C/C++实现Perigram属性

通常描述信号瞬时特征的物理量有：瞬时振幅、瞬时相位、及瞬时频率（“三瞬参数”），地震波的瞬时参数不仅可以直接用来研究岩性、构造等，而且也能够反演介质的品质因数等参数。在研究非平稳信号时，瞬时参数尤为重要。

假设原始信号为\(x(t)\)，通过Hilbert变换，将实信号转变为复信号\(S(t)=x(t)+iy(t)\)，并提取瞬时振幅、瞬时相位、瞬时频率三个参数。本文档将介绍使用C/C++代码，由三瞬属性得到Perigram以及Perigram×Cosine of Phase这两种属性的过程。

1 地震属性、意义及计算方法

(1)瞬时振幅 Instaneous Amplitude

(/振幅包络Amplitude Envelope/反射强度Reflection Strength)

瞬时振幅或反射强度是地震信号在某一时刻的总能量的平方根，它是地震轨迹的包络线。利用Hilbert变换提取复信号的虚部\(y(t)\)，瞬时振幅根据下式计算：

\[A(t)=\sqrt{x(t)^{2}+y(t)^{2}}
\]

瞬时振幅代表了声阻抗（或反射系数）的差别，因此能够识别亮点、气藏聚集位置、层序边界、不整合面、大的岩性和沉积相变化、孔隙度或其它岩性参数的空间变化、断层的侧向变化等。其值总为正。

(2)瞬时相位 Instaneous Phase

瞬时相位是地震剖面上同相轴连续性的量度，无论能量的强弱，它的相位都能显示出来。其计算式为：

\[\theta(t)=\tan ^{-1}[\frac{y(t)} {x(t)}]
\]

当波在各向异性的均匀介质中传播时，其相位是连续的；当波在有异常存在的介质中传播时，其相位将在异常位置发生显著变化，在剖面图中明显不连续。因此，利用瞬时相位能够较好地对地下分层和地下异常进行辨别，能够用来识别断层、尖灭、层序边界等。瞬时相位数据呈现不连续的锯齿状，其值域范围为\(-180°\sim180°\)。

(3)瞬时频率 Instaneous Frequency

瞬时频率表示瞬时相位随时间的变化率：

\[\omega(t)=\frac{d \theta(t)}{d t}
\]

瞬时频率能够在横向上有效判别地震资料的相关性，薄层的岩性和厚度的变化能够引起瞬时频率的变化。一般高瞬时频率反映了薄的泥岩沉积，低的瞬时频率反映富砂沉积，因此可以应用瞬时频率反映碎屑岩的沉积环境。

(4) 反射强度交流分量 Perigram

Perigram是去除了直流分量的反射强度，反射强度低频（直流）分量定义为给定时间窗口\(\Delta T\)内反射强度的平均值：

\[\bar{A}(t)=\frac{1}{\Delta T} \int_{(t-\Delta T / 2)}^{(t+\Delta T / 2)} A(\tau) d \tau
\]

当取一个极大的窗口时，\(A(t)\)接近直流分量，然后将其从反射强度中减去低频分量，得到Perigram：

\[g(t)=A(t)-\bar{A}(t)
\]

Perigram 与反射强度的用途基本相同。其能够加强反射强度峰值异常，并且更适合分析和处理，因为其有正负。

(5)反射强度交流分量与相位余弦之积

公式表示为：

\[G(t)=g(t)\cos \theta(t)
=[A(t)-\bar{A}(t)]\cos \theta(t)
=x(t)[1-\frac{\bar{A}(t)}{{A}(t)}],\quad if\quad g(t)>0\\
G(t)=0,\quad if\quad g(t)\le 0\\
\]

当Perigram>0时，该属性为反射强度交流分量与相位余弦的乘积，否则为0。换言之，当Perigram值为正时，Perigram×相位余弦的值等于输入数据\(x(t)\)乘以一个小于1的比例；而当Perigram值为负时，值为0，对应低反射强度。该属性可用于突出强振幅与连续相位，同反射强度的用途相似。例如，当周围低能反射体被减弱至0时，与含气砂有关的部分将被明显点亮。

2 代码实现

(1) Perigram

void Compute_perigram(float* dataInput, int nt, float* Perigram){

	float *Imaginary = NULL;	//定义变换后得到的虚部

	float *Amplitude_envelope = NULL;	//振幅包络

	float *DC_component = NULL;	//直流分量

	float sum = 0.0f;

	Imaginary = (float*)calloc(nt, sizeof(float));

	Amplitude_envelope = (float*)calloc(nt, sizeof(float));

	DC_component = (float*)calloc(nt, sizeof(float));

	hilbert(nt, dataInput, Imaginary);	//希尔伯特变换

	for (int i = 0; i < nt; i++)

	{

		Amplitude_envelope[i] = sqrt(pow(dataInput[i], 2) + pow(Imaginary[i], 2));	//实部、虚部平方和开根号求瞬时振幅

		sum += Amplitude_envelope[i];

	}

	float Avg = sum/nt;

	for (int i = 0; i < nt; i++)

	{

		DC_component[i] = Avg;	//直流分量为全局平均

		Perigram[i]=Amplitude_envelope[i]- DC_component[i];	//Perigram

	}

	free(Imaginary);

	free(Amplitude_envelope);

	free(DC_component);

}

(2) Perigram*Cosine of Phase

void Compute_perigramXcosine(float* dataInput, int nt, float* PerigramXcos) {

	float *Imaginary = NULL;	//定义变换后得到的虚部

	float *Amplitude_envelope = NULL;	//振幅包络

	float *DC_component = NULL;	//直流分量

	float sum = 0.0f;

	Imaginary = (float*)calloc(nt, sizeof(float));

	Amplitude_envelope = (float*)calloc(nt, sizeof(float));

	DC_component = (float*)calloc(nt, sizeof(float));

	hilbert(nt, dataInput, Imaginary);	//希尔伯特变换

	for (int i = 0; i < nt; i++)

	{

		Amplitude_envelope[i] = sqrt(pow(dataInput[i], 2) + pow(Imaginary[i], 2));

		sum += Amplitude_envelope[i];

	}

	float Avg = sum / nt;

	for (int i = 0; i < nt; i++)

	{

		DC_component[i] = Avg;

		if ((Amplitude_envelope[i] - DC_component[i]) > 0) {	//如果g(t)大于0

			PerigramXcos[i] = dataInput[i] * (1 - DC_component[i] / Amplitude_envelope[i]);

		}

		else {	//g(t)小于0，那乘积余弦就赋为0

			PerigramXcos[i] = 0;

		}

	}

	free(Imaginary);

	free(Amplitude_envelope);

	free(DC_component);

}

主函数

int compute_type = 1;	//选择计算属性类别，当前为Perigram

//遍历每一道

for (int itrace = 0; itrace < ncdp; itrace++)

{

    fread(&traceheader, sizetraceheader, 1, fpinput);

    fread(dataInput, nt * sizeof(float), 1, fpinput);

    if (compute_type==1)	//Perigram

    {

        float* Perigram = NULL;

        Perigram = (float*)calloc(nt, sizeof(float));

        Compute_perigram(dataInput, nt, Perigram);

        fwrite(&traceheader, sizetraceheader, 1, fpoutput);

        fwrite(Perigram, nt * sizeof(float), 1, fpoutput);

        free(Perigram);

    }

    else {		//Perigram*cos of phase

        float* Perigram_cos = NULL;

        Perigram_cos = (float*)calloc(nt, sizeof(float));

        Compute_perigramXcosine(dataInput, nt, Perigram_cos);

        fwrite(&traceheader, sizetraceheader, 1, fpoutput);

        fwrite(Perigram_cos, nt * sizeof(float), 1, fpoutput);

        free(Perigram_cos);

    }

}

完整代码

I hilbert.h

#include "convolution.h"

void hilbert(int n, float x[], float y[]);

II hilbert.cpp

#include "hilbert.h"

#define PI 3.14159265358

#define LHHALF 30	/* half-length of Hilbert transform filter*/

#define LH 2*LHHALF+1	/* filter length must be odd */

void hilbert(int n, float x[], float y[])

/*****************************************************************************

Compute Hilbert transform y of x

******************************************************************************

Input:

n		length of x and y

x		array[n] to be Hilbert transformed

Output:

y		array[n] containing Hilbert transform of x

*****************************************************************************/

{

	static int madeh = 0;

	static float h[LH];

	int i;

	float taper;

	/* if not made, make Hilbert transform filter; use Hamming window */

	if (!madeh) {

		h[LHHALF] = 0.0;

		for (i = 1; i <= LHHALF; i++) {

			taper = 0.54 + 0.46*cos(PI*(float)i / (float)(LHHALF));

			h[LHHALF + i] = taper * (-(float)(i % 2)*2.0 /

				(PI*(float)(i)));

			h[LHHALF - i] = -h[LHHALF + i];

		}

		madeh = 1;

	}

	/* convolve Hilbert transform with input array */

	convolve_cwp(LH, -LHHALF, h, n, 0, x, n, 0, y);

    //需要用到卷积函数，见https://www.cnblogs.com/GeophysicsWorker/p/16399209.html

    //convolution.cpp

}

III Perigram_Attributes.h

#include<stdio.h>

#include<stdlib.h>

#include<math.h>

#include<string.h>

#include"hilbert.h"

void Compute_perigram(float* dataInput, int nt, float* Perigram);

void Compute_perigramXcosine(float* dataInput, int nt, float* PerigramXcos);

IV Perigram_Attributes.cpp

#include "Perigram_Attributes.h"

void Compute_perigram(float* dataInput, int nt, float* Perigram){

	float *Imaginary = NULL;	//定义变换后得到的虚部

	float *Amplitude_envelope = NULL;	//振幅包络

	float *DC_component = NULL;	//直流分量

	float sum = 0.0f;

	Imaginary = (float*)calloc(nt, sizeof(float));

	Amplitude_envelope = (float*)calloc(nt, sizeof(float));

	DC_component = (float*)calloc(nt, sizeof(float));

	hilbert(nt, dataInput, Imaginary);	//希尔伯特变换

	for (int i = 0; i < nt; i++)

	{

		Amplitude_envelope[i] = sqrt(pow(dataInput[i], 2) + pow(Imaginary[i], 2));	//实部、虚部平方和开根号求瞬时振幅

		sum += Amplitude_envelope[i];

	}

	float Avg = sum/nt;

	for (int i = 0; i < nt; i++)

	{

		DC_component[i] = Avg;	//直流分量为全局平均

		Perigram[i]=Amplitude_envelope[i]- DC_component[i];	//Perigram

	}

	free(Imaginary);

	free(Amplitude_envelope);

	free(DC_component);

}

void Compute_perigramXcosine(float* dataInput, int nt, float* PerigramXcos) {

	float *Imaginary = NULL;	//定义变换后得到的虚部

	float *Amplitude_envelope = NULL;	//振幅包络

	float *DC_component = NULL;	//直流分量

	float sum = 0.0f;

	Imaginary = (float*)calloc(nt, sizeof(float));

	Amplitude_envelope = (float*)calloc(nt, sizeof(float));

	DC_component = (float*)calloc(nt, sizeof(float));

	hilbert(nt, dataInput, Imaginary);	//希尔伯特变换

	for (int i = 0; i < nt; i++)

	{

		Amplitude_envelope[i] = sqrt(pow(dataInput[i], 2) + pow(Imaginary[i], 2));

		sum += Amplitude_envelope[i];

	}

	float Avg = sum / nt;

	for (int i = 0; i < nt; i++)

	{

		DC_component[i] = Avg;

		if ((Amplitude_envelope[i] - DC_component[i]) > 0) {	//如果相减后仍大于0

			PerigramXcos[i] = dataInput[i] * (1 - DC_component[i] / Amplitude_envelope[i]);

		}

		else {	//相减后小于0，那乘积余弦也就赋为0

			PerigramXcos[i] = 0;

		}

	}

	free(Imaginary);

	free(Amplitude_envelope);

	free(DC_component);

}

V main.cpp

#include<stdio.h>

#include<stdlib.h>

#include<math.h>

#include<string.h>

#include<iostream>

#include<vector>

#include<algorithm>

#include"hilbert.h"

#include"segy.h"

#include"Perigram_Attributes.h"

int main() {

	const char *filenameInput = "test.sgy";

	const char *filenameOutput = "Output66.sgy";

	bhed fileheader;

	segy traceheader;

	unsigned int nt = 0;

	unsigned int sizefileheader = sizeof(fileheader);

	unsigned int sizetraceheader = sizeof(traceheader);

	unsigned int ncdp = 0;

	long long size_file = 0;

	long long size_trace = 0;

	FILE *fpinput = NULL;

	FILE *fpoutput = NULL;

	float *dataInput = NULL;

	float *dataOutput = NULL;

	fpinput = fopen(filenameInput, "rb");

	fpoutput = fopen(filenameOutput, "wb");

	if (fpinput == NULL) {

		printf("can not open %s file\n", filenameInput);

		return false;

	}

	if (fpoutput == NULL) {

		printf("can not open %s file\n", filenameOutput);

		return false;

	}

	fread(&fileheader, sizefileheader, 1, fpinput);

	nt = fileheader.hns;

	_fseeki64(fpinput, 0, SEEK_END);

	size_file = _ftelli64(fpinput);

	size_trace = nt * sizeof(float) + sizetraceheader;

	ncdp = (size_file - (long long)sizefileheader) / size_trace;

	_fseeki64(fpinput, sizefileheader, SEEK_SET);

	fwrite(&fileheader, sizefileheader, 1, fpoutput);

	dataInput = (float*)calloc(nt, sizeof(float));

	dataOutput = (float*)calloc(nt, sizeof(float));

    /************以上为segy读写，不再赘述***********/

	int compute_type = 2;

	//遍历每一道

	for (int itrace = 0; itrace < ncdp; itrace++)

	{

		fread(&traceheader, sizetraceheader, 1, fpinput);

		fread(dataInput, nt * sizeof(float), 1, fpinput);

		if (compute_type==1)	//Perigram

		{

			float* Perigram = NULL;

			Perigram = (float*)calloc(nt, sizeof(float));

			Compute_perigram(dataInput, nt, Perigram);

			fwrite(&traceheader, sizetraceheader, 1, fpoutput);

			fwrite(Perigram, nt * sizeof(float), 1, fpoutput);

			free(Perigram);

		}

		else {		//Perigram*cos

			float* Perigram_cos = NULL;

			Perigram_cos = (float*)calloc(nt, sizeof(float));

			Compute_perigramXcosine(dataInput, nt, Perigram_cos);

			fwrite(&traceheader, sizetraceheader, 1, fpoutput);

			fwrite(Perigram_cos, nt * sizeof(float), 1, fpoutput);

			free(Perigram_cos);

		}

	}

	free(dataInput);

	free(dataOutput);

	fclose(fpinput);

	fclose(fpoutput);

	return 0;

}

参考：

1 唐金炎. 基于地震属性参数的地震相识别方法研究[D].成都理工大学,2011.

2 高静怀,汪文秉,朱光明.小波变换与信号瞬时特征分析[J].地球物理学报,1997(06):821-832.

3 BASIC THEORY AND GEOLOGICAL MEANING OF SEISMIC ATTRIBUTES (linkedin.com)

4 B. Gelchinsky, E. Landa, and V. Shtivelman, (1985), "Algorithms of phase and group correlation," GEOPHYSICS 50: 596-608.https://doi.org/10.1190/1.1441935

5 SHTIVELMAN, V., LANDA, E. and GELCHINSKY, B. (1986), PHASE AND GROUP TIME SECTIONS AND POSSIBILITIES FOR THEIR USE IN SEISMIC INTERPRETATION OF COMPLEX MEDIA. Geophysical Prospecting, 34: 508-536.https://doi.org/10.1111/j.1365-2478.1986.tb00479.x

6 PostStack Family Software Reference Manual.Landmark Software & Services.