1. Geology 5660/6660 Applied Geophysics Last time: The Refraction Method: For a single horizontal layer over a halfspace, observed travel-times for direct and refracted arrivals: give the velocities and layer thickness from t = m i x + b i as: The crossover distance x co can be used in experiment design, to ensure adequate geophone sampling: For Wed 3 Feb: Burger 106-141 (§3.7-3.11) 01 Feb 2016 © A.R. Lowry 2016

2. V1V1 Direct arrival i1i1 h1h1 V2V2 V3V3 h2h2 Refracted arrival For multiple layers we take advantage of Snell’s Law: Via similar algebra & geometry to the one-layer case, or, generalizing to n -layers:

3. V1V1 Direct arrival Refracted arrival icic h V2V2 ?

4. V1V1 Direct arrival Refracted arrival icic huhu V2V2 Refraction from a dipping interface : Now thickness changes with offset, so slope is no longer = inverse of layer velocity. Refer to as “ apparent velocity ”, and this is why we reverse our shots! hdhd  After some trig & algebra, we get: t dd t ud mdmd mumu t0dt0d t0ut0u

5. Like before, we have the observed travel-times and want to know the model parameters. We know V 1 from direct arrival, & we know the slopes are: then we can solve for velocity V 2 :  and angle  : The normal thicknesses use intercept times t 0u, t 0d : (Can get vertical distance from shot to top of layer by dividing h d or h u by cos  ).

6. Aside: Solving for parameters in this way is a simple form of inversion of the data: Most geophysical (& geological, & other) problems can be expressed as: observations = some function of parameters, or (where the arrow denotes plural: a vector). Inverse theory, in which we seek to find is a very important part of geophysics!

7. Equations for multiple dipping layers can be derived using a similar (if slightly more algebraically complicated) approach to that we used for the multiple horizontal layer case [Adachi]: V1V1 V2V2 V3V3 zu1zu1 zu2zu2 u1u1 u2u2 i2i2 i1i1 d2d2 d1d1 22 33 (And these are the equations Craig Jones used in Refract…)

9. Some Limitations of the Refraction Method: It does not get returns from low velocity layers. This means that not only does your best-fitting model not include the low velocity layer, but also estimates of the depth to top of all subsequent layers will be overestimated. Thin layers may be aliased by geophone spacing, & in some cases will be missed regardless of spacing! As with the low velocity layer, this won’t affect velocity estimates for lower layers but will affect depth (in this case depth is underestimated ).

10. Is this real? :-\

12. water shale gas sand shale Example for a synthetic seismogram (Note however that just because the refraction isn’t the first arrival, doesn’t mean it isn’t there!)

13. What if velocity changes within layers? V 1 = 2000 m/s V 2 = 1000 m/s m = 1/V 1 m = 1/V 2

14. V 2 = 2000 m/s V 3 = 1000 m/s m = 1/V 2 m = 1/V 3 V 1 = 500 m/s m = 1/V 1

15. V 1 = 500 m/s V 2 = 2000 m/s What if the layer interface is not planar? z AB m = 1/V 2 m = 1/V 1

16. V 1 = 500 m/s V 2 = 2000 m/s Basically have a difference in the intercept times on one part of the geophone string relative to the other: z AB