A method and system for inversion of crop leaf area index

Abstract

The present invention relates to a crop leaf area index inversion method and system. The three-dimensional structure of crops in different growth periods is simulated by a crop growth model, and the canopy reflectivity under different planting parameters is simulated by a three-dimensional radiation transmission model. A neural network is trained by constructing a crop canopy reflectivity simulation data set. At the same time, the high spatial resolution crop leaf area index is inverted in combination with satellite remote sensing data. The method can effectively reduce the simulation time of the three-dimensional radiation transmission model, improve the inversion accuracy of the crop leaf area index, and realize near real-time monitoring of crop growth.

Description

Technical Field

[0001] The present invention relates to the technical field of crop monitoring, and in particular to a crop leaf area index inversion method and system. Background Art

[0002] With the increase of global population and per capita demand, the global demand for crops is increasing. However, with the increase of global average annual surface temperature and the increase of frequency and intensity of extreme weather events, problems such as water shortage and pest outbreak caused by climate change have a great impact on regional and global food security. In recent decades, remote sensing observation has been gradually applied to agricultural production as a fast and efficient means of crop monitoring, and has been widely used in breeding, pest and disease detection and other aspects.

[0003] Leaf area index (LAI) is defined as the sum of the single-surface areas of vegetation leaves per unit land surface. It is closely related to the growth status and ecological processes of crops and is an important parameter of crop canopy structure. Leaf area index products inverted by remote sensing play an important role in crop growth monitoring, crop yield estimation and growth models.

[0004] Methods for inverting crop canopy leaf area index using remote sensing data include statistical methods and physical methods. Statistical methods estimate parameters based on the statistical relationship between spectral characteristics and variables, but are often limited by observation conditions, locations, and vegetation types, and only use a small amount of band information. Physical methods estimate vegetation canopy parameters by establishing a physical model between various state variables of leaves and canopies and canopy reflectivity. Among them, the one-dimensional radiation transfer model simplifies the canopy into a horizontally uniform scene, and the geometric optical model simplifies the canopy into a simple geometric shape. However, the simple assumptions of the canopy structure in the one-dimensional model or the geometric optical model will lead to the loss of some information and cannot well reflect the true structure of many canopies, limiting the accuracy of parameter inversion.

[0005] Therefore, there is an urgent need in this field for a technical solution to accurately invert the leaf area index in the canopy structure of crops. Summary of the invention

[0006] The purpose of the present invention is to provide a crop leaf area index inversion method and system, which utilizes a three-dimensional radiation transfer model to simulate a variety of data, and effectively solves the problem of low leaf area index inversion accuracy in the prior art.

[0007] To achieve the above object, the present invention provides the following solutions:

[0008] A method for inverting leaf area index of crops, the method comprising:

[0009] Use crop growth models to generate three-dimensional canopy structures of crops at different growth stages;

[0010] Determine the leaf reflectance and transmittance of crops;

[0011] For each of the three-dimensional canopy structures of the crops, according to the corresponding leaf reflectivity and transmittance, the LESS three-dimensional radiation transmission model is used to calculate the canopy reflectivity of the leaves under different sun and observation angles;

[0012] Construct a canopy reflectance simulation data set; the canopy reflectance simulation data set includes multiple data sets Among them, A i represents the leaf area index in the three-dimensional canopy structure of the i-th crop in the canopy reflectance simulation data set, Represents the three-dimensional canopy structure of the i-th crop under the j-th sun and observation angle α j The corresponding canopy reflectance below;

[0013] The canopy reflectance simulation data set is divided into a training set and a validation set in a preset ratio, and a back-propagation neural network is trained to obtain a neural network model; the input of the back-propagation neural network is the canopy reflectance, the sun and the observation angle, and the output is the leaf area index in the three-dimensional canopy structure of the crop;

[0014] The reflectivity and angle data of the corresponding band in the remote sensing data to be inverted are input into the neural network model to obtain the corresponding leaf area index.

[0015] In some embodiments, determining the reflectivity and transmittance of the leaves of the crop specifically includes:

[0016] The reflectivity and transmittance of crop leaves are determined based on the absorption coefficient of crop leaves and the PROSPECT leaf radiation transmission model.

[0017] In some embodiments, the calculation formula of the absorption coefficient of the crop leaves is:

[0018] K=∑k ab (λ)C ab +k w (λ)C w +k dm (λ)C dm ;

[0019] Among them, k ab (λ), k w (λ) and k dm (λ) represents the absorption coefficients of leaf chlorophyll, water and dry matter respectively; C ab , C w and C dm represent leaf chlorophyll content, leaf water content and leaf dry matter content, respectively.

[0020] In some embodiments, the LESS three-dimensional radiation transmission model is used to calculate the canopy reflectivity of the leaves under different sun and observation angles according to the corresponding leaf reflectivity and transmittance, specifically including:

[0021] The leaf reflectance and canopy reflectance under the same sun and observation angle are fitted to obtain the fitting function;

[0022] Based on the fitting function, the canopy reflectivity corresponding to the leaf reflectivity under the sun and observation angle is determined.

[0023] In some embodiments,

[0024] The SmoothingSpline function is used to fit the leaf reflectivity and canopy reflectivity under the same sun and observation angle.

[0025] In some embodiments,

[0026] The preset ratio is: 70% for training set and 30% for validation set.

[0027] In another aspect, the present invention further provides a crop leaf area index inversion system, the system comprising:

[0028] A canopy structure generation module is used to generate three-dimensional canopy structures of crops at different growth stages using crop growth models;

[0029] Leaf reflectance and transmittance calculation module, used to determine the leaf reflectance and transmittance of crops;

[0030] A canopy reflectance calculation module is used to calculate the canopy reflectance of the leaves under different sun and observation angles using the LESS three-dimensional radiation transmission model for each of the three-dimensional canopy structures of the crop according to the corresponding leaf reflectance and transmittance;

[0031] The data set construction module is used to construct a canopy reflectance simulation data set; the canopy reflectance simulation data set includes multiple data sets Among them, A i represents the leaf area index in the three-dimensional canopy structure of the i-th crop in the canopy reflectance simulation data set, Represents the three-dimensional canopy structure of the i-th crop under the j-th sun and observation angle α j The corresponding canopy reflectance below;

[0032] A model training module is used to divide the canopy reflectance simulation data set into a training set and a validation set in a preset ratio, train a back-propagation neural network, and obtain a neural network model; the input of the back-propagation neural network is the canopy reflectance, the sun and the observation angle, and the output is the leaf area index in the three-dimensional canopy structure of the crop;

[0033] The inversion module is used to input the reflectivity and angle data of the corresponding band in the remote sensing data to be inverted into the neural network model to obtain the corresponding leaf area index.

[0034] In some embodiments, determining the reflectivity and transmittance of the leaves of the crop specifically includes:

[0035] The reflectivity and transmittance of crop leaves are determined based on the absorption coefficient of crop leaves and the PROSPECT leaf radiation transmission model.

[0036] In some embodiments, it is characterized in that the calculation formula of the absorption coefficient of the crop leaves is:

[0037] K=∑k ab (λ)C ab +k w (λ)C w +k dm (λ)C dm ;

[0038] Among them, k ab (λ), k w (λ) and k dm (λ) represents the absorption coefficients of leaf chlorophyll, water and dry matter respectively; C ab , C w and C dm represent leaf chlorophyll content, leaf water content and leaf dry matter content, respectively.

[0039] In some embodiments, the LESS three-dimensional radiation transmission model is used to calculate the canopy reflectivity of the leaves under different sun and observation angles according to the corresponding leaf reflectivity and transmittance, specifically including:

[0040] The leaf reflectance and canopy reflectance under the same sun and observation angle are fitted to obtain the fitting function;

[0041] Based on the fitting function, the canopy reflectivity corresponding to the leaf reflectivity under the sun and observation angle is determined.

[0042] According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0043] The present invention provides a crop leaf area index inversion method and system, which simulates the three-dimensional structure of crops in different growth periods through a crop growth model, and simulates the canopy reflectivity under different planting parameters using a three-dimensional radiation transmission model. A neural network is trained by constructing a canopy reflectivity simulation data set. At the same time, combined with satellite remote sensing data, a high spatial resolution crop leaf area index is inverted. The method can effectively reduce the simulation time of the three-dimensional radiation transmission model and improve the inversion accuracy of the crop leaf area index. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0045] Figure 1 This is a flow chart of the crop leaf area index inversion method provided in Example 1 of the present invention.

[0046] Figure 2 This is a block diagram of a crop leaf area index inversion system provided in Embodiment 2 of the present invention.

[0047] Figure 3 This is a flow chart of the crop leaf area index inversion method provided in Example 3 of the present invention.

[0048] Figure 4 This is a schematic diagram of a typical three-dimensional corn canopy structure provided in Example 3 of the present invention.

[0049] Figure 5 Schematic diagram of the monotonically increasing functional relationship between canopy reflectivity and leaf reflectivity in different scenes provided by the third embodiment of the present invention

[0050] Figure 6 A schematic diagram showing the comparison between the model inversion results provided in the third embodiment of the present invention and the ground measured values. DETAILED DESCRIPTION

[0051] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0052] The purpose of the present invention is to provide a crop leaf area index inversion method and system, which utilizes a three-dimensional radiation transfer model to simulate a variety of data, and effectively solves the problem of low leaf area index inversion accuracy in the prior art.

[0053] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0054] Embodiment 1:

[0055] like Figure 1 As shown, this embodiment provides a crop leaf area index inversion method, the method comprising:

[0056] S1. Generate three-dimensional canopy structures of crops at different growth stages using crop growth models.

[0057] S2. Determine the reflectivity and transmittance of crop leaves.

[0058] In this embodiment, many groups of different leaf reflectance and transmittance are generated during the simulation and applied to the canopy in each growth period.

[0059] Specifically, the relationship between crop canopy and leaf biochemical parameters (chlorophyll, water and dry matter content) is determined by the total leaf absorption coefficient (K). The calculation formula of K is:

[0060] K=∑k ab (λ)C ab +k w (λ)C w +k dm (λ)C dm

[0061] Among them, k ab (λ), k w (λ) and k dm (λ) represents the absorption coefficients of leaf chlorophyll, water and dry matter respectively; C ab , C w and C dm They represent leaf chlorophyll content, leaf water content and leaf dry matter content respectively. By inputting K into the PROSPECT leaf radiation transfer model, the leaf reflectance and transmittance corresponding to the input leaf biochemical parameters can be obtained. Since the total leaf absorption coefficient (K) has determined the relationship between the crop canopy and the leaf biochemical parameters, the leaf reflectance and transmittance of the crop can be determined here.

[0062] S3. For each of the three-dimensional canopy structures of the crops, according to the corresponding leaf reflectivity and transmittance, the LESS three-dimensional radiation transfer model is used to calculate the canopy reflectivity under different sun and observation angles, and a canopy reflectivity simulation data set is constructed.

[0063] Specifically include:

[0064] For each crop three-dimensional canopy structure, according to its corresponding leaf reflectivity and transmittance, under given sun and observation angle, the LESS three-dimensional radiation transfer model is input to calculate the leaf reflectivity [RL1,…,RL n ] canopy reflectance [RC1,…,RC n ].

[0065] The leaf reflectivity and canopy reflectivity under the same sun and observation angle are fitted using the Smoothing Spline function to obtain a fitting function; based on the fitting function, the canopy reflectivity corresponding to the leaf reflectivity under the sun and observation angle is determined.

[0066] Then, by simulating the canopy reflectivity corresponding to different canopy structures under different suns and observation angles, a canopy reflectivity simulation data set was constructed.

[0067] As an optional embodiment, based on the Smoothing Spline function, a monotonically increasing functional relationship between the canopy reflectivity and the leaf reflectivity is fitted, thereby obtaining the canopy reflectivity corresponding to any leaf reflectivity under a given canopy structure, sun, and observation angle, so as to simulate the canopy reflectivity corresponding to different canopy structures under different suns and observation angles.

[0068] Construct a canopy reflectance simulation data set, the canopy reflectance simulation data set includes multiple data sets Among them, A i represents the leaf area index in the three-dimensional canopy structure of the i-th crop in the canopy reflectance simulation data set, Represents the three-dimensional canopy structure of the i-th crop under the j-th sun and observation angle α j The corresponding canopy reflectance.

[0069] S4. Divide the canopy reflectance simulation data set into a training set and a validation set in a preset ratio, train a back-propagation neural network, and obtain a neural network model; the input of the back-propagation neural network is the canopy reflectance, the sun and the observation angle, and the output is the leaf area index in the three-dimensional canopy structure of the crop.

[0070] The neural network in this embodiment includes an input layer, a hidden layer and 5 neurons. The input data of the neural network are canopy reflectance, sun angle and observation angle in different bands, and the output data is leaf area index. The leaf area index is the sum of the single-sided area of ​​vegetation leaves per unit area, and is an important parameter for describing the canopy structure.

[0071] In this embodiment, the selected preset ratio is: 70% for the training set and 30% for the validation set.

[0072] S5, the corresponding bands (band 1, ..., band n ) is input into the neural network model to obtain the corresponding leaf area index. During the inversion, the number of bands required is not fixed. Only three bands of red, green and blue can be used, or five or even ten bands of red, green, blue, near infrared, short-wave infrared, etc. can be used.

[0073] This embodiment provides a method for calculating the crop leaf area index based on a three-dimensional radiation transfer model. The three-dimensional structure of crops in different growth periods is simulated by a crop growth model, and the canopy reflectivity under different planting parameters is simulated using a three-dimensional radiation transfer model. The neural network is trained by constructing a crop canopy reflectivity simulation data set. At the same time, the high spatial resolution crop leaf area index is inverted in combination with satellite remote sensing data. This method can effectively reduce the simulation time of the three-dimensional radiation transfer model, improve the inversion accuracy of the crop leaf area index, and realize near real-time monitoring of crop growth.

[0074] Embodiment 2:

[0075] like Figure 2 As shown, a crop leaf area index inversion system comprises:

[0076] The canopy structure generation module M1 is used to generate three-dimensional canopy structures of crops in different growth stages using crop growth models.

[0077] The leaf reflectivity and transmittance calculation module M2 is used to determine the leaf reflectivity and transmittance of crops.

[0078] The canopy reflectivity calculation module M3 is used to calculate the canopy reflectivity of the leaves under different sun and observation angles for each three-dimensional canopy structure of the crop according to its corresponding leaf reflectivity and transmittance, using the LESS three-dimensional radiation transmission model.

[0079] The data set construction module M4 is used to construct a canopy reflectance simulation data set; the canopy reflectance simulation data set includes multiple data sets Among them, A irepresents the leaf area index in the three-dimensional canopy structure of the i-th crop in the canopy reflectance simulation data set, Represents the three-dimensional canopy structure of the i-th crop under the j-th sun and observation angle α j The corresponding canopy reflectance.

[0080] The model training module M5 is used to divide the canopy reflectance simulation data set into a training set and a validation set in a preset ratio, train a back-propagation neural network, and obtain a neural network model; the input of the back-propagation neural network is the canopy reflectance, the sun and the observation angle, and the output is the leaf area index in the three-dimensional canopy structure of the crop.

[0081] The inversion module M6 is used to input the reflectivity and angle data of the corresponding band in the remote sensing data to be inverted into the neural network model to obtain the corresponding leaf area index.

[0082] Among them, the blade reflectivity and transmittance calculation module M2 is specifically used for:

[0083] The reflectivity and transmittance of crop leaves are determined based on the absorption coefficient of crop leaves and the PROSPECT leaf radiation transmission model.

[0084] The calculation formula of the absorption coefficient of the crop leaves is:

[0085] K=∑k ab (λ)C ab +k w (λ)C w +k dm (λ)C dm ;

[0086] Among them, k ab (λ), k w (λ) and k dm (λ) represents the absorption coefficients of leaf chlorophyll, water and dry matter respectively; C ab , C w and C dm represent leaf chlorophyll content, leaf water content and leaf dry matter content, respectively.

[0087] According to the corresponding leaf reflectivity and transmittance, the LESS three-dimensional radiation transmission model is used to calculate the canopy reflectivity under different sun and observation angles, and to construct a canopy reflectivity simulation data set, which specifically includes:

[0088] The leaf reflectance and canopy reflectance under the same sun and observation angle are fitted to obtain the fitting function;

[0089] Based on the fitting function, the canopy reflectivity corresponding to the leaf reflectivity under the sun and observation angle is determined.

[0090] Embodiment three:

[0091] like Figure 3 As shown, this embodiment provides a method for inverting the leaf area index of crops. The research object of this embodiment is the inversion of the leaf area index of corn based on Sentinel-2 satellite data, and its specific steps include:

[0092] A1. Determine the three-dimensional structure of corn at different growth stages

[0093] Based on the López-Lozano maize 3D model, different maize crop phenotypic parameters and field planting parameters (Table 1) were input to generate 650 maize canopy 3D structures, such as Figure 4 shown.

[0094] Table 1 Phenotypic parameters of corn crops and field planting parameters

[0095]

[0096] A2. Constructing a simulated corn canopy reflectance dataset

[0097] The total absorption coefficients of six leaves, K1 = 0.0045, K2 = 0.05, K3 = 0.12, K4 = 0.28, K5 = 0.46, and K6 = 1.09, were input into the PROSPECT model to obtain the corresponding leaf reflectance and transmittance. For each scene, the leaf reflectance and transmittance were input, and the corresponding canopy reflectance was simulated using the three-dimensional radiation transfer model LESS. Based on the SmoothingSpline function, the functional relationship between the canopy reflectance and the leaf reflectance, which increases monotonically, is fitted. The functional relationship is as follows: Figure 5 As shown. For 650 scenes, different sun angles were simulated, and the canopy reflectance under different leaf optical properties was calculated (see Table 2), and a simulated corn canopy reflectance data set was constructed. The data set includes 650 scenes multiplied by 1875 leaf and soil reflectance combinations, resulting in 33750 simulated data. Each data contains canopy reflectance data at 8 bands corresponding to Sentinel-2 data, sun angle data, and leaf area index corresponding to the scene.

[0098] Table 2 Sun angle and blade optical property parameters

[0099]

[0100] A3. Training the Neural Network

[0101] 70% of the data were randomly selected as the training data set, and the remaining 30% of the data were used as the validation data set to train the back-propagation neural network. The input data were the sun angle and observation angle, as well as the canopy reflectance data at 8 bands corresponding to the Sentinel-2 data, and the output data was the leaf area index.

[0102] A4. Inversion of Maize Leaf Area Index

[0103] The 10-meter reflectivity data of Sentinel-2 satellite from May to August 2018 (band 3, band 4, band 5, band 6, band 7, band 8a, band 11 and band 12) were input into the trained neural network to obtain the inversion result corn leaf area index, which was then verified using the ground measured values. The verification results are as follows: Figure 6 shown.

[0104] In this specification, each embodiment is described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the embodiments can be referred to each other. For the system disclosed in the embodiment, since it corresponds to the method disclosed in the embodiment, the description is relatively simple, and the relevant parts can be referred to the method part.

[0105] The principles and implementation methods of the present invention are described in this article using specific examples. The description of the above embodiments is only used to help understand the method and core idea of ​​the present invention. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting the present invention.

Claims

1. A method for inverting leaf area index of crops. It is characterized in that The method comprises: Use crop growth models to generate three-dimensional canopy structures of crops at different growth stages; Determine the leaf reflectance and transmittance of crops; For each of the three-dimensional canopy structures of the crops, according to the corresponding leaf reflectivity and transmittance, the LESS three-dimensional radiation transmission model is used to calculate the canopy reflectivity of the leaves under different sun and observation angles; Construct a canopy reflectance simulation data set; the canopy reflectance simulation data set includes multiple data sets Among them, A i represents the leaf area index in the three-dimensional canopy structure of the i-th crop in the canopy reflectance simulation data set, Represents the three-dimensional canopy structure of the i-th crop under the j-th sun and observation angle α j The corresponding canopy reflectance below; The canopy reflectance simulation data set is divided into a training set and a validation set in a preset ratio, and a back-propagation neural network is trained to obtain a neural network model; the input of the back-propagation neural network is the canopy reflectance, the sun and the observation angle, and the output is the leaf area index in the three-dimensional canopy structure of the crop; Inputting the reflectivity and angle data of the corresponding band in the remote sensing data to be inverted into the neural network model to obtain the corresponding leaf area index; The determining of the reflectivity and transmittance of the leaves of the crop specifically includes: According to the absorption coefficient of crop leaves corresponding to the three-dimensional canopy structure of each crop and the PROSPECT leaf radiation transmission model, the reflectivity and transmittance of crop leaves are determined; The calculation formula of the absorption coefficient of the crop leaves is: K=Σk ab (λ)C ab +k w (λ)C w +k dm (λ)C dm ; Among them, k ab (λ), k w (λ) and k dm (λ) represents the absorption coefficients of leaf chlorophyll, water and dry matter respectively; C ab , C w and C dm represent leaf chlorophyll content, leaf water content and leaf dry matter content respectively; The LESS three-dimensional radiation transmission model is used to calculate the canopy reflectivity of the leaves under different sun and observation angles according to the corresponding leaf reflectivity and transmittance, specifically including: The Smoothing Spline function is used to fit the leaf reflectance and canopy reflectance under the same sun and observation angle to obtain the fitting function; Based on the fitting function, the canopy reflectivity corresponding to the leaf reflectivity under the sun and observation angle is determined.

2. The crop leaf area index inversion method according to claim 1, It is characterized in that The preset ratio is: 70% for training set and 30% for validation set.

3. A crop leaf area index inversion system, It is characterized in that The system comprises: A canopy structure generation module is used to generate three-dimensional canopy structures of crops at different growth stages using crop growth models; Leaf reflectance and transmittance calculation module, used to determine the leaf reflectance and transmittance of crops; The determining of the reflectivity and transmittance of the leaves of the crop specifically includes: Determine the reflectivity and transmittance of crop leaves based on the absorption coefficient of crop leaves and the PROSPECT leaf radiation transmission model; The calculation formula of the absorption coefficient of the crop leaves is: K=Σk ab (λ)C ab +k w (λ)C w +k dm (λ)C dm ; Among them, k ab (λ), k w (λ) and k dm (λ) represents the absorption coefficients of leaf chlorophyll, water and dry matter respectively; C ab , C w and C dm represent leaf chlorophyll content, leaf water content and leaf dry matter content respectively; A canopy reflectance calculation module is used to calculate the canopy reflectance of the leaves under different sun and observation angles using the LESS three-dimensional radiation transmission model for each of the three-dimensional canopy structures of the crop according to the corresponding leaf reflectance and transmittance; The data set construction module is used to construct a canopy reflectance simulation data set; the canopy reflectance simulation data set includes multiple data sets Among them, A i represents the leaf area index in the three-dimensional canopy structure of the i-th crop in the canopy reflectance simulation data set, Represents the three-dimensional canopy structure of the i-th crop under the j-th sun and observation angle α j The corresponding canopy reflectance below; The LESS three-dimensional radiation transmission model is used to calculate the canopy reflectivity of the leaves under different sun and observation angles according to the corresponding leaf reflectivity and transmittance, specifically including: The Smoothing Spline function is used to fit the leaf reflectance and canopy reflectance under the same sun and observation angle to obtain the fitting function; Based on the fitting function, determining the canopy reflectivity corresponding to the leaf reflectivity under the sun and observation angle; A model training module is used to divide the canopy reflectance simulation data set into a training set and a validation set in a preset ratio, train a back-propagation neural network, and obtain a neural network model; the input of the back-propagation neural network is the canopy reflectance, the sun and the observation angle, and the output is the leaf area index in the three-dimensional canopy structure of the crop; The inversion module is used to input the reflectivity and angle data of the corresponding band in the remote sensing data to be inverted into the neural network model to obtain the corresponding leaf area index.

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