Gravity field of the earth - Physical Geodesy - page 91

Next we consider the gravity disturbance. Since

g =grad W,

(2-238)

γ =grad U,

the gravity disturbance vector (2-231) becomes

∂T
∂x ,

.

∂T
∂y ,

∂T
∂z

δ g =grad( W − U ) = grad T ≡ (2-239)

Then

∂W
∂n

∂U
∂n

∂U
∂n ,

=

g = − , γ = − − (2-240)

because the directions of the normals n and n almost coincide. Therefore,
the gravity disturbance is given by

∂W
∂n −

= ∂W

∂n −

∂U
∂n

∂U
∂n

δg = g P − γ P = − − (2-241)

or

∂T
∂n .

δg = − (2-242)

Since the elevation h is reckoned along the normal, we may also write

∂T
∂h .

δg = − (2-243)

Comparing (2-242) with (2-239), we see that the gravity disturbance δg ,be-
sides being the difference in magnitude of the actual and the normal gravity
vector, is also the normal component of the gravity disturbance vector δ g .
We now turn to the gravity anomaly ∆ g .Since

= γ Q + ∂γ γ P

∂h N,

(2-244)

we have

∂T
∂h

∂γ
∂h N.

− = δg = g P − γ P = g P − γ Q − (2-245)

Remembering the definition (2-228) of the gravity anomaly and taking into
account Bruns' formula (2-237), we find the following equivalent relations:

∂T
∂h

∂γ
∂h N,

− =∆ g − (2-246)

∂T

∂h + ∂γ

∆ g = −

∂h N,

(2-247)