CN110764135A - Irregular seismic data full-band reconstruction method - Google Patents
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Abstract
The invention discloses an irregular seismic data full-band reconstruction method, which comprises the following steps: step 1: collecting original irregular seismic data; step 2: obtaining a frequency domain data volume; and step 3: obtaining a frequency wave number domain data volume; and 4, step 4: determining a dominant frequency band and a threshold value; and 5: setting the components with the frequency spectrum amplitude smaller than the threshold value to zero, weighting in a frequency wave number domain, and performing Fourier inversion in a spatial direction to obtain a frequency domain reconstruction data volume; step 6: replacing the sampled data with a frequency domain data body to obtain a frequency domain updating data body; and 7: repeating the steps 3 to 6 on the frequency domain updating data body until the iteration times or the error termination condition is met; and 8: and iterating all frequency components, and outputting a time-space domain reconstruction gather after final interpolation. The method of the invention not only ensures the calculation precision, but also reduces the calculation amount, protects the weak reflection signals in the seismic data, and effectively improves the imaging quality of the small fault and the micro-garment structure.
Description
Technical Field
The invention belongs to the technical field of seismic data prestack preprocessing, and particularly relates to an irregular seismic data full-frequency band reconstruction method.
Background
The actual field seismic data acquisition process is limited by acquisition cost and complex surface conditions, the seismic data are often sparsely and irregularly sampled in the spatial direction, and the irregularly acquired data can cause the loss of underground response information so as to influence the quality of migration imaging.
Aiming at the regularization problem of seismic data, domestic and foreign scholars carry out deep research, for example, Spitz and the like research a single-step prediction filtering interpolation algorithm in a frequency domain, and the low-frequency part of the data is utilized to realize the anti-aliasing interpolation of high frequency; jakubotz and the like utilize a Sinc function to perform seismic channel interpolation and data reconstruction; kao et al discuss non-uniform sampling seismic data interpolation methods with aliasing; gulunay et al propose a generalized frequency wavenumber domain seismic data interpolation method; duijndam et al have studied the irregular seismic data reconstruction method based on non-uniform Fourier transform, can be suitable for the arbitrary irregular acquisition data; the MWNI (minimum Weighted Norm interpolation) method proposed by Liu et al assumes that seismic data sampling is random and regularizes the bandlimited seismic signals from an inverse problem; xu and the like propose an anti-leakage Fourier transform (ALFT) regularization algorithm aiming at the problem of frequency spectrum leakage existing in irregular acquisition; abma et al introduce an anti-aliasing Convex set projection POCS (projection Onto transform sets) method into seismic data regularization processing from the image processing field, and the method is an iterative threshold interpolation method based on Fourier transform.
The research of the interpolation algorithm promotes the development of sparse irregularly acquired seismic data reconstruction technology, but the interpolation algorithm generally needs to provide a threshold value to limit the frequency and wave number range of data to realize interpolation, so that the calculated amount can be greatly reduced, but the energy of weak reflection and weak signals is ignored, potential damage is caused to fault and micro-suit structures, and the fine depiction and description of fracture-cavity reservoirs and complex fracture systems are not facilitated.
Disclosure of Invention
In order to solve the problems, the invention provides an irregular seismic data full-band reconstruction method which can solve the reconstruction problem of weak signals and weak reflection data in sparse irregular sampling seismic data.
The invention provides an irregular seismic data full-band reconstruction method, which specifically comprises the following steps:
step 1: acquiring original irregular seismic data d (t, x);
step 2: carrying out Fourier transform on the original irregular seismic data d (t, x) in the time direction to obtain a frequency domain data volume d (f, x);
and step 3: then, Fourier transform in the space direction is carried out to obtain a frequency wave number domain data volume di(f,k)；
And 4, step 4: determining a dominant frequency band and a threshold value in a full frequency band wave number range;
and 5: setting the components with the frequency spectrum amplitude smaller than the threshold value to zero, weighting the reset frequency spectrum amplitude in a frequency wave number domain, and then performing Fourier inversion in the space direction to obtain a frequency domain reconstruction data volume drec(f,x)；
Step 6: reconstructing the frequency domain into a data volume drecThe sampled data in (f, x) is replaced by the original frequency domain data body d (f, x), and a frequency domain updated data body d is obtainedi(f,x)；
And 7: updating the frequency domain to data volume di(f, x) repeating the steps 3 to 6 until the iteration number or the error termination condition is met;
and 8: and (4) performing iterative processing on all frequency components, performing inverse Fourier transform in the time direction, and outputting a time-space domain reconstruction gather d (t, x) after final interpolation.
In one embodiment, the original seismic data actually acquired in step 1 is a sparse representation of ideal complete data, and the relationship between the original seismic data and the ideal complete data is as follows:
dobs＝Rd
wherein d isobsD is ideal complete data and R is a sparse matrix for actually acquiring seismic data; the reconstruction of the seismic data is to acquire data d from sparse irregularityobsAnd d is recovered.
In one embodiment, in step 2 and step 3, the original irregular seismic data d (t, x) is subjected to forward and backward fourier transform, and the formula is as follows:
wherein the content of the first and second substances,
fourier transform coefficients being frequency components w; Δ X is the sum of the sampling intervals, Δ X ═ Σ Δ Xl,xlTime sampling points; l represents the l-th sampling point; f (x)l) Time domain seismic data; f. ofw(xl) Fourier transform coefficients for frequency component w versus time sample point xlThe contribution of (1); j represents an imaginary symbol; and N is the total number of sampling points.
In one embodiment, in step 4, the amplitude integrated energy is analyzed in the full-band wavenumber range, so as to determine the dominant band.
In one embodiment, in step 4, the full-band wavenumbers are grouped according to the number of iterations, and the threshold τ of each iteration is determinedi。
In one embodiment, in step 4, the linear attenuation threshold τ under the current iteration condition is set according to the integral of the amplitude energy from large to smalliWherein, the iteration format is as follows:
di+1＝(I-RTR)F-1τiFdi+d0
wherein d isi+1An interpolation data volume after the (i + 1) th iteration; i is an identity matrix; rTTranspose for sparse matrix; f and F-1Respectively carrying out Fourier forward and inverse transformation; tau isiIs the threshold value of the ith iteration of the frequency wave number domain, namely taui(f,k)；diThe data volume after the ith iteration is taken as the data volume; d0Is an iteration initial value, where d0＝RTdobs。
In one embodiment, the amplitude energy integral sets a linear attenuation threshold from large to small, which is expressed as:
wherein: tau ismax(f, k) is a maximum threshold determined from the amplitude energy integral; tau ismin(f, k) is a minimum threshold determined from the amplitude energy integral; tau isiterAnd (f, k) is a frequency wave number domain threshold value of the iter iteration of the current frequency, and the nter is the total iteration number.
In one embodiment, in step 5, in order to suppress the interference of the alias to the wave number spectrum, the component with the spectrum amplitude smaller than the threshold value is set to zero, the reconstructed spectrum is weighted in the frequency wave number domain, and then the inverse fourier transform is performed in the frequency wave number domain to obtain the frequency domain reconstructed data volume drec(f,x)；
The inverse Fourier transform formula after weighting in the frequency wave number domain is as follows:
wherein k is the wave number; NK is the sum of wave numbers; m is the current wavenumber subscript; k is a radical ofmIs the current wave number;
for frequency-wavenumber-domain weighting operators, taui(f,km) A frequency wave number domain threshold value of the current wave number; f. of0,k0Is the frequency and wave number at the center point of the sliding window function; a (f)0,k0) Is the amplitude at the center point of the sliding window function; g (f-f)0,km-k0) Is a sliding gaussian window function.
In one embodiment, in said step 5, the frequency domain is reconstructed into a data volume drecThe sampled data in (f, x) is replaced by a frequency domain data body d (f, x), the data at the original sampling position is kept unchanged, and the data at the non-sampling position is reconstructed to obtain a frequency domain updating data body di(f,x)。
In one embodiment, in step 8, all frequency components are iterated, and the final interpolated frequency space domain of all frequency components is updated to data volume diAnd (f, x) performing inverse Fourier transform in the time direction to obtain a final interpolated time-space domain output gather d (t, x).
The invention has the advantages that: aiming at the reconstruction problem of weak signals and weak reflection data in sparse and irregular sampling seismic data, the invention provides an irregular seismic data full-frequency band reconstruction algorithm, creatively provides a frequency segmentation and centralized inversion algorithm, and utilizes all frequency components of signals to carry out interpolation reconstruction of data, thereby not only ensuring the calculation precision, but also reducing the calculation amount, protecting the weak reflection signals in the seismic data, ensuring the reconstruction of the seismic data to have better amplitude retention, and effectively improving the imaging quality of small faults and micro-garment structures.
Drawings
The invention will be described in more detail hereinafter on the basis of embodiments and with reference to the accompanying drawings. Wherein:
FIG. 1 is a flow chart of an irregular seismic data full band reconstruction method of the present invention;
FIGS. 2a to 2d are the results of model data testing, wherein FIG. 2a is the original complete model data; FIG. 2b is sparse irregular sampling data; FIG. 2c is a graph of reconstructed interpolated data and FIG. 2d is a graph of the error between the original data and the reconstructed data;
fig. 3a and 3b are coverage number attribute distribution diagrams before and after data interpolation reconstruction, respectively;
FIGS. 4a and 4b are comparison graphs of a common midpoint gather before and after data reconstruction, respectively;
FIGS. 5a and 5b are comparison graphs of the effect of the superimposed cross-section before and after data reconstruction, respectively; (ii) a
FIGS. 6a and 6b are graphs comparing the shift results before and after data reconstruction, respectively;
fig. 7a and 7b are graphs comparing the shift results before and after data reconstruction, respectively.
In the drawings like parts are provided with the same reference numerals. The figures are not drawn to scale.
Detailed Description
The invention will be further explained with reference to the drawings. Therefore, the realization process of how to apply the technical means to solve the technical problems and achieve the technical effect can be fully understood and implemented. It should be noted that the technical features mentioned in the embodiments can be combined in any way as long as no conflict exists. It is intended that the invention not be limited to the particular embodiments disclosed, but that the invention will include all embodiments falling within the scope of the appended claims.
The invention provides an irregular seismic data full-band reconstruction method, as shown in fig. 1, which is a flow chart of the irregular seismic data full-band reconstruction method, and specifically comprises the following steps:
step 1: acquiring original irregular seismic data d (t, x);
step 2: carrying out Fourier transform on the original irregular seismic data d (t, x) in the time direction to obtain a frequency domain data volume d (f, x);
and step 3: then, Fourier transform in the space direction is carried out to obtain a frequency wave number domain data volume di(f,k)；
And 4, step 4: determining a dominant frequency band and a threshold value in a full frequency band wave number range;
and 5: setting the components with the frequency spectrum amplitude smaller than the threshold value to zero, and resetting the reset frequency spectrum amplitude in the frequency wave number domainWeighting, and performing inverse Fourier transform in the space direction to obtain a frequency domain reconstruction data volume drec(f,x)；
Step 6: reconstructing the frequency domain into a data volume drecThe sampled data in (f, x) is replaced by the original frequency domain data body d (f, x), and a frequency domain updated data body d is obtainedi(f,x)；
And 7: updating the frequency domain to data volume di(f, x) repeating the steps 3 to 6 until the iteration number or the error termination condition is met;
and 8: and (4) performing iterative processing on all frequency components, performing inverse Fourier transform in the time direction, and outputting a time-space domain reconstruction gather d (t, x) after final interpolation.
Further, in fig. 1, nter is the total number of iterations; i is the current iteration number; error is the error of the set termination iteration; j is the error of the data after the current iteration; f is the frequency of the current iteration processing; fmax is the maximum frequency of the data.
Preferably, with the development of the seismic acquisition technology of 'two widths and one height', the seismic data volume is increased steeply, on one hand, densely sampled data is beneficial to more accurately obtaining underground structure information, on the other hand, due to the limitation of complex surface conditions, sparse and irregular sampling of seismic data in the space direction is caused, and the requirements of seismic data processing and imaging methods on seismic data space regularity are not met. The original seismic data actually acquired in the step 1 is sparse expression of ideal complete data, and the relation between the original seismic data and the ideal complete data is as follows:
dobs＝Rd
wherein d isobsD is ideal complete data and R is a sparse matrix for actually acquiring seismic data; the reconstruction of the seismic data is to acquire data d from sparse irregularityobsAnd d is recovered.
From nyquist sampling theorem, if the relation between the original seismic data and the ideal complete data expresses a band-limited signal sampled regularly, the basis function of the singe function (sinc function) satisfies the orthogonal condition, and the accurate reconstruction of the signal can be realized by using the sinc function. The orthogonality of Fourier basis functions (Fourier basis functions) is destroyed by irregularly sampled data, so that Fourier coefficients of a certain frequency component leak to other frequency components, a non-zero value appears at a zero value in an ideal frequency spectrum, the estimation of a data accurate spectrum is influenced, and the phenomenon caused by spatial irregular sampling is called a frequency spectrum leakage phenomenon. Meanwhile, irregular seismic data can cause spatial aliasing, and noise interference in the data is more and more serious along with the increase of the missing channel proportion. Therefore, it is necessary to adopt an interpolation algorithm with anti-leakage and anti-aliasing to realize data reconstruction. Therefore, the invention provides a seismic data full-band iterative weighting anti-alias interpolation method, which can keep the effective signal of each frequency wave number component and protect the weak signal in the data from loss.
Preferably, in step 2 and step 3, the original irregular seismic data d (t, x) is subjected to forward and backward fourier transform, and the formulas are respectively:
wherein the content of the first and second substances,
fourier transform coefficients being frequency components w; Δ X is the sum of the sampling intervals, Δ X ═ Σ Δ Xl,xlTime sampling points; l represents the l-th sampling point; f (x)l) Time domain seismic data; f. ofw(xl) Fourier transform coefficients for frequency component w versus time sample point xlThe contribution of (1); j represents an imaginary symbol; and N is the total number of sampling points.
In order to suppress the spectrum leakage phenomenon in the iterative solution process, the spectrum energy needs to be sequentially processed from large to small to eliminate the influence of the spectrum leakage on accurate spectrum estimation, so that the re-orthogonality of Fourier basis functions is realized. Different from the conventional data interpolation reconstruction algorithmIn order to fully utilize all information of seismic data to carry out data reconstruction, the invention provides that amplitude integral energy is analyzed in a full-band wave number range to determine a dominant frequency band, and meanwhile, a full-wave number band is grouped according to iteration times to determine an attenuation threshold tau of each iterationiThe loss of weak effective signals caused by improper threshold value determination is avoided, and the method has important significance for improving the micro-amplitude structure of a complex exploratory area and the imaging quality of small fractures.
Specifically, in the full-band wavenumber range, the amplitude integrated energy is analyzed, thereby determining the dominant band.
Specifically, the full-band wave number is grouped according to the iteration times, and the threshold value tau of each iteration is determinedi。
Preferably, the relation between the original seismic data and the ideal complete data in step 1 is an underdetermined equation, which can be solved in an iterative manner in which the threshold is gradually decreased, and the linear attenuation threshold τ under the current iterative condition is set according to the amplitude energy integral from large to smalliWherein, the iteration format is as follows:
di+1＝(I-RTR)F-1τiFdi+d0
wherein d isi+1An interpolation data volume after the (i + 1) th iteration; i is an identity matrix; rTTranspose for sparse matrix; f and F-1Respectively carrying out Fourier forward and inverse transformation; tau isiIs the threshold value of the ith iteration of the frequency wave number domain, namely taui(f,k)；diThe data volume after the ith iteration is taken as the data volume; d0Is an iteration initial value, where d0＝RTdobs。
Preferably, the amplitude energy integral sets a linear attenuation threshold from large to small, and the expression is as follows:
wherein: tau ismax(f, k) is a maximum threshold determined from the amplitude energy integral; tau ismin(f, k) is a minimum threshold determined from the amplitude energy integral; tau isiterAnd (f, k) is a frequency wave number domain threshold value of the iter iteration of the current frequency, and the nter is the total iteration number.
Preferably, in step 5, in order to suppress the interference of the spurious frequency to the wave number spectrum, the component with the spectrum amplitude smaller than the threshold value is set to be zero, the re-set spectrum is weighted in the frequency wave number domain, and then the inverse fourier transform is performed in the frequency wave number domain, so as to obtain the frequency domain reconstruction data volume drec(f,x)；
The inverse Fourier transform formula after weighting in the frequency wave number domain is as follows:
wherein k is the wave number; NK is the sum of wave numbers; m is the current wavenumber subscript; k is a radical ofmIs the current wave number;
for frequency-wavenumber-domain weighting operators, taui(f,km) A frequency wave number domain threshold value of the current wave number; f. of0,k0Is the frequency and wave number at the center point of the sliding window function; a (f)0,k0) Is the amplitude at the center point of the sliding window function; g (f-f)0,km-k0) Is a sliding gaussian window function.
Preferably, in the step 5, the frequency domain is reconstructed into a data volume drecThe sampled data in (f, x) is replaced by a frequency domain data body d (f, x), the data at the original sampling position is kept unchanged, and the data at the non-sampling position is reconstructed to obtain a frequency domain updating data body di(f,x)。
Preferably, in step 8, all frequency components are iterated, and the final interpolated frequency space domain of all frequency components is updated to the data volume di(f, x) obtained by performing inverse Fourier transform in the time directionThe final interpolated spatio-temporal domain outputs gather d (t, x).
To verify the validity of the algorithm, a numerical model trial calculation was performed. As shown in fig. 2a, the original complete model data is randomly thinned, and the obtained spatially sparse irregular sampling data is shown in fig. 2 b. The trace gather shown in fig. 2b is interpolated by the method of the present invention, and the data result obtained by reconstruction is shown in fig. 2c, and it can be known that the missing trace is well reconstructed and the continuity of the data in-phase axis becomes good compared with fig. 2 a. The error between the original data and the reconstructed data is shown in fig. 2d, and the error value is about 1% of the original data. The model trial calculation result shows that the method has better reconstruction effect and calculation precision on irregular data.
The method of the present invention is applied to actual data processing. FIG. 3 is a coverage number attribute distribution graph before and after processing, and the coverage number uniformity is improved after seismic data reconstruction, and the adaptability of the data to the migration imaging algorithm is improved. Fig. 4 shows a common midpoint gather before and after interpolation reconstruction (after dynamic correction), the near-end and far-end offset information in the reconstructed data is richer, the signal-to-noise ratio is also improved, and the improvement of the imaging quality of different structural forms is facilitated. Fig. 5 shows a comparison between the superimposed sections before and after the interpolation reconstruction process, as shown in the right side of fig. 5, the signal-to-noise ratio of the shallow effective signal is improved, and at about 2100 at cdp for about 5s, the oblique linear interference existing in the original superimposed section is suppressed. The reconstructed data is imaged with offset and the offset result is shown in fig. 6-7. Because full-band data is utilized for interpolation processing, weak signals in the data are protected, cdp is 1300 after regularization as shown in FIG. 6, and small fracture imaging within the depth range of 3.0km-3.5km is clearer; both the offset profile signal-to-noise ratio and the continuity are significantly improved in fig. 7. The practical data application result shows that the method provides offset data with more uniform coverage times and illumination for offset imaging, and has practical significance for improving the offset imaging precision.
The results of the application of the theoretical model and the data show that: the anti-alias interpolation method for iterative weighting in the full frequency band range of the seismic data realizes the accurate reconstruction of the data, protects weak effective signals in the seismic data from being lost, and effectively improves the quality of migration imaging.
While the present invention has been described with reference to the preferred embodiments as above, the description is only for the convenience of understanding the present invention and is not intended to limit the present invention. It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.
Claims (10)
1. The irregular seismic data full-band reconstruction method is characterized by comprising the following steps:
step 1: acquiring original irregular seismic data d (t, x);
step 2: carrying out Fourier transform on the original irregular seismic data d (t, x) in the time direction to obtain a frequency domain data volume d (f, x);
and step 3: then, Fourier transform in the space direction is carried out to obtain a frequency wave number domain data volume di(f,k)；
And 4, step 4: determining a dominant frequency band and a threshold value in a full frequency band wave number range;
and 5: setting the components with the frequency spectrum amplitude smaller than the threshold value to zero, weighting the reset frequency spectrum amplitude in a frequency wave number domain, and then performing Fourier inversion in the space direction to obtain a frequency domain reconstruction data volume drec(f,x)；
Step 6: reconstructing the frequency domain into a data volume drecThe sampled data in (f, x) is replaced by the original frequency domain data body d (f, x), and a frequency domain updated data body d is obtainedi(f,x)；
And 7: updating the frequency domain to data volume di(f, x) repeating the steps 3 to 6 until the iteration number or the error termination condition is met;
and 8: and (4) performing iterative processing on all frequency components, performing inverse Fourier transform in the time direction, and outputting a time-space domain reconstruction gather d (t, x) after final interpolation.
2. The irregular seismic data full-band reconstruction method according to claim 1, wherein the original seismic data actually acquired in step 1 is a sparse representation of ideal complete data, and the relationship between the original seismic data and the ideal complete data is:
dobs＝Rd
wherein d isobsD is ideal complete data and R is a sparse matrix for actually acquiring seismic data; the reconstruction of the seismic data is to acquire data d from sparse irregularityobsAnd d is recovered.
3. The irregular seismic data full-band reconstruction method according to claim 2, wherein in the step 2 and the step 3, the original irregular seismic data d (t, x) is subjected to forward and backward fourier transform, and the formula is as follows:
wherein the content of the first and second substances,
fourier transform coefficients being frequency components w; Δ X is the sum of the sampling intervals, Δ X ═ Σ Δ Xl,xlTime sampling points; l represents the l-th sampling point; f (x)l) Time domain seismic data; f. ofw(xl) Fourier transform coefficients for frequency component w versus time sample point xlThe contribution of (1); j represents an imaginary symbol; and N is the total number of sampling points.
4. The irregular seismic data full-band reconstruction method of claim 3, wherein in the step 4, the amplitude integrated energy is analyzed in the full-band wavenumber range, so as to determine the dominant band.
5. The method of claim 4, wherein in step 4, the full-band wavenumbers are grouped according to the number of iterations, and the threshold τ is determined for each iterationi。
6. The method as claimed in claim 5, wherein in step 4, the linear attenuation threshold τ under the current iteration condition is set according to the amplitude energy integral from large to smalliWherein, the iteration format is as follows:
di+1＝(I-RTR)F-1τiFdi+d0
wherein d isi+1An interpolation data volume after the (i + 1) th iteration; i is an identity matrix; rTTranspose for sparse matrix; f and F-1Respectively carrying out Fourier forward and inverse transformation; tau isiIs the threshold value of the ith iteration of the frequency wave number domain, namely taui(f,k)；diThe data volume after the ith iteration is taken as the data volume; d0Is an iteration initial value, where d0＝RTdobs。
7. The irregular seismic data full-band reconstruction method according to claim 6, wherein the amplitude energy integral is set to a linear attenuation threshold from large to small, and the expression is as follows:
wherein: tau ismax(f, k) is a maximum threshold determined from the amplitude energy integral; tau ismin(f, k) is a minimum threshold determined from the amplitude energy integral; tau isiterAnd (f, k) is a frequency wave number domain threshold value of the iter iteration of the current frequency, and the nter is the total iteration number.
8. The irregular number of earthquakes as set forth in claim 7In the step 5, in order to suppress interference of the alias to the wave number spectrum, the component whose spectrum amplitude is smaller than the threshold value is set to zero, the reconstructed spectrum is weighted in the frequency-wave number domain, and then inverse fourier transform is performed in the frequency-wave number domain to obtain the frequency-domain reconstructed data body drec(f,x)；
The inverse Fourier transform formula after weighting in the frequency wave number domain is as follows:
wherein k is the wave number; NK is the sum of wave numbers; m is the current wavenumber subscript; k is a radical ofmIs the current wave number;
for frequency-wavenumber-domain weighting operators, taui(f,km) A frequency wave number domain threshold value of the current wave number; f. of0,k0Is the frequency and wave number at the center point of the sliding window function; a (f)0,k0) Is the amplitude at the center point of the sliding window function; g (f-f)0,km-k0) Is a sliding gaussian window function.
9. The irregular seismic data full-band reconstruction method of claim 8, wherein in the step 5, the data volume d is reconstructed from the frequency domainrecThe sampled data in (f, x) is replaced by a frequency domain data body d (f, x), the data at the original sampling position is kept unchanged, and the data at the non-sampling position is reconstructed to obtain a frequency domain updating data body di(f,x)。
10. The irregular seismic data full-band reconstruction method of claim 9, wherein in the step 8, all frequency components are iterated and all frequencies are appliedFrequency-space domain update data volume d after final interpolation of rate componentsiAnd (f, x) performing inverse Fourier transform in the time direction to obtain a final interpolated time-space domain output gather d (t, x).
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