Optimization is achieved by fitting the
simulated spectrum (
) to the
experimental one 
by using the minimization of the
least-squares residuals, which are a function of the parameters 
of the Hamiltonian. The goodness of
least-squares fitting for one spectrum is defined as 
. For a global fitting of multiple spectra
with different experimental parameters, 
for the individual spectrum is weighted
by its own noise level. The global goodness of fitting is defined as:
where 
and 
are the noise variance and number of
points in spectrum k,
respectively. We assume that the
experimental envelope modulation is free of artifacts (for example microwave
phase drift, “glitches” above the rms noise level),
which is the case in our experiments [22; 23; 24], and that the noise variance
of each point in the ESEEM waveform is identical (dead time points
removed). In this normalized
representation and with the two assumptions, the variance of the simulation
error of all data points 
is estimated as follows:
=
where 
is the optimized minimum at the global minimum, N is the total number of experimental
points, as given by 
, and L is the number of independent
parameters for optimization. 
can be expanded
at 
, as:
where 
are the optimal parameters when 
. Assuming that the terms with order
higher than two are negligible, and that the variance of the
measurement errors follow a normal distribution, then the term:
follows a 
distribution with a freedom of L [25]. The simultaneous
confidence region of parameters 
can be determined by the following
expression:
The simultaneous confidence region gives an
estimation of the uncertainty of a selected optimization parameter with
respect to the other parameters. The probability of the simultaneous confidence
region containing the true values of 
is determined by 
and a,
as a cumulative distribution of from 
to a. For example, when L = 2 and a = 12,
22, or 32, the simultaneous confidence probability is
39.3%, 86.5% or 98.9%.