0 Introduction

Gravity exploration, with the advantages of lightweight, fast speed, less investment, is widely used in oil and gas exploration, mineral exploration and the study of deep earth structure and other fields (Nabighian M N et al., 2005). With the rapid development of computer and the increasing requirements of gravity exploration, the 3D inversion of gravity data has become a research hot, however, the 3D inversion still has two problems, one is the multi-solution problem, another is the calculation efficiency. In geophysical inversion, the generation of multiple solutions is due to the error of measured data, the discrete and sparse data, and the equivalence of field sources. Generally, increasing the field observation data and improving the observation accuracy can reduce the multiple solvability of the inversion. Regarding to the multiple solutions caused by the equivalence of field sources, we can solve the problem by adding constraints and prior information in inversion. Last & Kubik (1983) first considered the minimum gradient support function in compact gravity inversion. Guillen & Menichetti(1984) used the similar principle into gravity and magnetic inversion by minimizing the moment of inertia. Portniaguine & Zhdaov (1999) discussed the focusing inversion and showed that they could more precisely locate geophysical anomalies using this algorithm. Silva et al. (2001) discussed a variety if constraint methods. Zhdanov (2009) discussed the focusing inversion and its improvement in detail. Li & Oldenburg (1998) applied a "smooth inversion" method to gravity data inversion. Geng et al. (2014) applied variogram to the inversion of gravity gradient data. Liu et al. (1997) use genetic algorithm to 2D density interface inversion. CHEN Hua-gen et al. (2002) applied improved simulated annealing method into gravity inversion. Geng et al. (2013) used neural network method into 2D gravity inversion. Chen et al. (2013) applied pre-conjugate gradient method to the inversion of gravity gradient data. Feng et al. (2014) constrained the basin model by the total variation function, and retrieved the basin basement by gravity data. Chen et al. (2014) applied focusing inversion into inversion of gravity gradient. Liu Yan et al. (2015) conducted 3D gravity inversion using Bayesian method based on model reduction. Geng et al. (2016) applied cokriging method into gravity gradient data inversion.

3D Gravity inversion divides the observed subsurface media into small non-overlapping units, and then calculate the density of each discrete element by the inversion method to determine the actual distribution of field sources. With the expansion of measurement area and the improvement of precision, the mesh quantity of subdivision is massive, and the inversion calculation takes a lot of time. Yao et al. (2003) optimized the matrix storage and improved the inversion efficiency according to the special relationship between gridding survey points and underground subdivision. Chen et al. (2012) achieved the 3D fast forward modeling of gravity and gravity gradient using GPU. Hou et al. (2015) used GPU for fast speed inversion calculation of gravity gradient data.

In view of the above mentioned two gravity inversion technical problems, this paper applies the principle of focusing inversion to inverse gravity data using conjugate gradient method. In this paper, several key problems in the parallel computation of inversion are analyzed, and the 3D conjugate gradient gravity fast inversion based on GPU parallel technology of NIVIDIA CUDA platform is achieved, due to the good parallel improvement ability of conjugate gradient inversion. The experimental results show that the method can well determine the abnormal location, and the inversion efficiency is greatly improved.

1 Forward theory

By the law of gravity, the forces of any point in the geologic body acting in the outer space is:

(1)

where is a vector from observation point P(x,y,z) to Q(ξ,η,ζ) which is in the geological body, and G is gravitational constant.

从2008年，重庆对插秧机购机补贴实行中央与市级累加，按市场价85%的比例补贴，部分区县财政对购机补贴10%，购机者实际只支付5%的钱；个别区县的街道办事处、乡镇人民政府把剩下5%的购机款也进行补贴了；机插育秧秧盘市级和区县各补贴50%。这样购机者只需花很少的钱，甚至不花钱就可以得到插秧机，导致购机者不爱惜插秧机，有的根本不使用，有的用完之后也不维修保养，机器老化快，报废率高；插秧机生产季节性强，补贴过高，会使个别生产企业为扩大销量，抢工抢时，把不合格的产品投放市场[16]。

Like the underground space coordinate system shown in Fig.2, the x axis and y axis are in the horizontal direction, and the Z axis is vertical downward. Coordinate of observation point is (x,y,z), (ξ,η,ζ)∈[ξ1, ξ2]×[η1,η2]×[ζ1,ζ2]. It’s a volume element of any cuboid in space and gravity anomalies and gravity gradients produced by this element (Guo et al., 2004) is

In 3D inversion, underground space is divided into cuboid units of equal size and compact arrangement like Fig.1. Analytical formulas for gravity and gravity gradient forward modeling in rectangular space is shown as below:

通过对《滇南山遥三首》曲式的详细分析，我们了解到，虽然作者运用了大量的西方音乐元素，曲式结构并不复杂，“复三—单二—复三”，调式调性的运用上也较为简单，但是却更好的展现了中国的传统民族特色，表达了作者对儿时家乡的怀念，以及对自己民族浓浓的情怀。

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Underground space is divided into cuboid units of equal size and compact arrangement, and each unit is set as a fixed residual density value. We assume underground space is divided into M cuboid elements and the observation surface is arranged with N observation points. And the gravity anomaly of j th (j=1, …, M) element in i th (i=1, …, N) observation point is

(9)

Where mj is residual density value of jth element and Gij represent the geometric framework ith cuboid. According to the principle of potential field superposition, the total anomaly at the ith observation point is the sum of the anomalies of the underground j cuboid:

(10)

k=k+1

(11)

is same as

dαβ=Gαβm

(12)

while g0≠0

2 Inversion theory

The solution of geophysical inverse problems is often ill conditioned. Tikhonov regularization method is often used to solve the problem. The objective function can be expressed as

Safety and risk management of cosmetic products 11 1

φ(m)=φd+μφm

(13)

Where μ is regularization factor, φd is fitting difference function which usually expressed in terms of L2 norm like

刘家平表示，与华谊集团合作开发603地块，既是桃浦转型赋予临港集团的宝贵机遇，也是市国资委、普陀区、华谊集团对临港集团的信任。桃浦地区作为全市五大转型区域的先发区域，临港集团必将不辱使命、不负重托，坚持对标国际最高标准、最好水平，从建设高品质城区的定位进行整体规划建设，加快培育高质量产业，建设高品质物业，集聚高素质人才，打造高活力生态，建设有温度、有活力的卓越科创园区，为实现桃浦地区“脱胎换骨”式的转型发展而努力。

(14)

In the weighted focus inversion algorithm, the minimum support function is introduced as

(15)

The upper form can be rewritten as

为满足我国动力定位海工平台对电力系统的发展要求，本文研究挪威船级社(DNV GL)最新颁布的增强型DP附加标志(Dynamic Positioning System With Enhanced Reliability,DP-ER)附加标志的规范要求，结合已完成的母联闭合型海工平台电力系统设计成果，对该母联闭合型电力系统进行建模，仿真分析工况切换和故障条件下电力系统的动态响应特性。

(16)

so

(17)

There is an existence of a diagonal matrix

(18)

Where β is smaller constants to avoid generating singular values when mj equal to zero. At the same time, in order to avoid the skin effect, we add the sensitivity matrix based on weighted function

φ(m)=‖GWW-1m-d‖2+μ‖w-1m‖2

(19)

Assuming mw=w-1m, Gw=GW, then

李殿平说：“近年不少农资流通企业通过延伸产业链，从一买一卖的传统农资流通网点正在向现代农资综合服务商转型。他们加强了社会化服务体系建设，开展了测土配方、专家坐诊、农技培训、科技示范、农机作业、信息服务、金融保险等一系列增值服务。一些企业还建立了基于微信、微博、手机APP等农技服务体系，农资流通信息化、现代化水平明显提高。”

φ(m)=‖Gwmw-d‖2+μ‖mw‖2

(20)

The inversion problem is taken into account

(21)

Substituting into formula 21, the formula 21 can be written in the following form:

Amw=b

(22)

We use the conjugate gradient method for solving the above equation, and pseudo code is as follows:

Input: G, d, μ, k

Output: m

Begin

set m0=0

compute W, mw0, Gw0

k=0; g=2AT(Amw0-b)

else

Where dαβ is observed data, m is residual density, Gαβ is response of the j th cuboid at the i th observation point which is called sensitivity matrix.

express as matrix form

if k=1

p1=-g0

从此，他们两个经常到一起睡，每次睡过后，杨力生便给李秀花钱，只要给，少则几百，多则几千。李秀花概不推辞。

新规则对档号的编制原则、档号结构及编制要求更为详细。档号是档案实体的主要标识，赋予档号的过程也是档案从无序到有序的过程。规范、稳定、唯一的档号可以实现档案的规范化管理、提升检索效率，因此将“DA/T 13-1994档号编制原则”作为规范性引用文件有其必然性。

mwk=mwk-1+αpk

Update m, W, A, gk

End

where k is number of iterations, gk is gradient of objective function, pk is search direction, βkis step length of nonlinear exact line search.

3 Inversion results

It is assumed that there are zones with maximum buried depth of 1000 m, and axial and longitudinal range from -1 000 m to 1 000 m. Underground space has two residual density 0.5 g/cm3 cuboid anomaly with size of 300 m×400 m×500 m and anomaly centers located at (-450 m, 0 m, -450 m) and (450 m, 0 m, -450 m). Computational domain [-1 000 m, 1 000 m]×[-1 000 m, 1 000 m]×[-1 000 m, 0 m] is divided into 20×20×10 small units, each unit is 100 m×100 m×100 m. The observation point is set on the ground with z=0. There are total 20 lines designed with line interval of 100 m, point interval of 100 m, and there are 20×20 points in total. Fig.3 shows the plane inversion profile of inversion results of z=500 m and y=0 m. The abnormal body boundary position of the actual theoretical model is indicated by the black frame in the Fig 3. From the Figure, it is indicated that the weighted focus inversion has a good effect, which can reflect the location and burial depth of the abnormal body.

4 GPU parallel theory

In recent years, GPU technology has developed rapidly. CUDA(Compute Unified Devices Architecture)was Launched in 2006 by NVIDIA, the GPU of which can be used for general calculation. The programming mode is using CPU as the host, for logical and serial computing, and GPU, as a device, is for executing highly parallel thread tasks.

综上所述，颈动脉超声作为一种常用的检查方式，在颈动脉狭窄和脑卒中高危人群的筛查中都有着重要的作用；颈动脉狭窄与年龄、冠心病、下肢静脉疾病存在相关性，使用颈动脉超声针对＞60岁、冠心病和下肢静脉疾病患者进行颈动脉狭窄和脑卒中的重点筛查，能尽早发现无症状脑卒中患者，便于颈动脉狭窄和脑卒中的早发现和早治疗。

Matlab are usually used for calculation, which belong to interpretative language with low efficiency. C++ calculation that has higher computational efficiency than Matlab is used in this study. In the gravity inversion, because of the large sensitivity matrix, the corresponding calculation part of sensitivity matrix needs to consume a large amount of time. It is also the main reason for making inversion time consuming. The algorithm is shown in Fig.4. By parallel computing gk and Gwk, an obvious acceleration effect is achieved. In addition, in the parallel computation, due to repeated call for sensitivity matrix iteration, even if the GPU interface transmission bandwidth is large, it also needs to consume a lot of time. By storing the sensitivity matrix in the GPU memory, the whole inversion process only need to be passed from the host once therefore it accelerates the speed of inversion without multiple transmissions.

Table 1 Comparison of time consuming among GPU and Matlab

5 GPU parallel result analysis

The numerical test hardware environment used in this paper：i7 6700k CPU, 4 GHz main frequency and 8G memory. GPU model is Geforce GTX 1060 with 1280 processing cores, 6G memory and computing capacity of 6.1. Software environment is CUDA 8.0 version, Windows10 64 bits operating system and Visual Studio 2015 compiler. The calculating time spent in iterating 100 times is compared between Matlab and GPU parallel in scales of 1 000 m×1 000 m×1 000 m, 2 000 m×2 000 m×1 000 m, 3 000 m×3 000 m×1 000 m, 4 000 m×4 000 m×1 000 m and 5 000 m×5 000 m×1 000 m, and grid partition is dx=100 m, dy=100 m, dz=100 m. Because the GPU memory limit, we did not calculate the size of larger scale. If possible, we can use multiple GPU cards to calculate larger scale. It can be seen from Table 1 that when the scale is small, GPU parallel acceleration effect relative to the Matlab is not obvious, with the acceleration effect only ten times better, but with the increase of the size of the GPU model, calculation efficiency is obviously improved, finally accelerating effect can be more than 100 times better than Matlab.

结合高校的特殊性，充分利用好决算报表数据，通过增加新模块新功能，让软件整理的数据全面直接反映高校的办学情况。增加反映工作实际需要的表格和数据统计。如增加单独反映教学经费、科研经费的报表；单独反映学生奖学金、助学金、助学贷款、勤工俭学、学生活动费和就业指导等经费的报表；单独反映生均办学经费等办学指标的报表；反映现金流量情况等报表，这样才能实现统计出来的数据能运用，能运用的数据促进工作开展。

6 Conclusion

In this study, we firstly established a model with two separated abnormal bodies, then we got the inverson result from the gravity data. It is indicated that the gravity focusing inversion has a good effect to reflect the gravity abnormal. After that, we researched rapid inversion of gravity data and achieved fast gravity inversion based on conjugate gradient method using GPU parallel technique. The result showed that the algorithm has a high speedup ratio. The GPU parallel technique and its realization in gravity inversion provide technical support for fast 3D large-scale gravity inversion. It can also provide references for inversion methods like simulated annealing, genetic algorithm and artificial neural network algorithm.

没有没有，你不要急啊，不是车祸。晚上我和小涵一起吃饭，下楼的时候，小涵突然摔倒了。楼梯是有些滑，小涵的鞋跟也不矮。估计是摔到肚子上了，那里就见红了。去医院检查了，医生说要手术。

Acknowledgements

We are particularly grateful to Professor Huang Danian. Specially thanks to the project teams of Geophysical Comprehensive Survey, and to all the experts who have put efforts into this paper.

不为人所知的是，在两航起义前，1949年8月，周恩来在中南海接见了即将派往香港进行“两航起义”策反工作的吕明和查夷平。吕明是我党隐蔽战线的一名战士，加入中国共产党后在党的安排下投考国民党空军，并被派往美国学习，起义时任中国航空公司总经理的刘敬宜，便是吕明在美学习时的老师。而查夷平曾在中央航空公司担任过副总经理。吕明、查夷平曾在1946年到过南京梅园新村中共代表团驻地，聆听了周恩来的教诲。他们在起义前秘密到香港开展工作，并在周恩来的亲自领导下参与组织了两航起义。

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