Frequency expansion method and device based on fast sparse inversion, and readable storage medium

Abstract

The invention provides a frequency expansion method and device based on fast sparse inversion and a readable storage medium, and belongs to the technical field of seismic data processing. The method comprises the following steps: acquiring pre-stack seismic data; filtering the pre-stack seismic data by using a filtering operator to obtain filtered pre-stack seismic data; fourier transform is carried out on the filtered pre-stack seismic data to obtain transformed pre-stack seismic data; based on the filtered pre-stack seismic data and a sparse inversion algorithm, constructing a frequency inversion objective function by taking the transformed pre-stack seismic data as a constraint condition; and iterating the frequency inversion objective function by using a Lagrange multiplier method, updating the filtered pre-stack seismic data until the update amount of the updated filtered pre-stack seismic data is smaller than a preset update threshold or the number of iterations reaches a preset iteration threshold, and outputting the updated filtered pre-stack seismic data. The method has the advantages that the frequency inversion precision and the data signal-to-noise ratio are high, the calculation amount in the data processing process can be reduced, and the calculation cost is reduced.

Description

Technical Field

[0001] The present invention relates to the technical field of seismic data processing, and in particular to a frequency expansion method based on fast sparse inversion, a frequency expansion device based on fast sparse inversion, an electronic device and a readable storage medium. Background Art

[0002] In the process of land seismic data acquisition, the collected seismic data usually loses low-frequency components due to the filtering effect of the earth and the limitations of the artificial seismic source itself, as well as interference such as detector distortion and surface waves. In the processing of marine data, the data lacks low-frequency and some high-frequency information due to the lack of low-frequency components in the seismic source itself and the low-frequency response of the detector, as well as interference such as the ghost wave effect. The lack of low-frequency components will make the full waveform inversion results prone to local minima, greatly reducing the accuracy of seismic signal processing and interpretation. The lack of high-frequency components will also reduce the resolution of processing. Therefore, there is an urgent need for a new technology to perform low-frequency and high-frequency extension of seismic data to improve the quality of subsequent data processing. Summary of the invention

[0003] The purpose of the embodiments of the present invention is to provide a frequency expansion method, device and readable storage medium based on fast sparse inversion, so as to at least solve the above-mentioned problems that the lack of low-frequency components in seismic data will make the full waveform inversion result prone to local minima, greatly reducing the accuracy of seismic signal processing and interpretation, and the lack of high-frequency components will reduce the processing resolution.

[0004] In order to achieve the above object, the first aspect of the present invention provides a frequency extension method based on fast sparse inversion, the method comprising: Acquire pre-stack seismic data; Filtering the pre-stack seismic data using a pre-constructed filter operator to obtain filtered pre-stack seismic data; Performing Fourier transformation on the filtered pre-stack seismic data to obtain transformed pre-stack seismic data; Based on the filtered prestack seismic data and the sparse inversion algorithm, a frequency inversion objective function is constructed with the transformed prestack seismic data as a constraint condition; The frequency inversion objective function is iteratively solved using the Lagrange multiplier method, and the filtered pre-stack seismic data is updated until the update amount of the updated filtered pre-stack seismic data is less than a preset update threshold or the number of iterations reaches a preset iteration threshold, and the updated filtered pre-stack seismic data is output as the pre-stack seismic data after frequency expansion.

[0005] Optionally, before filtering the pre-stack seismic data using a pre-constructed filtering operator to obtain filtered pre-stack seismic data, the method further includes: Based on the effective frequency band of pre-stack seismic data, the frequency band where the noise is located and the frequency band to be inverted, a filtering operator is constructed; The expression of the filtering operator is: .

[0006] Optionally, it is characterized in that performing Fourier transform on the filtered pre-stack seismic data to obtain transformed pre-stack seismic data comprises: The transformed pre-stack seismic data is obtained using the following calculation formula: ;

[0007] in, is the transformed pre-stack seismic data; is the filtered pre-stack seismic data; n is the temporal number of the seismic data sampling points; j is an imaginary unit; k is the number of seismic data in the frequency domain, and .

[0008] Optionally, the sparse inversion algorithm is an L1 sparse inversion algorithm; The Lagrange multiplier method is a double augmented Lagrange multiplier method.

[0009] Optionally, the expression of the frequency inversion objective function is: ;

[0010] in, is the frequency inversion objective function; D is the data matrix of the filtered pre-stack seismic data.

[0011] Optionally, the constraint condition of the frequency inversion objective function is: ;

[0012] ;

[0013] ;

[0014] ;

[0015] in, A is the discrete Fourier transform matrix; D is the data matrix of filtered pre-stack seismic data; B is the data matrix of the transformed pre-stack seismic data; j Is an imaginary unit.

[0016] Optionally, the filtered pre-stack seismic data is updated using the following formula: ;

[0017] in, For the m The data matrix of filtered pre-stack seismic data after iterations; is the sparse inversion parameter; is the preset vector; For the m -Data matrix of filtered prestack seismic data after 1 iteration; For the general All values ​​greater than 1 are set to 1; is the inverse Fourier transform; is the Fourier forward transform; are fitting parameters, and ; B is the data matrix of the transformed pre-stack seismic data; N is the latitude of pre-stack seismic data, is a constant.

[0018] A second aspect of the present invention provides a frequency extension device based on fast sparse inversion, the device comprising: A data acquisition module, used for acquiring pre-stack seismic data; A data filtering module, used for filtering the pre-stack seismic data using a pre-built filtering operator to obtain filtered pre-stack seismic data; A data denoising module performs Fourier transform on the filtered pre-stack seismic data to obtain transformed pre-stack seismic data; An objective function establishment module is used to construct a frequency inversion objective function based on the filtered prestack seismic data and the sparse inversion algorithm, taking the transformed prestack seismic data as a constraint condition; The frequency expansion module is used to iteratively solve the frequency inversion objective function using the Lagrange multiplier method and update the filtered pre-stack seismic data until the update amount of the updated filtered pre-stack seismic data is less than a preset update threshold or the number of iterations reaches a preset iteration threshold, and then output the updated filtered pre-stack seismic data as the pre-stack seismic data after frequency expansion.

[0019] A third aspect of the present invention provides an electronic device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the above-mentioned frequency expansion method based on fast sparse inversion when executing the computer program.

[0020] On the other hand, the present invention further provides a readable storage medium having instructions stored thereon, the instructions being used to enable a machine to execute the above-mentioned frequency expansion method based on fast sparse inversion.

[0021] This technical solution can extend the frequency of missing data while keeping the original frequency components of the data unchanged. It has high frequency inversion accuracy and data signal-to-noise ratio, and can reduce the amount of calculation in the data processing process and reduce the calculation cost.

[0022] Other features and advantages of the embodiments of the present invention will be described in detail in the following detailed description. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] The accompanying drawings are used to provide a further understanding of the embodiments of the present invention and constitute a part of the specification. Together with the following specific embodiments, they are used to explain the embodiments of the present invention, but do not constitute a limitation on the embodiments of the present invention. In the accompanying drawings: Figure 1 is a flow chart of a frequency expansion method based on fast sparse inversion provided by the present invention; Figure 2 is a schematic diagram of pre-stack data and its spectrum provided by the present invention; Figure 3 is a schematic diagram of the pre-stack data and its spectrum after inversion provided by the present invention; Figure 4 is a schematic diagram of pre-stack data with noise and its spectrum provided by the present invention; Figure 5 It is a schematic diagram of pre-stack data and its spectrum after the noise band is padded provided by the present invention; Figure 6 It is a schematic diagram of single-shot model data and its spectrum provided by the present invention; Figure 7 is a schematic diagram of the filtered single-shot model data and its spectrum provided by the present invention; Figure 8 It is a schematic diagram of the single-shot model data and its spectrum after frequency expansion provided by the present invention; Fig. 9 It is a schematic diagram comparing the low-frequency component of the original data provided by the present invention and the low-frequency component after extension of the present solution; Fig.10 It is a structural schematic diagram of a frequency expansion device based on fast sparse inversion provided by an embodiment of the present invention.

[0024] Description of Reference Numerals 10-data acquisition module; 20-data filtering module; 30-data denoising module; 40-adjustment feedback device; 50-frequency extension module. DETAILED DESCRIPTION

[0025] The specific implementation of the present invention is described in detail below in conjunction with the accompanying drawings. It should be understood that the specific implementation described here is only used to illustrate and explain the present invention, and is not used to limit the present invention.

[0026] Figure 1 is a flow chart of a frequency expansion method based on fast sparse inversion provided by the present invention; Figure 2 is a schematic diagram of pre-stack data and its spectrum provided by the present invention; Figure 3 is a schematic diagram of the pre-stack data and its spectrum after inversion provided by the present invention; Figure 4 is a schematic diagram of pre-stack data with noise and its spectrum provided by the present invention; Figure 5 It is a schematic diagram of pre-stack data and its spectrum after the noise band is padded provided by the present invention; Figure 6 It is a schematic diagram of single-shot model data and its spectrum provided by the present invention; Figure 7 is a schematic diagram of the filtered single-shot model data and its spectrum provided by the present invention; Figure 8 It is a schematic diagram of the single-shot model data and its spectrum after frequency expansion provided by the present invention; Fig. 9 It is a schematic diagram comparing the low-frequency component of the original data provided by the present invention and the low-frequency component after extension of the present solution; Fig.10 It is a structural schematic diagram of a frequency expansion device based on fast sparse inversion provided by an embodiment of the present invention.

[0027] like Figure 1 As shown, an embodiment of the present invention provides a frequency extension method based on fast sparse inversion, the method comprising: Step 1: Obtain pre-stack seismic data; Specifically, in this embodiment, Figure 2As shown in (a), pre-stack seismic data refers to the original seismic record data obtained in seismic exploration. These data are obtained by sending source signals underground through seismic instruments and then receiving signals reflected and refracted by underground rocks. Pre-stack seismic data contains reflection and refraction information of underground rocks and can be used to study the distribution of underground structures, lithology, oil and gas resources, etc. Processing of pre-stack seismic data is one of the important steps in seismic exploration. During the processing, the original data needs to be denoised first to remove noise interference in the signal. Then, time-distance conversion is required to convert the time information in the seismic record into underground depth information. Next, stacking processing is required to stack multiple seismic records together to enhance the intensity of underground reflection information. Finally, velocity model establishment and migration processing are required to obtain more accurate underground structural information. Through the processing of pre-stack seismic data, seismic exploration personnel can obtain detailed information on underground structures, providing important basis for oil and gas exploration, geological disaster prediction, etc. At the same time, pre-stack seismic data can also be used for earthquake research and earthquake monitoring, helping to understand the laws and trends of earthquake activities and improve the accuracy of earthquake warning.

[0028] Step 2: filtering the pre-stack seismic data using a pre-constructed filter operator to obtain filtered pre-stack seismic data; In this embodiment, based on the effective frequency band of pre-stack seismic data, the frequency band where the noise is located and the frequency band to be inverted, a filter operator is constructed, and denoising is performed by the filter operator. The filter operator acts on the frequency domain data to filter out the frequency band containing noise in the data and suppress the abnormal energy noise to ensure the accuracy of the data. The frequency value of the effective signal is set to 1, and the frequency part of the noise is set to 0. The expression of the filtering operator is: .

[0029] Step 3, performing Fourier transform on the filtered pre-stack seismic data to obtain transformed pre-stack seismic data; In this embodiment, the following calculation formula is used to obtain the transformed pre-stack seismic data: ;

[0030] in, is the transformed pre-stack seismic data, such as Figure 2 (b) shows the spectrum of the prestack seismic data after transformation; is the filtered pre-stack seismic data, such as Figure 2 (a) n is the temporal number of the seismic data sampling points; j is an imaginary unit; k is the number of seismic data in the frequency domain, and .

[0031] Step 4: constructing a frequency inversion objective function based on the filtered prestack seismic data and the sparse inversion algorithm and taking the transformed prestack seismic data as a constraint condition; In this embodiment, the sparse inversion algorithm is an L1 sparse inversion algorithm, which has the advantages of high inversion accuracy, less multi-solution, and low requirements on the number of well logging and distribution uniformity.

[0032] Among them, the expression of the frequency inversion objective function is: ;

[0033] in, is the frequency inversion objective function; D is the data matrix of the filtered pre-stack seismic data.

[0034] The above method is used to solve the value of the minimum solution of the L1 norm of the original data, that is, to solve the sparsest solution of data D to invert the frequency of missing data. When the data D is sparser, the waveform of the seismic record is closer to the unit pulse and its frequency band is wider.

[0035] In this embodiment, the constraints of the frequency inversion objective function are: ;

[0036] ;

[0037] ;

[0038] ;

[0039] in, A is the discrete Fourier transform matrix, and the fast Fourier transform can be used to reduce the computational complexity from Reduce to , reduce the amount of calculation and improve the calculation efficiency; D is the data matrix of filtered pre-stack seismic data; B is the data matrix of the transformed pre-stack seismic data; is the transformed pre-stack seismic data; is the filtered pre-stack seismic data; n is the temporal number of the seismic data sampling points; j Is an imaginary unit.

[0040] Specifically, As a constraint condition of the frequency inversion objective function, it ensures that the effective frequency of the original data conforms to the relationship between the forward and inverse Fourier transforms and does not change the signal of the original data.

[0041] Step 5: Use the Lagrange multiplier method to iteratively solve the frequency inversion objective function and update the filtered pre-stack seismic data until the update amount of the updated filtered pre-stack seismic data is less than a preset update threshold or the number of iterations reaches a preset iteration threshold, and then output the updated filtered pre-stack seismic data as the pre-stack seismic data after frequency expansion.

[0042] Specifically, in this embodiment, the Lagrangian multiplier method is a double augmented Lagrangian multiplier method. There are many methods for L1 inversion in the prior art, among which the augmented Lagrangian method (ALM) has become a popular choice due to its robustness and low computational cost per iteration. However, the ALM usually requires a large number of iterations to converge. In contrast, the double augmented Lagrangian method (DALM) requires fewer iterations to converge, but its cost is higher due to the need to perform matrix inversion.

[0043] In addition, the following formula is used to update the filtered prestack seismic data: ;

[0044] in, For the m The data matrix of filtered pre-stack seismic data after iterations; is the sparse inversion parameter; is the preset vector; For the m -Data matrix of filtered prestack seismic data after 1 iteration; For the general All values ​​greater than 1 are set to 1, and the original signs are retained; is the inverse Fourier transform; is the Fourier forward transform; is the fitting parameter, and ; B is the data matrix of the transformed pre-stack seismic data; N is the latitude of pre-stack seismic data, is a constant.

[0045] By the above scheme, under the premise of ensuring the accuracy of frequency inversion, the computational cost of DALM L1 inversion is significantly reduced. Thereby, the elimination of noise frequency in the data is achieved, and the missing frequency is inverted based on L1 time domain sparsity. From the three examples provided in this application, it can be clearly seen that when the data meets the time domain sparsity assumption, the frequency of missing data can be extended well. When the DALM algorithm is used to realize frequency inversion, the amount of calculation in the iterative process can be reduced by optimizing the matrix operation. The constraint condition used when inverting the missing frequency is to keep the original frequency component of the data unchanged, so this technology can protect the original frequency component well. Before frequency inversion, the original data is filtered to suppress the abnormal energy noise in the data. In the inversion process, the effective frequency destroyed by the filtering effect can also be extended, so this technology can improve the data signal-to-noise ratio to a certain extent.

[0046] The present invention demonstrates that in this specific case, the Fast Fourier Transform (FFT) can be used to bypass the matrix inversion step (usually complexity), and the complexity of the corresponding matrix multiplication process in each iteration is changed from Reduce to , where N represents the number of samples.

[0047] In this embodiment, Figure 3 The results are shown after the missing frequencies are expanded based on the fast sparse inversion frequency. Figure 3 (a) It can be seen that the phase axis of the seismic data is suppressed. Figure 3 (b) It can be seen that the processed spectrum can not only expand the high-frequency components, but also remain completely consistent with the effective frequency band range of the original data. This is because the constraints used in this technology ensure that the original frequency of the data can always conform to the Fourier transform relationship.

[0048] By using model data with abnormal noise Figure 4 (a) To test, from the original data spectrum, such as Figure 4 (b) shows that the input data lacks low-frequency data below 4Hz and high-frequency data above 16Hz, and contains abnormal energy noise in the range of 9-12Hz. After being processed by this technology, the test results can be seen, such as Figure 5 (a) and Figure 5 As shown in (b), it can not only invert the missing low-frequency and high-frequency components, but also has a good suppression effect on abnormal energy noise.

[0049] In order to test the applicability of this method, more complex model data tests were added.

[0050] Figure 6 (a) is a simulated single shot record. Figure 6The blue solid line in (b) is the spectrum of the model data. A high-pass filter is used to filter out the low-frequency components below 5 Hz in the original data. Figure 7 As shown in (a), Figure 7 The green solid line in (b) is the spectrum of the model data after filtering. This technology is used to extend the low-frequency data below 5 Hz. Figure 8 (a) is the single shot data after frequency extension. Figure 8 The red solid line in (b) is the spectrum of the data after frequency extension. From the spectrum, we can see that the extended low frequency is very close to the low frequency of the original data, which verifies the correctness of this technology. Fig. 9 (a) and the low-frequency components inverted by this technique Fig. 9 (b) It can be seen that the two are also very close in the gun records, which further verifies the correctness of this method.

[0051] like Fig.10 As shown, an embodiment of the present invention provides a frequency extension device based on fast sparse inversion, the device comprising: A data acquisition module 10 is used to acquire pre-stack seismic data; A data filtering module 20, configured to filter the pre-stack seismic data using a pre-built filtering operator to obtain filtered pre-stack seismic data; A data denoising module 30 performs Fourier transformation on the filtered pre-stack seismic data to obtain transformed pre-stack seismic data; An objective function building module 40 is used to build a frequency inversion objective function based on the filtered prestack seismic data and the sparse inversion algorithm, taking the transformed prestack seismic data as a constraint condition; The frequency expansion module 50 is used to iteratively solve the frequency inversion objective function using the Lagrange multiplier method and update the filtered pre-stack seismic data until the update amount of the updated filtered pre-stack seismic data is less than a preset update threshold or the number of iterations reaches a preset iteration threshold, and then output the updated filtered pre-stack seismic data as the pre-stack seismic data after frequency expansion.

[0052] An embodiment of the present invention further provides an electronic device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the above-mentioned frequency expansion method based on fast sparse inversion when executing the computer program.

[0053] An embodiment of the present invention further provides a readable storage medium, on which instructions are stored, and the instructions are used to enable a machine to execute the above-mentioned frequency expansion method based on fast sparse inversion.

[0054] Those skilled in the art will understand that all or part of the steps in the method for implementing the above-mentioned embodiments can be completed by instructing the relevant hardware through a program, and the program is stored in a storage medium, including several instructions for making a single-chip microcomputer, a chip or a processor execute all or part of the steps of the method described in each embodiment of the present invention. The aforementioned storage medium includes: U disk, mobile hard disk, read-only memory (ROM, Read-Only Memory), random access memory (RAM, Random Access Memory), disk or optical disk and other media that can store program codes.

[0055] Those skilled in the art can clearly understand that for the convenience and simplicity of description, only the division of the above-mentioned functional units and modules is used as an example. In actual applications, the above-mentioned functions can be assigned to different functional units and modules as needed, that is, the internal structure of the device can be divided into different functional units or modules to complete all or part of the functions described above. The functional units and modules in the embodiments can be integrated into a processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The above-mentioned integrated unit can be implemented in the form of hardware or in the form of software functional units. In addition, the specific names of the functional units and modules are only for the convenience of distinguishing each other, and are not used to limit the scope of protection of this application.

[0056] The optional embodiments of the present invention are described in detail above in conjunction with the accompanying drawings. However, the embodiments of the present invention are not limited to the specific details in the above embodiments. Within the technical concept of the embodiments of the present invention, the technical scheme of the embodiments of the present invention can be subjected to a variety of simple modifications, and these simple modifications all belong to the protection scope of the embodiments of the present invention. It should also be noted that the various specific technical features described in the above specific embodiments can be combined in any suitable manner without contradiction. In order to avoid unnecessary repetition, the embodiments of the present invention will not further describe various possible combinations.

[0057] In addition, various embodiments of the present invention may be arbitrarily combined, and as long as they do not violate the concept of the embodiments of the present invention, they should also be regarded as the contents disclosed in the embodiments of the present invention.

Claims

1. A frequency expansion method based on fast sparse inversion, characterized in that: The method comprises: Acquire pre-stack seismic data; Filtering the pre-stack seismic data using a pre-constructed filter operator to obtain filtered pre-stack seismic data; Performing Fourier transformation on the filtered pre-stack seismic data to obtain transformed pre-stack seismic data; Based on the filtered prestack seismic data and the sparse inversion algorithm, a frequency inversion objective function is constructed with the transformed prestack seismic data as a constraint condition; The frequency inversion objective function is iteratively solved using the Lagrange multiplier method, and the filtered pre-stack seismic data is updated until the update amount of the updated filtered pre-stack seismic data is less than a preset update threshold or the number of iterations reaches a preset iteration threshold, and the updated filtered pre-stack seismic data is output as the pre-stack seismic data after frequency expansion.

2. The frequency expansion method based on fast sparse inversion according to claim 1, characterized in that: Before filtering the pre-stack seismic data using a pre-constructed filter operator to obtain filtered pre-stack seismic data, the method further includes: Based on the effective frequency band of pre-stack seismic data, the frequency band where the noise is located and the frequency band to be inverted, a filtering operator is constructed; The expression of the filtering operator is: 。 3. The frequency expansion method based on fast sparse inversion according to claim 1, characterized in that: Performing Fourier transformation on the filtered pre-stack seismic data to obtain transformed pre-stack seismic data includes: The transformed pre-stack seismic data is obtained using the following calculation formula: ； in, is the transformed pre-stack seismic data; is the filtered pre-stack seismic data; n is the temporal number of the seismic data sampling points; j is an imaginary unit; k is the number of seismic data in the frequency domain, and .

4. The frequency expansion method based on fast sparse inversion according to claim 1, characterized in that: The sparse inversion algorithm is an L1 sparse inversion algorithm; The Lagrange multiplier method is a double augmented Lagrange multiplier method.

5. The frequency expansion method based on fast sparse inversion according to claim 1, characterized in that: The expression of the frequency inversion objective function is: ； in, is the frequency inversion objective function; D is the data matrix of the filtered pre-stack seismic data.

6. The frequency expansion method based on fast sparse inversion according to claim 1, characterized in that: The constraints of the frequency inversion objective function are: ； ； ； ； in, A is the discrete Fourier transform matrix; D is the data matrix of filtered pre-stack seismic data; B is the data matrix of the transformed pre-stack seismic data; is the transformed pre-stack seismic data; is the filtered pre-stack seismic data; n is the temporal number of the seismic data sampling points; j Is an imaginary unit.

7. The frequency expansion method based on fast sparse inversion according to claim 1, characterized in that: The following formula is used to update the filtered prestack seismic data: ； in, For the m The data matrix of filtered pre-stack seismic data after iterations; is the sparse inversion parameter; is the preset vector; For the m -Data matrix of filtered prestack seismic data after 1 iteration; For the general All values ​​greater than 1 are set to 1; is the inverse Fourier transform; is the Fourier forward transform; are fitting parameters, and ; B is the data matrix of the transformed pre-stack seismic data; N is the latitude of prestack seismic data; is a constant.

8. A frequency expansion device based on fast sparse inversion, characterized in that: The device comprises: A data acquisition module, used for acquiring pre-stack seismic data; A data filtering module, used for filtering the pre-stack seismic data using a pre-built filtering operator to obtain filtered pre-stack seismic data; A data denoising module performs Fourier transformation on the filtered pre-stack seismic data to obtain transformed pre-stack seismic data; An objective function establishment module is used to construct a frequency inversion objective function based on the filtered prestack seismic data and the sparse inversion algorithm, taking the transformed prestack seismic data as a constraint condition; The frequency expansion module is used to iteratively solve the frequency inversion objective function using the Lagrange multiplier method and update the filtered pre-stack seismic data until the update amount of the updated filtered pre-stack seismic data is less than a preset update threshold or the number of iterations reaches a preset iteration threshold, and output the updated filtered pre-stack seismic data as the pre-stack seismic data after frequency expansion.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the computer program, the frequency expansion method based on fast sparse inversion described in any one of claims 1 to 7 is implemented.

10. A readable storage medium having instructions stored thereon, the instructions being used to enable a machine to execute the frequency expansion method based on fast sparse inversion as described in any one of claims 1 to 7.

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