## 1 Introduction  
A seismic array is a group of seismic receivers and/or sources planted in a systematic arrangement. In general, seismic arrays are used to suppress noise in the field. Many of the existing studies on seismic arrays assume ideal array conditions with perfect element (geophone) weights, perfect implementation conditions, and simple near-surface layers. However, these assumptions are rarely satisfied in the field, hence limiting the applicability of the results of these studies to field data. In particular, it is difficult to have similar ground coupling of all array elements, leading to variable element weights. Furthermore, element spacing is practically estimated by survey crew, leading to often slightly variable element positions. Local variations in the near-surface layer, such as topography and heterogeneity, can also lead to more non-ideal array conditions. All these conditions may decrease the effectiveness of the array's multi-channel processing and its waveform [1]. Originally, seismic arrays were developed to monitor nuclear explosions [2]. The studies of nuclear explosion identification have been employed to distinguish between a natural earthquake and nuclear explosion [3, 4]. Ref. [5] shows that seismic arrays improve the velocity models of the Earth's interior using large-number and dense spatial sampling of the seismic wavefield at several locations on the Earth's surface. More recently, seismic arrays have been implemented in high-resolution imaging [6, 7], shallow geology applications [8, 9, 10], monitoring induced seismicity [11], seismic imaging using ambient seismic noise [12], and seismic signal processing using distributed acoustic sensing (DAS) [13]. Ref. [14] proposed an efficient algorithm for ambient seismic noise beamforming without the need for explicit cross-correlations, while [15] presented a case study using a 10-month ambient DAS data set acquired on a fiber array to investigate long-term seasonal velocity variations. Seismic arrays are often integrated with other geophysical methods for a better understanding of the subsurface conditions [10, 16, 17, 18]  
In terms of reflection seismology, the main purpose of an array of geophones is to increase the signal-to-noise ratio (S/N) by enhancing the desired signals (reflections) and attenuating seismic noise during the acquisition of seismic data in the field. As the desired seismic signals (reflections) travel across the array with a much higher horizontal velocity, the signals produce a longer wavelength, as compared to that of the unwanted signals [19]. The desired signals are usually those that have been reflected from deep geologic horizons approaching the array at almost normal incidence. The unwanted signals are incoherent noise or horizontally traveling ground roll. Thus, a geophone array can also be considered as a directional filter [20]. Seismic array response is the arithmetic sum of traces recorded by geophones (elements) in the array. Ref. [21] presented general concepts of arrays, where they used harmonic analysis to explain the array response disrupted by variations in velocity, dip, and elevation within the array length. Ref. [22] described a method for computing the expected waveform from a closely spaced array of shots or geophones when the individual signal is a pulse and studied the effects of various array weighting functions on the performance of seismic arrays. Ref. [23] presented a study on the effects of errors in position and weighting on the array response, where they used plane harmonic waves with different weights. Ref. [24] investigated effects of errors in gain and position on the array response. Ref. [25] explained the wavelet response due to errors in weights and positions of the elements of the geophone arrays with respect to several variables. These variables included the error amount, number of array elements, and weighting function. They found that the degradation of the wavelet response increased nonlinearly with the error's value, number of elements, and weighting function complexity.