f-k filters

1

Mitch Withers, Res. Assoc. Prof., Univ. of Memphis

2

Consider a 2-d delta function, 𝛿 𝑥, 𝑦 = &1, 𝑥 𝑎𝑛𝑑 𝑦 = 00, 𝑥 𝑜𝑟 𝑦 ≠ 0

3

Consider a 2-d delta function, 𝛿 𝑥, 𝑦 = &1, 𝑥 𝑎𝑛𝑑 𝑦 = 00, 𝑥 𝑜𝑟 𝑦 ≠ 0

/!"

"/!"

"𝛿 𝑥, 𝑦 𝑑𝑥𝑑𝑦 = 1 A point source

4

Consider a 2-d delta function, 𝛿 𝑥, 𝑦 = &1, 𝑥 𝑎𝑛𝑑 𝑦 = 00, 𝑥 𝑜𝑟 𝑦 ≠ 0

/!"

"/!"

"𝛿 𝑥, 𝑦 𝑑𝑥𝑑𝑦 = 1

Similarly, the 2-d sifting property is, /

!"

"/!"

"𝛿 𝑥 − 𝑥#, 𝑦 − 𝑦# 𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦 = 𝑓 𝑥#, 𝑦#

𝑦

𝑥#𝑥

𝛿 𝑥 − 𝑥#

A point source

a line source sifts values of 𝑓 𝑥, 𝑦 at 𝑥# for all 𝑦.

5

Consider a 2-d delta function, 𝛿 𝑥, 𝑦 = &1, 𝑥 𝑎𝑛𝑑 𝑦 = 00, 𝑥 𝑜𝑟 𝑦 ≠ 0

/!"

"/!"

"𝛿 𝑥, 𝑦 𝑑𝑥𝑑𝑦 = 1

Similarly, the 2-d sifting property is, /

!"

"/!"

"𝛿 𝑥 − 𝑥#, 𝑦 − 𝑦# 𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦 = 𝑓 𝑥#, 𝑦#

𝑦

𝑥#𝑥

𝛿 𝑥 − 𝑥#

A point source

a line source sifts values of 𝑓 𝑥, 𝑦 at 𝑥# for all 𝑦. /

!"

"𝛿 𝑥 − 𝑥# 𝑓 𝑥, 𝑦 𝑑𝑥 = 𝑓 𝑥#, 𝑦

𝑥#𝑥

𝑦

𝑓 𝑥#, 𝑦

6

The mathematics of the 2-d FT is ambivalent to our choice of units so one choice can be time and space, a seismic reflection line for example.

Φ 𝑓, 𝑘 = 𝐹[𝜙 𝑡, 𝑥 ] = /!"

"/!"

"𝜙(𝑡, 𝑥)𝑒!$%&(()*+,)𝑑𝑡𝑑𝑥

7

The mathematics of the 2-d FT is ambivalent to our choice of units so one choice can be time and space, a seismic reflection line for example.

Φ 𝑓, 𝑘 = 𝐹[𝜙 𝑡, 𝑥 ] = /!"

"/!"

"𝜙(𝑡, 𝑥)𝑒!$%&(()*+,)𝑑𝑡𝑑𝑥

Where 𝑓 is the familiar frequency in cycles/second and 𝑘 is spatial frequency in cycles/length.

𝑥

𝜆𝜆 is the wavelength

𝑘 = ./

is the spatial frequency

Wavenumber is 𝑘∗ = 2𝜋𝑘 = %&/

similar to 𝜔 = 2𝜋𝑓.

8

The mathematics of the 2-d FT is ambivalent to our choice of units so one choice can be time and space, seismic reflection line for example.

Φ 𝑓, 𝑘 = 𝐹[𝜙 𝑡, 𝑥 ] = /!"

"/!"

"𝜙(𝑡, 𝑥)𝑒!$%&(()*+,)𝑑𝑡𝑑𝑥

Where 𝑓 is the familiar frequency in cycles/second and 𝑘 is spatial frequency in cycles/length.

𝑥

𝜆𝜆 is the wavelength

𝑘 = ./

is the spatial frequency

Wavenumber is 𝑘∗ = 2𝜋𝑘 = %&/

similarly to 𝜔 = 2𝜋𝑓.

We use 𝑘∗ here to distinguish wavenumber from spatial frequency 𝑘 though you may encounter 𝑘 for both. Know which it is by context.

Spatial frequency not wavenumber.

9

A few familiar properties in 2-d

𝐹 𝑔 𝑥 − 𝑥#, 𝑡 − 𝑡# = 𝐺(𝑘, 𝑓)𝑒!$%& +,!*()! Shifting property

10

A few familiar properties in 2-d

𝐹 𝑔 𝑥 − 𝑥#, 𝑡 − 𝑡# = 𝐺(𝑘, 𝑓)𝑒!$%& +,!*()! Shifting property

𝐹 𝑔 𝑥, 𝑡 + 𝑎𝑔 𝑥 − 𝑥#, 𝑡 − 𝑡#= 𝐺(𝑘, 𝑓) 1 + 𝑎𝑒!$%& +,!*()!

Where 𝑎 is a real scalar.Replication property (an echo)

11

A few familiar properties in 2-d

𝐹 𝑔 𝑥 − 𝑥#, 𝑡 − 𝑡# = 𝐺(𝑘, 𝑓)𝑒!$%& +,!*()! Shifting property

𝐹 𝑔 𝑥, 𝑡 + 𝑎𝑔 𝑥 − 𝑥#, 𝑡 − 𝑡#= 𝐺(𝑘, 𝑓) 1 + 𝑎𝑒!$%& +,!*()!

Where 𝑎 is a real scalar.Replication property (an echo)

𝐹 𝑔.(𝑥, 𝑡) ∗ 𝑔%(𝑥, 𝑡) = 𝐺.(𝑘, 𝑓)𝐺%(𝑘, 𝑓)

where

Convolution property (think f-k filter)

𝑔. 𝑥, 𝑡 ∗ 𝑔% 𝑥, 𝑡 = /!"

"/!"

"𝑔. 𝑢, 𝑣 𝑔% 𝑥 − 𝑢, 𝑡 − 𝑣 𝑑𝑢𝑑𝑣

12

A few familiar properties in 2-d

𝐹 𝑔 𝑥 − 𝑥#, 𝑡 − 𝑡# = 𝐺(𝑘, 𝑓)𝑒!$%& +,!*()! Shifting property

𝐹 𝑔 𝑥, 𝑡 + 𝑎𝑔 𝑥 − 𝑥#, 𝑡 − 𝑡#= 𝐺(𝑘, 𝑓) 1 + 𝑎𝑒!$%& +,!*()!

Where 𝑎 is a real scalar.Replication property (an echo)

𝐹 𝑔.(𝑥, 𝑡) ∗ 𝑔%(𝑥, 𝑡) = 𝐺.(𝑘, 𝑓)𝐺%(𝑘, 𝑓)

where

Convolution property (think f-k filter)

𝑔. 𝑥, 𝑡 ∗ 𝑔% 𝑥, 𝑡 = /!"

"/!"

"𝑔. 𝑢, 𝑣 𝑔% 𝑥 − 𝑢, 𝑡 − 𝑣 𝑑𝑢𝑑𝑣

𝐹 𝑔.(𝑥, 𝑡)𝑔%(𝑥, 𝑡) = 𝐺.(𝑘, 𝑓) ∗ 𝐺%(𝑘, 𝑓) Windowing

13

Plane wave fronts

𝜆�⃗�

𝑐, = apparent velocity = 1234 5

𝜃

Incidence angle

14

Plane wave fronts

𝜆�⃗�

𝑐, = apparent velocity = 1234 5

𝜃

Incidence angle

𝜃 → 90° Horizontally traveling wave (ground roll and surface waves)

𝜃 → 0° Vertically traveling wave (body waves and reflections)

15

Plane wave fronts

𝜆�⃗�

𝑐, = apparent velocity = 1234 5

𝜃

Incidence angle

𝜃 → 90° Horizontally traveling wave (ground roll and surface waves)

𝜃 → 0° Vertically traveling wave (body waves and reflections)

𝑣 =𝜆𝑇 =

M1 𝑘M1 𝑓

=𝑓𝑘

16

Plane wave fronts

𝜆�⃗�

𝑐, = apparent velocity = 1234 5

𝜃

Incidence angle

𝜃 → 90° Horizontally traveling wave (ground roll and surface waves)

𝜃 → 0° Vertically traveling wave (body waves and reflections)

𝑣 =𝜆𝑇 =

M1 𝑘M1 𝑓

=𝑓𝑘

𝑐, =𝑓𝑘,

→Δ𝑡Δ𝑥

!.

=Δ𝑓Δ𝑘

A horizontal seismic line measures 𝑐, & 𝑘, not 𝑣& 𝑘.

17

Plane wave fronts

𝜆�⃗�

𝑐, = apparent velocity = 1234 5

𝜃

Incidence angle

𝜃 → 90° Horizontally traveling wave (ground roll and surface waves)

𝜃 → 0° Vertically traveling wave (body waves and reflections)

𝑣 =𝜆𝑇 =

M1 𝑘M1 𝑓

=𝑓𝑘

𝑐, =𝑓𝑘,

→Δ𝑡Δ𝑥

!.

=Δ𝑓Δ𝑘

Slope in t-d plot

Slope in f-k plot

A horizontal seismic line measures 𝑐, & 𝑘, not 𝑣& 𝑘.

18

d

t

Refractions

P-wave

S-wave

Surface waves

Air wave

19

d

t

Refractions

P-wave

S-wave

Surface waves

Air wave

High 𝑐,Large 𝜃

Low 𝑐,Small 𝜃

20

d

t

Refractions

P-wave

S-wave

Surface waves

Air wave

k

f

RefractionsP-wave

S-wave

Surface waves (ground roll)

Air wave

High 𝑐,Large 𝜃

Low 𝑐,Small 𝜃

High 𝑐,Large 𝜃

Low 𝑐,Small 𝜃

21

d

t

Refractions

P-wave

S-wave

Surface waves

Air wave

k

f

RefractionsP-wave

S-wave

Surface waves (ground roll)

Air wave

High 𝑐,Large 𝜃

Low 𝑐,Small 𝜃

High 𝑐,Large 𝜃

Low 𝑐,Small 𝜃

We can apply a fan filter to remove ground roll (surface waves) and keep signals of interest based on incidence angle and horizontal velocity.

22

d

t

Refractions

P-wave

S-wave

Surface waves

Air wave

k

f

RefractionsP-wave

S-wave

Surface waves (ground roll)

Air wave

High 𝑐,Large 𝜃

Low 𝑐,Small 𝜃

High 𝑐,Large 𝜃

Low 𝑐,Small 𝜃

We can apply a fan filter to remove ground roll (surface waves) and keep signals of interest based on incidence angle and horizontal velocity.

But be careful, all the rules on windowing apply.

23

d

t

Refractions

P-wave

S-wave

Surface waves

Air wave

k

f

RefractionsP-wave

S-wave

Surface waves (ground roll)

Air wave

High 𝑐,Large 𝜃

Low 𝑐,Small 𝜃

High 𝑐,Large 𝜃

Low 𝑐,Small 𝜃

We can apply a fan filter to remove ground roll (surface waves) and keep signals of interest based on incidence angle and horizontal velocity.

But be careful, all the rules on windowing apply.

Can still filter specific frequencies (e.g. low pass, high pass). Filters can also be designed to remove updipsignals (right side) or downdip (left side) and many other features.

Many additional possible filters introduce greater opportunity for misinterpreting filter artifacts as structure.

24

Consider a rectangular surface array recording of a horizontal plane wave traveling in the 𝑥% direction.

𝑥.

𝑥%

This looks like a DC offset in the 𝑥.direction and a constant frequency sine wave in the 𝑥% direction.

Wav

e fro

nts

25

Consider a rectangular surface array recording of a horizontal plane wave traveling in the 𝑥% direction.

𝑥.

𝑥%

This looks like a DC offset in the 𝑥.direction and a constant frequency sine wave in the 𝑥% direction.

Wav

e fro

nts

𝑘.

𝑘%

−𝑘#

+𝑘#

The FT of a constant frequency sine wave is a delta function. When traveling in the 𝑥% direction the impulses lie on the 𝑘% axis.

26

𝑥.

𝑥%

𝑘.

𝑘%

−𝑘# +𝑘#

Wave fronts

Similarly

27

𝑥.

𝑥%

𝑘.

𝑘%

−𝑘# +𝑘#

Wave fronts

Similarly

𝑥.

𝑥% Wave f

ronts

𝑘.

𝑘%

−𝑘#

+𝑘#

28

There is a great deal more that can be done with F-K analysis (e.g. reflection seismology) and with arrays (they don’t always have to be rectangular or evenly spaced; array analysis) and it would be easy to spend an entire semester on just one of those topics.