ElasticBorn¶
sweep.equations.ElasticBorn ¶
Bases: sweep.equations.base.FirstOrderEquation
Born modelling for the 2-D first-order elastic equation (velocity-stress, staggered grid).
Two coupled copies of :class:Elastic: the background field
(vx, vz, sxx, szz, sxz) propagating through (vp, vs, rho), and the
scattered field (svx, svz, ssxx, sszz, ssxz) driven by the perturbation
(dvp, dvs, drho) acting on the background field. Each field has its
own ten CPML memories. Defaults: source on the background stresses
(explosion), receivers on the scattered velocities -- the layout of elastic
least-squares RTM and reflection waveform inversion.
The perturbations are absolute (same units as the background). Their gradients are the elastic LSRTM images; the background gradients are the reflection-waveform-inversion (RWI) ones of Li et al. (2019).
With impl='c' the backward is the :class:Elastic one: the scattered
field's adjoint imaged against the background field, which is exactly the
gradient of (dvp, dvs, drho), in every memory mode. The gradients of
the background (vp, vs, rho) and of the wavelet are impl='eager'
only (autograd); the compiled path refuses them.
Reference: Y. Li, Q. Guo, Z. Li and T. Alkhalifah, 2019, Elastic reflection waveform inversion with variable density, Geophysics 84(4), R553-R567, 10.1190/geo2017-0722.1.
Models (constructor input order)
vp(m/s): Elastic P-wave velocity model.vs(m/s): Elastic S-wave velocity model.rho(kg/m^3): Density model.dvp(m/s): P-wave velocity perturbation.dvs(m/s): S-wave velocity perturbation.drho(kg/m^3): Density perturbation.
Wavefields
vx(aliases:velocity_x): Particle velocity in the x direction.vz(aliases:velocity_z): Particle velocity in the z direction.sxx(aliases:stress_xx): Normal stress in the x direction; default source.szz(aliases:stress_zz): Normal stress in the z direction; default source.sxz(aliases:stress_xz,shear_xz): Shear stress component.m_vxx: CPML memory variable for dvx/dx (internal).m_vxz: CPML memory variable for dvx/dz (internal).m_vzx: CPML memory variable for dvz/dx (internal).m_vzz: CPML memory variable for dvz/dz (internal).m_txxx: CPML memory variable for dsxx/dx (internal).m_txxz: Reserved elastic auxiliary field (internal).m_tzzx: Reserved elastic auxiliary field (internal).m_tzzz: CPML memory variable for dszz/dz (internal).m_txzx: CPML memory variable for dsxz/dx (internal).m_txzz: CPML memory variable for dsxz/dz (internal).svx(aliases:scattered_vx): Scattered particle velocity in the x direction; default receiver.svz(aliases:scattered_vz): Scattered particle velocity in the z direction; default receiver.ssxx(aliases:scattered_sxx): Scattered normal stress in the x direction.sszz(aliases:scattered_szz): Scattered normal stress in the z direction.ssxz(aliases:scattered_sxz): Scattered shear stress component.sm_vxx: Scattered-field cPML memory variable for dvx/dx (internal).sm_vxz: Scattered-field cPML memory variable for dvx/dz (internal).sm_vzx: Scattered-field cPML memory variable for dvz/dx (internal).sm_vzz: Scattered-field cPML memory variable for dvz/dz (internal).sm_txxx: Scattered-field cPML memory variable for dsxx/dx (internal).sm_txxz: Scattered-field reserved elastic auxiliary field (internal).sm_tzzx: Scattered-field reserved elastic auxiliary field (internal).sm_tzzz: Scattered-field cPML memory variable for dszz/dz (internal).sm_txzx: Scattered-field cPML memory variable for dsxz/dx (internal).sm_txzz: Scattered-field cPML memory variable for dsxz/dz (internal).
Defaults
source_type:['sxx', 'szz']receiver_type:['svx', 'svz']pml_type:'cpmls'
Build the elastic Born equation operator.
Parameters:
-
spatial_order–FD accuracy order of the staggered first-derivative operator, as :class:
Elastic. Defaults to 4. -
device–Device for the operator's static gradient kernels. Defaults to
'cpu'. -
backend–'torch'or'jax'. Defaults to'torch'.
C_NAME
class-attribute
¶
str(object='') -> str str(bytes_or_buffer[, encoding[, errors]]) -> str
Create a new string object from the given object. If encoding or errors is specified, then the object must expose a data buffer that will be decoded using the given encoding and error handler. Otherwise, returns the result of object.str() (if defined) or repr(object). encoding defaults to sys.getdefaultencoding(). errors defaults to 'strict'.
cuda_layout
property
¶
:class:Elastic's layout with the scattered field added to the
forward: bind order [bg vx..sxz][sc vx..sxz][bg 10 memories][sc 10
memories] (csrc/cuda/equations/elastic_born2d/driver_traits.cuh).
Everything the backward binds -- adjoint fields, checkpoint snapshots
and replay state, boundary strips, last state, history -- is the
background's, in Elastic's layout: the backward is elastic2d's.
supports_batched_models
class-attribute
¶
bool(x) -> bool
Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.
supports_bs_free_surface
class-attribute
¶
bool(x) -> bool
Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.
supports_image_topography
class-attribute
¶
bool(x) -> bool
Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.
supports_image_topography_c
class-attribute
¶
bool(x) -> bool
Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.
supports_per_edge_free_surface
class-attribute
¶
bool(x) -> bool
Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.
supports_per_edge_free_surface_c
class-attribute
¶
bool(x) -> bool
Returns True when the argument x is true, False otherwise. The builtins True and False are the only two instances of the class bool. The class bool is a subclass of the class int, and cannot be subclassed.
c_refuse_grads ¶
c_refuse_grads(
wavelet_grad,
model_grad_mask,
boundary_saving=False,
boundary_dtype=None,
checkpoint=False,
)
impl='c' returns the (dvp, dvs, drho) gradients only.
interior_substeps ¶
The two half-steps of :meth:func (background and scattered
together) with pml zeroed and the free-surface BC kept, for the
eager boundary-saving reverse driver: composed in reverse order at
-dt they invert the step exactly.
Gradients¶
| gradient of | impl='eager' |
impl='c' |
|---|---|---|
dvp, dvs, drho (elastic LSRTM) |
autograd | every memory mode (Full, BoundarySaving, Ckpt) |
vp, vs, rho (reflection waveform inversion) |
autograd | refused |
| wavelet | autograd | refused |
On impl='c' the backward is the Elastic one: the adjoint of the
scattered field obeys the background elastic equation, and its imaging
condition against the background field is exactly the gradient of the
perturbation. A background receiver (vx, ..., recorded next to the scattered
svx, ...) does not depend on the perturbation, so its residual adds nothing
to these gradients.
from sweep.equations import ElasticBorn
from sweep.propagator.torch import PropTorch
eq = ElasticBorn(spatial_order=4, device="cuda")
prop = PropTorch(eq, shape=(nz, nx), dh=dh, dt=dt, abcn=40, free_surface=True)
dvp, dvs, drho = (torch.zeros_like(vp).requires_grad_() for _ in range(3))
d_sc = prop(wavelet, sources, receivers, models=[vp, vs, rho, dvp, dvs, drho])
loss = ((d_sc - d_obs) ** 2).sum()
loss.backward() # dvp.grad, dvs.grad, drho.grad