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空间探测信号经过对流层时延后具有较强的空间相关特性,也具有一定的规律性。在区域精密对流层建模问题中,经度在区域对流层建模中影响较小,纬度和高程信息是影响对流层分布的主要因素,建立了天顶对流层延迟与高程差及纬度的函数关系[3]。
$$ZTD(h,f(x)) = \sum {Ahf(x)} $$ (1) 式中:f(x)=cos2x;h为观测站所在大地高度;x的取值
为观测站纬度;A为矩阵参数。该模型通过建立天顶时延函数提高预测精度,缺点是在海拔较高处矩阵出现严重病态。 -
在经验模型中,网络模型精度较高,文中选用GPT2及GPT2w与提出的方法进行比较,气温、加权平均温度等气象参数[4]采用GPT2w提供的气压参数。这两种模型计算天顶湿延迟(Zenith Wet Delay, ZWD)的方法类似,方法如下:
$$ZWD = {10^{ - 6}}(a + b/T)\frac{R}{{(1 + q)g}}p$$ (2) 式中:a和b为经验参数;T为加权平均气温;R为干气常数;g为重力参数;p为环境水汽压;q为水汽变化梯度。GPTw2网络是目前精度较高的模型之一,但由于没有充分利用局部地区对流层参数,精度上略低于区域对流层模型。文中GPTw及GPTw2模型采用分辨率[5]为0.5°的网格。
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为了消除天线差异带来的时延预测误差,采用单一天线通信系统[6],接收机端的带噪声信号可以描述为:
$$x(t) = s(t) + u(t) + v(t)$$ (3) 式中:v(t)为复高斯白噪声,具有0均值与固定方差;s(t)为发送的信号;u(t)为多径干扰信号。为方便讨论,文中采用BFSK作为基带信号[7]。因此,s(t)可描述为:
$$s(t) = {s_1}(t) + {s_2}(t)$$ (4) $$\begin{split} {s_i}(t) =\; & A\sum\limits_{n = 1}^k {M{\rm{rect}}\left[\frac{{t - {T_i}/2 - (m - 1){T_i}}}{{{T_i}}}\right]}{\rm{×}}\\ &\exp (2\pi jt{f_n}) \end{split} $$ (5) 式中:A为信号幅值;M为系数矩阵;T为符号周期;f为基带信号频率。将信号采样离散化后可以得到信号格式为:
$$x[n]{\rm{ = }}s[n] + u[n] + v[n]$$ (6) 设信道传输矩阵为H,则接收端收到的复采样信号为:
$$R = H(X + U) + \bar V$$ (7) 文中采用单一GPS天线模拟[8]MIMO信号,发送信号为多个s(t)信号的叠加。
$$S = {[{s^1}(t),{s^2}(t),\cdots,{s^k}(t)]^{\rm{T}}}$$ (8) 式中:S为MIMO信号发送矩阵;k为信号通道数。在对流层延迟预测模型中,由于该模拟信号由单一天线发出,则对流层延迟参数对于每个信号通道来说相等。
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预测信号在空间传播过程中,经过大气层会发生信号衰减与折射,到达地面接收机时将产生时延误差。对流层由于高度低于电离层且接近地面[9],大气状变化更为明显,理想情况下卫星信号在真空中传播的速度为
$c = 299 792.458\;{\rm{km/s}}$ ,假设地面接收机测得的第$i$ 颗卫星的伪距为${D_i}$ ,第$i$ 颗卫星信号至接收机到达时间TOA为${t_i}$ ,接收机与卫星时钟偏差为$\Delta t$ ,则${D_i}$ 可表示为:$${D_i} = c{\rm{×}}({t_i} - \Delta t)$$ (9) 对于MIMO传输信号,第j个通道中卫星i的传播时延为
${t_{i,j}}$ ,则信号到达接收机的伪距误差如下:$${\Delta _{i,j}} = {t_{i,j}} \times c + {u_{i,j}}$$ (10) 式中:u为多径干扰项。对j个通道误差取算术平均,可以得到第i个卫星进行对流层延迟预测的平均误差err[i],如公式(11)所示:
$${\rm{err}}[i] = \sum\limits_j^k {{\Delta _{i,j}}} $$ (11) -
由于参与对流层时延的每个卫星传播路径及距离均不相同,文中求取最终时延结果时对不同传输路径的MIMO预测结果进行加权平均,采取自适应卡尔曼滤波的方式确定权重系数。构建自适应卡尔曼滤波器[10],引入时间因子p,在时刻p第i颗卫星测量信号的传播时延记为
${\Delta _{i,j}}(p)$ ,系统状态方程如下:$${\hat g_i}(p{\rm{ + 1}}){\rm{ = }}F\left[ {{g_i}(p)} \right]{\rm{ + }}{w_i}(p){\rm{ }}$$ (12) 其中,
${\hat g_s}(p{\rm{ + 1}})$ 为p+1时刻第i颗卫星时延预测权重的估计量;函数F为:$$F\left[ {{g_i}(p)} \right] = {g_i}(p) - {\Delta _{i,j}}(p)$$ (13) ${w_i}(p)$ 为该卡尔曼滤波器的过程噪声,并假设为高斯白噪声。则$p$ 时刻的时延预测误差为:$${\rm{err}}(p) = \frac{1}{{kQ}}\sum\limits_{i = 1}^Q {\sum\limits_{j = 1}^k {{g_i}(p)×{\Delta _{i,j}}(p)} } $$ (14) -
基于广汉机场2019年气象观测站点采样率为3 h的对流层数据进行区域对流层延迟建模,为改善数据完整性,实际建模中使用年积日1~80天的数据进行模型参数的初始化,采用观测站的大地坐标和ZTD数据进行模型精度评定。MIMO预测模型采用的GPS卫星经过RAIM(Receiver Autonomous Integrity Monitoring)完好性检测[11],自适应卡尔曼滤波器在1 h内完成预测模型的权重计算,并输出最终时延预测结果。
文中实验中一共用到4颗卫星,单次实验计算的归一化权重系数如下:
$({\tilde \omega _{1,1}},{\tilde \omega _{1,2}},{\tilde \omega _{1,3}},{\tilde \omega _{1,4}})$ =(0.125,0.233,0.619,0.012)$({\tilde \omega _{2,1}},{\tilde \omega _{2,2}},{\tilde \omega _{2,3}},{\tilde \omega _{2,4}})$ =(0.013,0.331,0.469,0.204)$({\tilde \omega _{3,1}},{\tilde \omega _{3,2}},{\tilde \omega _{3,3}},{\tilde \omega _{3,4}})$ =(0.047,0.028,0.875,0.106)$({\tilde \omega _{{\rm{4}},1}},{\tilde \omega _{{\rm{4}},2}},{\tilde \omega _{{\rm{4}},3}},{\tilde \omega _{{\rm{4}},4}})$ =(0.579,0.328,0.009,0.117)式中:
${\tilde \omega _{i,j}}$ 为第i颗卫星第j通道MIMO预测信号的权重系数,归一化权重系数后取值范围在[0, 1]区间内。重复10次实验,自适应滤波的迭代次数为5 000,权重系数MSE如图1所示。分析可知当迭代次数大于3 000时,算法能够收敛到稳定的权重系数。实验计算滤波器达到稳态情况下,当自适应MIMO算法通道数目为3、6、9时,权重系数均方误差,经过相应次数迭代计算,产生的权重系数误差如图2所示。
由图2实验结果可知,10次实验中权重系数更新产生的误差接近于稳态,这是因为在1 h的实验周期内,对流层参数基本恒定不变,当滤波器收敛后得到的权重系数更接近真实的情况。将文中提出的算法与GPT2、GPT2w、UNB3以及EGNOS模型[10]进行比较,结果如表1所示。
表 1 不同模型计算误差比较
Table 1. Comparison of calculation errors of different models
Error distribution Average STD/mm Average BIAS/mm GPT2 6.012 0.028 GPT2w 5.725 0.034 UNB3 7.132 0.012 EGNOS 6.249 0.076 Adaptive MIMO 5.683 0.083 比较结果可以看出,5种模型的BIAS都较小,说明模型的稳定性较好,文中提出自适应MIMO算法的误差区间在[−10 cm,10 cm]范围内,优于GPT2、GPT2w、EGONS和UNB3模型,但系统稳定性不及这4种模型。5种模型时延预测误差进行比较如图3所示。
由于权重系数的加权导致MIMO信号不同通道的信号能量不一致,接收机端总的信噪比(Sigal to Noise Ratio,SNR)为:
$$SNR = \frac{{\sum\limits_k {{E_{2FSK}}} }}{{{E_v} + {E_J}}}$$ (15) 在不同SNR下根据自适应MIMO信号观测值计算对流层延迟误差,结果如图4所示。
实验结果表明,自适应MIMO算法的时延预测模型精确度随信噪比SNR的增大而提高,且当通道数目更大时,预测误差将显著下降。
Deep space detection tropospheric delay prediction based on adaptive MIMO technology
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摘要: 在深空探测中探测信号经过对流层延迟后在接收机端信号将出现一定程度的时延,影响探测精度。现有方法主要通过网格模型、空间模型实现时延预测,但由于区域差异导致模型准确度受限,预测精度仍有改进空间。提出了一种基于自适应多输入多输出(MIMO)信号的深空探测对流层延迟预测模型。基于单一收发天线模拟卫星信号MIMO传输方式,然后构建自适应卡尔曼滤波器,通过自适应调整MIMO信号分量权重系数的方法选取最优传输路径以实现对流层延迟量的预测。参与测量的卫星数目为4颗,在不同信噪比以及改变MIMO通道数目情况下开展实验,研究自适应MIMO模型的准确度和实际测量误差。实验结果表明,新方法相对于GPT2模型、GPT2w模型以及实时导航定位中常用的UNB3模型、EGNOS模型的预测精度有较大提高。Abstract: In the deep space exploration, the signal is delayed by the troposphere and a certain delay will occur at the receiver end, which affects the detection accuracy. The existing methods mainly realize the delay prediction by the grid model and the space model, but the accuracy of the model is limited due to the regional difference, and the prediction accuracy still has room for improvement. In this paper, a deep-space tropospheric delay prediction model based on adaptive multiple-input multiple-output (MIMO) signals was proposed. The satellite signal MIMO transmission mode was simulated based on a single transceiver antenna, and then an adaptive Kalman filter was constructed. The optimal transmission path was selected by adaptively adjusting the weight coefficient of the MIMO signal component to predict the tropospheric delay. The number of satellites participating in the measurement was four. Experiments were carried out under different signal-to-noise ratios and changing the number of MIMO channels to study the accuracy and actual measurement error of the adaptive MIMO model. The experimental results show that the prediction accuracy of the new method is much higher than that of the GPT2 model, GPT2w model and the commonly used UNB3 model and EGNOS model in real-time navigation and positioning.
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Key words:
- deep space exploration /
- tropospheric delay /
- MIMO /
- adaptive Kalman filter
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表 1 不同模型计算误差比较
Table 1. Comparison of calculation errors of different models
Error distribution Average STD/mm Average BIAS/mm GPT2 6.012 0.028 GPT2w 5.725 0.034 UNB3 7.132 0.012 EGNOS 6.249 0.076 Adaptive MIMO 5.683 0.083 -
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