Aerosol extinction coefficient inversion method combining CALIOP and Mie scattering laser radar

Abstract

The invention belongs to the technical field of aerosol extinction, and relates to a CALIOP and Mie scattering laser radar combined aerosol extinction coefficient inversion method, which comprises the following steps of: calculating a first-order gradient according to a Mie scattering laser radar echo signal, and judging whether a strong positive and negative gradient pair layer exists or not according to whether a strong gradient exists in the first-order gradient; calculating the relative increase amplitude of the strong gradient, comparing the relative increase amplitude with a threshold value to judge the type of a strong positive and negative gradient pair layer, and setting an atmospheric boundary layer height range; a discrete wavelet covariance transformation function is adopted to invert the height of the atmospheric boundary layer, and in the Gaussian process machine learning network training process, scale factor parameters are selected for the discrete wavelet covariance transformation function so as to accurately invert the height of the atmospheric boundary layer; dividing the atmospheric layer into an atmospheric boundary layer and an atmospheric boundary layer top to a troposphere top according to the inversion atmospheric boundary layer height; according to the invention, the inversion precision of the aerosol extinction coefficient can be improved.

Description

Technical Field

[0001] The invention belongs to the technical field of aerosol measurement, and in particular relates to an aerosol extinction coefficient inversion method combining CALIOP and Mie scattering laser radar. Background Art

[0002] Atmospheric aerosols are particulate matter suspended in the atmosphere, produced by a variety of natural and man-made processes. Aerosols affect the atmospheric radiation balance by scattering and absorbing solar radiation, and have an important impact on climate change, environmental pollution, and human health. The extinction coefficient is the most typical optical property of aerosols, and plays a very important role in studying the impact of aerosols on climate change and environmental pollution.

[0003] Lidar is an active remote sensing detection tool with the advantages of high spatial resolution, long detection distance, and strong continuous observation capability. It has significant technical advantages in atmospheric aerosol research. By solving the lidar equation, the vertical distribution characteristics of the aerosol extinction coefficient can be inverted. Existing lidar aerosol extinction coefficient detection technologies include Mie scattering lidar, Raman scattering lidar, and hyperspectral lidar, among which Mie scattering lidar is the most widely used aerosol extinction coefficient detection method. However, the Mie scattering lidar equation contains two unknown quantities, the aerosol extinction coefficient and the backscattering coefficient. When solving the equation using the classic Klett or Fernald method, it is necessary to assume the relationship between the two unknown quantities, namely the lidar ratio (the ratio of the aerosol extinction coefficient to the backscattering coefficient). However, the lidar ratio is a quantity related to aerosol characteristics, and this assumption will affect the accuracy of aerosol extinction coefficient inversion. The echo signal intensity of the Raman lidar is only related to the aerosol extinction coefficient, and the aerosol extinction coefficient can be directly inverted using the Raman frequency shift signal received by it. Since there is no need to set parameters in advance, the accuracy of the inverted aerosol extinction coefficient is significantly improved. Based on this, Song et al. from Xi'an University of Technology used the Raman lidar echo signal to invert the aerosol extinction coefficient. Based on this, they used the BP neural network to establish a relationship model between the Mie scattering lidar echo signal and the Raman extinction coefficient, and realized the inversion of the Mie scattering extinction coefficient. Shen et al. from Nanjing University of Information Science and Technology used the Mie-Raman lidar to detect atmospheric aerosols and measured the lidar ratio curve using the Raman channel echo signal, which provided a reference for the selection of the lidar ratio when the Mie channel uses the Fernald method to invert the extinction coefficient.

[0004] The laser radar echo signal contains various other signals, such as background light noise, atmospheric turbulence noise and device noise. The echo signal collected by the receiving end is relatively weak and needs to be amplified. The presence of these noise signals will seriously affect the accuracy of laser radar detection. The Raman scattering echo signal is a weaker signal. When the echo signal is amplified, the background light noise signal will also be amplified. In order to prevent the background light noise signal from damaging the photon counter, a threshold height will be set to make the signal under the threshold height zero. Therefore, the Raman scattering laser radar cannot detect aerosols in the near-ground area (<2km). In addition, since sunlight is also a very strong background light noise signal, the Raman laser radar generally works at night. According to the above analysis, although the aerosol extinction coefficient inverted by the Raman laser radar has advantages in accuracy, its detection distance and use period are significantly different from those of the Mie scattering laser radar. Therefore, in the prior art, using the aerosol extinction coefficient inverted by the Raman laser radar as a reference value for the aerosol extinction coefficient of the Mie scattering laser radar has certain limitations.

[0005] CALIOP (Cloud-Aerosol Lidar with Orthogonal Polarization, CALIOP) is an orthogonally polarized cloud-aerosol lidar carried on a satellite. It detects from top to bottom. Since the air quality at high altitudes is significantly higher than that at low altitudes, the attenuation of the laser is smaller, so it can more accurately measure the high-altitude distribution characteristics of aerosols. The products provided by CALIOP include vertical profile information such as scattering coefficient and extinction coefficient. In particular, when its V4 version inverts the extinction coefficient, it uses a value inversely proportional to the average inverted extinction amount in the layer to adjust the lidar ratio, taking into account the vertical distribution characteristics of the lidar ratio, overcoming the limitations of the Fernald method, and achieving accurate inversion of the extinction coefficient.

[0006] The methods for retrieving the atmospheric boundary layer height based on lidar include the gradient method, the maximum variance method, the curve fitting method, and the wavelet covariance transform method. Studies have shown that under ideal atmospheric conditions, these four methods can all invert the atmospheric boundary layer height well. However, when there are clouds and residual layers, the use of these methods is limited. For example, in the commonly used wavelet covariance transform method, the scale factor a will seriously affect the inversion accuracy of the boundary layer height.

[0007] The aerosol extinction coefficient inversion method based on neural network usually uses radial basis function neural network, BP neural network and feedforward neural network to construct network models. Neural network has good nonlinear description ability and can approximate any relationship, but the performance of the network is related to its structure. The three-layer neural network used in the existing technology has a simple structure and limited generalization ability of the network. Deep network can make fuller use of its nonlinear characteristics, extract the basic features of samples, and better express the relationship between samples. However, as the depth of the network increases, the convergence speed of the network will be affected and overfitting will easily occur. However, the residual connection of the network can solve this problem. This method has achieved remarkable results in the fields of image processing and machine vision. Summary of the invention

[0008] The present invention aims to overcome the deficiencies in the prior art and to provide an aerosol extinction coefficient inversion method combining CALIOP and Mie scattering lidar.

[0009] In order to achieve the purpose of the present invention, the present invention will be implemented by adopting the technical solution described below.

[0010] A method for inverting aerosol extinction coefficient by combining CALIOP and Mie scattering lidar comprises the following steps:

[0011] S1. Calculate the first-order gradient of the echo signal according to the echo signal of the Mie scattering laser radar, and judge the existence of strong positive and negative gradient pairs by whether there is a strong gradient in the first-order gradient; wherein: the strong gradient is a gradient with a gradient value greater than 2 or less than -2;

[0012] If it does not exist, it means there is no cloud layer or the residual layer is an ideal atmosphere, then the atmospheric boundary layer height z m = argminD(z);

[0013] If yes, it means there is a cloud layer or a residual layer, then go to step S2;

[0014] S2. Calculate the relative growth amplitude of the strong gradient, and judge the type of the strong positive and negative gradient layer by comparing it with the threshold of the relative growth amplitude, and set the atmospheric boundary layer height range according to the type; the types are divided into residual layer and cloud layer, among which, if the strong positive and negative gradient layer is a residual layer, the upper limit of the atmospheric boundary layer height is (z m ) max =z th , z th For the strong positive and negative gradient layer, at the minimum gradient value, z th =arg min D(z); if the strong positive and negative gradients are clouds, then at (0, z th ) Range Search At this time, the atmospheric boundary layer height z mThe range is [z m ',z th ];

[0015] S3. Within the atmospheric boundary layer height range obtained in step S2, the discrete wavelet covariance transformation function of the laser radar is used to invert the atmospheric boundary layer height, and during the Gaussian process machine learning network training process, the Gaussian process machine learning method is used to select the optimization parameters of the scale factor for the discrete wavelet covariance transformation function of the laser radar to accurately invert the atmospheric boundary layer height; wherein: the discrete wavelet covariance transformation function of the laser radar is:

[0016]

[0017] Where b is the translation factor of the Haar function, and a is the scale factor; here, a = nΔz, n = 2, 4, 6, ..., Δz represents the vertical resolution of the lidar;

[0018] S4, according to the atmospheric boundary layer height inverted in step S3, the atmosphere is divided into two regions: a short-distance region and a long-distance region, wherein the short-distance region is the atmospheric boundary layer; and the long-distance region is from the top of the atmospheric boundary layer to the top of the troposphere; wherein:

[0019] In the long-range area, the extinction coefficient profile is obtained using the CALIOP V4 data product. and backscatter coefficient profile Calculate LiDAR Ratio Profile And using the Mie scattering lidar echo signal Retrieving the atmospheric aerosol extinction coefficient in close-range areas in:

[0020] The laser radar profile The expression is:

[0021]

[0022] The atmospheric aerosol extinction coefficient The expression is:

[0023]

[0024] Where X(z) = P(z)·z 2 is the distance squared signal, λ 0 represents the laser wavelength, is the boundary value of aerosol particle extinction coefficient, is the atmospheric molecule extinction backscattering ratio, is the atmospheric molecular extinction coefficient, is the boundary value of the atmospheric molecule extinction coefficient; the relevant value of the atmospheric molecule can be obtained by the American standard atmospheric molecule model, and the boundary value of the aerosol particle extinction coefficient can be obtained by referring to the CALIOP extinction coefficient profile;

[0025] In the close-range area, the deep residual autoencoder network is used to invert the aerosol extinction coefficient, where:

[0026] The deep residual autoencoder network is based on the atmospheric boundary layer height and the spatial resolution of the Mie scattering lidar. The rule of l being a regulation constant of 0-10 is used to determine the number of input layer network nodes m and the number of hidden layer network nodes n. Based on Pascal Vincent's layer-by-layer greedy training principle, when setting the number of hidden layer nodes, the previous layer is used as the input layer.

[0027] The training data set of the deep residual autoencoder network is the echo signal of Mie scattering lidar. and the aerosol extinction coefficient constructed;

[0028] The loss function of the deep residual autoencoder network adopts the mean square error term, which is expressed as:

[0029]

[0030] Where W (L) Represents the connection weight between the L+1 and L layers of the network;

[0031] The training process of the deep residual autoencoder network includes the following steps:

[0032] S41. Use stochastic gradient descent method to update the network parameters W in the loss function (L) , network parameters W (L) Expressed as:

[0033]

[0034] S42: Update the noise label according to the training data set Noise Label The expression is:

[0035]

[0036] Where, τ is the aerosol optical depth measured by the global automatic observation network AERONET;

[0037] S43, return to S41 until the maximum number of iterations is reached, or Until the threshold condition is met;

[0038] S5, combined with the long-distance regional extinction coefficient inverted by CALIOP, realizes the inversion of the aerosol extinction coefficient from the ground to the top of the troposphere. The inversion results are:

[0039]

[0040] In the formula, represents the aerosol extinction coefficient in the atmospheric boundary layer inverted by the Mie scattering lidar using the Klett method, It represents the aerosol extinction coefficient inverted by CALIOP using the Klett method from the top of the boundary layer to the top of the troposphere.

[0041] As a preferred embodiment of the present invention, the relative growth rate is expressed as:

[0042] [X(z+Δz)-X(z)] / X(z),

[0043] Where X(z) represents the laser radar echo signal intensity at height z, and Δz represents the vertical resolution of the laser radar.

[0044] As a preferred solution of the present invention, the threshold value is 55%.

[0045] As a preferred embodiment of the present invention, the Gaussian process machine learning method comprises the following steps:

[0046] S41. Define performance function:

[0047] F(a)=max[W X (a,b)],

[0048] S42, constructing a training data set D = {p; F(p)}, p = {ai}, i = 1, ..., N, N represents the number of sample points;

[0049] S43, using Gaussian process to build a model to predict the distribution of performance function F(p*) within parameter set p*

[0050]

[0051] Where K(p,p) represents the covariance function, represents the variance of Gaussian white noise, μ(p*) and cov(p*) represent the mean function and covariance function of F(p*), respectively.

[0052] S44. Select the optimal parameter popt*(a), whose expression is:

[0053]

[0054] In the formula, Ω represents the set of all possible values ​​of parameter a.

[0055] S45. Calculate the atmospheric boundary layer height: z m = argmax[W X (p opt * (a),b)].

[0056] Beneficial Effects

[0057] 1. Inversion of atmospheric boundary layer height based on Gaussian-wavelet covariance. In complex atmospheric conditions, when wavelet covariance transform method is used to invert the atmospheric boundary layer height, the scale factor a of the wavelet will seriously affect the inversion accuracy. Based on this, the Gaussian process is used to select the optimal parameter a and accurately invert the atmospheric boundary layer height.

[0058] 2. Aerosol extinction coefficient inversion method based on deep residual autoencoder network. The expression ability of shallow neural networks is limited, while the convergence speed of deep neural networks is slow and overfitting is prone to occur. Based on this, a relationship model between the Mie scattering lidar echo signal and the aerosol extinction coefficient is established using a deep residual neural network. Due to the influence of related parameters, there is a certain error in the initial expected output. Therefore, in the process of network training, the present invention not only adjusts the parameters of the network, but also adjusts the expected output, thereby achieving high-precision inversion of the aerosol extinction coefficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 Technical roadmap for the inversion method of aerosol extinction coefficient by combining CALIOP and Mie scattering lidar;

[0060] Figure 2 Atmospheric boundary layer height inversion based on Gaussian-wavelet covariance;

[0061] Figure 3 Aerosol extinction coefficient inversion method based on residual autoencoder network;

[0062] Figure 4 Deep residual autoencoder network structure. DETAILED DESCRIPTION

[0063] The present invention is further described in conjunction with the accompanying drawings and embodiments.

[0064] As an embodiment of the present invention, Figures 1 to 4 As shown in FIG. 1 , a method for inverting aerosol extinction coefficient by combining CALIOP and Mie scattering lidar includes the following steps:

[0065] Step 1: Inversion of atmospheric boundary layer height based on Gaussian-wavelet covariance, such as Figure 2 shown.

[0066] (1) Calculate the first-order derivative of the lidar echo signal:

[0067] D(z)=dX(z) / dz

[0068] Where X(z) = P(z)z 2 , P(z) is the laser radar echo signal, and z is the distance.

[0069] (2) Determine whether there is a strong positive and negative gradient layer. If not, it means that the atmosphere is ideal, then the atmospheric boundary layer height z m =arg min D(z); if it exists, it means there are clouds and residual layers, and then go to the next step.

[0070] (3) Calculate the relative growth rate of the strong gradient [X(z+Δz)-X(z)] / X(z). If the growth does not exceed 55%, it means that the strong gradient layer is a residual layer. At this time, the upper limit of the atmospheric boundary layer height is (z m ) max =z th , zth is the minimum gradient value in the strong positive and negative gradient layer. If the increase exceeds 55%, it means that the strong gradient layer is a cloud layer, then look for the value in the range [0, zth].

[0071] z m '=arg min D(z), then the boundary layer height range is [z m ',z th ].

[0072] (4) Within the boundary layer height range, the Gaussian-wavelet covariance method is used to invert the atmospheric boundary layer height.

[0073] Discrete wavelet covariance transformation function based on laser radar

[0074]

[0075] Where b is the translation factor of the Haar function and a is the scale factor. Here, a = nΔz, n = 2, 4, 6, ..., Δz represents the vertical resolution of the lidar. Under complex atmospheric conditions, the value of parameter a will seriously affect the inversion accuracy of the boundary layer height. Based on this, the Gaussian process machine learning optimization method is used to select the optimal parameter a.

[0076] a) Define performance function

[0077] F(a)=max[W X (a,b)]

[0078] b) Construct a training data set D = {p; F(p)}, p = {ai}, i = 1, ..., N, where N represents the number of sample points.

[0079] c) Use Gaussian process to build a model to predict the distribution of performance function F(p*) within parameter set p*

[0080]

[0081] Among them, K(p,p) represents the covariance function, represents the variance of Gaussian white noise, μ(p*) and cov(p*) represent the mean function and covariance function of F(p*), respectively.

[0082] d) Select the optimal parameter popt*(a)

[0083]

[0084] Ω represents the set of all possible values ​​of parameter a.

[0085] e) Calculate the boundary layer height

[0086] z m = argmax[W X (p opt * (a),b)]

[0087] Step 2: Inversion of atmospheric aerosol extinction coefficient based on deep residual autoencoder neural network, such as Figure 3 shown.

[0088] (1) Obtain the extinction coefficient profile based on the CALIOP V4 data product and backscatter coefficient profile Calculate LiDAR Ratio Profile

[0089]

[0090] (2) According to the laser radar comparison provided by CALIOP Inversion of atmospheric aerosol extinction coefficient in the boundary layer using Mie scattering lidar

[0091]

[0092] Where X(z) = P(z)·z 2 is the distance squared signal, λ 0 represents the laser wavelength, is the boundary value of aerosol particle extinction coefficient, is the atmospheric molecule extinction backscattering ratio, is the atmospheric molecular extinction coefficient, is the boundary value of the atmospheric molecule extinction coefficient. The relevant values ​​of atmospheric molecules can be obtained from the American Standard Atmospheric Molecular Model, and the boundary value of the aerosol particle extinction coefficient can be obtained by referring to the CALIOP extinction coefficient profile.

[0093] (3) Construct a training data set. Select the Mie scattering lidar signal and the aerosol extinction coefficient in the boundary layer inverted in the previous step to construct a training data set. in Represents the Mie scattering lidar echo signal.

[0094] (4) Construct the network. The number of input layer network nodes m is determined according to the boundary layer height and the spatial resolution of the Mie scattering lidar, and the number of hidden layer network nodes n is determined according to the following rules: l is a regulation constant of 0-10. Based on Pascal Vincent's layer-by-layer greedy training principle, when setting the number of hidden layer nodes, the previous layer is used as the input layer. This network uses three hidden layers, and the residual autoencoder network structure is as follows Figure 4 shown.

[0095] (5) Define the loss function. The performance of the network is related to the selected loss function. The mean square error term is used as the loss function, which can be expressed as

[0096]

[0097] Among them, W (L) Represents the connection weight between the L+1 and L layers of the network.

[0098] (6) Training network

[0099] a) Use stochastic gradient descent method to update the network parameters W (L)

[0100]

[0101] b) Update the noise label

[0102]

[0103] Here, τ is the aerosol optical depth measured by the global automatic observation network AERONET.

[0104] c) Return to a) until the maximum number of iterations is reached, or until the threshold condition is met.

[0105] Step 3: Combine the long-distance regional extinction coefficient inverted by CALIOP to invert the aerosol extinction coefficient from the ground to the top of the troposphere.

[0106]

[0107] in, represents the aerosol extinction coefficient retrieved from the Mie scattering lidar in the atmospheric boundary layer, It represents the aerosol extinction coefficient retrieved by CALIOP in the region from the top of the boundary layer to the top of the troposphere.

Claims

1. A method for inverting aerosol extinction coefficient by combining CALIOP and Mie scattering lidar, characterized by: The steps include: S1. Calculate the first-order gradient of the echo signal according to the echo signal of the Mie scattering laser radar, and judge the existence of strong positive and negative gradient pairs by whether there is a strong gradient in the first-order gradient; wherein: the strong gradient is a gradient with a gradient value greater than 2 or less than -2; If it does not exist, it means there is no cloud layer or the residual layer is an ideal atmosphere, then the atmospheric boundary layer height z m = arg min D(z); If yes, it means there is a cloud layer or a residual layer, then go to step S2; S2. Calculate the relative growth amplitude of the strong gradient, and judge the type of the strong positive and negative gradient layer by comparing it with the threshold of the relative growth amplitude, and set the atmospheric boundary layer height range according to the type; the types are divided into residual layer and cloud layer, among which, if the strong positive and negative gradient layer is a residual layer, the upper limit of the atmospheric boundary layer height is (z m ) max =z th , z th For the strong positive and negative gradient layer, at the minimum gradient value, z th =arg min D(z); if the strong positive and negative gradients are clouds, then at (0, z th ) Range Search At this time, the atmospheric boundary layer height z m The range is [z m ',z th ]; S3. Within the atmospheric boundary layer height range obtained in step S2, the discrete wavelet covariance transformation function of the laser radar is used to invert the atmospheric boundary layer height, and during the Gaussian process machine learning network training process, the Gaussian process machine learning method is used to select the optimization parameters of the scale factor for the discrete wavelet covariance transformation function of the laser radar to accurately invert the atmospheric boundary layer height; wherein: the discrete wavelet covariance transformation function of the laser radar is: Where b is the translation factor of the Haar function, and a is the scale factor; here, a = nΔz, n = 2, 4, 6, ..., Δz represents the vertical resolution of the lidar; S4, according to the atmospheric boundary layer height inverted in step S3, the atmosphere is divided into two regions: a short-distance region and a long-distance region, wherein the short-distance region is the atmospheric boundary layer; and the long-distance region is from the top of the atmospheric boundary layer to the top of the troposphere; wherein: In the long-range area, the extinction coefficient profile is obtained using the CALIOP V4 version data product and backscatter coefficient profile Calculate LiDAR Ratio Profile And using the Mie scattering lidar echo signal Retrieving the atmospheric aerosol extinction coefficient in close-range areas in: The laser radar profile The expression is: The atmospheric aerosol extinction coefficient The expression is: Where X(z) = P(z)·z 2 is the distance square signal, λ0 represents the laser wavelength, is the boundary value of aerosol particle extinction coefficient, is the atmospheric molecule extinction backscattering ratio, is the atmospheric molecular extinction coefficient, is the boundary value of the atmospheric molecule extinction coefficient; the relevant value of the atmospheric molecule can be obtained by the American standard atmospheric molecule model, and the boundary value of the aerosol particle extinction coefficient can be obtained by referring to the CALIOP extinction coefficient profile; In the close-range area, the deep residual autoencoder network is used to invert the aerosol extinction coefficient, where: The deep residual autoencoder network is based on the atmospheric boundary layer height and the spatial resolution of the Mie scattering lidar. The rule of l being a regulation constant of 0-10 is used to determine the number of input layer network nodes m and the number of hidden layer network nodes n. Based on Pascal Vincent's layer-by-layer greedy training principle, when setting the number of hidden layer nodes, the previous layer is used as the input layer. The training data set of the deep residual autoencoder network is the echo signal of Mie scattering lidar. and the aerosol extinction coefficient constructed; The loss function of the deep residual autoencoder network adopts the mean square error term, which is expressed as: Where W (L) Represents the connection weight between the L+1 and L layers of the network; The training process of the deep residual autoencoder network includes the following steps: S41. Use stochastic gradient descent method to update the network parameters W in the loss function (L) , network parameters W (L) Expressed as: S42: Update the noise label according to the training data set Noise Label The expression is: Where, τ is the aerosol optical depth measured by the global automatic observation network AERONET; S43, return to S41 until the maximum number of iterations is reached, or Until the threshold condition is met; S5, combined with the long-distance regional extinction coefficient inverted by CALIOP, realizes the inversion of the aerosol extinction coefficient from the ground to the top of the troposphere. The inversion results are: In the formula, represents the aerosol extinction coefficient in the atmospheric boundary layer inverted by the Mie scattering lidar using the Klett method, It represents the aerosol extinction coefficient inverted by CALIOP using the Klett method from the top of the boundary layer to the top of the troposphere.

2. The aerosol extinction coefficient inversion method combining CALIOP and Mie scattering lidar according to claim 1, characterized in that: The relative growth rate is expressed as: [X(z+Δz)-X(z)] / X(z), Where X(z) represents the laser radar echo signal intensity at height z, and Δz represents the vertical resolution of the laser radar.

3. The aerosol extinction coefficient inversion method combining CALIOP and Mie scattering lidar according to claim 1 is characterized by: The threshold is 55%.

4. The aerosol extinction coefficient inversion method combining CALIOP and Mie scattering lidar according to claim 1, characterized in that: The Gaussian process machine learning method comprises the following steps: S41. Define performance function: F(a)=max[W X (a,b)], S42, constructing a training data set D = {p; F(p)}, p = {ai}, i = 1, ..., N, N represents the number of sample points; S43, using Gaussian process to build a model to predict the distribution of performance function F(p*) within parameter set p* Where K(p,p) represents the covariance function, represents the variance of Gaussian white noise, μ(p*) and cov(p*) represent the mean function and covariance function of F(p*), respectively; S44. Select the optimal parameter popt*(a), whose expression is: In the formula, Ω represents the set of all possible values ​​of parameter a; S45. Calculate the atmospheric boundary layer height: z m = argmax[W X (p opt * (a),b)].

Images