def _phir(delta, tau):
"""Residual part of the dimensionless Helmholtz potential."""
psi = _exp(-_C*(delta-1)**2 - _D*(tau-1)**2)
theta = (1-tau) + _A*((delta-1)**2)**(1/(2*_beta2))
Delta = theta**2 + _B*((delta-1)**2)**_a
return util.sum(_n[0:7]*delta**_d[0:7]*tau**_t[0:7]) \
+ util.sum(_n[7:51]*delta**_d[7:51]*tau**_t[7:51]*\
_exp(-delta**_c[7:51])) \
+ util.sum(_n[51:54]*delta**_d[51:54]*tau**_t[51:54] * \
_exp(-_alpha*(delta-_epsilon)**2 - _beta1*(tau-_gamma)**2)) \
+ util.sum(_n[54:56]*Delta**_b*delta*psi)
def _phir_tau(delta, tau):
"""Partial derivative of phi_r wrt tau."""
psi = _exp(-_C*(delta-1)**2 - _D*(tau-1)**2)
theta = (1-tau) + _A*((delta-1)**2)**(1/(2*_beta2))
Delta = theta**2 + _B*((delta-1)**2)**_a
return util.sum(_n[0:7]*_t[0:7]*delta**_d[0:7]*tau**(_t[0:7]-1)) +\
util.sum(_n[7:51]*_t[7:51]*delta**_d[7:51]*tau**(_t[7:51]-1) * \
_exp(-delta**_c[7:51])) + \
util.sum(_n[51:54]*delta**_d[51:54]*tau**_t[51:54] * \
_exp(-_alpha*(delta-_epsilon)**2-_beta1*(tau-_gamma)**2) * \
(_t[51:54]/tau-2*_beta1*(tau-_gamma))) + \
util.sum(_n[54:56]*delta*(_pder_Deltab_tau(Delta, theta)*psi + \
Delta**_b*_pder_psi_tau(tau, psi)))
def _phir_delta(delta, tau):
"""Partial derivative of phi_r wrt delta."""
psi = _exp(-_C*(delta-1)**2 - _D*(tau-1)**2)
theta = (1-tau) + _A*((delta-1)**2)**(1/(2*_beta2))
Delta = theta**2 + _B*((delta-1)**2)**_a
return util.sum(_n[0:7]*_d[0:7]*delta**(_d[0:7]-1)*tau**_t[0:7]) + \
util.sum(_n[7:51]*_exp(-delta**_c[7:51]) * \
delta**(_d[7:51]-1)*tau**_t[7:51] * \
(_d[7:51]-_c[7:51]*delta**_c[7:51])) + \
util.sum(_n[51:54]*delta**_d[51:54]*tau**_t[51:54] * \
_exp(-_alpha*(delta-_epsilon)**2 - \
_beta1*(tau-_gamma)**2) *
(_d[51:54]/delta - 2*_alpha*(delta-_epsilon))) +\
util.sum(_n[54:56]*(Delta**_b*(psi+delta*_pder_psi_delta(psi, delta)) +
_pder_Deltab_delta(Delta, delta, theta)*delta*psi))
def model(self, params):
tau,y0,y8 = params
x_data = self.info['x data (s)']
x0 = x_data[0] # raw times in seconds are too far from the epoc
a = 1 - y0/y8
self._model_data[:] = y8*(1-a*_exp(-(x_data-x0)/tau))
return self._model_data
def _dtheta_Br_for_ml_complex(self, r, th, ph, c, m, l, a=6371.2):
"""
Calculates the d_theta(Br)/R contribution for one set of m,l, using the potential field.
Inputs
------
r:
radius location (km)
th:
latitude location (radians)
ph:
longitude location (radians)
c:
complex gauss coefficient
m:
Order of calculation
l:
Degree of calculation
a:
Radius (km) at which Gauss coefficients are calculated
Returns
-------
d_theta(Br)/R in Tesla at a particular point from a particular degree and order.
"""
return (l+1.)*a**(l+2.)/abs(r)**(l+2.)*c*_exp(1j*m*ph)*self._dtheta_Pml(_cos(th), l, m)/r
def model(self, params):
tau,y0,y8 = params
x_data = self.info['x data (s)']
x0 = x_data[0] # raw times in seconds are too far from the epoc
a = 1 - y0/y8
self._model_data[:] = y8*(1-a*_exp(-(x_data-x0)/tau))
return self._model_data
def exp(x):
if isinstance(x, DataRef):
return DataRef("exp({})".format(x))
else:
return _exp(x)
def _phio_tautau(tau):
"""Second partial derivative of phi_o wrt delta."""
return -_no[2]*tau**-2 - \
util.sum(_no[3:8]*_gammao[3:8]**2*_exp(-_gammao[3:8]*tau)*\
(1-_exp(-_gammao[3:8]*tau))**(-2))
def _phio_tau(tau):
"""Partial derivative of phi_o wrt tau."""
return _no[1] + _no[2]/tau + \
util.sum(_no[3:8]*_gammao[3:8]*((1-_exp(-_gammao[3:8]*tau))**(-1) - 1))
def _phio(delta, tau):
"""Ideal gas part of the dimensionless Helmholtz potential."""
return numpy.log(delta) + _no[0] + _no[1]*tau + _no[2]*numpy.log(tau) + \
util.sum(_no[3:8]*numpy.log(1-_exp(-_gammao[3:8]*tau)))