{i} left( {x_{i} } right) + frac{1}{2}h_{i} f_{t}^{2} left( {x_{i} } right)} right] + Omega left( {f_{t} } right) $$
(8)
Final objective function:$$ {hat{text{L}}}^{ i} left( q right) = – frac{1}{2}mathop sum limits_{j = 1}^{T} frac{{(mathop sum nolimits_{{i in I_{j} }} g_{i} )^{2} }}{{mathop sum nolimits_{{i in I_{j} }} h_{i} + lambda }} + gamma T $$
(9)
Equation (9) can be used as the fraction of tree cotyledons, and the tree structure is directly proportional to the fraction. If the result after splitting is less than the maximum value of the given parameter, the cotyledon depth stops growing24,28.AL solves the problems of limited number and high cost of grass hyperspectral labeling samples. The default model of traditional AL is logistic regression, which is mostly studied on the ideal public dataset. However, the actual data has more uncertain noise, which still poses a certain challenge to AL. Consequently, we propose the extreme active learning (EAL) framework to minimize the classification cost of visible-NIR hyperspectral. The framework replaces the logistic regression model with XGBoost. Taking advantage of AL, XGBoost can improve performance with less training marker samples. By jointing of XGBoost and AL, EAL provides significantly better results than AL in field Grassl dataset recognition. Additionally, based on the characteristics of XGBoost, EAL more intuitively enhances the physical essence behind spectral classification than AL. Algorithm 1 summarizes the workflow of EAL framework.Random forest (RF) and decision tree (DT) were used to compare with EAL. RF and DT are frequently used in the field of grassland remote sensing9,29. Furthermore, RF, DT and XGBoost have the same point is that are learning algorithms based on tree structure. DT determines the direction by judging the conditions of the decision node12. RF is an integrated learning of multiple decision trees30. More

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Stability analysis of the deterministic modelSolving (left( x(t) times left( r_{p}x(t)^{(alpha )}left( 1-big (frac{x(t)}{K}big )^{beta }right) – nx(t)^{(delta )} right) right) =0), we obtain two stable and one unstable equilibrium points for the model. One stable equilibrium is trivial, i.e., (x(t)=0), another stable equilibrium point being the non-zero satisfying (left( r_{p}x(t)^{(alpha )}left( 1-big (frac{x(t)}{K}big )^{beta }right) – nx(t)^{(delta )} right) =0). Figure 1a shows three different equilibrium points of the model. In addition to the equilibrium, the model has two inflection points (Fig. 1a). At these inflection points the absolute growth rates are minimum and maximum. The density vs relative proliferation rate (RPR) profile of the model shows that the model can attain negative RPR for a positive cell density, suggesting that the model can portray the Allee phenomenon (Fig. 1b). Figure 1c,d portray the proliferation and decay phases, respectively through the model.Figure 1Growth dynamics of the proposed model: (a) Absolute proliferation rate (APR) profile considering (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99) and (delta =0.2); (b) RPR profiles for different n and other same constant model parameters; (c) Cell population survive for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99) and (delta =0.2) with the initial cell density 0.1; (d) The population goes to extinction for the initial cell density 0.06 with the same constant parameters.Full size imageThe solution of the deterministic model finally provides two theorems.
Theorem 1

(x^{*}approx K -Kleft( frac{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )-sqrt{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )^{2}-2 left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) nK^{delta }}}{left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) }right)) is the conditional MSSCD for the intercellular-interaction-induced proliferative cells. The conditional threshold density for cell-proliferation upon interaction is (x^{*}=K -Kleft( frac{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )+sqrt{Big (beta r_{p}K^{alpha }+n delta K^{delta }Big )^{2}-2 left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) nK^{delta }}}{left( 2 alpha beta r_{p}K^{alpha } +beta (beta -1)r_{p}K^{alpha }+delta (delta -1)nK^{delta } right) }right)) (proof is in the supplementary information).
Allee and cooperation models are the only extended logistic law other than our model to provide a threshold population size for growth or proliferation. Our proposed model is superior to the Allee and cooperation model as it can detect the conditional threshold cell density for proliferation and regulate the density by its different parameters. For example, One may reduce the conditional threshold density by either regulating the interaction between growth-inhibiting molecules and cells ((delta)) or reducing the inhibiting molecule concentration (n).The conditional MSSCD from Theorem 1 is lower than the carrying capacity of the conventional logistic model due to growth-inhibiting molecules; it provides the expected cell density during culture in a given environment. Theorem 1 also states the set of parameters to control the cell proliferation and get the desired density during such cell cultures. A further question arises knowing this set of parameters: which one of the parameters in the expression is crucial in terms of application purpose? Since the (r_{p}) is the constant proliferation rate for a given cell line, controlling the conditional MSSCD is not possible through (r_{p}). We simulate the distribution of conditional MSSCD for other parametric planes to answer this question. For this, we use the parameter values obtained from the data.

Theorem 2

The RPR is maximum at the cell density (x^{*}= K-Kleft( frac{r_{p}beta K^{alpha -1}+ndelta K^{delta -1}}{2r_{p}alpha beta K^{alpha -1}+r_{p}beta (beta -1)K^{alpha -1}+ndelta (delta -1)K^{delta -1}}right)) for the concave downward profile under the condition (r_{p}alpha (alpha -1){x^{*}}{}^{(alpha -2)}-frac{r_{p}}{K^{beta }}(alpha +beta )(alpha +beta -1){x^{*}}{}^{(alpha +beta -2)}-ndelta (delta -1){x^{*}}{}^{(delta -2)}n) (see the supplementary information). The cell population sustain with any positive initial cell density x(t) and try to stabilize at (x(t)= K(1-frac{n}{r_{p}})^frac{1}{beta }). Therefore, bimodality vanishes and unimodality is observed for the case (alpha =delta) (r_{p} >n). The RPR profile will be concave downward always with the maximum RPR value is at the inflection point (x(t)= K(frac{(r_{p}-n)alpha }{r_{p}(alpha +beta )})^frac{1}{beta }). The deterministic potential function in this case is (U(x)=-Big [(r_{p}-n)frac{x^{(alpha +2)}}{(alpha +2)}-frac{r_{p}}{K^{beta }}frac{x^{(alpha +beta +2)}}{(alpha +beta +2)} Big ]). The minima of this effective potential function will be at (x(t)= K(1-frac{n}{r_{p}})^frac{1}{beta }) which is the maximum stable cell density for (r_{p} >n).
Parameter estimationThe density-RPR and time-density fitting to the scratch assay datasets show a lower RSS for our model than the logistic one for each of the three seeding conditions. The estimated parameters from the RPR fitting through the grid-search are in Table 2. Although the RSS for the RPR fitting of the seeding 2 is very low, the data itself is too scattered in both the upper and lower range for the small cell density. Therefore, there is a chance that regardless of the low RSS value, the fitting for seeding 2 may not reflect the actual estimates of the parameters with the bias in the data set (Fig. 2b). Nevertheless, the density-RPR fittings to the other two seeding density datasets do not suffer from bias (Fig. 2a,c).Table 2 Estimated model parameters from density-RPR fitting of our model.Full size table
Figure 2Our proposed model best fitted the cell density-RPR datasets for all of the seeding conditions generated through the grid-search method.Full size image
Jin et al.1 suggested that their two phase logistic model may share similarities with the Allee effect. However, they did not fit the Allee model stating seeding 2 and 3 were large enough seeding densities. We calculated the conditional threshold density, conditional MSSCD, density at the minimum and maximum RPR for the model from our estimated parameters (Table 3). The conditional threshold cell density calculated from our estimated parameters confirms that the smallest initial seeding density of the dataset was greater than the conditional threshold cell density.Table 3 Calculated cell densities from estimated parameters from our model fitting.Full size tableFigure 3 compares the portrayal of the data through our model with the fitting by Jin et al.1. The blue dashed line is the time-series fitting of the proposed model, and the red-colored line is the time-series fitting of the logistic model to the scratch assay data sets in the Fig. 3. The carrying capacity values are unexpectedly very high in the logistic fit, keeping the model near the exponential phase for the entire dataset. Thus the overall and two phase logistic fits are unrealistic compared to the highest cell density observed in the assay. Also, logistic fitting of the RPR profiles to the data after 18 h does not capture the whole scenario. The green solid and the violet dashed line represent the logistic time-density fit after and before 18 h density profiles respectively. The orange-colored lines in the Fig. 3 are the expected population density as per estimated parameters from the RPR fitting after 18 h data sets. Table 4 enlists all parameters for a comparison between logistic and our model fitting.Figure 3Time series solution of the proposed model and logistic law with comparative RSS for all three seeding conditions.Full size imageTable 4 Logistic model fitting with the Jin et al.1 estimates used in Fig. 3 with the specific colors.Full size tableTrends in cell densities under deterministic set upThe (r_{p}) is fixed for a cell line among all the determining parameters of the conditional MSSCD. n and K vary together with the culture media, flask, and environmental setup. On the other hand, the (alpha), (beta), and (delta) vary together with intercellular-interactions and cellular-interaction with growth-inhibitory molecules, which depend on the medium’s initial cell density per well and fluidity. We observe that the distribution of the conditional MSSCD depends more on the K than the n. There is a chance of overproliferation in the deterministic setup under low n but high K. The cells may die under high n. The cell density at maximum RPR also depends more on K than n (Fig. 4). So the cells should be cultured in the larger flask to achieve maximum proliferativeness.Figure 4The distribution of conditional MSSCD and cell density at maximum RPR in n-K parametric plane.Full size imageThe conditional MSSCD depends more on (beta) than (alpha) (Fig. 5a). The cells may tend to overproliferate under both high (alpha) and (beta). The conditional MSSCD does not exist for a high (delta) and low (beta) depending more on (delta) than (beta). The cells may overproliferate only under a high (beta) and low (delta) (Fig. 5b). The conditional MSSCD also depends more on (delta) than (alpha) showing mostly underproliferation of cells in the (delta ~-alpha) parametric plane. Therefore, the proliferation can be controlled via regulating the interaction between the growth-inhibitory molecules and cells followed by density-regulation through contact-inhibition and cell-cell cooperation (Fig. 5c).Figure 5The distribution of the conditional MSSCD in parametric plane of regulators in the growth law: (a) dependence of the conditional MSSCD on (alpha) and (beta) parameters; (b) dependence of the conditional MSSCD on (delta) and (beta) parameters; (c) dependence of the conditional MSSCD on (alpha) and (delta) parameters.Full size imageThe new cell fitness measure, i.e. cell density at maximum RPR depends more on the (alpha) than the (beta) (Fig. 6a). The cells achieve maximum RPR at a great cell density under the high value of these two parameters. Figure 6b,c suggest that cell density depends only a little on the (delta) under high (alpha) and (beta). Under the low value of these two regulators, a high (delta) always reduces the cell density attaining the maximum RPR, resulting a poor cell-fitness.Figure 6The distribution of cell density at maximum RPR in parametric plane of regulators in the growth law: (a) dependence on (alpha) and (beta) parameters; (b) dependence on (alpha) and (delta) parameters; (c) dependence on (delta) and (beta) parameters.Full size imageStochastic model analysisOur proposed stochastic model (3) can be compared with the general stratonovich stochastic differential equation (frac{dx}{dt}=f(x)+g_{1}(x)epsilon (t)+g_{2}(x)Gamma (t)). Comparing it with our proposed stochastic model we obtain (g_{1}(x)=-x^{delta +1}) and (g_{2}(x)=1). Using the help of47, we get noise induced drift (A(x)=r_{p}x^{alpha +1}left( 1-Big (frac{x}{K}Big )^{beta } right) -nx^{(delta +1)}+D(delta +1)x^{(2delta +1)}-lambda sqrt{DQ}(delta +1)x^{delta }) and noise induced diffusion coefficient (B(x)=Dx^{(2delta +2)}-2lambda sqrt{DQ}x^{(delta +1)}+Q). The cell density at long run can be obtained from the steady state probability density function (SSPDF). The analytical expression of the SSPDF is obtained from the Fokker-Planck equation. The Fokker-Planck equation is (frac{partial P(x, t)}{partial t} =- frac{partial big [ A(x) P(x, t)big ]}{partial x}+ frac{partial ^{2} big [B(x) P(x, t)big ]}{partial x^{2}}), where P(x,t) is the probability density function of the cell population at the time point t. Solving the Fokker-Planck equation we get the SSPDF as (P_{st} (x)= frac{N^{prime }}{B(x)} exp left( int _{x} frac{A(x^{prime })}{B(x^{prime })} dx^{prime }right)) with the normalizing constant (N^{prime }). The value of (N^{prime }) can be obtained from (int _{0}^{infty } P_{st} (x)dx=1).This SSPDF (P_{st} (x)) helps to understand the validity of the proposed stochastic model. Since the number of the data points is too low to fit the stochastic model to the data directly, validation of the stochastic model is challenging in this case. The dataset we used is a time series with 15 data points with three replicates only. An experiment must have many replicates to have a sample with a large sample size so that the SSPDF of cell densities obtained from theoretical findings can be validated with the real observation of cell densities at the steady state. Such datasets with many replicates are rare.So, we generate 2000 sample paths with the help of numerical simulation based on stochastic model 3. We use the parameter values estimated from the fittings of the deterministic model to the seeding condition 1, and we consider some particular values for the two noise intensities and correlation strength ((lambda)) to get a simulated dataset. To achieve the stationary state, we consider sufficiently large time points, and the cell densities at the final time point are used as the data set for the stationary state. We compare the frequency density of cell densities at steady-state of a simulated dataset of 2000 sample paths with the SSPDF obtained from the analytical solution. This comparison shows that the cell density distribution at the steady state matches the steady state probability density function obtained analytically (Fig. 7).In addition, we illustrated the time series generated with the help of stochastic model 3 through numerical technique (Fig. 8). We have plotted the time series data thus obtained for each of the three seeding conditions and in the same figure we also plotted the observed cell densities. The red dots (o) represent the original/experimental dataset of Jin et al.1. The blue dots ((*)) represent the simulated dataset obtained from the stochastic model. This Fig. 8 clarifies our claim that the proposed stochastic model is in good agreement with the actual observation.Figure 7The histogram shows the distribution of cell densities at steady state under additive and multiplicative noises. The blue curve is the SSPDF. The function SSPDF and the distribution of cell densities matches to each other.Full size imageFigure 8The red dots (o) in each sub-figures represent the experimental data of Jin et al.1. The blue dots ((*)) are obtained from the stochastic model (3) considering: (a) The seeding 1 estimated model parameters with (D= 0.002), (Q= 0.06) and (lambda = 0.4). (b) The seeding 2 estimated model parameters with (D= 0.01), (Q= 0.15) and (lambda = 0.6). (c) The seeding 3 estimated model parameters with (D= 0.002), (Q= 0.2) and (lambda = 0.4).Full size imageFigures 7 and 8 suggest that the stochastic model is valid. So the model can be further analyzed to meet the first objective. Differentiating (P_{st} (x)), we obtain (frac{dP_{st} (x)}{dx}=frac{N^{prime }}{[B(x)]^2} exp left( int frac{A(x)}{B(x)}dx right) left( A(x)-frac{dB(x)}{dx} right)) and (frac{d^{2}P_{st} (x)}{dx^{2}}= frac{N^{prime }}{[B(x)]^{2}}exp left( int frac{A(x)}{B(x)}dx right) left( frac{dA(x)}{dx}-frac{d^{2}B(x)}{dx^{2}} right) +frac{N^{prime }}{[B(x)]^{2}} left( A(x)-frac{dB(x)}{dx} right) exp left( int frac{A(x)}{B(x)}dx right) frac{A(x)}{B(x)}-frac{2}{[B(x)]^3}N^{prime } exp left( int frac{A(x)}{B(x)}dx right) left( A(x)-frac{dB(x)}{dx} right) frac{dB(x)}{dx}). At the extrema of the SSPDF, we must have (frac{dP_{st} (x)}{dx}=0) i.e. (left( A(x)-frac{dB(x)}{dx} right) =0).

Theorem 3

(x^{*}approx K-K left( frac{nK^{delta +1}+D(delta +1) K^{2delta +1}-lambda sqrt{DQ}(delta +1)K^{delta }}{beta r K^{alpha +1}+n(delta +1) K^{(delta +1)}+D(delta +1) (2delta +1)K^{(2delta +1)}-lambda sqrt{DQ}delta (delta +1)K^{delta }} right)) is the conditional MSSCD due to the correlated additive and multiplicative noises under the condition (r_{p}(alpha +1)x^{*}{}^{alpha }-frac{r_{p}}{K^{beta }}(alpha +beta +1)x^{*}{}^{(alpha +beta )} -n(delta +1)x^{*}{}^{delta }-D(delta +1)(2delta +1)x^{*}{}^{(2delta )}+lambda sqrt{Dalpha }delta (delta +1)x^{*}{}^{(delta -1)} < 0) (proof is in the supplementary information). Figure 9 visualizes the effect of noise strength and correlation strength on the conditional MSSCD. The conditional MSSCD increases with the additive noise strength (Q) and decreases with the multiplicative noise strength (D) when the other model parameters are fixed (Fig. 9a). There is a high chance of overproliferation for a low D and a high Q (Fig. 9a). Again, there is a high chance of extinction for the low Q and high D. The conditional MSSCD depends more on D than (lambda) (Fig. 9b), and more on (lambda) than Q (Fig. 9c). The conditional MSSCD increases with (lambda) and Q; there is a high chance of overproliferation for high (lambda) and Q. The extinction risk of cells from the culture increases with low (lambda) and Q.Figure 9The change in the conditional MSSCD value for different noise strengths and correlation strength using the parameters estimated for seeding 1: (a) the conditional MSSCD values in (D-Q) noise strength plane with highest correlation ((lambda =1)); (b) the conditional MSSCD values in (D-lambda) noise plane with (Q=0.01); (c) the conditional MSSCD values in (Q-lambda) noise plane with (D=0.01).Full size imageDue to the difficulty and complicated expression of the analytical expression of the SSPDF, we use numerical simulation to study the steady-state behavior in the long run under correlated noises. We draw a histogram of the cell densities based on 500 normal sample paths at the final time points. We use seeding 1 fitting estimates as the initial parameter values for this simulation. The cell population is stable and steady at either 0 cell density or at the conditional MSSCD. The distribution is symmetric around the conditional MSSCD for (lambda =1) (Fig. 10a). There is a loss in the symmetry for the decreasing (lambda). For (lambda =0.5), there is a mode at the zero states with another mode at conditional MSSCD (Fig. 10b). The histogram shows a bi-modality for low values of (lambda). The mode at the zero state is highest for (lambda =0) (Fig. 10c). Therefore, the extinction chance increases for zero noise correlation between the additive and the multiplicative noises.Figure 10Distribution of cell density for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99), (delta =0.2), (D=0.01), (Q=0.01), and variable correlation between additive and multiplicative noises: (a) (lambda =1), (b) (lambda =0.5) and (c) (lambda =0).Full size imageThe sustainability of the cell population depends on the strength of the two noises, like the correlation strength between them. For the zero strength multiplicative noise, the population has the mode at around the conditional MSSCD value (Fig. 11). Therefore, the population sustains in this case and tries to stabilize at the conditional MSSCD value. For (D=0.02), there is a bimodality, where the highest mode is at the zero cell density. For (D=0.05), we observe only one mode at (x=0). Therefore, with the increasing values of the multiplicative noise strengths (D), the chance of extinction increases for (lambda =0.5), (Q=0.01), and other constant model parameters for the seeding condition 1. Similar things happen for increasing Q values considering (D=0.01), (lambda =0.5), and other constant model parameters (Fig. 12).Figure 11Distribution of cell density for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99), (delta =0.2), (lambda =0.5), (Q=0.01), and variable strength of multiplicative noise: (a) (D=0.05), (b) (D=0.02) and (c) (D=0).Full size imageFigure 12Distribution of cell density for (r_{p}=0.13), (K=1.43), (n=0.0095), (alpha =1.15), (beta =0.99), (delta =0.2), (lambda =0.5), (D=0.01), and variable correlation between multiplicative noise: (a) (Q=0.05), (b) (Q=0.02) and (c) (Q=0).Full size image Remark 5 We have previously discussed the scenario for (alpha =delta) for deterministic case in Remark 4. It is important to understand the scenario under stochastic case too. For (alpha =delta) the proposed stochastic model 3 becomes (frac{dx(t)}{dt}=r_{p}x(t)^{(alpha +1)}left( 1-big (frac{x(t)}{K}big )^{beta }right) - nx(t)^{(alpha +1)}-x(t)^{(alpha +1)} epsilon (t)+ Gamma (t)). For this stochastic model (g_{1}(x)=-x^{alpha +1}) and (g_{2}(x)=1). We get, (A(x)=r_{p}x^{alpha +1}left( 1-Big (frac{x}{K}Big )^{beta } right) -nx^{(alpha +1)}+D(alpha +1)x^{(2alpha +1)}-lambda sqrt{DQ}(alpha +1)x^{alpha }) and (B(x)=Dx^{(2alpha +2)}-2lambda sqrt{DQ}x^{(alpha +1)}+Q). The extrema of the SPDF (big (x(t)=x^{*}big )) must satisfy the growth equation (r_{p}{x^{*}}^{alpha +1}-frac{r_{p}}{K^{beta }}(x^{*})^{alpha +beta +1}-n(x^{*})^{alpha +1}-D(alpha +1)(x^{*})^{2alpha +1}+lambda sqrt{D~Q}(alpha +1)(x^{*})^{alpha }=0). Therefore, for (alpha =delta) the conditional MSSCD is (x^{*}= K-Kfrac{nK^{(alpha +1)}+D(alpha +1)K^{(2alpha +1)}-lambda sqrt{DQ}(alpha +1)K^{alpha }}{beta r_{p}K^{(alpha +1)}+nK^{(alpha +1)}(alpha +1)+D(alpha +1)(2alpha +1)K^{(2alpha +1)}-alpha lambda sqrt{DQ}(alpha +1)K^{alpha }}) under the condition ((r_{p}-n)(alpha +1)(x^{*})^{alpha }-frac{r_{p}}{K^{beta }}(alpha +beta +1)(x^{*})^{(alpha + beta )}-(alpha +1)(2alpha +1)D(x^{*})^{2alpha }+lambda sqrt{DQ}(alpha +1)alpha (x^{*})^{(alpha -1)} More

SubjectsThe study comprised 34 men (19–26 years, mean 23 years; body mass index 19–31 kg/m2, mean 24 kg/m2) with no physical disabilities or mental disorders, including color blindness and claustrophobia30,31. All subjects were informed of the aims, the study procedure, and the financial compensation for participation, and were asked to follow the rules of the study. Prior to each experiment, subjects underwent short-term starvation31,54 (18–20 h; no food except pure water after lunch (12–1 pm) or dinner (6–7 pm), no later than 1 pm or 7 pm, respectively, one the day before the test), no medical treatments, and normal sleep (at least 6 h, between 10 pm the day before the test day to 8 am on the test day)31. Prior to starting each experiment, subjects were stabilized on a chair for ~ 10 min in a room next to the testing room. Based on an assessment with a pre-experiment questionnaire and the first blood glucose level, measured before starting the experiment (see “Geomagnetic orientation assay” section below), subjects who had not followed these rules were not allowed to take the test on the day and the test was postponed. The study was approved by the Institutional Review Board of Kyungpook National University and all the procedures followed the regulations for human subject research. Informed consent was obtained from all subjects.Modulation of GMFThe ambient GMF in the testing room had a total intensity 45.0 μT, inclination 53°, and declination − 7° (Daegu city, Republic of Korea); the total intensity of 50.0 μT in our previous study31 was changed due to a reconstruction of the building; 45.0 μT was maintained throughout the period of this study. To provide the subjects with various GMF-like magnetic fields (i.e., by modulating of total intensity, inclination, or direction of magnetic north), the coil system from our previous studies6,7,31 was used. It comprised three double-wrapped, orthogonal, rectangular Helmholtz coils (1.890 × 1.890 m, 1.890 × 1.800 m, and 1.980 × 1.980 m for the north–south, east–west, and vertical axes, respectively), electrically-grounded with copper mesh shielding. The subject was seated on a rotatable plastic chair with no metal components, at the center of the three-dimensional coils with his head positioned in the middle space of the vertical axis of the coils. To modulate the geomagnetic north, each pair of coils was supplied with direct current from a power supply (MK3003P; MKPOWER, Republic of Korea). The magnetic field was measured using a 3-axis magnetometer (MGM 3AXIS; ALPHALAB, USA); the field homogeneity at the position of the subject’s head was found to be 95%. The testing room was shielded by a six-sided Faraday cage comprising 10 mm thick aluminum plates, and was grounded during the entire experiment40. Background electromagnetic noise was measured inside the coils at the start and the end of each experimental day. It was attenuated by the Faraday cage more than 200-fold over the range from 500 Hz to 100 MHz as described in detail in our previous study31. The 60 Hz power frequency magnetic field was no more than 2 nT (3D NF Analyzer NFA 1000; Gigahertz Solutions, Germany). All electronic devices were placed outside the Faraday cage during the experiments, with the exception of the switch button module for GMF modulation and the antenna for generating the oscillating magnetic fields. The temperature experienced by the subjects was maintained at 25 ± 0.5 °C (Data logger 98,581; MIC Meter Industrial, Taiwan) by air circulation through the honeycomb on the ceiling of the Faraday cage31.Geomagnetic orientation assayAdopting a two-alternative forced choice (2-AFC) paradigm33,34, a geomagnetic orientation assay was conducted similar to our previous study31. Experiments were performed at 09:30–11:30 am or 1:00–5:00 pm (local time, UTC + 09:00) (each experiment: 50 min–1 h 10 min; mean ≈ 1 h, which was shorter by approximately 30 min than that in the previous study: 1 h 20 min–1 h 40 min; mean ≈ 1 h 30 min). Depending on the experiment, starved or unstarved subjects were tested individually. Prior to each experiment, the subjects were asked to remain with their heads facing the front, with eyes closed and earmuffs on during the experiment. In particular, they were asked to concentrate on sensing, if they could, the ambient geomagnetic north during the association phase, and to use the sensed information, depending on the experiment, to orient toward one of the two modulated magnetic norths (0°/180° for magnetic north–south axis or 90°/270° for magnetic east–west axis, rotated clockwise with respect to the ambient geomagnetic north) during the test phase. Subjects were instructed to avoid distracting thoughts and to think immediately “which direction is modulated magnetic north?” whenever they were distracted during the test phase, or felt they were being biased by experiences from earlier experiments. While seated on the rotatable chair, the subject’s blood glucose level was measured shortly before the first session and immediately after each session with eyes open except in the ‘dark’ experiment (Accu-Chek Guide; Roche, Germany)31. If the determined value before the first session varied by more than 15% relative to the mean (Table S2)31, the experiment was postponed and repeated at a later date (approximately 2% of experiments). The subjects were stabilized with eyes closed for 2 min before the first trial in the absence of visual, auditory, olfactory, and haptic sensory cues. For the ‘dark’ experiment (light intensity ≈ 0 lx), subjects wore home-made ‘blind’ goggles and were stabilized with eyes closed for 5 min55,56, and then asked whether they could see any light. If they could, the goggles were adjusted to prevent leakage of light, and the subject then had another 5 min of stabilization with eyes closed before starting the experiment. The subjects were illuminated with light from a filtered/non-filtered diffused light-emitting diode, depending on the experiment (Table S1). The home-made filter goggles were worn throughout the experiment, including the association and test phase, when required. The goggles contained glass filters (Tae Young Optics, Republic of Korea) to provide the eyes with particular wavelengths of light (Spectrometer USB4000-UV-VIS, Ocean Optics, USA) (Fig. S1). Each experiment consisted of 16 sequential trials for ‘no-association’ and ‘food-association’. For the food-association, a subject facing toward the ambient geomagnetic north was gently provided with a chocolate chip31 on his right palm by an experimenter, and given 30 s to eat it, while during no-association trials, food was not provided during the association phase. After a subsequent 5 s interval, the experimenter gently touched the subject’s right thenar area using a paper rod, as the cue to start the test. One of the two modulated magnetic north directions, depending on the experiment, was randomly provided 3 s before the cue for the test. Each of the modulated magnetic north directions was provided eight times for the no-association and food-association sessions. Subjects were informed of the nearly equal probability for each of the modulated magnetic north directions before each experiment. With the touch cue, subjects were asked to rotate freely toward any direction (clockwise or counterclockwise) by themselves (1–4 cycles of two-thirds rotation) and try to sense the direction of the modulated magnetic north during a 1 min period. Rotation was allowed within the rotation angle (− 30° to 210° for the magnetic north–south axis or − 120° to 120° for the east–west axis, depending on experiments, with respect to the ambient magnetic north), which was confined by the plastic stool (Fig. 1A) touching the left or right ankle of the subjects. When subjects determined the direction of the magnetic north, they stopped rotating to face toward the direction and lifted their right hand to indicate the direction to the experimenter. The direction was measured by the experimenter at 10° intervals using the scale on the walls of the Faraday cage31. A prerequisite for correct orientation was that the subject indicated the direction within the range of 30° to the both sides with respect to the magnetic cardinal directions, which was instructed to the subjects before each experiment. When the direction the subjects indicated was out of the 30° range, the trial was not included in the data and was repeated (approximately 0.63% of trials). Before the subsequent trial, the subject was gently rotated to face toward the ambient geomagnetic north and then rested for 5 s. For the ‘dark’ experiment, subjects were asked whether they could see any leaked light immediately after the last measurement of blood glucose level at the end of experiment. If the subject could see leaked light, the experiment was nullified and repeated later on (approximately 3% of experiments; 2/68). All experiments were performed in a double-blind fashion. The experimenter who conducted the orientation assay knew whether a subject was starved or not, wearing filter goggles, and food-associated or not, but did not know the random magnetic north sequences that were controlled by the personal computer (PC) system. Another experimenter responsible for analyzing the data did not know whether the subject was starved or not, the experimental conditions, including light wavelengths, or whether an oscillating magnetic field had been provided to the subjects. Thus, none of the experimenters were aware of all the subject information and data during the experiments and data analysis. The correct orientation rate was calculated by (the number of correct orientation trials/total number of trials) (raw data, Appendix S3). All the subjects participated in all the experiments performed in random order with an interval of at least 3 days between experiments. After each experiment, the subjects were asked to answer a post-experiment questionnaire about whether they closed their eyes when required during the entire period of the experiment. In cases when a subject did not maintain closed eyes, the experiment was repeated (approximately 1% of experiments). For each subject, a preliminary experiment on the “magnetic north–south axis” was conducted twice (unstarved and starved for each) with no goggles for adaption to the experimental procedure. These data were not included in the results.Experiments with oscillating magnetic fieldsExperiments with oscillating magnetic fields were performed using the standard geomagnetic orientation assay described above. To produce the oscillating magnetic fields, oscillating currents from a function generator (AFG3021; Tektronix, USA. For each magnetic field, sweep of 500 ms; interval of 1 ms. See Fig. S6A) were amplified (ENI 2100L RF power amplifier; Bell Electronics, USA) and fed into a calibrated coil antenna (30 cm diameter, 6509 loop antenna; ETS-LINDGREN, USA) mounted on a wooden frame, comprised of a single winding of coaxial cable. The oscillating magnetic fields were measured daily, before the first and after the last experiment of the day, using a spectrum analyzer (SPA-921TG; Com-Power, USA) with a calibrated loop antenna (48 cm diameter, AL-130R; Com-Power, USA) and a calibrated magnetometer (Probe HF 3061, NBM-550; Narda, Germany). Magnetic field intensities were measured on the glabella of the subjects; variations in intensity between subjects due to different seating heights were less than 10% of the average values (Table S3). The function generator and amplifier were placed outside the Faraday cage, and switched on during the dummy load control experiments with no signal from the PC system. The band widths of the monochromatic magnetic fields, i.e., 1.260 and 1.890 MHz were 0.020 and 0.019 MHz (“average”, √3 kHz), respectively, at the bottoms of the peaks. During the test phase, the maximum values of magnetic noise on the glabella of subjects including the dummy load did not exceed the following values: (1) 5 Hz–9 kHz; 2 nT/√ 2 kHz of “average” and 8 nT/√ 9 kHz of “max-hold” (0.05 nT/√ 2 kHz of “average” and 5 nT/√ 9 kHz of “max-hold” in the dummy load) (3D NF Analyzer NFA 1000; Gigahertz Solutions, Germany); (2) 9 kHz–500 kHz; 5 nT/√ 3 kHz of “average” and 8 nT/√ 3 kHz of “max-hold” (≈ 0 nT/√ 3 kHz of “average” and ≈ 1 nT/√ 3 kHz of “max-hold” in the dummy load) (the AL-130R antenna) (Fig. S6C); and (3) 500 kHz–30 MHz; 0.006 nT of 3.780 MHz harmonic in the 1.260 MHz, 0.03 nT of 5.670 MHz harmonic in the 1.890 MHz, and ≈ 0 nT in the dummy load (/√ 10 kHz of “average”) (Fig. S6B), and 0.15 nT/√ 10 kHz of “max-hold” at the same frequencies above and ≈ 0 nT in the dummy load (the AL-130R antenna).Statistical analysisTo determine the significance of orientation data from the 2-AFC paradigm, a one-sample t-test (test mean: 0.5), paired sample t-test, or two-sample t-test was performed using Origin software 11 (Origin, USA). To verify the reasonability of the t-tests, all data sets were checked using the Anderson–Darling test if the data follow a normal distribution (Appendix S4). To evaluate the difference between the means of two data sets when at least one of them did not show a normal distribution, the percentile bootstrap method57 was used (95% confidence interval, see Fig. S2, Appendices S1 and S2 for raw data). To analyze the blood glucose data, a paired sample t-test was used. Based on the results of previous study31, to describe different response groups of magnetic orientation in the 2-AFC paradigm, a principal component analysis36,37 was conducted on correct orientation rates by starved subjects, with no association/food-association under the full wavelength or  > 400 nm light conditions using SPSS 23 (IBM, USA). Following the principal component analysis calculation, the k-means clustering algorithm—one of the unsupervised learning methods—was used to objectively classify the groups58. The number of groups was two, and the distance between the center of the cluster and all points was Euclidean distance. The classification boundary was marked with the perpendicular bisector from the centers of the two groups. The first two principal components accounted for a significant portion of the total variance (73.1%; PC1 = 40.8%, PC2 = 32.3%). Statistical values are presented as mean ± SEM. More

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