def setUp(self):
nC = 20
M = discretize.TensorMesh([nC, nC])
y = np.linspace(0.0, 1.0, nC // 2)
rlocs = np.c_[y * 0 + M.vectorCCx[-1], y]
rx = tomo.Rx(locations=rlocs)
srcList = [
tomo.Src(loc=np.r_[M.vectorCCx[0], yi], rxList=[rx]) for yi in y
]
survey = tomo.Survey(srcList)
problem = tomo.Simulation(M, slownessMap=maps.IdentityMap(M))
problem.pair(survey)
self.M = M
self.problem = problem
self.survey = survey
def run(plotIt=True):
nC = 40
de = 1.0
h = np.ones(nC) * de / nC
M = discretize.TensorMesh([h, h])
y = np.linspace(M.vectorCCy[0], M.vectorCCx[-1], int(np.floor(nC / 4)))
rlocs = np.c_[0 * y + M.vectorCCx[-1], y]
rx = tomo.Rx(rlocs)
source_list = [
tomo.Src(location=np.r_[M.vectorCCx[0], yi], receiver_list=[rx])
for yi in y
]
# phi model
phi0 = 0
phi1 = 0.65
phitrue = utils.model_builder.defineBlock(M.gridCC, [0.4, 0.6], [0.6, 0.4],
[phi1, phi0])
knownVolume = np.sum(phitrue * M.vol)
print("True Volume: {}".format(knownVolume))
# Set up true conductivity model and plot the model transform
sigma0 = np.exp(1)
sigma1 = 1e4
if plotIt:
fig, ax = plt.subplots(1, 1)
sigmaMapTest = maps.SelfConsistentEffectiveMedium(nP=1000,
sigma0=sigma0,
sigma1=sigma1,
rel_tol=1e-1,
maxIter=150)
testphis = np.linspace(0.0, 1.0, 1000)
sigetest = sigmaMapTest * testphis
ax.semilogy(testphis, sigetest)
ax.set_title("Model Transform")
ax.set_xlabel("$\\varphi$")
ax.set_ylabel("$\sigma$")
sigmaMap = maps.SelfConsistentEffectiveMedium(M,
sigma0=sigma0,
sigma1=sigma1)
# scale the slowness so it is on a ~linear scale
slownessMap = maps.LogMap(M) * sigmaMap
# set up the true sig model and log model dobs
sigtrue = sigmaMap * phitrue
# modt = Model.BaseModel(M);
slownesstrue = slownessMap * phitrue # true model (m = log(sigma))
# set up the problem and survey
survey = tomo.Survey(source_list)
problem = tomo.Simulation(M, survey=survey, slownessMap=slownessMap)
if plotIt:
fig, ax = plt.subplots(1, 1)
cb = plt.colorbar(M.plotImage(phitrue, ax=ax)[0], ax=ax)
survey.plot(ax=ax)
cb.set_label("$\\varphi$")
# get observed data
data = problem.make_synthetic_data(phitrue,
relative_error=0.03,
add_noise=True)
dpred = problem.dpred(np.zeros(M.nC))
# objective function pieces
reg = regularization.Tikhonov(M)
dmis = data_misfit.L2DataMisfit(simulation=problem, data=data)
dmisVol = Volume(mesh=M, knownVolume=knownVolume)
beta = 0.25
maxIter = 15
# without the volume regularization
opt = optimization.ProjectedGNCG(maxIter=maxIter, lower=0.0, upper=1.0)
opt.remember("xc")
invProb = inverse_problem.BaseInvProblem(dmis, reg, opt, beta=beta)
inv = inversion.BaseInversion(invProb)
mopt1 = inv.run(np.zeros(M.nC) + 1e-16)
print("\nTotal recovered volume (no vol misfit term in inversion): "
"{}".format(dmisVol(mopt1)))
# with the volume regularization
vol_multiplier = 9e4
reg2 = reg
dmis2 = dmis + vol_multiplier * dmisVol
opt2 = optimization.ProjectedGNCG(maxIter=maxIter, lower=0.0, upper=1.0)
opt2.remember("xc")
invProb2 = inverse_problem.BaseInvProblem(dmis2, reg2, opt2, beta=beta)
inv2 = inversion.BaseInversion(invProb2)
mopt2 = inv2.run(np.zeros(M.nC) + 1e-16)
print("\nTotal volume (vol misfit term in inversion): {}".format(
dmisVol(mopt2)))
# plot results
if plotIt:
fig, ax = plt.subplots(1, 1)
ax.plot(data.dobs)
ax.plot(dpred)
ax.plot(problem.dpred(mopt1), "o")
ax.plot(problem.dpred(mopt2), "s")
ax.legend(["dobs", "dpred0", "dpred w/o Vol", "dpred with Vol"])
fig, ax = plt.subplots(1, 3, figsize=(16, 4))
im0 = M.plotImage(phitrue, ax=ax[0])[0]
im1 = M.plotImage(mopt1, ax=ax[1])[0]
im2 = M.plotImage(mopt2, ax=ax[2])[0]
for im in [im0, im1, im2]:
im.set_clim([0.0, phi1])
plt.colorbar(im0, ax=ax[0])
plt.colorbar(im1, ax=ax[1])
plt.colorbar(im2, ax=ax[2])
ax[0].set_title("true, vol: {:1.3e}".format(knownVolume))
ax[1].set_title("recovered(no Volume term), vol: {:1.3e} ".format(
dmisVol(mopt1)))
ax[2].set_title("recovered(with Volume term), vol: {:1.3e} ".format(
dmisVol(mopt2)))
plt.tight_layout()
model_mapping = maps.ReciprocalMap() # Define starting model starting_model = background_velocity * np.ones(mesh.nC) ############################################## # Define the Physics # ------------------ # # Here, we define the physics of the 2D straight ray tomography problem by # using the simulation class. # # Define the forward simulation. To do this we need the mesh, the survey and # the mapping from the model to the slowness value on each cell. simulation = tomo.Simulation(mesh, survey=survey, slownessMap=model_mapping) ####################################################################### # Define the Inverse Problem # -------------------------- # # The inverse problem is defined by 3 things: # # 1) Data Misfit: a measure of how well our recovered model explains the field data # 2) Regularization: constraints placed on the recovered model and a priori information # 3) Optimization: the numerical approach used to solve the inverse problem # # Define the data misfit. Here the data misfit is the L2 norm of the weighted # residual between the observed data and the data predicted for a given model. # Within the data misfit, the residual between predicted and observed data are