T 2 = T X 2 /V 2. It is a hyperbola with apex at X = 0 and T 0 = 2H/V. – –V and H are the layer velocity and thickness. T 2 -X 2 plot is a straight line whose slope = 1/V 2 and intercept = T 0 2. T 2 -X 2 plot can be used to find V and H. Normal moveout (NMO) – –the difference between traveltimes at offsets X and 0   T NMO (X)  X 2 /(2T 0 V 2 ) – –used to flatten the T-X curve before stacking We usually know T, T 0, and X from the seismic section and we want to know V and H. Time-distance (T-X) curves Single horizontal layer

 T NMO (X=1000) = T(X=1000)-T 0 T(X=0)=T 0 T(X=1000) Time-distance (T-X) curves Single horizontal layer

V = (slope) -1/2, H = T 0 V/2 Time-distance (T-X) curves Single horizontal layer Calculating layer velocity and thickness

T 2 = T 0 2 cos 2  + (X+2H sin  ) 2 /V 2 – –  : layer dip angle T-X curve is a hyperbola with apex at X a = -2H sin  and T a =T 0 cos , [T 0 =2H/V]. We usually know T, T 0, and X from the seismic section and we want to know , V, and H. Dip moveout (DMO): the difference between traveltimes at offsets +X and -X divided by X   T DMO (X)  2sin  /V Time-distance (T-X) curves Single dipping layer

T(X=1000) T(X=-1000)  T DMO = [T(X=1000)-T(X=-1000)]/1000 (T a,X a ) T0T0 Time-distance (T-X) curves Single dipping layer

Calculating layer velocity, dip, and thickness -We read T a, T 0, and  T DMO from the seismic record. -Then, we use them as follows: cos  = T a /T 0 V  2sin  /  T DMO H = V T 0 /2 Time-distance (T-X) curves Single dipping layer

T-X curve is NOT a hyperbola. It resembles a hyperbola at near offsets (X/Z < 1). We best-fit a hyperbola to the T-X curve at near offsets. This means that we lumped the multiple layers into one single layer. We use the single-layer approach to find the stacking velocity. We use Dix formula to calculate the interval velocities of the layers. We use the interval velocities and vertical traveltimes to calculate the layer thicknesses. Time-distance (T-X) curves Multiple layers

T 01 T 02 T 03 Time-distance (T-X) curves Multiple layers

V 1 = V s1, V 2 = {[V s2 2 T 02 - V s1 2 T 01 ]/(T 02 - T 01 )} 1/2, V 3 = {[V s3 2 T 03 - V s2 2 T 02 ]/(T 03 - T 02 )} 1/2, V s1 = (slope 1) -1/2, V s2 = (slope 2) -1/2, V s3 = (slope 3) -1/2 V si : stacking velocity to the bottom of the i th layer H 1 = V 1 T 01 /2 H 2 = V 2 (T 02 - T 01 )/2 H 3 = V 3 (T 03 - T 02 )/2 Time-distance (T-X) curves Multiple layers Calculating layer interval velocities and thicknesses 1. Compute stacking velocities from T 2 -X 2 curves 2. Compute layer interval velocities using Dix formula 3. Compute layer interval thicknesses