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.. _rtm_introduction:
**************************
RTM Introduction
**************************
Reverse Time Migration (RTM) is an important seismic imaging technique used
for producing an accurate representation of the subsurface.
It has been adopted by the oil and gas industry for many years.
However, the high computation demand, the high memory storage requirement
and the less cache-friendly memory access pattern present a significant
challenge.
The region under detection, in general, is a 3 dimentional field with
each direction few kilometers.
In some special situation, the problme is simplified to a 2 dimentional field by
assuming the uniformity along one direction.
Mathematics in RTM
===================
1. Wave equation and the finite difference method
--------------------------------------------------
*Equation (1)* is the basic mathematical equation for elastic wave propagation.
.. math::
\frac{1}{v(\mathbf{x})^2}\frac{\partial^2p(\mathbf{x}, t)}{\partial t^2} = \nabla^2 p(\mathbf{x}, t)\hspace{50mm}(1)
where :math:`\nabla^2` is the Laplace operator, :math:`p(\mathbf{x}, t)` is the wave field and :math:`v(\mathbf{x})` is the wave velocity.
In 2D-RTM, :math:`\mathbf{x} = \{x, z\}`, which is the length and depth of the imaging area respectively.
In 3D-RTM, :math:`\mathbf{x} = \{x, y, z\}`, which is the length, width and depth of the imaging area respectively.
Due to the non-uniform structures in the subsurface, there is no theoretical solution for *Equation 1*.
Hence, RTM is based on a numerical solution called a finite difference method.
All spacial dimensions and time domains are discretized properly.
The partial derivative with respect to time and the Laplace operator also have
corresponding transformation according to the discretization scheme.
For instance, the Pluto synthetic seismic velocity model which was used to evaluation our 2D-RTM implementation
has been discretized into a regular grid both spatially and temporally. The number of partiitions are listed in
*Table 1*.
.. csv-table:: Table 1: Pluto model discretization
"Parameter", "Length (:math:`N_{x}`)", "Depth (:math:`N_{z}`)", "Time (:math:`N_{t}`)"
"No. grids", "6960", "1201", "12860"
where :math:`N_x`, :math:`N_z`, :math:`N_t` denote the number of grids along :math:`x`, :math:`z`, :math:`t` respectively.
Similar to the approaches presented in many research publications, we also adopted Order-2 temporal scheme and Order-8 spatial scheme.
Following these discretization schemes, the 2D wave equation can be re-interpreted by *Equation (2) and (3)* shown below. **Equation (2*) and (3)** are for 3D RTM.
.. math::
\nabla^2p = \sum_{n=-4}^4 c_n(\frac{p(x+n\Delta x)}{\Delta x^2} + \frac{p(z+n\Delta z)}{\Delta z^2})\hspace{20mm}(2)
.. math::
\nabla^2p = \sum_{n=-4}^4 c_n(\frac{p(x+n\Delta x)}{\Delta x^2} + \frac{p(y+n\Delta y)}{\Delta y^2}+ \frac{p(z+n\Delta z)}{\Delta z^2})\hspace{20mm}(2*)
.. math::
p(t+1) = 2p(t) - p(t-1) + v^2 \Delta t^2 \nabla^2p \label{eq:temperal}\hspace{30mm}(3)
Where all :math:`c_n` are the coeffients shown in *Table 2* below.
.. csv-table:: Table 2: Finete Difference Method Order-8 Coefficients
":math:`c_{4}`, :math:`c_{-4}`", ":math:`c_{3}`, :math:`c_{-3}`", ":math:`c_{2}`, :math:`c_{-2}`", ":math:`c_{1}`, :math:`c_{-1}`", ":math:`c_{0}`"
":math:`-\frac{1}{560}`", ":math:`\frac{8}{315}`", ":math:`-\frac{1}{5}`", ":math:`\frac{8}{5}`", ":math:`-\frac{1}{560}`"
2. Imaging
-----------
The seismic imaging is via cross-correlation between a source wavefield and a receiver wavefield.
Mathematically, it can be expressed as *Equation 4*.
.. math::
I(\mathbf{x}) = \sum_{i=1}^{N_s}\sum_{t_n=1}^{N_t}p_s^i(\mathbf{x}, t_n)p_r^i(\mathbf{x}, t_n)\hspace{40mm}(4)
where :math:`N_s` is the total number of shots,
:math:`p_s^i(\mathbf{x}, t_n)` is a source wavefield
and :math:`p_r^i(\mathbf{x}, t_n)` is a receiver wavefield.
3. Boundary saving scheme
--------------------------
To correctly reconstruct the source wavefield in the backward process, it is necessary to extend the boundary and
save the boundaries value in a forward process.
For 2D-RTM, there are 4 boundaries to deal with.
Dussaud [1]_ et al. proposed saving half-order
length of all the 4 boundaries with absorbing boundary condition.
In this case, the reconstruction source wavefield is definitely perfect. However, a huge amount of storage is required.
In our project, we adopt the scheme proposed by Liu [2]_
which saved only half-order length of the upper boundaries' values with absorbing boundary conditions
and used random boundary condition for the other three boundaries in order to balance the accuracy and storage requirement.
Design information of L1 primitives
====================================
1. Stencil2D
--------------
Template class **Stencil2D** provides the implementation of *Equation (2) and (3)*, in which function **laplacian** realizes the *Equation (2)* and function **propagate** realizes the *Equation (3)*.
The **laplacian** function (also called stencil in *Figure 1*) applies line buffers and shifting registers to achieve high spatial parallism.
By assigning a value e.g. 2 to the template variable *t_nPE*,
the HLS will synthesize a compute IP with the ability to
compute the laplacian operation for 2 grid points in one clock cycle.
The **propagate** function (also called time upate in *Figure 1*) processes multiple data streams in parallel to compute multiple data points of the next wavefield simultaneously.
2. RTM2D
---------
Template class **RTM2D** provides the implementation of time step iteration in 2D-RTM. In this class, function **forward** emulates a single time step in the forward propagation path, function **backword** emulates a single time step in the backward propagation path.
The boundary condition adopted here is Hybrid Boudary Condition (HBC).
.. _forward-streaming-module:
Forward streaming module
^^^^^^^^^^^^^^^^^^^^^^^^^
The block diagram of **forward** function given in *Figure 1* shows the achieved temporal parallelims, namely computing the wavefield at time step :math:`t` and :math:`t+1` simultaneouly. The streaming interfaces of the basic modules allow them to be connected directly by FIFOs, which hold the wavefield data without storing them back to external memory. This streamlined architecture is the key to achieve high parallelism both spatially and temporally. Each Forward streaming module labeled with **F** basically takes the wavefield data from the last time stamp then computes the wavefield for current time stamp and outputs both the data and boundary data to other modules in streams.
.. figure:: /images/rtm2DStreaming.png
:align: center
:alt: 2D-RTM forward streaming module
Figure 1. Forward streaming module
Red line: data stream of wavefield :math:`p(t)`; Blue line: data stream of velocity model :math:`v^2 \Delta t^2`; Black line: data stream of boundaries
.. _backward-streaming-module:
Backward streaming module
^^^^^^^^^^^^^^^^^^^^^^^^^^
As shown in *Figure 2*, the **backward** function is composed of three modules connected by FIFOs.
Two of them labeled with **R** and **S**, similar to the forward streaming module,
are designed to compute Receiver wavefield and reconstruct Source wavefield respectively.
The module labeled with **I** (Imaging streaming module) realizes **Equation (4)**, whic is used to compute the cross correlation between the two wavefields (Receiver and Source wavefields).
.. figure:: /images/rtm2DBwdStr.png
:align: center
:alt: 2D-RTM backward streaming module
Figure 2. Backward streaming module
3. Stencil3D
-------------
Similarly, the template class **Stencil3D** provides the implementation of **Equation (2*) and (3)** by function **laplacian** and function **propagate** respectly.
The **laplacian** function (also called stencil in *Figure 1*) applies multiple buffers and shifting registers to achieve high spatial parallism.
By assigning values to the template variable *t_nPEX* and *t_nPEZ*,
the HLS will synthesize a compute IP with the ability to
compute the laplacian operation for multiple grid points in one clock cycle.
The **propagate** function (also called time upate in *Figure 1*) processes multiple data streams in parallel to compute multiple data points of the next wavefield simultaneously.
4. RTM3D
---------
Similarly, template class **RTM3D** provides the implementation of time step iteration in 3D-RTM.
In this class, function **forward** emulates a single time step in the forward propagation path.
There are multiple overload of this function for various boundary conditions e.g.
Random Boudnayr Condition (RBC).
The block diagram of **forward** function given in *Figure 1* shows the achieved temporal parallelims, namely computing the wavefield at time step :math:`t` and :math:`t+1` simultaneouly. The streaming interfaces of the basic modules allow them to be connected directly by FIFOs, which hold the wavefield data without storing them back to external memory. This streamlined architecture is the key to achieve high parallelism both spatially and temporally. Each Forward streaming module labeled with **F** basically takes the wavefield data from the last time stamp then computes the wavefield for current time stamp and outputs both the data and boundary data to other modules in streams.
.. [1] E.Dussaud,W.W.Symes,P.Williamson,L.Lemaistre,P.Singer, B. Denel, and A. Cherrett, “Computational strategies for reversetime migration,” in SEG Technical Program Expanded Abstracts 2008. Society of Exploration Geophysicists, 2008, pp. 2267– 2271.
.. [2] H. Liu, R. Ding, L. Liu, and H. Liu, “Waveﬁeld reconstruction methods for reverse time migration,” Journal of Geophysics and Engineering, vol. 10, no. 1, p. 015004, 2012.