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Study areaWith the advanced observational techniques, abundant data accumulation, and ability to carry on multi-scale experiments, the Huailai Remote Sensing Station and around (for short Huailai Station), located in Huailai, Hebei province, China (40.349°N, 115.785°E), becomes one of the ideal study areas for the observation and validation of the LAI27. The Huailai Station is mainly covered by corn and some weeds. So, we mainly use LAIS to monitor the growth cycle of corn (in April 2015, we submitted an application for plant collection permission to Huailai Remote Sensing Station and obtained approval.)Huailai WSN vegetation monitoring system includes 6 sets of monitoring equipment, and its distribution is shown in Fig. 1 as follows, in which red dot represents LAIS Node, purple frame represents MODIS pixel, red frame represents observation area. The observation system is designed for the application of remote sensing pixel scale authenticity tests. The observation scale is a 1 km MODIS pixel on the pixel scale, and the actual coverage area is 2 km * 1.5 km. The six sets of equipment cover the core area of the test station and the surrounding typical growth plot, which is a good representative of the 1 km pixel scale.Figure 1Equipment distribution of WSN vegetation monitoring network in Huailai (red dot represents LAIS Node; purple frame represents the footprint of a MODIS pixel.Full size imageEach piece of equipment consists of two cameras which were only one camera with two different angles in previous work23 set up at a height of 2.5–4 m above the ground (Fig. 2), one for vertical downward observation and the other for inclined observation, which can take canopy photos regularly every day at its fixed position. The observation system obtained the photos of the corn canopy from May to August, but the corn did not grow in August. Therefore, in this study, we selected the photos taken by the vertical observation camera of the corn sample plot in the experimental station from May to July 2015.Figure 2The design of the LAIS node.Full size imageRelated work—data acquisitionData collection using LAISThe data collection complies with the plant guidelines statement: “LAI-2000 Plant Canopy Analyzer Instrution Manual” (Supplementary Information 2) (https://www.licor.com/env/, Last visit time: 21 October 2021). Existing facilities such as the high poles and the wireless sensor network in the experimental station have proved convenient for the installation of the LAI measurement system. LAIS uses the GEO001 digital serial camera that is suitable for a variety of embedded image acquisition modes. The specification of the camera includes: the total field of view is 120°, the maximum image size is 2176 * 1920 (approximately 5 million pixels), mounted at a height of 3 m, the spatial resolution at ground level is about 3 mm. The acquired image is simultaneously stored in a flash card in two formats: the JPEG format merits in less file size thus suitable for quick wireless transfer; the RAW format, which is the user data in our analysis, contains 3 channel binary image in 10 bits bit-depth. Compared to our previous work, an important new feature of this camera is the programmable cut-off filter. As we know, unlike scientific sensor which has the precise spectral response to each band, the digital camera is cheap and can only acquire the so-called RGB image. Usual digital cameras have one NIR cut-off filter to exclude the near-infrared light. The GEO001 camera, which was a commercial camera produced by Zhongshan Yunteng Photographic Equipment Co., Ltd, has two cut-off filters: one is the NIR cut-off filter, another is a blue cut-off filter. Switching on the NIR cut-off filter results in an ordinary color image as in a usual household digital camera. While the blue cut-off filter is switched on and NIR cut-off filter is switched off, near-infrared light is allowed to reach the detector array and blue light is blocked, resulting in false-color images as in Fig. 3b. Adding near-infrared light can increase illumination in the shadow area, and blocking blue light can alleviate the disturbance of sun glint, so, switching to a blue cut-off filter helps to improve the image quality when the direct sunlight is strong such as around noon time.Figure 3Three images on July 2 of site 1: (a) and (c) are true-color images obtained at 05:31 a.m. and 6:32 p.m., and (b) is a false-color image when the blue filter is removed at 1:28 p.m.Full size imageTo acquire an image in the best illumination condition and avoid the influence of rain or other unsuitable weather, the image acquisition device based on WSN was set up to acquire images three times per day: 5:30 a.m., 1:30 p.m., and 6:30 p.m. According to our experience, when the canopy is open (sparse vegetation), usually images acquired at 6:30 p.m. are the best for classification because the direct sunlight is weak; when the canopy is closed (dense vegetation), the illumination on the soil background is very poor in all time, and classification is difficult. So, the camera is programmed to switch to a blue cut-off filter when acquiring images at 1:30 p.m., while the images acquired at other times were with NIR cut-off filter, resulting in true color images, as shown in Fig. 3.LAILLW data and LAI2000 dataTo evaluate the accuracy of the improved finite length averaging method proposed in this study, a field experiment was carried out to measure LAI by manual sampling (Supplementary Information 3,4). A field sampling scheme covering the corn growing season (late May to early July) was designed (Supplementary Information 1). The LAI of corn in the experimental area was measured by the quadrat harvesting method, and the validation data of LAI of corn in each growth period were obtained. Considering the rapid growth of the corn, the sampling experiment period was set as 1 week, but due to the actual work in summer and the influence of rainfall, six effective measurements were carried out in the field experiment: May 30, June 7, June 13, June 20, July 4 and July 16.The LAILLW method, which is also known as the shape factor method, involves outdoor and indoor measurements. The formulas are:$${text{L}} = {text{S}}*{text{N}}$$
(1)
$${text{f}} = {{text{S}} /{left( {sumlimits_{i = 1}^m {{text{len}}*{text{wid}}} } right)}}$$
(2)

where L represents the leaf area index, S refers to the area of a single plant, and N refers to the number of plants in a unit area. The shape factor ƒ is the ratio of the S to the value multiplied by the length and width of all leaves in the plant.To reduce measurement errors, 10 plants were selected in the sample, and the length and width of each leaf on each corn were recorded with a ruler. To obtain the shape factor, representative corn plants were cut next to the sample (not in the image coverage area) and the true area of each leaf was obtained by software, and the shape factor was derived from this23. Through the length and width of 10 strains measured in the field, and the shape factor obtained, the total leaf area of 10 corns can be calculated, and the average leaf area of one plant is finally obtained. The LAI value under the LAILLW method is obtained.Using the difference between the solar radiation values of the upper and lower canopies, the LAI2000 canopy analyzer can obtain LAI and set up a corresponding point folder to save the measured data for subsequent collation. 10 measurement points were selected for each site, and the average value was the final result for each site. To reduce the effects of the solar altitude angle on measurement accuracy, the experiments were repeated every two hours.To make it easier to record the date of data acquisition, the data were summarized in the order day of the year (DOY). For example, 30 May 2015 is the 150th day in the year and its DOY is 150. The DOY information of data acquisition using the LAILLW method and LAI2000 is specifically shown in Table 1.Table 1 The DOY information of data acquisition using the LAILLW and LAI2000.Full size tableMODIS LAI dataMODIS leaf area index data was downloaded from the United States Geological Survey (https://modis.gsfc.nasa.gov/data/dataprod/mod15.php), named MCD15A2Hv006. It is an 8-day composite dataset with a 500-m pixel size. The algorithm chooses the best pixel available from all the acquisitions of both MODIS sensors located on NASA’s Terra and Aqua satellites from within the 8 days.In the comparison of MODIS LAI data, as the pixel of the satellite product is in 500 m resolution, it is not recommended to directly compare single node LAIS measurement with the MODIS LAI product because of the scale mismatch. Though complicated upscaling approaches have been discussed and implemented in Huailai station for other parameters28, it is not the purpose of this study So, we simply averaged the LAI in all the LAIS nodes to compare to the average MODIS LAI product in the 3 * 3 nearest pixels (1.5 km * 1.5 km), referred to as MODIS LAI_Mean in a later context, which approximately covers the area of all LAIS nodes. Time matching was carried out by selecting the date of the MODIS product closest to the date of the handheld LAI2000 measurement. The following Table 2 is obtained by taking 3 * 3 pixels closest to the LAIS Nodes.Table 2 MODIS leaf area index of 3 * 3 pixels around Huailai experimental station.Full size tableImproved LAIS methodsIn previous work, we have deployed sensors and cameras, and also have an automatic image processing and preliminary method of calculating LAI23. Figure 4 is a flow chart of our work. The previous articles focused on hardware and system implementation but did not pay much attention to performance. On this basis, we upgrade the image classification method and LAI calculation method, which will be explained in detail below.Figure 4Flow chart of leaf area index measurement system based on WSN.Full size imageImage preprocessing and classification methodsBecause of weather-related factors such as water vapor and dust or inaccurate exposure, a small number of the photographs are not clear. Besides, some of the image data cannot be decoded because of unstable communications and other factors. Therefore, it is necessary to check and select the photographs that meet the processing requirements before binary image processing. Currently, the selection process is carried out by human visual inspection based on the following principles: (1) when the canopy is open (sparse vegetation), the image at 6:30 p.m. is preferred, when the vegetation the canopy is closed (sparse vegetation), the image at 1:30 p.m. is preferred; (2) if the preferred image is not clear, other clear image acquired on the same day should be used; if all the images are not clear, then this day is marked as a failure.If we decided to use the image acquired at 1:30 p.m. It is also necessary to convert it from a false-color image to a true-color-like image (as shown in Fig. 3b) in which the leaves are shown in green color. The conversion is carried out by multiplying the vector of DN (digital number) of 3 bands with a coefficient matrix which is provided by the camera manufacturer. Another preprocessing is to choose the near nadir-view area of the image for further processing. As the off-nadir-view area of the image is subject to large geometric distortion as well as saturation of fraction of vegetation cover (FVC), they are not used in this study. The images are clipped to an ROI (region of interest) of about 2 * 2 square meters in ground area, with a maximum view zenith angle less than 30°.The study of the color spatial distributions of the crop images is helpful for the classification of the images and extraction of the image information. The color of the image pixel is the most direct and effective element that can be used to describe the image29. Because the red–green–blue (RGB) color space has the characteristic of a clear and convenient expression of information. When corn leaves are small, the crops in the fields are sparse, and most of them are soil background in the images. The soil in a lower hue is similar to the corn in terms of R and B components, while it has an overlap with the corn in G components when soil is in a higher hue. This makes it difficult to classify sparse corn scenes only by RGB space, so it is necessary to consider the characteristics of hue, luminosity, and saturation (HLS) spatial components.Statistical analysis showed that the component values of the crop leave in the RGB color space were in the ranges of G  > R and G  > B while the corresponding values for the soil follow the law that B  More

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Stochastic age-specific transmission modelWe formulate a stochastic age-specific transmission model in the general Susceptible(S)-Exposed(E)-Reported(I)-Unreported(U)-Recovered(R) framework. For a particular age group (i) at time (t-1) ((i=1) corresponding to the 0–17 years, (i=2) to 18–44, (i=3) to 45–64 and (i=4) to 65+), we have$$begin{array}{l}{S}_{i}(t)= {S}_{i}(t-1)-{n}_{S{E}_{i}}(t)\ {E}_{i}(t)= {E}_{i}(t-1)+{n}_{S{E}_{i}}(t)-\ {n}_{E{I}_{i}}(t)-{n}_{E{U}_{i}}(t)\ {I}_{i}(t)= {I}_{i}(t-1)+{n}_{E{I}_{i}}(t)-{n}_{I{R}_{i}}(t)\ {U}_{i}(t)= {U}_{i}(t-1)+{n}_{E{U}_{i}}(t)-{n}_{U{R}_{i}}(t)\ {R}_{i}(t)= {R}_{i}(t-1)+{n}_{I{R}_{i}}(t)+{n}_{U{R}_{i}}(t),end{array}$$
(1)
where ({n}_{{XY}_{i}}(t)) represents number of transitions between a class X and class Y for age group (i) at time (t).The number of transitions from the susceptible to exposed class for group (i) at time (t) is modelled by$$begin{aligned}{n}_{S{E}_{i}}(t)&sim Poi({S}_{i}(t-1)times {gamma }_{i}(t)times \ & quad sum_{j=1}beta (t)times {c}_{j,i}(t)times {{I}_{j}(t-1)+{U}_{j}(t-1)}).end{aligned}$$
(2)
Here, (beta (t)) denotes the average infectiousness of an infectious individual and ({c}_{j,i}(t)) is the average number of contacts per day made by age group (j) to (i). Also note that the product (beta (t)times {c}_{j,i}(t)) may represent age-specific transmissibility (of age group (j)) accounting for contacts. We allow and infer two change points of (beta (t)) (one potentially correlates to changes due to the implementation of lockdown and another one to changes due to the lifting of lockdown), i.e.,$$beta left(tright)=left{begin{array}{ll}{beta }_{0},&quad if; tle {T}_{1}\ {beta }_{1}={omega }_{1}times {beta }_{0},&quad if ;{T}_{1}{T}_{2},end{array}right.$$
(3)
where ({T}_{1}) and ({T}_{2}) are the two change points to be inferred (({T}_{2}ge {T}_{1})). ({gamma }_{i}(t)) denotes the susceptibility of group (i) relative to the oldest age group (i.e., ({gamma }_{4}=1)), which is also allowed to change proportionally after lifting the lockdown. Note that ({gamma }_{i}(t)) implicitly incorporates any behavioral effects (e.g., potential reduction of risk of getting infection due to facemask wearing). Transitions between other classes are modelled as:$$begin{aligned}{n}_{E{U}_{i}}(t)sim & Bin({n}_{S{E}_{i}}(t-{D}_{EU}),{p}_{{U}_{i}}(t-{D}_{EU}))\ {n}_{E{I}_{i}}(t)=& {n}_{S{E}_{i}}(t-{D}_{EI})-{n}_{E{U}_{i}}(t)\ {n}_{I{R}_{i}}(t)=& {n}_{E{I}_{i}}(t-{D}_{IR})\ {n}_{U{R}_{i}}(t)=& {n}_{E{U}_{i}}(t-{D}_{UR}),end{aligned}$$
(4)
where ({D}_{EI}), ({D}_{EU}), ({D}_{IR}) and ({D}_{UR}) denote the mean waiting times between the indicated two classes. We assume that ({D}_{EI})= ({D}_{EU})=7 days and ({D}_{IR})= ({D}_{UR})=14 days. ({p}_{{U}_{i}}(t)) represents probability that an infection is unreported at times (t) for age group (i), we assume$${p}_{{U}_{i}}(t)=1-frac{{e}^{{f}_{i}(t)}}{1+{e}^{{f}_{i}(t)}}.$$
(5)
({f}_{i}(.)) is an increasing function with ({f}_{i}(t)={a}_{i}+{b}_{i}times t), where (-infty More