IEEE Signal Processing - March 2018 - 115

Normalized Telemetry Signal

1

0

0
-1

0

0.1

0.2

0.3

0.4

0.5 0.6
Time

0.7

0.8

0.9

1

-1

0

0.1

Normalized Mechanical Power Contributions (N.U.)

1
0
-1

Normalized Telemetry Signal

1

0.2

0.3

0.4

0.5 0.6
Time

0.7

0.8

0.9

1

Normalized Mechanical Power Contributions (N.U.)
0.02
0
-0.02

0

0.1

0.2

0.3

0.4

0.5

Magnetic

0.6

0.7

Viscosity

0.8

0.9

1

0

0.3

0.4

0.5

Magnetic

0.6

0.7

Viscosity

0.8

Time

Time
Normalized Electrical Current (N.U.)

0

0

-1
0.2

0.3

0.4

0.5

Iq(t )

0.6

0.7

0.8

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1

-0.1

0

0.1

0.2

0.3

|Iq(t )|

0.4

0.5

Iq(t )

0.6

0.9

1

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1

Inertia

Normalized Electrical Current (N.U.)

0.1

0.1

0.2

Inertia

1

0

0.1

0.7

0.8

|Iq(t )|

Time
(b)

Time
(a)

Figure 3. An illustration of the mechanical and electrical power necessary to create a QPSK telemetry signal (12-Hz carrier frequency) using a mud siren.
The green and red vertical lines demarcate the start and end of the transition period in signal phase. (a) The transition is 10% of the symbol duration.(b)
The transition is 50% of the symbol duration. All quantities are normalized with respect to peak values for 10% symbol transition. Please note the change
in y-scale between (a) and (b).

and inform on the system state. Based on these notations, the
dynamic state-space of the filtering problem can be written
as follows:
V

dx (t)
= f (t, x (t), d (t))
dt

(2)

y (t) = h (t, x (t), v (t)) .

(3)

The transition function f (.) in (2) represents the dynamics of the system. It describes the evolution of the system in
terms of the time instant t, the current state vector x (t), and
the dynamic vector process noise d (t) . The observation function h (.) in (3) represents the relationship between the observable and nonobservable data. The quantity v (t) represents the
observation noise that is present on the measurements.
The idea behind this model is to exploit the link existing
between the state dynamic and the observation to evaluate the
state vector at the current time. At each time instance, the evolution of the system is ruled by the superposition of a deterministic term, which can be modeled with physics, and a stochastic
term, which cannot be predicted.
Assuming that the mud pump has a global constant shaft
acceleration with local zero mean white Gaussian jitter, the
dynamic model of the phase is given by
0
0 1 0 i m (t)
i m (t)
d ~ (t) = 0 0 1 ~ (t) + 0 ,
m
m
>
H
>
H
>
H
H
>
dt
0 0 0 a m (t)
d am (t)
a m (t )

(4)

Drill String

Insulating Gap

I

Drill Bit

Figure 4. A simplified diagram of the EM telemetry system for drilling.

where ~ m (t) is the angular velocity, a m (t) is the angular
acceleration, and d am (t) is the stochastic acceleration component. Using a linear model for the amplitude a m, k (t), we
showed that the estimation of the mud pump parameters
can typically be achieved using a Bayesian filter [13], and
the estimated noise is subtracted in real time from the initial measurements.

IEEE Signal Processing Magazine

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March 2018

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115
Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018
