# Minimum phase

Two input wavelets, wavelet 1: (1, - 12) and wavelet 2: (- 12, 1), were used for numerical analyses of the inverse filter and least-squares inverse filter in this section. The results indicate that the error in converting wavelet 1 to a zero-lag spike is less than the error in converting wavelet 2 (Tables 2-8 and 2-10).

 Input: $(1,\ -{\frac {1}{2}})$ Desired Output: (1, 0, 0) Actual Output Error Energy Inverse Filter (1, 0, −0.25) 0.063 Least-Squares Filter (0.95, −0.09, −0.19) 0.048
 Input: $(-{\frac {1}{2}},\ 1)$ Desired Output: (1, 0, 0) Actual Output Error Energy Inverse Filter (1, 0, −4) 16 Least-Squares Filter (0.24, −0.38, −0.19) 0.792

Is this also true when the desired output is a delayed spike (0, 1, 0)? The cumulative energy of the error L associated with the application of a two-term least-squares filter (a, b) (Table 2-11) to convert the input wavelet (1, - 12) to a delayed spike (0, 1, 0) is

 ${L=a^{2}+\left[\left(b-{\frac {a}{2}}\right)-1\right]^{2}+\left(-{\frac {b}{2}}\right)^{2}}.$ (20)
 Filter Design Convolution of the filter (a, b) with input wavelet $(1,\ -{\frac {1}{2}})$ : 1 - 12 Actual Output Desired Output b a a 0 b a b − a/2 1 b a −b/2 0 Filter Application Least-Squares Filter (−0.09, 0.76) Input Wavelet $(1,\ -{\frac {1}{2}})$ Actual Output (−0.09, 0.81, −0.38) Desired Output (0, 1, 0)
 Input Wavelet: $(1,\ -{\frac {1}{2}})$ Desired Output Actual Output Error Energy (1, 0, 0) (0.95, −0.09, −0.19) 0.048 (0, 1, 0) (−0.09, 0.81, −0.38) 0.190

By simplifying equation (20), taking the partial derivatives of quantity L with respect to a and b, and setting the results to zero, we obtain

 ${{\frac {5}{2}}a-b=-1},$ (21a)

and

 ${-a+{\frac {5}{2}}b=2}.$ (21b)

Combine equations (21a,21b) into a matrix form

 ${\begin{pmatrix}5/2&-1\\-1&5/2\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}-1\\2\end{pmatrix}}.$ (22)

By solving for the filter coefficients, we obtain (a, b): (−0.09, 0.76). The design and application of this filter are summarized in Table 2-11.

Table 2-12 shows the results of the least-squares filtering to convert the input wavelet (1, - 12) to zero-lag (Table 2-7) and delayed spikes (Table 2-11). Note that the input wavelet is converted to a zero-lag spike with less error, and the corresponding actual output more closely resembles a zero-lag spike desired output.

 Filter Design Convolution of the filter (a, b) with input wavelet (1, - 12): 1 - 12 Actual Output Desired Output b a a 1 b a b − a/2 0 b a −b/2 0 Filter Application Least-Squares Filter (0.95, 0.38) Input Wavelet (1, −0.5) Actual Output (0.95, −0.09, −0.19) Desired Output (1, 0, 0)
 Filter Design Convolution of the filter (a, b) with input wavelet $(-{\frac {1}{2}},\ 1)$ : - 12 1 Actual Output Desired Output b a −a/2 0 b a −b/2 + a 1 b a b 0 Filter Application Least-Squares Filter (0.76, −0.09) Input Wavelet (−0.5, 1) Actual Output (−0.38, 0.81, −0.09) Desired Output (0, 1, 0)

We now examine the performance of the least-squares filter with the input wavelet (- 12, 1). The cumulative energy of the error L associated with the application of a two-term least-squares filter (a, b) (Table 2-13) to convert the input wavelet (- 12, 1) to a delayed spike (0, 1, 0) is

 $L=\left(-{\frac {a}{2}}\right)^{2}=\left[\left(-{\frac {b}{2}}+a\right)-1\right]^{2}+b^{2}.$ (23)

By simplifying equation (23), taking the partial derivatives of quantity L with respect to a and b, and setting the results to zero, we obtain

 ${{\frac {5}{2}}a-b=2},$ (24a)

and

 ${-a+{\frac {5}{2}}b=-1}.$ (24b)

Combine equations (24a,24b) into a matrix form

 ${\begin{pmatrix}5/2&-1\\-1&5/2\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}2\\-1\end{pmatrix}}.$ (25)

By solving for the filter coefficients, we obtain (a, b): (0.76, −0.09). The design and application of this filter are summarized in Table 2-13.

Table 2-14 shows the results of the least-squares filtering to convert the input wavelet (- 12, 1) to zero-lag (Table 2-9) and delayed spikes (Table 2-13). Note that the input wavelet is converted to a delayed spike with less error, and the corresponding actual output more closely resembles a delayed spike desired output.

 Input Wavelet: $(-{\frac {1}{2}},\ 1)$ Desired Output Actual Output Error Energy (1, 0, 0) (0.24, −0.38, 0.19) 0.762 (0, 1, 0) (−0.38, 0.81, −0.09) 0.190

Now, evaluate the results of the least-squares inverse filtering summarized in Tables 2-12 and 2-14. Wavelet 1: (1, - 12) is closer to being a zero-delay spike (1, 0, 0) than wavelet 2: (- 12, 1). On the other hand, wavelet 2 is closer to being a delayed spike (0, 1, 0) than wavelet 1. We conclude that the error is reduced if the desired output closely resembles the energy distribution in the input series. Wavelet 1 has more energy at the onset, while wavelet 2 has more energy concentrated at the end.

Figure 2.2-2 shows three wavelets with the same amplitude spectrum, but with different phase-lag spectra. As a result, their shapes differ. (From the 1-D Fourier transform, we know that the shape of a wavelet can be altered by changing the phase spectrum without modifying the amplitude spectrum.) The wavelet on top has more energy concentrated at the onset, the wavelet in the middle has its energy concentrated at the center, and the wavelet at the bottom has most of its energy concentrated at the end.

We say that a wavelet is minimum phase if its energy is maximally concentrated at its onset. Similarly, a wavelet is maximum phase if its energy is maximally concentrated at its end. Finally, in all in-between situations, the wavelet is mixed phase. Note that a wavelet is defined as a transient waveform with a finite duration — it is realizable. A minimum-phase wavelet is one-sided — it is zero before t = 0. A wavelet that is zero for t < 0 is called causal. These definitions are consistent with intuition — physical systems respond to an excitation only after that excitation. Their response also is of finite duration. In summary, a minimum-phase wavelet is realizable and causal.

These observations are quantified by considering the following four, three-point wavelets :

Wavelet A : (4, 0, -1)
Wavelet B : (2, 3, -2)
Wavelet C : (-2, 3, 2)
Wavelet D : (-1, 0, 4)

Compute the cumulative energy of each wavelet at any one time. Cumulative energy is computed by adding squared amplitudes as shown in Table 2-15. These values are plotted in Figure 2.2-3. Note that all four wavelets have the same amount of total energy — 17 units. However, the rate at which the energy builds up is significantly different for each wavelet. For example, with wavelet A, the energy builds up rapidly close to its total value at the very first time lag. The energy for wavelets B and C builds up relatively slowly. Finally, the energy accumulates at the slowest rate for wavelet D. From Figure 2.2-3, note that the energy curves for wavelets A and D form the upper and lower boundaries. Wavelet A has the least energy delay, while wavelet D has the largest energy delay.

 Wavelet 0 1 2 A 16 16 17 B 4 13 17 C 4 13 17 D 1 1 17

Given a fixed amplitude spectrum as in Figure 2.2-4, the wavelet with the least energy delay is called minimum delay, while the wavelet with the most energy delay is called maximum delay. This is the basis for Robinson’s energy delay theorem: A minimum-phase wavelet has the least energy delay.

Time delay is equivalent to a phase-lag. Figure 2.2-5 shows the phase spectra of the four wavelets. Note that wavelet A has the least phase change across the frequency axis; we say it is minimum phase. Wavelet D has the largest phase change; we say it is maximum phase. Finally, wavelets B and C have phase changes between the two extremes; hence, they are mixed phase.

Since all four wavelets have the same amplitude spectrum (Figure 2.2-4) and the same power spectrum, they should have the same autocorrelation. This is verified as shown in Table 2-16, where only one side of the autocorrelation is tabulated, since a real time series has a symmetric autocorrelation (The 1-D Fourier transform).

Note that zero lag of the autocorrelation (Table 2-16) is equal to the total energy (Table 2-15) contained in each wavelet — 17 units. This is true for any wavelet. In fact, Parseval’s theorem states that the area under the power spectrum is equal to the zero-lag value of the autocorrelation function (A mathematical review of the Fourier transform#A.1 The 1-D Fourier Transform|the 1-D Fourier Transform).

The process by which the seismic wavelet is compressed to a zero-lag spike is called spiking deconvolution. In this section, filters that achieve this goal were studied — the inverse and the least-squares inverse filters. Their performance depends not only on filter length, but also on whether the input wavelet is minimum phase.

 Wavelet A 4 0 −1 Output 4 0 −1 17 4 0 −1 0 4 0 −1 −4 Wavelet B 2 3 −2 Output 2 3 −2 17 2 3 −2 0 2 3 −2 −4 Wavelet C −2 3 2 Output −2 3 2 17 −2 3 2 0 −2 3 2 −4 Wavelet D −1 0 4 Output −1 0 4 17 −1 0 4 0 −1 0 4 −4

The spiking deconvolution operator is strictly the inverse of the wavelet. If the wavelet were minimum phase, then we would get a stable inverse, which also is minimum phase. The term stable means that the filter coefficients form a convergent series. Specifically, the coefficients decrease in time (and vanish at t = ∞); therefore, the filter has finite energy. This is the case for the wavelet (1, - 12) with an inverse $\left(1,\ {\frac {1}{2}},\ {\frac {1}{4}},\ \cdots \right).$ The inverse is a stable spiking deconvolution filter. On the other hand, if the wavelet were maximum phase, then it does not have a stable inverse. This is the case for the wavelet (- 12, 1), whose inverse is given by the divergent series (−2, −4, −8, …). Finally, a mixed-phase wavelet does not have a stable inverse. This discussion leads us to assumption 7.

Assumption 7. The seismic wavelet is minimum phase. Therefore, it has a minimum-phase inverse.

Now, a summary of the implications of the underlying assumptions for deconvolution stated in the convolutional model and inverse filtering is appropriate.

1. Assumptions 1, 2, and 3 allow formulating the convolutional model of the 1-D seismogram by equation (2).
2. Assumption 4 eliminates the unknown noise term in equation (2a) and reduces it to equation (3a).
3. Assumption 5 is the basis for deterministic deconvolution — it allows estimation of the earth’s reflectivity series directly from the 1-D seismogram described by equation (3a).
4. Assumption 6 is the basis for statistical deconvolution — it allows estimates for the autocorrelogram and amplitude spectrum of the normally unknown wavelet in equation (3a) from the known recorded 1-D seismogram.
5. Finally, assumption 7 provides a minimum-phase estimate of the phase spectrum of the seismic wavelet from its amplitude spectrum, which is estimated from the recorded seismogram by way of assumption 6.

 ${x(t)=w(t)\ast e(t)+n(t)},$ (2a)
 ${x(t)=w(t)\ast e(t)}.$ (3a)
 ${e(t)=f(t)\ast x(t)}.$ (4)

Once the amplitude and phase spectra of the seismic wavelet are statistically estimated from the recorded seismogram, its least-squares inverse — spiking deconvolution operator, is computed using optimum Wiener filters. When applied to the wavelet, the filter converts it to a zero-delay spike. When applied to the seismogram, the filter yields the earth’s impulse response (equation 4). In optimum wiener filters, we show that a known wavelet can be converted into a delayed spike even if it is not minimum phase.