Fundamentals of Seismic Wave Propagation

generality that can be achieved
with more sophisticated mathematics and notation. Naturally this is a very sub-
jective choice, but we have tried to achieve a compact notation while only assum-
ing vector and matrix algebra. We use bold symbols to denote vectors, matrices
and tensors, without any over-arrow or under-score. The dimension of the object
should be obvious from the context, e.g. we use I for the identity matrix whatever
its dimension. A vector is normally equivalent to a column matrix, e.g.
x =

 xy
z

 . (0.1.1)
In the text we sometimes write a vector as, for instance, x = (x, y, z) but this is still
understood as a column matrix. A row matrix would be written x = ( x y z ).
Unit or normalized vectors are denoted by a hat, e.g. xˆ, and we generalize the sign
0.1 Nomenclature xiii
function (sgn) of a scalar so
xˆ = sgn(x) = x/|x| . (0.1.2)
For a matrix, sgn is the signature of the matrix, i.e. the number of positive eigen-
values minus the number of negative eigenvalues.
An advantage of vector-matrix algebra is that it is straightforward to check
that the dimensions in an equation agree. Thus if a is an (l × m) matrix, and b
is (m × n), the matrix c = a b is (l × m) × (m × n) = (l × n), i.e. the repeated
dimension m cancels in the product. If a and b are (3 × 1) vectors, then aTb is
(1 × 1), i.e. this is the scalar product, and abT is a (3 × 3) matrix. It is probably
worth commenting that this useful chain-dimension rule is widely broken with
scalars. Thus we write c = ab, where a is a scalar and b and c vectors or matrices
with the same dimension. Some care is sometimes needed to maintain the chain-
dimension rule if a scalar is obtained from a scalar product. We do not use the
notation a b to represent a tensor (in vector-matrix algebra, the dimensions would
be inconsistent).
We use an underline to indicate a Green function, i.e. solutions for elementary,
point sources. Thus u is the particle displacement while u is the particle displace-
ment Green function and typically contains solutions for three unit-component,
body-force sources. The underline indicates an extra dimension, i.e. u is 3 × 3.
For cartesian vectors we use the notations x , y and z, and xi , i = 1 to 3 inter-
changeably. The former is physically more descriptive, whereas the latter is math-
ematically more useful as we can exploit the Einstein summation convention, i.e.
ai bi with a repeated index means a1b1 + a2b2 + a3b3. Similarly, we use ıˆ, jˆ and kˆ
for unit cartesian vectors or ıˆi , i = 1 to 3.
Sometimes it is useful to consider a restricted range of components. We follow
the standard practice of using a Greek letter for the subscript, i.e. pν with ν = 1
to 2 are two components of the vector p with components pi , i = 1 to 3. Thus
pν pν = p21 + p22. The two-dimensional vector formed from these components is
denoted by p, a sub-space vector in sans serif font. Thus pTp = pν pν . More gener-
ally we use the sans serif font to indicate variables in which the dimensionality is
restricted, in some sense, e.g. we use S and R to indicate the source and receiver,
as in xS and xR, as they are normally restricted to lie on a plane or line.
In general, a matrix of any dimension is denoted by a bold symbol, e.g. A.
The dimension should be obvious from the context and is described as m × n,
where the matrix has m rows and n columns. An element of the matrix is Ai j , or
occasionally (A)i j where i = 1 to m and j = 1 to n. We frequently form larger
matrices from vectors or smaller matrices. Thus, for instance, the 3 × 3 identity
matrix can be formed from the unit cartesian vectors, i.e. I = ( ıˆ jˆ kˆ ). Similarly
xiv Preliminaries
we write
A =
(
A11 A11
A21 A22
)
, (0.1.3)
where A is 2n × 2n, and Ai j are n × n sub-matrices. Conversely, we sometimes
need to extract sub-matrices. The notation we use is
A(i1...iµ)×( j1... jν) , (0.1.4)
for a µ × ν matrix formed from the intersections of the i1-th, i2-th to iµ-th rows
and j1-th, j2-th to jν-th columns of A. If all the rows or columns are included, we
abbreviate this as (.). Thus A(.)×( j) is the j-th column of the matrix A, A(.)×(.) is
the same as A, and A(i)×( j) is the element Ai j .
0.1.4 Identity matrices
We make frequent use of the identity matrix which we denote by I whatever the
size of the square matrix, i.e. the elements are (I)i j = δi j , where δi j is the Kro-
necker delta. Frequently we need matrices which change the order of rows and
columns and possibly the signs. For this purpose, it is useful to define the matrices
I1 =
(
0 I
− I 0
)
, I2 =
(
0 I
I 0
)
and I3 =
( − I 0
0 I
)
, (0.1.5)
where we have used the notation (0.1.3).
0.1.5 Gradient operator
The gradient operator ∇ can be considered as a vector (we break the above rule
and do not use a bold symbol here). Thus
∇ =

∂/∂x1∂/∂x2
∂/∂x3

 . (0.1.6)
Treating ∇ as a 3 × 1 vector, ∇φ, the gradient, is also a (3 × 1) × (1 × 1) = (3 ×
1) vector. The divergence ∇ · u can be rewritten ∇Tu which is (1 × 3) × (3 × 1) =
(1 × 1), i.e. a scalar. Similarly ∇uT is a 3 × 3 matrix with elements ∂ui/∂x j . Thus
∇ can be treated as a vector satisfying the normal rules of vector-matrix algebra.
However, algebra using this notation is limited to scalars, vectors and second-order
tensors. Thus we can write
∇ (aTb) = (∇aT) b + (∇bT) a , (0.1.7)
0.1 Nomenclature xv
for the 3 × 1 gradient of a scalar product. The 3 × 3 second derivatives of a scalar
φ can be written ∇ (∇φ)T but we cannot expand it for a scalar product, for
∂2
∂xi∂x j
(aTb) = ak ∂
2bk
∂xi∂x j
+ bk ∂
2ak
∂xi∂x j
+ ∂ak
∂xi
∂bk
∂x j
+ ∂ak
∂x j
∂bk
∂xi
, (0.1.8)
and the right-hand side involves third-order tensors which are contracted with a
vector. When such expressions arise, e.g. in Chapter 10, we must use the full
subscript notation rather than the compact vector-matrix notation. We note that the
invariant Gibbs notation (Dahlen and Tromp, 1998, §A.3) is elegant for vectors and
second-order tensors, but also becomes unwieldy for higher-order tensors, when
subscript notation is preferred (Dahlen and Tromp, 1998, pp. 821–822).
Partial derivatives are normally written as, for instance, ∂φ/∂t . We avoid the
notation φ,t to prevent confusion with other subscripts. In general we try to avoid
subscripts by using vector-matrix notation.
0.1.6 The vertical coordinate
Where possible we try to avoid specific coordinates by using vector notation, but
frequently it is necessary to use coordinates, usually cartesian. In the Earth, we
invariably define a vertical axis, even though gravity is neglected throughout this
book, as interfaces, including the surface of the Earth, are approximately horizon-
tal. We follow the usual practice of using the z coordinate for the vertical direction,
and the horizontal plane is defined by the x and y coordinates. Much confusion is
caused by the choice of the direction of positive z, and we can do nothing to avoid
this as both choices are common in the literature. All we can do is to be consistent
(justifying including these comments here). Throughout this book, z is measured
positive in the upwards direction, i.e. depths are in the negative z direction. The
choice is arbitrary and we only justify this by noting that when the sphericity of the
Earth is important, it is convenient that radius is measured in the same direction as
the vertical, so the signs of gradients are the same. Drawing z vertically upwards in
diagrams, and x positive to the right, means that in a right-handed system of axes,
y is measured into the page.
Further confusion arises from the convention used for numbering interfaces and
layers in a horizontally layered model. Again, all we can do is to be consistent
throughout this book. We define the �-th interface as being at z = z�, and number
the interfaces