""" TEM: AEMR TEM-FAST 48 system ============================ **In this example we compute the TEM response from the TEM-FAST 48 system.** This example was contributed by Lukas Aigner (`@aignerlukas `_), who was interested in modelling the TEM-FAST system, which is used at the TU Wien. If you are interested and want to use this work please have a look at the corresponding paper Aigner et al. (2024). The modeller ``empymod`` models the electromagnetic (EM) full wavefield Greens function for electric and magnetic point sources and receivers. As such, it can model any EM method from DC to GPR. However, how to actually implement a particular EM method and survey layout can be tricky, as there are many more things involved than just computing the EM Greens function. What is not included in ``empymod`` at this moment (but hopefully in the future), but is required to model TEM data, is to **account for arbitrary source waveform**, and to apply a **lowpass filter**. So we generate these two things here, and create our own wrapper to model TEM data. See also the example :ref:`sphx_glr_gallery_tdomain_tem_walktem.py`, on which this example builds upon. **References** - **Aigner, L., D. Werthmüller, and A. Flores Orozco, 2024**, Sensitivity analysis of inverted model parameters from transient electromagnetic measurements affected by induced polarization effects; *Journal of Applied Geophysics*, Volume 223, Pages 105334, doi: `10.1016/j.jappgeo.2024.105334 `_. """ import empymod import numpy as np import matplotlib.pyplot as plt from scipy.special import roots_legendre from matplotlib.ticker import LogLocator, NullFormatter from scipy.interpolate import InterpolatedUnivariateSpline as iuSpline plt.style.use('ggplot') # sphinx_gallery_thumbnail_number = 2 ############################################################################### # 1. TEM-FAST 48 Waveform and other characteristics # ------------------------------------------------- # # The TEM-FASt system uses a "time-key" value to determine the number of gates, # the front ramp and the length of the current pulse. # We are using values that correspond to a time-key of 5. turn_on_ramp = -3.0E-06 turn_off_ramp = 0.95E-06 on_time = 3.75E-03 injected_current = 4.1 # Ampere time_gates = np.r_[4.060e+00, 5.070e+00, 6.070e+00, 7.080e+00, 8.520e+00, 1.053e+01, 1.255e+01, 1.456e+01, 1.744e+01, 2.146e+01, 2.549e+01, 2.950e+01, 3.528e+01, 4.330e+01, 5.140e+01, 5.941e+01, # time-key 1 7.160e+01, 8.760e+01, 1.036e+02, 1.196e+02, # time-key 2 1.436e+02, 1.756e+02, 2.076e+02, 2.396e+02, # time-key 3 2.850e+02, 3.500e+02, 4.140e+02, 4.780e+02, # time-key 4 5.700e+02, 6.990e+02, 8.280e+02, 9.560e+02, # time-key 5 ] * 1e-6 # from us to s waveform_times = np.r_[turn_on_ramp - on_time, -on_time, 0.000E+00, turn_off_ramp] waveform_current = np.r_[0.0, injected_current, injected_current, 0.0] plt.figure() plt.title('Waveform') plt.plot(np.r_[-9, waveform_times*1e3, 2], np.r_[0, waveform_current, 0]) plt.xlabel('Time (ms)') plt.xlim([-4, 0.5]) ############################################################################### # 2. ``empymod`` implementation # ----------------------------- def waveform(times, resp, times_wanted, wave_time, wave_amp, nquad=3): """Apply a source waveform to the signal. Parameters ---------- times : ndarray Times of computed input response; should start before and end after `times_wanted`. resp : ndarray EM-response corresponding to `times`. times_wanted : ndarray Wanted times. wave_time : ndarray Time steps of the wave. wave_amp : ndarray Amplitudes of the wave corresponding to `wave_time`, usually in the range of [0, 1]. nquad : int Number of Gauss-Legendre points for the integration. Default is 3. Returns ------- resp_wanted : ndarray EM field for `times_wanted`. """ # Interpolate on log. PP = iuSpline(np.log10(times), resp) # Wave time steps. dt = np.diff(wave_time) dI = np.diff(wave_amp) dIdt = dI/dt # Gauss-Legendre Quadrature; 3 is generally good enough. # (Roots/weights could be cached.) g_x, g_w = roots_legendre(nquad) # Pre-allocate output. resp_wanted = np.zeros_like(times_wanted) # Loop over wave segments. for i, cdIdt in enumerate(dIdt): # We only have to consider segments with a change of current. if cdIdt == 0.0: continue # If wanted time is before a wave element, ignore it. ind_a = wave_time[i] < times_wanted if ind_a.sum() == 0: continue # If wanted time is within a wave element, we cut the element. ind_b = wave_time[i+1] > times_wanted[ind_a] # Start and end for this wave-segment for all times. ta = times_wanted[ind_a]-wave_time[i] tb = times_wanted[ind_a]-wave_time[i+1] tb[ind_b] = 0.0 # Cut elements # Gauss-Legendre for this wave segment. See # https://en.wikipedia.org/wiki/Gaussian_quadrature#Change_of_interval # for the change of interval, which makes this a bit more complex. logt = np.log10(np.outer((tb-ta)/2, g_x)+(ta+tb)[:, None]/2) fact = (tb-ta)/2*cdIdt resp_wanted[ind_a] += fact*np.sum(np.array(PP(logt)*g_w), axis=1) return resp_wanted ############################################################################### def get_time(time, r_time): """Additional time for ramp. Because of the arbitrary waveform, we need to compute some times before and after the actually wanted times for interpolation of the waveform. Some implementation details: The actual times here don't really matter. We create a vector of time.size+2, so it is similar to the input times and accounts that it will require a bit earlier and a bit later times. Really important are only the minimum and maximum times. The Fourier DLF, with `pts_per_dec=-1`, computes times from minimum to at least the maximum, where the actual spacing is defined by the filter spacing. It subsequently interpolates to the wanted times. Afterwards, we interpolate those again to compute the actual waveform response. Note: We could first call `waveform`, and get the actually required times from there. This would make this function obsolete. It would also avoid the double interpolation, first in `empymod.model.time` for the Fourier DLF with `pts_per_dec=-1`, and second in `waveform`. Doable. Probably not or marginally faster. And the code would become much less readable. Parameters ---------- time : ndarray Desired times r_time : ndarray Waveform times Returns ------- time_req : ndarray Required times """ tmin = np.log10(max(time.min()-r_time.max(), 1e-10)) tmax = np.log10(time.max()-r_time.min()) return np.logspace(tmin, tmax, time.size+2) ############################################################################### def temfast(off_time, waveform_times, model, square_side=12.5): """Custom wrapper of empymod.model.bipole. Here, we compute TEM-FAST data using the ``empymod.model.bipole`` routine as an example. This function is based upon the Walk TEM example. We model the big source square loop by computing only half of one side of the electric square loop and approximating the finite length dipole with 3 point dipole sources. The result is then multiplied by 8, to account for all eight half-sides of the square loop. The implementation here assumes a central loop configuration, where the receiver (1 m2 area) is at the origin, and the source is a square_side x square_side m electric loop, centered around the origin. Note: This approximation of only using half of one of the four sides obviously only works for central, horizontal square loops. If your loop is arbitrary rotated, then you have to model all four sides of the loop and sum it up. Parameters ---------- off_time : ndarray times at which the secondary magnetic field will be measured waveform_times : ndarray Depths of the resistivity model (see ``empymod.model.bipole`` for more info.) depth : ndarray Depths of the resistivity model (see ``empymod.model.bipole`` for more info.) res : ndarray Resistivities of the resistivity model (see ``empymod.model.bipole`` for more info.) square_side : float sige length of the square loop in meter. Returns ------- TEM-FAST waveform : EMArray TEM-FAST response (dB/dt). """ if 'm' in model: depth = model['depth'] res = model del res['depth'] else: res = model['res'] depth = model['depth'] # === GET REQUIRED TIMES === time = get_time(off_time, waveform_times) # === GET REQUIRED FREQUENCIES === time, freq, ft, ftarg = empymod.utils.check_time( time=time, # Required times signal=1, # Switch-on response ft='dlf', # Use DLF ftarg={'dlf': 'key_601_2009'}, verb=2, ) # === COMPUTE FREQUENCY-DOMAIN RESPONSE === # We only define a few parameters here. You could extend this for any # parameter possible to provide to empymod.model.bipole. hs = square_side / 2 # half side length EM = empymod.model.bipole( src=[hs, hs, 0, hs, 0, 0], # El. bipole source; half of one side. rec=[0, 0, 0, 0, 90], # Receiver at the origin, vertical. depth=depth, # Depth-model, including air-interface. res=res, # if with IP, res is a dictionary with # all params and the function freqtime=freq, # Required frequencies. mrec=True, # It is an el. source, but a magn. rec. strength=8, # To account for 4 sides of square loop. srcpts=3, # Approx. the finite dip. with 3 points. htarg={'dlf': 'key_401_2009'}, # Short filter, so fast. ) # Multiply the frequecny-domain result with # \mu for H->B, and i\omega for B->dB/dt. EM *= 2j*np.pi*freq*4e-7*np.pi # === Butterworth-type filter (implemented from simpegEM1D.Waveforms.py)=== cutofffreq = 1e8 # determined empirically for TEM-FAST h = (1+1j*freq/cutofffreq)**-1 # First order type h *= (1+1j*freq/3e5)**-1 EM *= h # === CONVERT TO TIME DOMAIN === delay_rst = 0 # unknown for TEM-FAST, therefore 0 EM, _ = empymod.model.tem(EM[:, None], np.array([1]), freq, time+delay_rst, 1, ft, ftarg) EM = np.squeeze(EM) # === APPLY WAVEFORM === return waveform(time, EM, off_time, waveform_times, waveform_current) ############################################################################### def pelton_res(inp, p_dict): """ Pelton et al. (1978). code from: https://empymod.emsig.xyz/en/stable/examples/time_domain/ cole_cole_ip.html#sphx-glr-examples-time-domain-cole-cole-ip-py """ # Compute complex resistivity from Pelton et al. # print('\n shape: p_dict["freq"]\n', p_dict['freq'].shape) iwtc = np.outer(2j*np.pi*p_dict['freq'], inp['tau'])**inp['c'] rhoH = inp['rho_0'] * (1 - inp['m']*(1 - 1/(1 + iwtc))) rhoV = rhoH*p_dict['aniso']**2 # Add electric permittivity contribution etaH = 1/rhoH + 1j*p_dict['etaH'].imag etaV = 1/rhoV + 1j*p_dict['etaV'].imag return etaH, etaV ############################################################################### # 3. Computation non-IP # --------------------- depths = [8, 20] rhos = [25, 5, 50] model = {'depth': np.r_[0, depths], 'res': np.r_[2e14, rhos]} # Compute conductive model response = temfast(off_time=time_gates, waveform_times=waveform_times, model=model) ############################################################################### # 4. Computation with IP # ---------------------- depths = [8, 20] rhos = [25, 5, 50] charg = np.r_[0, 0.9, 0] taus = np.r_[1e-6, 5e-4, 1e-6] cs = np.r_[0, 0.9, 0] eta_func = pelton_res depth = np.r_[0, depths] model = {'depth': depth, 'res': np.r_[2e14, rhos], 'rho_0': np.r_[2e14, rhos], 'm': np.r_[0, charg], 'tau': np.r_[1e-7, taus], 'c': np.r_[0.01, cs], 'func_eta': eta_func} # Compute conductive model response_ip = temfast(off_time=time_gates, waveform_times=waveform_times, model=model) ############################################################################### # 5. Comparison # ------------- plt.figure(figsize=(5, 5), constrained_layout=True) # Plot result of model 1 ax1 = plt.subplot(111) plt.title('TEM-FAST responses') # empymod plt.plot(time_gates, response, 'r.--', ms=7, label="response") plt.plot(time_gates, abs(response_ip), 'kx:', ms=7, label="response with IP") sub0 = response_ip[response_ip < 0] tg_sub0 = time_gates[response_ip < 0] plt.plot(tg_sub0, abs(sub0), marker='s', ls='none', mfc='none', ms=8, mew=1, mec='c', label="negative readings") # Plot settings plt.xscale('log') plt.yscale('log') plt.xlabel("Time(s)") plt.ylabel(r"$\mathrm{d}\mathrm{B}_\mathrm{z}\,/\,\mathrm{d}t$") plt.grid(which='both', c='w') plt.legend(title='Data', loc=1) # Force minor ticks on logscale ax1.yaxis.set_minor_locator(LogLocator(subs='all', numticks=20)) ax1.yaxis.set_minor_formatter(NullFormatter()) plt.grid(which='both', c='w') ############################################################################### empymod.Report()