A method and system for removing speckle noise spectrum of a satellite-borne ocean spectrometer

Abstract

The present invention discloses a method and system for removing speckle noise spectrum of a satellite-borne ocean spectrometer, the method comprising: performing ocean wave spectrum detection by a satellite-borne ocean spectrometer based on an electronic scanning method, the spectrometer sequentially emitting 0°, 8° and 10° beams within an ocean wave spectrum detection macrocycle, fixing the observation direction of the spectrometer within each beam cycle T to receive backscattered signals from the sea surface within the corresponding beam cycle and transmitting them to a ground server; the ground server processes the received signal to obtain a modulated signal and intercepts two modulated signal segments corresponding to different time periods in the modulated signal, performs a cross-spectrum operation, and obtains a signal modulation spectrum that does not contain a speckle noise spectrum. The satellite-borne spectrometer ocean wave spectrum designed based on an electronic scanning method for realizing a non-uniform dwell time scheme provides favorable conditions for using the cross-spectrum method to remove the speckle noise spectrum, and can significantly improve the effect of the cross-spectrum method in removing the speckle noise spectrum along the track, thereby improving the ocean wave spectrum inversion performance.

Description

Technical Field

[0001] The present invention belongs to the technical field of ocean wave spectrum detection by a satellite-borne ocean spectrometer, and more specifically, relates to a method and a system for removing speckle noise spectrum of a satellite-borne ocean spectrometer. Background Art

[0002] As one of the important hydrological elements of the ocean, waves have always received close attention from marine scientific researchers. The study of many dynamic processes in the ocean requires accurate grasp of wave information, and the wave conditions are closely related to the energy and material exchange between the sea and the air. The wave directional spectrum, referred to as the wave spectrum, is the power spectrum of the wave surface function. It can not only characterize the distribution of wave energy in various directions, but also be used to calculate wave parameters including effective wave height and main wave direction. Obtaining accurate wave spectrum information is one of the key tasks in wave research. The spectrometer is a microwave sensor specifically used for wave directional spectrum detection. It works in the Ku band and performs 360° scanning through the antenna at a small incident angle. The wave spectrometer (Surface Waves Investigation and Monitoring, SWIM) is the world's first satellite-borne microwave radar that focuses on wave spectrum detection, and is currently the only sensor that can achieve long-term and continuous detection of wave spectrum on a global scale.

[0003] The current working mode of SWIM is that within one macrocycle (220ms), the radar antenna sequentially irradiates the sea surface with 6 beams with incident angles of 0°, 2°, 4°, 6°, 8° and 10°. After each beam is emitted, the corresponding echo signal is received and processed on the satellite before a modulated signal is transmitted back to the ground. During this period, the radar antenna adopts a mechanical rotation method, that is, the radar antenna rotates at a constant speed at a certain angular velocity, and its pointing residence time in all azimuths is consistent, which is called the uniform residence time scheme. In SWIM using the uniform residence time scheme, the beams with the same incident angle in adjacent macrocycles are delayed by 220ms, and are 7.5° apart in azimuth, resulting in irrelevant parts between the two modulation spectra. Therefore, when the ground server modulates the beams with the same incident angle, it cannot meet the requirements of using the cross-spectrum method to remove the speckle noise spectrum, and the analytical method and the empirical method cannot accurately estimate the speckle noise spectrum, resulting in a decrease in the accuracy of wave spectrum detection. Summary of the invention

[0004] In view of the above defects or improvement needs of the prior art, the present invention provides a method and system for removing speckle noise spectrum of a satellite-borne ocean spectrometer, which aims to effectively remove the speckle noise spectrum in the wave spectrum through the cross-spectrum method and improve the wave spectrum inversion performance.

[0005] To achieve the above object, according to one aspect of the present invention, a method for removing speckle noise spectrum of a spaceborne ocean spectrometer is provided, comprising:

[0006] The ocean wave spectrum is detected by a satellite-borne ocean spectrometer based on an electronic scanning method. The satellite-borne ocean spectrometer sequentially transmits 0°, 8° and 10° beams within the ocean wave spectrum detection macrocycle. The observation direction of the satellite-borne ocean spectrometer is fixed within each beam cycle T to receive the backscattered signal of the sea surface within the corresponding beam cycle and transmit it to the ground server.

[0007] The ground server processes the received signal to obtain a modulated signal and intercepts modulated signal segments corresponding to two different time periods in the modulated signal, performs a cross-spectrum operation, and obtains a signal modulation spectrum that does not contain a speckle noise spectrum.

[0008] In one embodiment, the beam period of the spectrometer when the relative observation angle is 0° is larger than the beam period at other relative observation angles.

[0009] In one of the embodiments, the beam periods of different beams at a relative observation angle of 0°≤φ≤15° are: the observation time required to ensure that the signal-to-spot ratio is greater than or equal to 1 under small sea conditions.

[0010] In one of the embodiments, the beam periods of different beams at a relative observation angle φ>15° are set to 38ms-42ms.

[0011] In one embodiment, the method of intercepting modulation signal segments corresponding to two different time periods in the modulation signal and performing cross-spectrum operation includes:

[0012] When the relative observation angle is 0°≤φ≤15°, the corresponding modulated signal is divided into two segments according to the time length, and the two are respectively used as the coherent signals m(x,φ,t) and m(x,φ,t+δt);

[0013] When the relative observation angle φ>15°, the corresponding modulated signal is divided into three segments according to the time length, and the first and third segments are taken as the coherent signals m(x,φ,t) and m(x,φ,t+δt) respectively;

[0014] The signal modulation spectrum is obtained by the cross-spectral method:

[0015] P m (k,φ)=Re{FT(m(x,φ,t))·FT * (m(x,φ,t+δt))}

[0016] Where t represents time, φ represents the relative observation angle, x represents the ground distance, and k represents the wave number.

[0017] In one of the embodiments, the observation azimuth angles when transmitting beams twice are separated by 5°.

[0018] In one of the embodiments, the satellite-borne ocean spectrometer uses a phased array to control antenna pointing to fix the antenna pointing within a beam period and adjust the antenna pointing at set azimuth intervals.

[0019] According to another aspect of the present invention, a system for removing speckle noise spectrum of a spaceborne ocean spectrometer is provided, comprising:

[0020] A satellite-borne ocean spectrometer based on an electronic scanning method is used for detecting ocean wave spectra. The satellite-borne ocean spectrometer sequentially transmits 0°, 8° and 10° beams in the ocean wave spectrum detection macrocycle and fixes the observation direction in each beam cycle T to receive backscattered signals from the sea surface in the corresponding beam cycle and transmit them to a ground server.

[0021] The ground server is used to process the received signal to obtain a modulated signal and intercept the modulated signal segments corresponding to two different time periods in the modulated signal, perform cross-spectrum operation, and obtain a signal modulation spectrum without a speckle noise spectrum.

[0022] In one embodiment, the spaceborne ocean spectrometer satisfies:

[0023] The beam period of different beams at the relative observation angle of 0°≤φ≤15° is: the observation time required to ensure that the signal-to-spot ratio is greater than or equal to 1 under small sea conditions;

[0024] The beam period of different beams at a relative observation angle φ>15° is set to 38ms~42ms.

[0025] In one embodiment, the ground server includes:

[0026] The segmentation unit is used for dividing the corresponding modulated signal into two segments according to the time length when the relative observation angle is 0°≤φ≤15°, and taking the two segments as the coherent signals m(x,φ,t) and m(x,φ,t+δt) respectively; when the relative observation angle φ>15°, dividing the corresponding modulated signal into three segments according to the time length, taking the first segment and the third segment, and taking the two segments as the coherent signals m(x,φ,t) and m(x,φ,t+δt) respectively;

[0027] The processing unit is used to obtain the signal modulation spectrum by cross spectrum method:

[0028] P m (k,φ)=Re{FT(m(x,φ,t))·FT * (m(x,φ,t+δt))}

[0029] Where δt represents the time interval, t represents time, φ represents the relative observation angle, x represents the ground distance, and k represents the wave number.

[0030] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects compared with the prior art:

[0031] The present invention uses a satellite-borne ocean spectrometer to detect the wave spectrum by means of electronic scanning. Compared with the traditional mechanical scanning, the radar antenna is artificially designed to have different observation durations under different pointing directions, and the observation direction is fixed in each beam cycle by means of electronic scanning. After the previous beam cycle ends, the relative observation angle is changed according to the azimuth interval and the next beam cycle begins. This control scheme is called a non-uniform dwell scheme. Compared with the uniform dwell scheme, the fixed observation direction can enhance the coherence of the received signal. At the same time, after obtaining a modulated signal through the non-uniform dwell scheme, the present invention directly intercepts two modulated signal segments corresponding to different time periods from a received modulated signal for cross-spectrum processing, which can effectively remove the speckle noise spectrum and obtain the signal modulation spectrum without the speckle noise spectrum. Moreover, using the above processing method, the detection of the wave spectrum can be achieved by only transmitting 0°, 8° and 10° beams in sequence.

[0032] Furthermore, since the speckle noise spectrum value is very large when the relative observation angle is 0°≤φ≤15°, setting the beam period when the relative observation angle is 0°≤φ≤15° to be larger than the wave velocity period at other relative observation angles can obtain a higher signal-to-spot ratio, thereby satisfying the premise of applying the cross-spectrum method to remove speckle noise.

[0033] Furthermore, the relative observation angle is divided into two intervals of 15°, and different beam periods are set for different observation angles, so as to obtain a compromise between time and signal-to-spot ratio.

[0034] Furthermore, after obtaining the modulated signal based on the non-uniform dwell scheme, the modulated signal is processed in different segments according to different observation angles and used as a coherent signal for cross-spectral denoising, which can further improve the coherence of the signal and effectively remove the speckle noise spectrum. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 A flowchart of the steps of a method for removing speckle noise spectrum of a satellite-borne ocean spectrometer according to an embodiment;

[0036] Figure 2 The result of cross-spectrum processing of the modulation signal in a mechanical scanning mode of an embodiment;

[0037] Figure 3 The cross-spectrum processing result of the modulation signal in the electronic scanning mode of an embodiment is shown;

[0038] Figure 4 The inversion result of cross-spectrum despeckle noise in an electronic scanning mode of an embodiment under a uniform dwell time scheme along the track;

[0039] Figure 5 The inversion result of despeckle noise using the cross-spectral method under the electronic scanning mode of one embodiment with a non-uniform dwell time scheme along the track. DETAILED DESCRIPTION

[0040] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0041] like Figure 1 The figure is a flowchart of a method for removing speckle noise spectrum of a satellite-borne ocean spectrometer in one embodiment, which mainly includes two steps:

[0042] The ocean wave spectrum is detected by a satellite-borne ocean spectrometer based on an electronic scanning method. The satellite-borne ocean spectrometer sequentially transmits 0°, 8° and 10° beams within the ocean wave spectrum detection macrocycle. The observation direction of the satellite-borne ocean spectrometer is fixed within each beam cycle T to receive the backscattered signal of the sea surface within the corresponding beam cycle and transmit it to the ground server.

[0043] The ground server processes the received signal to obtain a modulated signal and intercepts modulated signal segments corresponding to two different time periods in the modulated signal, performs a cross-spectrum operation, and obtains a signal modulation spectrum that does not contain a speckle noise spectrum.

[0044] The beam period is the duration of receiving the corresponding backscattered signal from the sea surface after transmitting any beam.

[0045] The satellite-borne ocean spectrometer of the present invention adopts an electronic scanning method, which refers to a mechanical rotation method relative to a traditional satellite antenna, and adopts a phased array antenna to realize the function of freely selecting the direction of the radar antenna. In the existing satellite-borne ocean spectrometer technology, the radar antenna is a mechanical rotation method, that is, the radar antenna rotates at a constant speed at a certain angular velocity, and its dwell time in all directions is consistent, that is, a uniform dwell time scheme. The present invention realizes a non-uniform dwell time scheme through electronic scanning, that is, the function of freely selecting the direction can be used with the help of the electronic scanning method, and the different observation times of the radar antenna under different directions are artificially selected and designed. After the beam is transmitted, the echo signal (backscattered signal of the sea surface) is received under a fixed direction to obtain better detection performance, and a coherent modulated signal with a signal-to-spot ratio of not less than 1 is obtained.

[0046] Among them, the spectrometer can adjust the length of the corresponding beam period according to different relative observation angles. The relative observation angle refers to the angle between the observation direction of the radar antenna and the flight direction of the radar. It has been found that when the relative observation angle is 0°≤φ≤15°, the speckle noise spectrum value is very large, which is greater than the speckle noise spectrum value at other relative observation angles. Therefore, in one embodiment, the beam period of the spectrometer when the relative observation angle is 0°≤φ≤15° is increased so that it is larger than the wave velocity period of other relative observation angles. By setting a larger beam period to obtain a higher signal-to-spot ratio, the premise of applying the cross-spectrum method to remove speckle noise is met.

[0047] Specifically, this embodiment divides the relative observation angle φ into two major intervals, and sets different beam periods according to each relative observation angle, namely:

[0048] The beam period of different beams at the relative observation angle of 0°≤φ≤15° is: the observation time required to ensure that the signal-to-spot ratio is greater than or equal to 1 under small sea conditions, where the beam period of φ=0° is set to the maximum;

[0049] The beam period of different beams when the relative observation angle φ>15° is set to 38ms-42ms, and specifically may be 40ms.

[0050] The along-track area and the non-along-track area are divided by 15°, that is, within the range of 15°, the speckle noise spectrum will change with the change of sea surface conditions, and outside 15°, the speckle noise spectrum will not change with the change of sea surface conditions. By setting different beam periods for different observation angles, a compromise between time and signal-to-speckle ratio can be obtained.

[0051] In one embodiment, the method of using the cross-spectral method for processing includes:

[0052] When the relative observation angle is 0°≤φ≤15°, the corresponding modulated signal is divided into two segments according to the time length, and the two are respectively used as the coherent signals m(x,φ,t) and m(x,φ,t+δt);

[0053] When the relative observation angle φ>15°, the corresponding modulated signal is divided into three segments according to the time length, and the first and third segments are taken as the coherent signals m(x,φ,t) and m(x,φ,t+δt) respectively;

[0054] The signal modulation spectrum is obtained by the cross-spectral method:

[0055] P m (k,φ)=Re{FT(m(x,φ,t))·FT * (m(x,φ,t+δt))}

[0056] Where δt represents the time interval, t represents time, φ represents the relative observation angle, x represents the ground distance, and k represents the wave number.

[0057] In one embodiment, the satellite-borne ocean spectrometer uses a phased array to control the antenna pointing to fix the antenna pointing within the beam period and adjust the antenna pointing according to the set azimuth interval. Specifically, the observation azimuth angle interval between the two consecutive beam transmissions is 5°, where the observation azimuth angle is the angle between the projection of the radar observation direction and the range direction, and the range direction is the line between the radar projection on the ground and the detection target.

[0058] Accordingly, the present invention also relates to a system for removing speckle noise spectrum of a satellite-borne ocean spectrometer, comprising:

[0059] A satellite-borne ocean spectrometer based on an electronic scanning method is used for detecting ocean wave spectra. The satellite-borne ocean spectrometer sequentially transmits 0°, 8° and 10° beams in the ocean wave spectrum detection macrocycle and fixes the observation direction in each beam cycle T to receive backscattered signals from the sea surface in the corresponding beam cycle and transmit them to a ground server.

[0060] The ground server is used to process the received signal to obtain a modulated signal and intercept the modulated signal segments corresponding to two different time periods in the modulated signal, perform cross-spectrum operation, and obtain a signal modulation spectrum without a speckle noise spectrum.

[0061] Specifically, the satellite-borne ocean spectrometer meets the following requirements:

[0062] The beam period of different beams at the relative observation angle of 0°≤φ≤15° is: the observation time required to ensure that the signal-to-spot ratio is greater than or equal to 1 under small sea conditions;

[0063] The beam period of different beams at a relative observation angle φ>15° is set to 38ms~42ms.

[0064] The ground server comprises:

[0065] The segmentation unit is used for dividing the corresponding modulated signal into two segments according to the time length when the relative observation angle is 0°≤φ≤15°, and taking the two segments as the coherent signals m(x,φ,t) and m(x,φ,t+δt) respectively; when the relative observation angle φ>15°, dividing the corresponding modulated signal into three segments according to the time length, taking the first segment and the third segment, and taking the two segments as the coherent signals m(x,φ,t) and m(x,φ,t+δt) respectively;

[0066] The processing unit is used to obtain the signal modulation spectrum by cross spectrum method:

[0067] P m (k,φ)=Re{FT(m(x,φ,t))·FT *(m(x,φ,t+δt))}

[0068] Where δt represents the time interval, t represents time, φ represents the relative observation angle, x represents the ground distance, and k represents the wave number.

[0069] The advantages of the present invention are described below through specific embodiments.

[0070] The first group: under the conditions of wind wave, wind wave inverse wave age Ω = 0.84, effective wave height of 1m, wind speed U = 10m / s, peak wave number of 0.0314rad / m, wave direction of 90°, and incident angle of 10°, wave spectrum detection simulation was carried out.

[0071] Comparative example: Using mechanical scanning, consider two signals with a detection time interval of 220ms. According to the rotation speed of the SWIM antenna, the antenna rotates 7.5°. The cross spectrum method is used to remove the speckle noise spectrum of the two signals to obtain Figure 2 It can be seen that the cross-spectrum method cannot obtain the required inversion slope spectrum. This is because there is a large difference between the slope spectra at the relative observation angles of 90° and 97.5°, and the signal coherence is poor.

[0072] Embodiment: The electronic scanning method is used, which can control the observation direction to be unchanged, and the observation time is selected to be 40ms. The cross spectrum method is used to remove the speckle noise spectrum of the two signals obtained by segmentation to obtain Figure 3 The results are shown in Figure 2. It can be seen that the inversion slope spectrum obtained by the cross spectrum method fluctuates around the theoretical slope spectrum. This shows that the electronic scanning method can obtain coherent signals, thus creating the conditions for using the cross spectrum method to remove the speckle noise spectrum.

[0073] The second group: under the conditions of wind wave, wind wave inverse wave age Ω = 0.84, effective wave height of 1m, wind speed U = 10m / s, peak wave number of 0.0314rad / m, wave direction of 0°, and incident angle of 10°, wave spectrum detection simulation was carried out.

[0074] Comparative example: The speckle noise spectrum along the track (relative observation angle is 0°) is analyzed by using the electronic scanning method, the observation time is selected as 40ms, and the speckle noise spectrum is removed by the cross spectrum method from the segmented signal. Figure 4 It can be seen that although the coherent signal is obtained by electronic scanning, the speckle noise spectrum along the track is large, and the speckle noise has completely submerged the useful signal, and the inversion slope spectrum cannot be effectively extracted.

[0075] Example: For the electronic scanning method, under the same sea conditions, the observation time is selected as 440ms, and the cross spectrum method is used to remove the speckle noise spectrum of the segmented signal to obtain Figure 5It can be seen that the inverted wave spectrum results basically fluctuate around the theoretical spectrum results. This shows that compared with the non-along-track area, designing a longer observation time in the along-track area can effectively improve the signal-to-spot ratio, and then the cross-spectrum method can be used to obtain the inverted slope spectrum.

[0076] In summary, the satellite-borne spectrometer wave spectrum designed based on the electronic scanning method to realize the non-uniform dwell time scheme provides favorable conditions for using the cross-spectrum method to remove the speckle noise spectrum, and can significantly improve the effect of the cross-spectrum method to remove the speckle noise spectrum along the track, thereby improving the wave spectrum inversion performance.

[0077] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for removing speckle noise spectrum of a satellite-borne ocean spectrometer, characterized in that: include: The ocean wave spectrum is detected by a satellite-borne ocean spectrometer based on an electronic scanning method. The satellite-borne ocean spectrometer sequentially transmits 0°, 8° and 10° beams within the ocean wave spectrum detection macrocycle. The observation direction of the satellite-borne ocean spectrometer is fixed within each beam cycle T to receive the backscattered signal of the sea surface within the corresponding beam cycle and transmit it to the ground server. The ground server processes the received signal to obtain a modulated signal and intercepts modulated signal segments corresponding to two different time periods in the modulated signal, performs a cross-spectrum operation, and obtains a signal modulation spectrum that does not contain a speckle noise spectrum.

2. The method for removing speckle noise spectrum of a satellite-borne ocean spectrometer according to claim 1, characterized in that: The beam period of the spectrometer when the relative observation angle is 0° is larger than the beam period at other relative observation angles.

3. The method for removing speckle noise spectrum of a satellite-borne ocean spectrometer according to claim 2, characterized in that: The beam period of different beams at a relative observation angle of 0°≤φ≤15° is: the observation time required to ensure that the signal-to-spot ratio is greater than or equal to 1 under small sea conditions.

4. The method for removing speckle noise spectrum of a satellite-borne ocean spectrometer according to claim 2, characterized in that: The beam period of different beams at a relative observation angle φ>15° is set to 38ms~42ms.

5. The method for removing speckle noise spectrum of a satellite-borne ocean spectrometer according to claim 1, characterized in that: Intercepting modulation signals corresponding to two different time periods from the modulation signal and performing a cross-spectrum operation, including: When the relative observation angle is 0°≤φ≤15°, the corresponding modulated signal is divided into two segments according to the time length, and the two are respectively used as the coherent signals m(x,φ,t) and m(x,φ,t+δt); When the relative observation angle φ>15°, the corresponding modulated signal is divided into three segments according to the time length, and the first and third segments are taken as the coherent signals m(x,φ,t) and m(x,φ,t+δt) respectively; The signal modulation spectrum is obtained by the cross-spectral method: P m (k,φ)=Re{FT(m(x,φ,t))·FT * (m(x,φ,t+δt))} Where t represents time, φ represents the relative observation angle, x represents the ground distance, and k represents the wave number.

6. The method for removing speckle noise spectrum of a satellite-borne ocean spectrometer according to claim 1, characterized in that: The observation azimuth angle interval between the two consecutive beam transmissions is 5°.

7. The method for removing speckle noise spectrum of a satellite-borne ocean spectrometer according to claim 1, characterized in that: The satellite-borne ocean spectrometer uses a phased array to control the antenna pointing so as to fix the antenna pointing within a beam period and adjust the antenna pointing according to a set azimuth interval.

8. A speckle noise spectrum removal system for a satellite-borne ocean spectrometer, characterized in that: include: A satellite-borne ocean spectrometer based on an electronic scanning method is used for detecting ocean wave spectra. The satellite-borne ocean spectrometer sequentially transmits 0°, 8° and 10° beams in the ocean wave spectrum detection macrocycle and fixes the observation direction in each beam cycle T to receive backscattered signals from the sea surface in the corresponding beam cycle and transmit them to a ground server. The ground server is used to process the received signal to obtain a modulated signal and intercept the modulated signal segments corresponding to two different time periods in the modulated signal, perform cross-spectrum operation, and obtain a signal modulation spectrum without a speckle noise spectrum.

9. The system for removing speckle noise spectrum of a satellite-borne ocean spectrometer according to claim 8, characterized in that: The spaceborne ocean spectrometer meets the following requirements: The beam period of different beams at the relative observation angle of 0°≤φ≤15° is: the observation time required to ensure that the signal-to-spot ratio is greater than or equal to 1 under small sea conditions; The beam period of different beams at a relative observation angle φ>15° is set to 38ms~42ms.

10. The system for removing speckle noise spectrum of a satellite-borne ocean spectrometer according to claim 8, characterized in that: The ground server comprises: The segmentation unit is used for dividing the corresponding modulated signal into two segments according to the time length when the relative observation angle is 0°≤φ≤15°, and taking the two segments as the coherent signals m(x,φ,t) and m(x,φ,t+δt) respectively; when the relative observation angle φ>15°, dividing the corresponding modulated signal into three segments according to the time length, taking the first segment and the third segment, and taking the two segments as the coherent signals m(x,φ,t) and m(x,φ,t+δt) respectively; The processing unit is used to obtain the signal modulation spectrum by cross spectrum method: P m (k,φ)=Re{FT(m(x,φ,t))·FT * (m(x,φ,t+δt))} Where δt represents the time interval, t represents time, φ represents the relative observation angle, x represents the ground distance, and k represents the wave number.

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