تركيب روش‏هاي سيگنال تحليلي و واهماميخت اويلر (Euler Deconvolution) (AN_EUL) براي تفسير داده‏هاي مغناطيسي دوبُعدي

Combination of analytic signal and Euler Deconvolution methods for interpretation of 2-D magnetic data

روش AN-EUL روش خودكار براي برآورد مكان و محاسبه عمق و انديس ساختاري (Structural index) (شكل) چشمه‏هاي مغناطيسي است. اين روش تركيبي از دو روش‏ سيگنال تحليلي و واهماميخت اويلر است و معادلات آن از جايگذاري مشتق‏هاي مناسب معادله اويلر در معادله سيگنال تحليلي ميدان به‌دست مي‏آيد. در اين مقاله روش AN-EUL براي برآورد مكان و محاسبه عمق و انديس ساختاري ساختارهاي دوبُعدي به‌كار گرفته شده است. به‌‌اين‌‌ترتيب كه، ابتدا با استفاده از مدل‌‌سازي به روش پيشرو (Forward modeling) براي برخي مدل‏هاي دوبُعدي مغناطيسي ساده مانند، دايك نازك، دايك ضخيم واستوانه افقي، كه همه پارامترهاي آنها معين است داده مصنوعي توليد شده است. سپس در مرحله بعد همه كميت‏هاي موردنياز در روش AN-EUL براي اين داده‏هاي توليد شده محاسبه شده و درنهايت با استفاده از روابط كلي روش AN-EUL مقادير عمق و انديس ساختاري براي هركدام از اين مدل‏ها به‌دست آمده و با مقادير واقعي‏شان مقايسه شده است. درصد خطا و تفاوت مقادير محاسبه شده با مقادير واقعي پارامترها نشان مي‏دهد كه اين روش دقت قابل‌‌ملاحظه‏اي در تعيين عمق و انديس ساختاري چشمه‏هاي مغناطيسي دارد. همه مراحل محاسباتي در اين مقاله، اعم از توليد داده مصنوعي، فيلترهاي لازم و روابط اصلي تعيين عمق و انديس ساختاري با استفاده از برنامه‏هاي d كه در محيط مطلب نوشته شده، صورت گرفته است.

AN_EUL is a new automatic method for simultaneous approximation of location, depth, and structural index (geometry) of magnetic sources. In the 2D case analytic signal is defined as a conjugate function whose imaginary part is the Hilbert transform of its real part. Since the first vertical derivative of the magnetic field is the Hilbert transform of its horizontal derivative, vertical and horizontal derivatives of the magnetic field can produce an analytic signal function whose amplitude is equal to the root of the summation of square of the horizontal and vertical derivatives. Analytic signal has some useful properties. For 2D sources the amplitude shape of the analytic signal is an even and symmetric function whose maximum determines the location of the source. Moreover, the shape of the amplitude does not depend on the orientation of magnetization, strike, inclination and declination of the magnetic field. The Euler Deconvolution method is an automatic procedure to approximate the geometry, depth and location of the magnetic sources. In this method there is no need for reduction to the pole and remanent magnetization is not an interfering factor. The principle of the Euler Deconvolution method is based on the Euler Homogeneous differential equation. The AN_EUL method is a combination of analytic signal and Euler Deconvolution methods, and its main equations are derived by substituting appropriate derivatives of the Euler homogeneous equation into the expression of analytic signal of the potential field. AN_EUL equations are calculated at the source location that is approximated by the location of the maximum of analytic signal amplitude. In this paper, the AN- EUL method has been used for the determination of location, depth and structural index of 2D magnetic structures. At the first step, by using the forward modeling for some ideal 2D magnetic models, such as thin dike, thick dike and horizontal cylinder with given parameters, the synthetic data has been produced. At the next step, all of the required quantities in the AN-EUL method have been calculated for these series of data. In the final step, the depth and structural index of these models are calculated using the general formulas of AN-EUL and are compared with their real values. The levels of error and difference between the calculated values and real values of depth and structural index show that this method has an acceptable accuracy in approximating the structural index and depth of sources. Because AN_EUL equations use high order magnetic derivatives, the noises are amplified intensely; consequently for better resolution an upward continuation filter should be used. For some models like thick dikes, the analytic signal amplitude may have two maximums. For solving this problem the data should be continued to a higher level by upward continuation filter. In addition, since the derivatives are calculated by Fourier transform, it is necessary to taper the data before using this transform to avoid Gibbs effect. All of the computational steps in this paper, such as creation of synthetic data, necessary filters and the main equations of AN-EUL have been done by codes written in MATLAB.