The possible complexity of the subsurface layers makes possible the presence
of multiple coherent interferences. This can occur in a simple sense because
acoustic energy conversion occurs when signals arrive at the sensor borehole,
and coherent signals29_{are launched vertically into the borehole which usually}

contains water. Alternatively, multipath propagation due to multiple reflec- tors and waveguide effects can also lead to the presence of multiple coherent sources. In this section, the number of coherent interference sources is ex- tended to P and an optimisation problem formulated with which to solve for the optimal array filter.

3.2.1 Signal Model

Each sensor signal contains: the desired component x; P coherent interfer- ences uj; and the random sensor noisew, i.e.

yn(t) =anx(t−ξn) + P

X

j=1

bnjuj(t−ρnj) +wn(t) (3.22) where the quantities {bnj, ρnj} have aj subscript to identify the correspond- ing interference source. After magnitude normalisation, time shifting and sampling, the sensor signals to be processed are

sn(k) =x(k) + P

X

j=1

αnjuj(k−Knj) +vn(k) (3.23) where {αnj, Knj} are defined for each coherent interference in the same way as for a single source. In the work that follows, the random sensor noise term

vn is ignored30.

3.2.2 Derivation of Objective Function

The minimisation of the rejection response for each coherent source retains
the same form as for the single source problem. Because the desired look-
direction response is common to the minimisation of each jth _{component of}
the coherent interferences, matrices{S, A, V, Z}remain unchanged but there
are now unique matrices {Lj, Rj} associated with each jth interference and
itsαi,j, Ki,j. If all interference sources are considered equally damaging, then

30_{As mentioned in the introduction, a subsequent section considers the case where a}

the optimisation problem may be stated J(u) = min u P X j=1 kLj[fI +Zu]k2 (3.24)

subject to the constraint

R0_{j}u= 0 for all j = 1. . . P (3.25)
Observe that, since each component j of the overall Pth _{order problem is}
subject to the same look-direction constraint for the desired signal, then the

same initial solution fI obtained from (2.59) may be used in (3.24). The P sets of constraints (3.25) then allow the general filter parameterf =fI+Zu to be selected by varying f in the right null space ofall Rj matrices so that the optimisation procedure does not allow any output due to the coherent interferences. By defining

˜

L0 = [L0_{1}L0_{2}. . . L0_{P}] (3.26)
the quadratic optimisation problem may be precisely stated as follows:

J∗(u) = min
u k(fI+Zu)
0_{˜}
L0L˜(fI+Zu)k2 (3.27)
subject to
Z0L˜0u= 0 (3.28)

wherefIis defined from (2.59). However, there is no guarantee that a solution exists which simultaneously nulls all interferences. Matrix ˜L has dimension

α×N M where α=P M+ P X j=1 max n ¯ Knj (3.29)

so any solution which nulls all interferences is satisfyingM look-direction re- sponse constraints plusαinterference wavefront nulling constraints in (3.28). Thus, a ‘necessary’ condition that the array processor null P coherent inter- ferences is (P + 1)M + P X j=1 max n ( ¯Knj)≤N M (3.30)

The optimal filter design may be stated as a generalisation of Theorem 2.2

3.2.3 Optimal filter for nulling multiple coherent sources

Theorem 3.2

If there exist no esoteric multiple equivalent constraintsLjf = 0 causing rank loss ˜L, and if ˜R= ˜LZ has full rankα defined in (3.29), thenone minimising solutionf∗(γ) is given by

f∗(γ) = ˜T γ (3.31)

where

˜

T =I−ZR˜0( ˜RR˜0)−1L˜S (3.32) The minimum cost is

J∗(γ) = 0 (3.33)

Proof

See Theorem 2.2 with {R, L} are replaced by {R,˜ L}˜ .

• • •

plete nulling and numerical methods may be required to check rank condi- tions.

However, if one dominant coherent source must be nulled with simulta-

neous attenuation of other coherent interferences, then the initial solutionfI used in (3.27) may be replaced byf∗ derived according to Theorem 2.2. This alternative quadratic programming problem may be stated as follows:

J∗(u) = min u P X j=2 βjk(f∗+Zu)0Lj0Lj(f∗+Zu)k2 (3.34) subject to (3.25) andf∗ defined by (2.83,2.84), and where the selection ofβj allows weighting of each cost component. Nulling of the dominant coherent interference is, however, guaranteed.

Recall that correlation based identification methods or, given suitable wavelet processing, standard time and magnitude ‘picking’ do require that the interference be uniquely present at certain points in the trace so that mea- surements unique to an interference may be made. When multiple coherent interferences are present, it may become difficult to estimate individual inter- ference parameters. Fortunately, coherent interference such as ‘tube waves’ occurs across many traces and it is usually possible to measure magnitude and time delay information at points in the data trace where only one such interference is present.