b'FeatureEstimating Nth-order vertical gradientsEstimating Nth-order vertical gradients from Nth-order horizontalgradientsMichael D. OConnellRichard S. Smith Ottawa, Ontario, Canada Laurentian University, Sudbury, CanadaE moconnellphys@gmail.comE RSSmith@Laurentian.caSummaryOne can compute the 4th-order horizontal gradient in the wavenumber domain, F 4hor () using:Recently, there has been considerable interest in calculating higher-order derivatives, like the fourth-order vertical gradientsF 4hor () = (i )4 F()2but the Fourier-domain method of calculating these derivatives(Bracewell 1965), where i is the square root of -1. Asi2= -1, can be unstable. We present a new approach that starts with the Nth-order horizontal gradient (calculated in the space domain)F 4hor () = 4 F().3and converts this to the Nth-order vertical gradient without any unstable wave-number multiplication required. If we assume that the geology is two dimensional, so there is no change in the field in the y direction, then Laplaces equation (in a source free region) gives us a relation between the second Introduction order vertical and horizontalderivativesThere is currently interest in the fourth-vertical gradient toF 2hor +F 2ver= 0,4resolve near-surface geology (FitzGerald 2023) but the standardwhich is true in the space or wavenumber domain. Taking the spatial-domain method for calculating this gradient is unstablesecond-order horizontal derivative of both these, switching the as the method requires multiplying the real and imaginary components by 4, which can become quite large as theorder of double integration and substituting equation 4, givesmultiplication approaches the Nyquist frequency/wavenumberf 4ver (x) = f 4hor (x),5and this amplifies noise at these high frequencies (FitzGerald 2023). The Cauchy integral (FitzGerald, 2023; Thurston andand hence thatFornberg, 2024) is proposed as a more stable approach. F 4ver () = F 4hor ().6We are proposing a simple alternative that involves convertingThe Hilbert transform relation between the horizontal and the Nth-order horizontal gradients to Nth-order verticalvertical derivatives (Nabighian 1972) is gradients. The method is quite robust and only requires the user to select the appropriate spatial-domain horizontalF 1ver () = i sgn() F 1hor ()7gradient operator to match the desired wavelengths or depth of investigation. The data need not be smoothed by any filterswhere sgn() is +1 for positive arguments, -1 for negative other than the removal of spikes and micropulsations (magneticarguments, and zero for= 0. Using similar arguments to those data) (OConnell, 2001). above, the general case isF Nth-ver () = ( i )N [sgn()] F Nth-hor ().8The method The [sgn()] is only required for N odd; for N even, it should Let f(x) be a profile of magnetic data. Let CN be an Nth-orderbe replaced with unity. We can see that equation8 reduces to horizontal gradient operator that is applied as a convolution inequation 6 (an all-pass filter) when order = 4n. Equation 8 does not the space domain. There are multitudes of such operators, oneamplify the high frequencies, so is stable, provided that the relevant example is the Savitzky-Golay smoothing filters (Savitzky andhorizontal derivative has been calculated in a stable manner. Golay 1964; Press et al. 1992, section 5.7), which fit polynomialsThe derivatives, fi, fii and fiii are from Beyer (1984). We get the to the data and can differentiate these polynomials to the4th-derivative operator (fiv ) when we apply a convolution of the required order. A less sophisticated operator is the standardfii onto itself (Beyer 1984). We thenobtain:4th-difference operator, (1, -4, 6, -4, 1) (Abramowitz and Stegun 1972, Table 25.2). Better operators, fi, fii and fiii can be found infi = (-1, 9, -45, 0, 45, -9, 1)/609Beyer (1984). fii = (2, -27, 270, -490, 270, -27, 2)/18010Using these convolution filters to operate on f(x),fiii = (1, -8, 13, 0, -13, 8, -1)/811f4hor (x) = C4* f(x),1 fiv = (1, -9, 56, -154, 212, -154, 56, -9, 1)/1812where * is the convolution operator and f4hor (x) is the fourth- Here, we truncated fiv from thirteen points to nine, as the first order horizontal derivative of f(x). two and last two points in the thirteen point operator were Let F() be the forward Fourier transform of f(x) whereis theessentially zero. We normalised the denominator of fiv using wavenumber in the Fourier transform space and we denote thedata from the simple test function, y = x 4, and evaluating the Fourier transform of f4hor (x) as F 4hor (). derivative at x=0.JUNE 2024PREVIEW 45'