A deep sea shadow zone target positioning method considering horizontal array direction finding error

Abstract

The application relates to a deep sea shadow area target positioning method considering horizontal array direction finding errors. First, a conventional beam forming method is used to respectively perform target direction estimation on array receiving signals of two different sub-arrays of a large-aperture horizontal line array; then, according to the marine environment, a Bellhop ray model is used to model and simulate a sound field to obtain a direction estimation value distribution diagram at possible existing positions of the target; secondly, a cost function is established, and the possible existing position points of the target relative to the two sub-arrays are extracted by combining the target direction estimation results of the two sub-arrays; the possible existing position points of the target are converted into rectangular coordinate systems to perform straight line fitting, and two straight line equations are obtained; finally, the intersection coordinates of the two straight lines are the target positions obtained by the application. Compared with a traditional two-station direction finding cross positioning method, the method has higher positioning precision.

Description

Technical Field

[0001] This invention belongs to the fields of marine engineering, underwater acoustic engineering, array signal processing and sonar technology, and relates to a deep-sea target localization method that takes into account the direction finding error of the horizontal array. It is applicable to the passive localization problem of near-sea surface targets using a large-aperture horizontal array in the deep sea. Background Technology

[0002] Underwater target localization technology has always been one of the most important and challenging research directions in underwater acoustic detection. In actual underwater acoustic experiments, when the sound source and receiving array are located at relatively shallow depths, the deep-sea sound field can be spatially divided into: the direct sound zone, the shadow zone, and the convergence zone. Existing research mainly focuses on the direct sound zone and the convergence zone, and target localization methods are relatively mature. However, because the shadow zone occupies a large proportion of the deep-sea sound field and there is limited research on target localization methods for the shadow zone, research on target localization methods for the shadow zone is of great significance.

[0003] Among existing underwater target localization methods, towed horizontal arrays are an important form of passive detection. Positioning the towed array far from the mother ship not only significantly reduces the impact of towed platform noise but also significantly improves the signal-to-noise ratio, thereby enhancing sonar detection capabilities. Furthermore, the size of the towed array in a towed horizontal array sonar is not limited by the ship's size, allowing for the installation of more hydrophones compared to hull-mounted sonars.

[0004] However, when using a towed horizontal array for target detection in the shadow area, a direction-finding error caused by the physical characteristics of underwater acoustic propagation (sound refraction) occurs. This is because the main contributors in the shadow area are surface and seabed reflected waves, and the horizontal array cannot distinguish the angle of arrival of the sound signals on the horizontal and vertical planes, thus leading to direction-finding errors. In this case, when using the traditional two-station cross-location method (the basic principle of two-station cross-location is: two reconnaissance stations perform direction finding on the same radiation source target, and according to geometric knowledge, the intersection of the two direction finding lines is the position of the radiation source target), positioning errors will occur because the angle calculated by the towed array is not the physical azimuth angle.

[0005] Because of the inherent direction-finding error in the shadow area, and because the variation pattern of this error is predictable, this variation pattern can be utilized, combined with geometric knowledge, to improve target positioning accuracy. Therefore, this invention proposes a deep-sea shadow area target positioning method that considers acoustic field characteristics. Summary of the Invention

[0006] Technical problems to be solved

[0007] To avoid the shortcomings of existing technologies, this invention proposes a deep-sea target positioning method that takes into account the direction finding error of the horizontal array, thereby improving the accuracy of target positioning in the shadow area.

[0008] Technical solution

[0009] A method for locating targets in deep-sea shadow areas that takes into account the orientation error of the horizontal array, characterized by the following steps:

[0010] Step 1: Using conventional beamforming methods, beamforming is performed on the array received signals of the two subarrays 1 and 2 of the large-aperture horizontal line array to obtain the target azimuth estimate. and

[0011] The first M1 array elements are taken and defined as subarray 1, with the array element numbers being... Take the last M2 array elements, define them as subarray 2, and the array element numbers are:

[0012] Step 2: Based on the marine environment, model the sound field using the Bellhop ray model, and simulate the distribution of azimuth estimates θ for different target positions relative to the two subarrays. h1 (θ1, r1) and θ h2 (θ2, r2);

[0013] The horizontal distance r is selected from the center of the first element of each subarray to the sound source. The angle between the sound source and the center of the first element of each subarray is denoted as θ, where θ1 and θ2 are the azimuth angles of the target relative to subarray 1 and subarray 2, respectively, and r1 and r2 are the horizontal distances of the target from subarray 1 and subarray 2.

[0014] When the target's position is L hi =[θ ι r ι When [the target's position is determined], the estimated azimuth of the target is a function θ of the target's position. hi =g(L hi ), where θ ι Let r be the azimuth angle of the target relative to subarray i. i The horizontal distance from the target to subarray i;

[0015] Step 3: Estimate the angle based on the target's azimuth. and The distribution of azimuth estimates θ obtained by combining sound field modeling h1 and θ h2 Extract the target location sets L1 and L2 relative to the two subarrays, and establish the cost function:

[0016]

[0017] Where ε represents the error control factor;

[0018] Step 4: Transform the discrete point sets L1 and L2 of the target location from the azimuth-range domain to the same Cartesian coordinate system domain;

[0019] Step 5: Apply Ax+By+C=0 to the discrete point sets L1 and L2 in the Cartesian coordinate system domain after the transformation in Step 4 to perform linear fitting, and obtain the equation parameters A1, B1, C1 and A2, B2, C2 of the two lines.

[0020] Step 6: Substitute the obtained parameters A1, B1, C1 and A2, B2, C2 into the following formula to calculate the positioning result:

[0021]

[0022] The estimated target coordinates are (x e ',y e ').

[0023] The large-aperture horizontal line array has L elements, a spacing of d, and element numbers of n. l l = 1 to L, where the array element closest to the mother ship is array element 1, denoted as n1, and the array element farthest from the mother ship is defined as n. L The starboard direction is 0 to 180°, and the port direction is 0 to -180°.

[0024] The conventional beamforming method is calculated using the following formula:

[0025]

[0026] Where σ ι R represents the beam direction response of subarray i. xi Let w represent the covariance matrix of the received signal from subarray i. ci Let H represent the beam weighting vector of subarray i, and H represent the transpose sign. The azimuth estimate corresponding to the maximum value of the beam directional response is denoted as .

[0027] The covariance matrix R xi =E[x i H x i ], where x i Let E[.] represent the received signal of subarray i, and E[.] denotes the expectation.

[0028] The beam weighting vector w ci =P(θ) / M i ,in 0°≤θ≤180°, d represents the element spacing, M iLet k be the number of elements in subarray i; k = 2πf / c, where c represents the speed of sound and f represents the signal frequency.

[0029] The g(.) is obtained by Bellhop simulation of the ray model and conventional beamforming methods to calculate θ. The function that obtains θ by ray model simulation and conventional beamforming methods is called the g function.

[0030] The error control factor ε = 0.001.

[0031] The formula for transforming the discrete point sets L1 and L2 of the target location from the azimuth-range domain to the same Cartesian coordinate system domain in step 4 is as follows:

[0032] Beneficial effects

[0033] This invention proposes a deep-sea target location method that takes into account the direction-finding error of a horizontal array. First, conventional beamforming methods are used to estimate the target's azimuth from the received signals of two different subarrays of a large-aperture horizontal linear array. Then, based on the marine environment, a Bellhop ray model is used to model and simulate the sound field, obtaining a distribution map of the estimated azimuth values ​​at possible target locations. Next, a cost function is established, and combined with the target azimuth estimation results from the two subarrays, to extract the possible locations of the target relative to the two subarrays. These possible locations are then converted to a Cartesian coordinate system, and straight line fitting is performed to obtain the equations of two lines. Finally, the coordinates of the intersection of these two lines represent the target location obtained by this invention. Compared to traditional two-station cross-location methods, the proposed method has higher positioning accuracy.

[0034] This invention is applicable to application scenarios where the direction finding error changes monotonically in the deep-sea shadow area. It utilizes this monotonically changing pattern and combines it with geometric knowledge to improve the target positioning accuracy.

[0035] This invention is applicable not only to conventional beamforming methods and horizontal linear arrays, but also to other beamforming methods and non-uniform linear arrays.

[0036] This invention proposes an underwater target localization method based on the inherent direction-finding error of horizontal arrays in deep-sea shadow areas. The proposed method utilizes conventional beamforming to estimate the target's azimuth from the received signals of two different subarrays of a large-aperture horizontal array. Then, based on the marine environment, a Bellhop ray model is used to model and simulate the sound field, obtaining the azimuth estimate distribution at the target's possible locations. Next, a cost function is established, and combined with the target azimuth estimation results from the two subarrays, the possible locations of the target relative to the two subarrays are extracted. These possible locations are then converted to a Cartesian coordinate system, and straight line fitting is performed to obtain two straight line equations. Finally, the coordinates of the intersection of these two lines represent the target location obtained by this invention. Compared to traditional direction-finding cross-location methods, the proposed method offers higher accuracy. The basic principles and implementation schemes of the method proposed in this invention have been verified by computer numerical simulation. The results show that, in a typical deep-sea environment, the method proposed in this invention can effectively estimate the position of near-sea surface targets. In the given typical embodiments, the method proposed in this invention (compared to traditional methods) can reduce the distance error from 3.14km to 0.33km. Attached Figure Description

[0037] Figure 1 This is a schematic diagram of the sound velocity profile in a simulated scene.

[0038] Figure 2 It is a simulation scene sound trajectory diagram

[0039] Figure 3 It is a diagram showing the positional relationships of targets in a simulation scene.

[0040] Figure 4 This is a flowchart of a deep-sea target localization method that takes into account the orientation-finding error of the horizontal array.

[0041] Figure 5 It is the array beam output response

[0042] Figure 6 The azimuth estimation angle is and The target locations may exist in the map. Subgraph (a) shows the possible locations of target 1 in subarray, and subgraph (b) shows the possible locations of target 2 in subarray.

[0043] Figure 7 For target location estimation in the method proposed in this invention Detailed Implementation

[0044] The present invention will now be further described in conjunction with the embodiments and accompanying drawings:

[0045] The technical solution adopted by this invention to solve its technical problem is a method for locating targets in deep-sea shadow areas that takes into account the orientation error of the horizontal array, characterized by the following steps:

[0046] Step 1: For a large-aperture horizontal uniform linear array, assuming the number of array elements is L and the spacing is d, define the array element numbers as n. l l = 1 to L, where the array element closest to the mother ship is array element 1, denoted as n1, and the array element farthest from the mother ship is defined as n. L Ignoring port and starboard ambiguity, and defining the starboard direction as 0–180° and the port direction as 0–-180°, we take M1 array elements at the bow, defining them as subarray 1, with element numbers as follows: Take the last M2 array elements, define them as subarray 2, and the array element numbers are: Extract the array received signals of subarray 1 and subarray 2 respectively.

[0047] Step 2: Perform conventional beamforming on the received signals of each subarray to obtain the target's azimuth estimate relative to subarray 1 and subarray 2. and

[0048] Conventional beamforming methods can be calculated using the following formula:

[0049]

[0050] Where σ ι R represents the beam direction response of subarray i. xi Let w represent the covariance matrix of the received signal from subarray i. ci Let H represent the beam weighting vector of subarray i, and H denote the transpose sign. The azimuth estimate corresponding to the maximum value of the beam directional response is denoted as . Unless otherwise specified, in this patent, i∈{1,2}.

[0051] The R xi It is calculated using the following formula.

[0052] R xi =E[x i H x i (2)

[0053] Where, x i Let E[.] represent the received signal of subarray i, and E[.] denotes the expectation.

[0054] The w ci Calculated using the following formula

[0055] w ci =P(θ) / M i (3)

[0056] in 0°≤θ≤180°, d represents the element spacing, M iLet i be the number of elements in subarray i.

[0057] The value of k is calculated using the following formula.

[0058] k = 2πf / c (4)

[0059] Where c represents the speed of sound and f represents the signal frequency.

[0060] Step 3: Based on the known marine environment and assuming that the sound source depth is consistent with the array receiving depth, the sound field is modeled using the Bellhop ray model, and the azimuth estimate distribution θ of different target positions relative to the two subarrays is obtained through simulation. h1 (θ1, r1) and θ h2 (θ2, r2), the horizontal distance r is selected from the center of the first element of each subarray to the sound source, and the angle of the line connecting the sound source and the center of the first element of each subarray is denoted as θ. Where θ1 and θ2 are the azimuth angles of the target relative to subarray 1 and subarray 2, respectively, and r1 and r2 are the horizontal distances of the target from subarray 1 and subarray 2.

[0061] Assume the target's location is L hi =[θ ι r ι ], where θ1 is the azimuth angle of the target relative to subarray i, r i Let be the horizontal distance between the target and subarray i. The target's azimuth estimate is a function of its position.

[0062] θ hi =g(L hi (5)

[0063] Where g(.) is determined by the marine environment and can be calculated using Bellhop ray model simulation, in this invention, θ hi It is a discrete distribution value of azimuth estimation obtained by scanning a finite azimuth-distance space.

[0064] Step 4: Estimate the azimuth angle of the target from Step 2. and The azimuth estimate θ obtained from the sound field modeling in step three is combined with this. h1 and θ h2 To extract the set of possible target locations L1 and L2 relative to the two subarrays, and to establish the cost function.

[0065]

[0066] Where ε represents the error control factor, which is set manually.

[0067] Step 5: Transform the discrete point set L1 and L2 of the target position estimated in Step 4 from the azimuth-range domain to the same Cartesian coordinate system domain using the following formula.

[0068]

[0069] Step Six: For the discrete point sets L1 and L2 in the Cartesian coordinate system domain, use the following formula

[0070] Ax + By + C = 0 (8)

[0071] By performing linear fitting, we obtain the parameters A1, B1, C1 and A2, B2, C2 of the two straight line equations.

[0072] Step 7: Substitute the parameters A1, B1, C1 and A2, B2, C2 obtained in Step 6 into the following formula to calculate the result, which is the positioning result of the method mentioned in this invention.

[0073]

[0074] That is, the target coordinates estimated by this invention are (x e ',y e ').

[0075] 1. Deep-sea environment configuration in specific embodiments

[0076] To verify the effectiveness of the method of the present invention, a computer simulation experiment was conducted. This embodiment considers a typical deep-sea environment, and the Munk sound velocity profile is shown in the attached figure. Figure 1 As shown, the sound speed at the sea surface is 1550 m / s, the axial depth of the sound channel is 1300 m, the sound speed at the axial depth of the sound channel is 1500 m / s, the critical depth is 4430 m, and the sound speed on the seabed is 1552 m / s. Furthermore, the seabed is modeled as a uniform infinite half-space, with a sound speed of 1550 m / s, a density of 1500 kg / m³, and an attenuation coefficient of 0.14 dB / λ.

[0077] 2. Send and receive configuration

[0078] For a large-aperture horizontal linear array, the number of array elements is L = 360, the spacing is d = 4m, and the array depth is z. r =200m. Ignoring port and starboard ambiguity, the starboard direction is defined as 0–180°, and the port direction as 0–-180°. Assume the sound source signal is a single-frequency CW pulse signal with a frequency of f = 125Hz, a pulse length of 1s, and an amplitude of 1. The sound source depth is consistent with the array receiving depth, z. s =200m. The sound ray trajectory diagram obtained by simulation using the Bellhop ray model is shown in the attached figure. Figure 2 As shown, the sound shadow zone is roughly within a range of 10 to 50 km.

[0079] 3. A method for target localization in deep-sea shadow areas, taking into account the direction-finding error of the horizontal array.

[0080] As attached Figure 4 As shown, the specific implementation process of the deep-sea shadow area target positioning method proposed in this invention, which takes into account the horizontal array direction finding error, is as follows:

[0081] Step 1: In a typical deep-sea environment, select the first 91 elements of a large-aperture horizontal array, denoted as subarray 1; and the last 91 elements, denoted as subarray 2. Ignore port and starboard ambiguity, and define the starboard direction as 0–180° and the port direction as 0–-180°. Determine the horizontal distance r from the center of the first element of each subarray to the sound source, and denot θ as the angle between the sound source and the center of the first element of each subarray. At time t, the target is located in subarray 1 at θ1 = 150°, with r1 = 30 km; the target is located in subarray 2 at θ2 = 149.02°, with a distance r2 = 29.14 km. The actual position coordinates at this time are E(25.98, 15) km; the center coordinates (x1, y1) of the first element n1 of subarray 1 are (0, 0) km; the center coordinates of the first element n1 of subarray 2 are... 271 The center location coordinates (x2, y2) are (1.08, 0) km. The orientation diagram is attached. Figure 3 As shown, the array received signals of subarray 1 and subarray 2 are extracted respectively.

[0082] Step 2: Perform conventional beamforming on the received signals of each subarray to obtain the target's azimuth estimate relative to subarray 1 and subarray 2. and

[0083] Conventional beamforming methods can be calculated using the following formula:

[0084]

[0085] Where σ ι R represents the beam direction response of subarray i. xi Let w represent the covariance matrix of the received signal from subarray i. ci Let H represent the beam weighting vector of subarray i, and H denote the transpose sign. The azimuth estimate corresponding to the maximum value of the beam directional response is denoted as . Unless otherwise specified, in this patent, i∈{1,2}.

[0086] The R x It is calculated using the following formula.

[0087] R xi =E[x i H x i (11)

[0088] Where, x i Let E[.] represent the received signal of subarray i, and E[.] denotes the expectation.

[0089] The w ci Calculated using the following formula

[0090] w ci =P(θ) / M i (12)

[0091] in 0°≤θ≤180°, d represents the element spacing, and Mi is the number of elements in subarray i.

[0092] The value of k is calculated using the following formula.

[0093] k = 2πf / c (13)

[0094] Where c = 1530 m / s, f = 125 Hz.

[0095] The output diagrams for conventional beamforming in subarrays 1 and 2 can be obtained from this calculation and are shown in the attached diagram. Figure 5 As shown. The location of the maximum value of the target response can be obtained in... At this point, traditional methods assume that the target's true location is at 145.65° in subarray 1 and at 144.45° in subarray 2.

[0096] Step 3: Based on the known marine environment and assuming that the sound source depth is consistent with the array receiving depth, the sound field is modeled using the Bellhop ray model, and the azimuth estimate distribution θ of different target positions relative to the two subarrays is obtained through simulation. h1 (θ1, r1) and θ h2 (θ2, r2), where θ1 and θ2 are the azimuth angles of the target relative to subarray 1 and subarray 2, respectively, and r1 and r2 are the horizontal distances of the target from subarray 1 and subarray 2.

[0097] Assume the target's location is L hi =[θ ι r ι ], where θ1 is the azimuth angle of the target relative to subarray i, r i Let be the horizontal distance between the target and subarray i. The target's azimuth estimate is a function of its position.

[0098] θ hi =g(L hi (14)

[0099] Where g(.) is determined by the marine environment and can be calculated using Bellhop ray model simulation, in this invention, θ hi It is a discrete distribution value of azimuth estimation obtained by scanning a finite azimuth-distance space.

[0100] Step 4: Estimate the azimuth angle of the target from Step 2. The azimuth estimate θ obtained from the sound field modeling in step three is combined with this. h1 and θ h2 To extract the set of possible target locations L1 and L2 relative to the two subarrays, and to establish the cost function.

[0101]

[0102] Where i∈{1,2}, ε represents the error control factor, and ε=0.001 is taken.

[0103] Step 5: Transform the discrete point set L1 and L2 of the target position estimated in Step 4 from the azimuth-range domain to the same Cartesian coordinate system domain using the following formula.

[0104]

[0105] Step Six: For the discrete point sets L1 and L2 in the Cartesian coordinate system domain, use the following formula

[0106] Ax + By + C = 0 (17)

[0107] Linear fitting is performed to obtain the parameters A1, B1, C1, A2, B2, and C2 of the two line equations. The specific process is as follows:

[0108] The transformed result was linearly fitted using Ax + By + C = 0, yielding two linear equations with parameters A1 = 0.7748, B1 = -1, C1 = -5.1464; and A2 = 0.8065, B2 = -1, C2 = -5.9532. The graphs of the two lines are attached. Figure 6 As shown in (a) and (b).

[0109] Step 7: Substitute the parameters A1, B1, C1, A2, B2, and C2 obtained in Step 6 into the following formula to calculate the positioning result, which is the positioning result of the method mentioned in this invention.

[0110]

[0111] Substituting the obtained straight line equation parameters into the equation yields the target coordinates estimated by this invention.

[0112] x e = 25.72km, y e = 14.80km.

[0113] The location results are attached. Figure 7 As shown, there is one and only one intersection point, with coordinates E. *The actual target coordinates are E(25.72, 14.80) km, while the true target coordinates are E(25.98, 15) km. The corrected ranging error is 0.33 km. Before correction, when using the two-station direction finding cross-positioning method, the target ranging error was too large, i.e., using the following formula...

[0114]

[0115] The cross-location result obtained from the two-station direction finding is E(22.92, 15.7) km, with a ranging error of 3.14 km. This demonstrates that the present invention effectively improves the target positioning accuracy.

Claims

1. A method for locating deep-sea targets in shadow areas, taking into account the direction-finding error of the horizontal array, characterized in that... The steps are as follows: Step 1: Using conventional beamforming methods, beamforming is performed on the array received signals of the two subarrays 1 and 2 of the large-aperture horizontal line array to obtain the target azimuth estimate. and ; The number of elements in the large-aperture horizontal linear array is L The spacing is d The array element numbers are as follows: n l , l =1~ L The element closest to the mother ship is element number 1, denoted as... n 1. The array element furthest from the mothership is defined as... n L ; Take the head M One array element is defined as subarray 1, and the array element number is... Take the tail section M Two array elements, defined as subarray 2, with element numbers as follows: ; Step 2: Based on the marine environment, the sound field is modeled using the Bellhop ray model, and the azimuth estimates of different target positions relative to the two subarrays are obtained through simulation. ; The horizontal distance is defined as the distance from the center of the first element of each subarray to the sound source. r Let the angle between the sound source and the center of the first element of each subarray be denoted as . ,in and These are the azimuth angles of the target relative to subarray 1 and subarray 2, respectively. r 1 and r 2 represents the horizontal distance between the target and subarrays 1 and 2. When the target's position is L hi = At that time, the target's azimuth estimate is a function of the target's position. ,in For the target relative to the subarray i azimuth angle, r i For the target distance subarray i Horizontal distance; Step 3: Estimate the angle based on the target's azimuth. and Distribution of azimuth estimates obtained by combining sound field modeling Extract the target location sets L1 and L2 relative to the two subarrays, and establish the cost function: in, Indicates the error control factor; Step 4: Transform the target location point sets L1 and L2 from the azimuth-range domain to the same Cartesian coordinate system domain; Step 5: Utilize the target location point set L1 and L2 transformed in Step 4... Ax + By + C =0, perform linear fitting to obtain the parameters of the two line equations. A 1, B 1, C 1 and A 2, B 2, C 2; Step 6: Obtain A 1, B 1, C 1 and A 2, B 2, C The two parameters, when substituted into the following formula, yield the positioning result: The estimated target coordinates are .

2. The deep-sea shadow area target localization method considering horizontal array direction finding errors according to claim 1, characterized in that: The number of elements in the large-aperture horizontal linear array is L The spacing is d The array element numbers are as follows: n l , l =1~ L The element closest to the mother ship is element number 1, denoted as... n 1. The array element furthest from the mothership is defined as... n L The starboard direction is 0°~180°, and the port direction is -180°~0°.

3. The deep-sea shadow area target localization method considering horizontal array direction finding errors according to claim 1, characterized in that: The conventional beamforming method is calculated using the following formula: in Subarray i Beam direction response, R xi Subarray i The covariance matrix of the received signal, w ci Subarray i The beam weighting vector, H Indicates the transpose sign; the azimuth estimate corresponding to the maximum value of the beam directional response is . ; .

4. The deep-sea shadow area target positioning method considering horizontal array direction finding errors according to claim 3, characterized in that: The covariance matrix R xi =E[ x i H x i ],in, x i For the sub-array i The received signal, E[.] represents the expectation.

5. The deep-sea shadow area target positioning method considering horizontal array direction finding errors according to claim 3, characterized in that: The beam weighting vector , in , , d Indicates the spacing between array elements. M i For the sub-array i The number of array elements; the ,in, c Indicates the speed of sound. f Indicates the signal frequency.

6. The deep-sea shadow area target localization method considering horizontal array direction finding errors according to claim 1, characterized in that: The g (.) was obtained through Bellhop simulation using the ray model and conventional beamforming methods. The results will be obtained through ray model simulation and conventional beamforming methods. The function is called function.

7. The deep-sea shadow area target localization method considering horizontal array direction finding errors according to claim 1, characterized in that: The error control factor .

8. The deep-sea shadow area target localization method considering horizontal array direction finding errors according to claim 1, characterized in that: The formula for transforming the target location point sets L1 and L2 from the azimuth-range domain to the same Cartesian coordinate system domain in step 4 is as follows:

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