应用约束反演

应用约束反演
指导教师：成世琦研究生：夏勇
摘要
众所周知，测井能提供地下介质较准确的岩性参数和具有较高的分辨率，但只局限于井的附近，而地震资料有较密的横向采样，对岩性变化敏感，本文以测井数据为约束，以地面地震数据为基础，在提高地震勘探反演精度的条件下，对宽带约束反演技术作了改进，把井中数据扩展到井间，能够大面积提高检测地层岩性参数和构造参数的分辨率和精度．该法也使用于薄互层的研究．
宽带约束反演根据波形相似原则用试探法迭代反演求取层速度等参数的方法．我们知道，各种地震反演方法都不可避免地存在多解性，只有对所采用的反演方法的实质与产生多解性的原因有透切的了解才能正确使用，得到逼近真解的全局极小解．本文有效地解决了多解性问题，使对实际地震资料经处理后得到的地震参数与声波测井曲线基本吻合．
在论文过程中，查阅了大量的资料，所以我能抓住当今前缘．首先，将约束反演的理论发展成一套完整的系统；其次，根据经典共轭梯度法创造了广义共轭梯度法．发展了宽带约束反演．
综合起来，在约束反演领域内获得了以下一些成果：
１）从约束反演的基本理论出发，在高斯概率密度等具体的假设条件下，推导出随机反演的理论体系，深入阐述了随机反演的宽带特性．
２）从经典的共轭梯度法着手，推导出广义共轭梯度法，从而解决了我们在约束反演问题中数值运算的最关键的难题．
３）利用先验知识和测井资料的约束来具体解决约束反演问题的多解性，实际上证明了约束条件如何恢复高低频成分，及其可靠性．
4）将传统的宽带约束反演的目标函数修改，使阻尼因子变化为逐点噪声方差和模型方差之比，无论从理论还是实践上都已证明，可得到分辨率很高的剖面．
5）由于初次在微机上涉及这样的研究工作，存在许多困难，一切绘图软件和应用程序都必须亲手编制，所以我不得不开发滤波，频谱分析和压噪等辅助软件，形成了一个软件包，这在国内的论文答辩上还是很少的．
经理论模型和实际资料表明,这种方法具有精度高，稳定性好,适应性强,具有优良的抗噪性能.用它制作的合成记录具有较高的分辨率.
据了解，该反演算法居于国内领先水平．
一、储层物性参数反演的基本流程图
利用测井资料和地质先验知识的约束来反演地下储层的细微变化结构，属于多参数约束反演的范畴，以下是ＨＧＳ公司的储层物性参数反演的基本流程图
储层物性参数约束反演图
二、建立约束反演方程式的理论
反演前,搜集有关模型参数的先验知识、从而使模型参数的理论响应朝着先验信息方向偏置、目标函数： φβ=--+--()()()() d Gm d Gm Dm h Dm h T T 2
式中: d ：实际地震记录
m ：模型参数
G ：Jacobi 矩阵
D ：对角线矩阵
β：Lagrange 常数
为了加快运算速度，可以对解估计考虑长度约束，使其缩小搜索范围，目标函数如下： φββ=+=+-q q e e x x L T T 12
02 ()
式中：
e=d-G m
L 0 x 的摄动边界(长度)
三、随机反演 随机反演就是不断迭代以下非线性方程：
∆ m G G C C G S d T m n T =+---()()221 式中：
C m 模型协方差矩阵
C n 噪声协方差矩阵 ∆ m 模型参数的矫正值
这方程有个特点：D 直接依赖于解，反过来m 又极大地受到D 的影响，阻尼因子是根据数据的噪声方差和模型方差自动修改的，由概率论出发，可以证明输出的估计解有稀疏特征。

噪声方差和模型方差
式中定义相邻道之间不相干的部分为噪音,据此可以求得噪音方差.
模型方差必须在不断迭代过程中求取，随着迭代过程的不断进行，模型方差不断增大，导致反演有高的分辨率。

四.广义的共轭梯度法
方程组：
Ax b =
由经典的共轭梯度法可知：
αααββj j j j j j j j j
j j j j
j j j j j
j j j j
g g p Ap x x p
g g Ap g g g g p g p
==+=-==+++++++++(,)(,)
(,)(,)
11111111
式中：g 梯度向量
p 共轭梯度向量
x 解向量
αβj j ,+1 分别表示x 、p 的修正因子.
j 迭代序号
由此，可证对方程组
()A A Id x A b T T +=
其推广的共轭梯度法：
αααββj j j
j j j j j j j j
j j j j
j T j j j j j j j j j j
j
T g g Ap Ap p dp x x p h h Ap g A h y g g g g p g p
h b Ax g p A h =+=+=-
=-==+=-==-+++++++++++
,(,)(,)
(,),)
1111111111100
000y 0
对不同的情况、可将参数表示成两种不同的形式，会有不同的计算效果。

易证：
(,)(,)Ap Ap p A Ap j j j T j =
五.宽带约束反演
子波反演：目标函数
φ()|()()|min w x t y t p =-−→−
x(t) 井旁地震道.
y(t) 合成地震记录.
迭代格式
w w J R J R x y n n T n n n +-=+-11()()
J=∂∂y w / 为Jacobi 矩阵
R n+1=diag{|r k n |p-2
r J w x y k n k j j j n n
k =∑--,()∆
y n 第n 次迭代的合成记录
如果精度不要求太高,直接取L 2模.
子波内插：根据层间模型,用井位作控制,井间各道子波经过内插得出，也可
以采取通常的多道反演,取相邻多道子波叠加的平均效应作为各道子波.
反射系数的反演:
由宽带约束反演的目标函数,采取延迟脉冲模型
r r A A I s s i i T i i +-=++-11()()λ
Jacobi 矩阵：
A S t r w t ij i i
i j =
=-∂∂τ()() 式中:
r ：反射系数幅值.
BCI 具体实施 根据地震资料的初步解释建立一个地质模型.其次确定层间类型,用测井数据进行简单的距离加权内插,这样建立的模型作为宽带约束反演的初始输入模型.
在工区内岩性变化比较稳定的地方,可以辅之以迭代技巧,将前一道的结果作为后一道的初始输入,这样可以减小摄动长度,使速度成倍增长.这类似加上平滑约束.其效果可以和多道反演比拟.实践证明,此迭代过程能够正确地修改初始模型以逼近真实模型.
宽带约束反演的目标函数
φ()||||||||||||m D F W M M W M M p I I I pri p C x I x I pri p =-+-+∇-∇................5.1
D 、F 分别实际地震记录和合成地震记录.
M I 波阻抗模型参数
M i pri 波阻抗模型参数的先验值
∇x 横向梯度
W I ,W C 为波阻抗模型先验值以及波阻抗横向连续性的约束权系数.
||·||P p 模.
式5-1目标函数的地质意义：
第一项表示记录残差，第二项表示先验约束，第三项表示求模型参数的平滑解。

如果我们用前述的模型方差和噪声方差作约束，并且假设随机序列服从0 均值的高斯概率密度
分布时,则有：
∆M G G D F i T n m i =+--()()σσ22
式中模型参数受长度约束： L M U i ≤≤
数值计算结果分析：
1)模型方差对反演结果的影响：当阻尼因子逐渐变小时,这时模型方差逐渐变大,其分辨率增高,当没有模型方差作为约束时,阻尼因子减小的结果导致方程失去稳定。

2)反演子波的影响：当子波反演准确时，反演精度很高，当偏离实际子波太远时，反演结果不可靠．
3)初始模型的影响：对中间速度比上下围岩都高或都低的薄层，结果有可能出现畸变，如果上下界层选厚层作为控制，结果就会非常可靠．
4)剖面的形式：定义不同的阻尼因子，可以输出不同强度以上的反射系数剖面．只要信噪比大于１的地方，结果基本可靠．
5)时窗平滑处理的影响：许多情况不平滑处理时窗，结果反而较好．
Abstract
It is well known that logging can provide relatively reliable lithological parameters of the subsurface medium and has a higher vertical resolution in the vicinity of wellbore whereas seismic data has a denser lateral sample rate and is sensitive to lateral lithological variations.In this paper,we use logging data as constrains and seismic data as a base to extrapolate borehore parameter to crosshle by means of the BCI technique improved under the condition of no damaging to the precision of seismic inversion.By sodoing,the resolution and precision in detecting formation lithological and structural paramrters in larger area are inproved. the techniques is also suitable for the study of thin and inter_bedded reservoirs.
Broad-band constraint invertion is a mehod to obtain parameters such as interval veolocities,which utilizes trial iterative on the basis of waveform similarity.it is a widely solving method of the nonlinear geophysical problem .it is well known that there are no seismic inversion which can overcome the nonuniqueness of solutions,only when the nature of the adopted inversion method and the factors leading to the nonuniqueness are thoroughly understood,can the inversion method be correctly used and the global minimum solution appoximating the truth be derived.
In this paper,some skills ,principles and ways are proposed to effectively decrease the nonuniqueness of the solution and to make the inteval parameters derived from real data through BCI basically consistent with the acoustic logging curves 。

A lot of material is looked over in the course of searching a subject of constrained inversion ,so that I can grip the leading edge.firstly ,the paper develops the theory of constrain into a set of system;Secondly ,a Generalized Conjugate Gradient Method is created on the base of bible Conjugate Gradient Method 。

I achieve the following results in the constrained inversion field:
1)begining with the founded theory of constrained inversion ,using the Gauss density hypothesis values ,I extrapolate a set of theorical system of stochastic inversion,and set forth the broad-band behave of stochastic inversion。

2)Begining with bible conjugate gradient,I extrapolate the Generalized Conjugate gradient with which the paper resolve the problem of slow calculating velocity.
3)Using the prior information and well-logging constrians to resolve nonuniqueness of inversion ,the paper set forth how to renew the low frequency and high frequency from logging constrains in details ,which is proved in practice ,too。

4)this thesis change traditional damping factor of broad-band into rate between noise variance
and variance in modeling space each point. It is verified in practice and point by point 2
n
theory that it can have high resolution profile.
5)Because the thesis is the first research in the micro_apparatus,it lie in lots of questions,such as map-making etc.,we can’t use any ready-made software,so the paper develope filter,spectral anolysis,and compress noise program so on.which forms a set of software-wrap,it is known that is scarce in the thesis reply in my nature。

Theoretical modeling and data processing show that this method is characteried by high accuracy ,good stability,excellent adaptability and nice noise -resistant ability and that is results in high-resolution synthetic seismograms.
it is said that the inversion method is ahead of my nature。

一、chart of reservior parameter of physical property Array
chart of reservior parameter of physical property
二、Constrained Inversion ‘s Objecu function
Before inversion ,we collect a lot of prior information relative to model parameters,and give the response of model parameter a bias to prior information.
Object function:
φβ=--+--()()()() d Gm d Gm Dm h Dm h T T 2
where ： d ：seismic data. m ：model parameter.
G ：Jacobi matrix.
D ：diagonal matrix.
β：Lagrange constant
In order to fasten calculating veolocity and substract searching field ,may consider length constrans of solution estimate.Object function is following :
φββ=+=+-q q e e x x L T T 1202 ()
where ：
e=d-G m
L 0 the perterbation length of x.
3.Stochastic Inversion
Stochastic inversion is contineously iterate following nonlinear function ∆ m G G C C G S d T m n T =+---()()221
where ：
C m － model convariance matrix.
C n －noise convariance matrix. ∆ m － corrected value of model parameters.
It is charactoristic of Cm directly depending on solution m.On the contrary ,m is influenced greatly by Cm matrix .so lon as we give out enough noise variance ,the solution must be seeked out .It can be proved that the soution estimate has sparse behave.
Noise variance &Model variance
we definite the noncoherence part between trace and trace as noise on which we may derive out noise variance 。

Model variance merely can be achieved in the couse of continueously iteration. while the couse of iteration is contineous ,model variance gradually increase to result to sparse solution ,so that it has higher resolution 。

4.Generilized Conjugate gradient Metheod
equation group Ax b =
it’s bible conjugate gradient calaulating method is following:
αααββj j j j j j j j j
j j j j
j j j j j j j j j
g g p Ap x x p g g Ap g g g g p g p ==+=-==+++++++++(,)(,)
(,)(,)
11111111 where:
g —gradient rector.
p —conjugate gradient vector
x —solution vector
αβj j ,+1—modifying factor of x,pvector.
j —iterating ordinal
On the base of above-mentioned theory,we may calculate out the solution of following equation group. ()A A Id x A b T T +=
The solution of generalized conjugate gradient method is: αααββj j j
j j j j j j j j
j j j j
j T j j j j j j j j j j j
T g g Ap Ap p dp x x p h h Ap g A h y g g g g p g p h b Ax g p A h =+=+=-=-==+=-==-+++++++++++ ,(,)(,)
(,)(,)
1111111111100
000y 0
we may have two forms to parametor for different condition.it can be proved: (,)(,)Ap Ap p A Ap j j j T j =
5.Broad-band Constrained Inversion
5.1 wavelet inversion:object function is following
φ()|()()|min w x t y t p =-−→−
x(t)—the seismic record nearby well
y(t)—the synthesized seismograph
the iterating form is following:
w w J R J R x y n n T n n n +-=+-11()()
J=∂∂y w / Jacobi matrix
R n+1=diag{|r k n |p-2
r J w x y k n k j j
j n n k =∑--,()∆
y n —the n itrating synthsizing record
if we don’t request h igh accuracy we may let p equal to 2.
5.2 wavelet interpolation:On the condition of being constrained by the logging
data,crosshole wavelet may be interpolated out .Of course,wavelet may use multitrace inversion technique which take place wavelet each trace with multitrace average wavelet 。

5.3 reflection coefficient Inversion
On the basis of object function of broad-band inversion,the paper adopt delaying pulse model 。

r r A A I s s i i T i i +-=++-11()()λ
Jacobi matrix ：
A S t r w t ij i i
i j ==-∂∂τ()() where:
r —the amplitude value of reflect coefficient.
5.4 Specific implement of BCI
Firstly ,a geological model is founded on the base of the migration section of seimic
material preliminary interpretation.Secondly,we found cross-layers model arcording to the result of interprertation then simply interpolate crosshole data through wellbore data. The model is used for preliminary model of BCI.
Where jib site change continously,we may adopt the technical know-how which is the results of ahead trace as initial value of rear trace.it may reduce the perturbation of model parameters,thus result in velocity to double.In fact ,this is similar to a sort of smooth solution of geological model .it is often more effective than multi-trace inversion .Practice verify that iteration procedure can correctly modify initial model to real model 。

Object function of BCI:
φ()||||||||||||m D F W M M W M M p I I I pri p C x I x I pri p =-+-+∇-∇................5.1 D,F — seismic data and synthetic seismograms
M I —wave imperdance model parameter
M i pri —prior value of imperdance model parameter
∇x —lateral gradient
W I ,W C —constraints weight coeffient of imperrdance model and constraints weight coeffient imperdance lateral continuity.
||·||P —L p module
means of the object function:
item1 express recording residual error;
item2 express prior constraints
item3 express smoothing model constraints
When model parameter is random series obeying 0 average Gauss probability density distribution,the object function may be simplified arcording to above-mentioned damping factor :
∆M G G D F i T n m i =+--()()σσ22
Here model parameter is constrained by length:：
L M U i ≤≤
5.5 The results of numerical analysis:
1)The effect of damping factor on inversion results:The resolution increse while damping factor gradually reduce and model variance constraint gradually enhance.if inversion lose constrains of damping factor,equation tend to unstable 。

2)The effect of wavelet pricision on the inversion result.The higher pricision is,the more accurate inversion results is.When wavelet deviate real wavelet,the resuts is bader and bader and discredulous 。

3)the effect of initial modeling on inversion :when the mid-layer velocity is higher than that above it or below it ,the result may be bad 。

4)high resolution section:defineing differient pamping factor,we may have diffrent coeffient section ,so long as the S/N is greater than 1the results is credulous.
5)The effect of smooth time-window:it may be better if don’t process the edge in the time-window.。